Black Holes in Multidimensional Spacetimes Mgr

Silesian University in Opava Faculty of Philosophy and Science Institute of Physics

Black Holes in Multidimensional Spacetimes A thesis presented to Silesian University in Opava in partial fulfillment of the requirement for the degree of Doctor of Philosophy in Physics by

Mgr. Martin Blaschke

Opava, Czech Republic, 2017

Annotation In this thesis we discuss basic multidimensional ideas and models, like Kaluza–Klein and braneworld model. We also discuss some black hole solutions for these models. Especially, we focus on (4+1)-dimensional braneworld Kerr–Newman black hole and naked singularity spacetimes. We discuss the general properties of these solutions and also several new features, like the influence of fifth dimension, represented by the tidal parameter, on Aschenbach effect. We also discuss new, classical instability of naked singularities that we call “mining” instability. Key worlds: Braneworld scenario, multidimensional black holes, Aschenbach effect, mining instability.

Anotace V této disertaci se zabýváme základními myšlenkami vícedimenzionálních modelu, ˚ jako jsou modely Kaluzy–Kleina a modely bránových svˇetu. ˚ Také se zabýváme nˇekterými cˇ ernodˇerovými rˇešeními tˇechto modelu. ˚ Speciálnˇe se soustˇredíme na (4+1)-dimenzionální prostoroˇcasy s Kerrovou–Newmanovou cˇ ernou dírou nebo nahou singularitou v rámci scénáˇre bránových svˇetu. ˚ Diskutujeme obecné vlastnosti tohoto rˇešení a také nˇekolik nových vlastností, jako je vliv páté dimenze, pˇredstavovaný pˇrílivovým parametrem, na Aschenbachuv ˚ efekt. Také zkoumáme novou, klasickou nestabilitu nahých singularit, kterou nazýváme “tˇežební” nestabilita. Klíˇcová slova: scénáˇr bránových svˇetu, ˚ vícedimenzionální cˇ erné díry, Aschenbachuv ˚ efekt, tˇežební nestabilita.

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Declaration

I hereby declare that this thesis was written by myself under the supervision of Prof. RNDr. Zdenˇek Stuchlík CSc. and that I used only the sources listed in the bibliography and in the list of figures. The thesis work was finished in July 2017.

In Opava, July 2017

.................. Martin Blaschke

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Acknowledgment

I would like to express gratitude to my supervisor Prof. Zdenˇek Stuchlík for his guidance, patience and support during the years of my study of physics. Also, I would like to thank my colleagues and teachers at Silesian University in Opava for providing perfect study environment, many opportunities to study abroad and participations on interesting projects. Same goes to my dear friends and family. Without them nothing of this would be possible. To my future wife, Jana, I would like to thank for the patience and support during the difficult times.

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Notation and Conventions The Einstein summation convention is understood on coordinate indices. Round brackets indicate symmetrization, square brackets indicate anti-symmetrization of tensor indices. To further simplify notation, especially in more technical parts (chapter 4), we use completely dimensionless units c = G = M = 1 , where M stands for the mass of a central object in discussion. Symbol ADD ADM EM ED GL GR HJ KN KK MP PUFT RN SR ST G(d+1) κd2 κ M(d+1) Mew y , x5 Γ

Description Arkani-Hamed, Dimopoulos and Dvali Arnowitt, Deser and Misner electromagnetic extra dimensions Gregory–Laflamme general relativity Hamilton–Jacobi Kerr–Newman Kaluza–Klein Myers–Perry Projective Unified Field Theory Reissner–Nordström special relativity string theory gravitation constant in d spacial dimensions gravitation coupling constant ≡ 8π/G(d+1) κ5 /κ4 Plank constant in d spacial dimensions electro-weak scale fifth dimension Gamma function

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List of Figures

1.1 1.2 1.3 1.4

Mysterium Cosmographicum: Wikipedia . . . . . . . . . . . . . . The cover to Flatland, first edition: Wikipedia . . . . . . . . . . . Gunnar Nordström, Theodor Kaluza and Oskar Klein: Wikipedia Illustration of Klein idea. . . . . . . . . . . . . . . . . . . . . . . .

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D-brane . . . . . . . . . . . . . . . . . . . . . . . . Illustration of ADD braneworld model with n = 1. Illustration of RS braneworld model I. . . . . . . . Illustration RS braneworld model II. . . . . . . . .

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3.1 3.2 3.3

Spherical coordinate system: Wikipedia . . . . . . . . . . . . . . . . . . . . . Illustration of parallel transport on the sphere: Wikipedia . . . . . . . . . . . The hyperspace Σ divides the spacetime M into two regions M + and M − .

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4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15

Contour plot for radii of boundary of the causality . . . . . . . . Examples of causality violation region. . . . . . . . . . . . . . . . Kretschmann’s scalar . . . . . . . . . . . . . . . . . . . . . . . . . Polar slice through the braneworld KN spacetime (a = 1) . . . . . Polar slice through the braneworld KN spacetime (b = 0.9) . . . . Ergosphere and causality violation region. . . . . . . . . . . . . . Classes according to the properties of circular photon geodesics . Functions aSol± and aH defining location of horizons . . . . . . . Six different types behavior for function VEff (r, a) . . . . . . . . . Function a2extr(ex) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions bex and bdiv . . . . . . . . . . . . . . . . . . . . . . . . Parameter space (a − b) divided into eight separate regions I-VIII Behavior of effective potential with zero angular momentum . . . Behavior of effective potential with zero angular momentum . . . Mapping of existence of the marginally stable circular geodesics .

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LIST OF FIGURES 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 4.45 4.46

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Classification of the braneworld KN spacetimes . . . . . . . . . . . . . . Classification of the braneworld KN spacetimes . . . . . . . . . . . . . . E and L for Kerr black hole and naked singularities. . . . . . . . . . . . L, E and effective potential for class I. . . . . . . . . . . . . . . . . . . . . L, E and effective potential for class II. . . . . . . . . . . . . . . . . . . . L, E and effective potential for class IIIa. . . . . . . . . . . . . . . . . . . Energy measured by the LNRF observers . . . . . . . . . . . . . . . . . . Absolute value of covariant energy . . . . . . . . . . . . . . . . . . . . . L, E and effective potential for class IIIb. . . . . . . . . . . . . . . . . . . L, E and effective potential for class IVa. . . . . . . . . . . . . . . . . . . L, E and effective potential for class IVb. . . . . . . . . . . . . . . . . . . L, E and effective potential for class Va. . . . . . . . . . . . . . . . . . . . L, E and effective potential for class Vb. . . . . . . . . . . . . . . . . . . L, E and effective potential for class Vc. . . . . . . . . . . . . . . . . . . L, E and effective potential for class VI. . . . . . . . . . . . . . . . . . . . L, E and effective potential for class VII. . . . . . . . . . . . . . . . . . . L, E and effective potential for class VIII. . . . . . . . . . . . . . . . . . . L, E and effective potential for class IX. . . . . . . . . . . . . . . . . . . . L, E and effective potential for class X. . . . . . . . . . . . . . . . . . . . Energetic efficiency η(a, b) . . . . . . . . . . . . . . . . . . . . . . . . . . Energy efficiency of the Keplerian accretion . . . . . . . . . . . . . . . . Keplerian velocity profiles related to the LNRF . . . . . . . . . . . . . . . Non-monotonic plus-family orbital velocity profiles . . . . . . . . . . . . Non-monotonic plus-family orbital velocity profiles . . . . . . . . . . . . Minus-family orbital velocity profiles . . . . . . . . . . . . . . . . . . . . The loci of the photon circular orbits aph (r, b) . . . . . . . . . . . . . . . Naked singularity regions with potentially retrograde plus-family orbits. Classification of the braneworld Kerr spacetimes . . . . . . . . . . . . . . Spin range of braneworld Kerr black holes . . . . . . . . . . . . . . . . . Non-monotonic LNRF-related velocity profiles . . . . . . . . . . . . . . . (ϕ) Function VK,ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ϕ)

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84 85 87 91 91 92 93 93 94 95 95 96 96 97 97 98 98 99 99 101 102 107 108 109 110 111 112 113 114 115 116

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4.47 Function VK (r = rms , a, b) − VK (r = rmin,a,b ) . . . . . . . . . . . . . . . . . 117 (ϕ)

4.48 Function ∆VK,ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.49 Sequence of non-monotonic LNRF-related velocity profiles . . . . . . . . . . 119

List of Tables

4.1 4.2 4.3

All kinds of extrema of function aph (r, b). . . . . . . . . . . . . . . . . . . . . 68 Ten possible divisions of braneworld KN spacetimes . . . . . . . . . . . . . . 70 Classification of parameter space b − a with respect to ISCO. . . . . . . . . . 89

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Contents

Annotation

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Declaration

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Acknowledgment

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Notation and Conventions

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List of Figures

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List of Tables

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I

The overview

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Why black holes and why in extra dimensions? 1.1 The cultural ramification of higher dimensions . . . 1.2 The quest for unification . . . . . . . . . . . . . . . 1.3 First tries . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Gunnar Nordström scalar theory of gravity 1.3.2 Weyl field theory . . . . . . . . . . . . . . . 1.3.3 Kaluza–Klein theory . . . . . . . . . . . . . 1.3.4 Simple KK black holes . . . . . . . . . . . . 1.3.5 Nonrotating charged black holes . . . . . . 1.3.6 General Kaluza–Klein black holes . . . . . . 1.3.7 Induced-matter theory . . . . . . . . . . . . 1.3.8 Brans–Dicke theory . . . . . . . . . . . . . . 1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . xiii

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CONTENTS

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2 Braneworld models 2.1 Motivation . . . . . . . . . . . . . . . . . . . . 2.2 Higher-dimensional black holes . . . . . . . . 2.2.1 The Kerr–Newman metric . . . . . . . 2.2.2 Schwarzschild–Tangherlini black holes 2.2.3 Myers–Perry black holes . . . . . . . . 2.2.4 Other solutions . . . . . . . . . . . . . 2.2.5 ADD braneworlds . . . . . . . . . . . . 2.2.6 Randall–Sundrum braneworlds . . . . 2.2.7 Randall–Sundrum I . . . . . . . . . . . 2.2.8 Randall–Sundrum II . . . . . . . . . . 2.2.9 Black holes in RSII . . . . . . . . . . . 2.2.10 DGP braneworld . . . . . . . . . . . . 2.2.11 Summary . . . . . . . . . . . . . . . .

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3 A General Covariant Approach 3.1 Gauss–Godazzi equations . . . . . . . . . . . 3.1.1 Israel’s formalism . . . . . . . . . . . 3.2 Model with one extra dimension . . . . . . . 3.3 Classes of vacuum solutions on the brane . . 3.3.1 Braneworld Kerr–Newmann solution 3.4 Summary . . . . . . . . . . . . . . . . . . . .

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4 Braneworld Kerr–Newman spacetime 4.1 General circular geodesic motion of axially symmetric ... 4.1.1 E , L and Ω of circular geodesics . . . . . . . . . 4.2 Effective potential . . . . . . . . . . . . . . . . . . . . . . 4.3 Causality violation region . . . . . . . . . . . . . . . . . . 4.4 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Ergosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Circular geodesics of photons . . . . . . . . . . . . . . . 4.7 Equatorial geodesics with zero angular momentum . . . 4.7.1 Kerr spacetimes . . . . . . . . . . . . . . . . . . . 4.7.2 Braneworld Kerr–Newman spacetimes . . . . . . 4.8 Radial fall from infinity . . . . . . . . . . . . . . . . . . . 4.9 Stable circular geodesics . . . . . . . . . . . . . . . . . . 4.9.1 Marginally stable circular geodesics . . . . . . . . 4.9.2 Innermost stable circular geodesics . . . . . . . . 4.10 Analytical proof . . . . . . . . . . . . . . . . . . . . . . . 4.11 Classification of braneworld KN spacetimes . . . . . . .

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CONTENTS

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4.13 4.14 4.15

4.11.1 Case b=0: Kerr black hole and naked singularity spacetimes 4.11.2 Case a=0: RN black hole and naked singularity spacetimes 4.11.3 Characteristic points of the KN spacetime classification . . 4.11.4 Character of circular geodesics in the KN spacetimes . . . . Efficiency of the Keplerian accretion . . . . . . . . . . . . . . . . . . 4.12.1 Locally non-rotating frames and orbital motion . . . . . . . 4.12.2 Energy measured in LNRF . . . . . . . . . . . . . . . . . . . 4.12.3 Future-oriented particle motion . . . . . . . . . . . . . . . . Aschenbach effect for braneworld KN spacetimes . . . . . . . . . . Velocity profiles for the Keplerian distribution of ` . . . . . . . . . 4.14.1 Aschenbach effect in braneworld spacetimes . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Conclusion 121 5.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Bibliography

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Individual papers

6 Paper 1: Non-monotonic Keplerian velocity profiles around near-extreme braneworld Kerr black holes 7

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Paper 2: Efficiency of the Keplerian accretion in braneworld Kerr–Newman spacetimes and mining instability of some naked singularity spacetimes 159

Part I The overview

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Chapter

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Why black holes and why in extra dimensions? The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all. Aristotle, On Heaven

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he existence of black holes is an interesting prediction of general theory of relativity (GR) on its own. They are, so far, the only macroscopic objects in the known universe that are completely described by some theory. By Wheeler’s famous no hair theorem, a classical black hole is fully determined by a few macroscopic characteristics (mass, charge), with no apparent room for additional microscopic states. Because their properties can be influenced by the number of dimensions of the universe1 , the physics of black holes can be crucial in studies and falsifications of those theories with additional spacial or temporal dimensions. This is probably reason why black holes are often analyzed in the framework of modern Kaluza–Klein (KK) and string theories (ST). Also, they are the main topic of this thesis. One of the most intriguing and perhaps most fundamental question we can ask about nature of the universe is why we are living in four dimensional spacetime – with three dimensions of space and one of time. General relativity and Newton’s law of universal gravitation seems to be among the first theories which tackle this problem, because four is the minimum number of dimensions in which a gravitation can exist in an empty space. Also, three spacial dimensions, from the point of view of Newton’s theory alone, are necessary for the existence of stable orbits for planets. Therefore, we can have interesting physics 1 Regarding

of spelling see article “Universe Or universe? It All Depends On The Multiverse”[Gleiser, 2013]

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1.1. THE CULTURAL RAMIFICATION OF HIGHER DIMENSIONS

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or/and quite possibly even life [Tegmark, 1997], and the answer may lie in the domain of the anthropic principle. For example, it can be argued that in universes with more than one time dimension, differential equations mostly used in quantum mechanics become chaotic and therefore useless in predicting a future [Tegmark, 1998]. Then, one can say that in such conditions evolutionary processes cannot support the development of living creatures with something we can call brain. But is it really true? Is it not possible that there are in fact more dimensions, somehow hidden from us, creating the illusion that we exist in just four dimensions? These questions seems to be more philosophical than something a physicist might ask. But nothing can be far from the truth. Nowadays, the theories which postulate extra spacial dimensions represent a promising attempt to find quantum gravity or/and a unified theory of interactions. Indeed, the string theory is often regarded as a candidate to a final theory of everything or sometimes even as a theory of anything which, surprisingly, can be perceived as an advantage in modern cosmological theories including multiverse [Albrecht, 1995]. More recent discussions about dimensionality of the universe have involved the stability of planetary orbits and atomic ground states, the use of wave propagation for information transmission, the fundamental constants of nature, wormhole effects [Volovich, 1989], the cosmological constant [Gasperini, 1989], string theories, and nucleation probabilities in quantum cosmology [Embacher, 1996]. There are many reasons why we bother with extra dimensions (EDs), though our everyday experience tells us that our environs are quite adequately described by Aristotle’s observation. For some it can be just a curiosity, the cornerstone of basic research. But the main reason which drives this endeavor is that theories with EDs usually have ability to unify different physical laws into one, more coherent theory. But before we approach the main reason why we study higher-dimensions and their tantalizing power of unification, let us make a small caveat and look upon this issues from a historical perspective.

1.1

The cultural ramification of higher dimensions

To study the historical development of an idea that higher dimensions exist would be on it’s own an endeavor you can spend a lifetime on2 . Here I will try to insinuate some philosophical, psychological and physical connotations that have been attributed to higherdimensional theories. The first who pondered and posed the question: “Why does space have just three dimensions?” was philosopher Aristotle (384–322 B.C.E.) in his De Caelo et Mundo (On the Heavens). As we can say from the quote at the beginning of this chapter, Aristotle was convinced that three-dimensional solids are perfect and cannot be improved or surpassed 2 The

following section is mainly inspired by the philosophical lectures by physicist Jiˇrí Langer.


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by any other object with a different number of dimensions. He considered the number three more plausible for the number of dimensions because of the old Pythagorean notion that this number is special. After all, everything has a beginning, middle and an end. Claudius Ptolemy (90–168) is believed to be the author of now lost study called On Dimensionality where he argued that more than three spatial dimensions are impossible, by pointing out our inability to construct more than three mutually perpendicular lines in a single point. Plato’s theory (in The Republic) of perfect “Ideas” or “Forms” can also be considered as an early proposal of a reality based on higher-dimensions. Indeed, Henry More (1614–1687) the Cambridgian Platonist named the location of Plato’s Ideas the “fourth dimension” and was the first man who published this term (in Enchiridion Metaphysicum, 1671). French mathematician Jean le Rond d’Alembert (1717–1783) made the first published suggestion that time is the fourth dimension in 1754, although he allegedly attributed this Figure 1.1: Kepler’s Platonic solid idea to another French mathematician Joseph Louis model of the Solar system from Lagrange (1736–1813)3 . Almost one hundred years later Mysterium Cosmographicum. in 1895 Herbert George Wells (1866–1946) wrote that “there are really four dimensions, three which we call the three planes of Space, and a fourth, Time” (The Time Machine). Another interesting boost to the belief that three dimensions of space are a fundamental property of the universe, were the so called Platonic solids, equal-sided polyhedra. In three dimensions there are only five of them, while in two dimension there are an infinite number of equal-sided polygons (triangle, square, and so forth). This disproportionation between two and three dimensions drawn attention to Platonic solids in hope that they can provide a pure mathematical explanation for the dimensionality of the universe. For example Johannes Kepler (1571–1630) was trying to solve the problem of the motion of planets around the Sun by using Platonic solids and spheres in his Mysterium Cosmographicum, 1596. In same book, Kepler also speculated that the nature of the Holy Trinity might be responsible for number of dimensions. Platonic solids can be generalized to higher dimensions: In the case of four dimensions, the number of Platonic solids (regular polytopes) is six and in any higher there are exactly three. In the nineteenth century, a discussion about higher dimensions took place in the realm 3 He

actually did not use Lagrange’s name explicitly but he spoke about “enlightened man he knows”. Some authors and historians of science doubt that d’Alembert by this mysterious reference meant Lagrange because he was at that time only eighteen. On the other hand, it was Lagrange who published an idea that time can be considered as another coordinate in 1796 in his text The Theory of Analytical Functions.


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` Lobachevsky (1793–1856) of mathematics. It started with Russian mathematician Nicolaj who showed that Euclidean geometry is not unique and introduced his hyperbolic geometry which was logically consistent but did not obey the so-called fifth postulate of Euclidean geometry. Georg Friedrich Bernhard Riemann (1826–1866) followed some of this ideas and in his famous lecture on June 10, 1854 On the Hypotheses which lie at the Bases of Geometry [Riemann, 1873] he proposed a more general kind of geometry, called differential geometry, with an arbitrary number of dimensions. These new mighty tools (manifold, metric, curvature and embedding) were developed by Riemann in hope of unifying all known physics, gravitation, electricity and thermodynamics under one mathematical principle. Unfortunately, because his medium was space rather than space-time, he inevitably failed. But his new mathematics were used after quite long time by Albert Einstein (1873–1955) in GR theory of gravitation that superseded Newton’s old theory. Einstein proposed a completely new view on gravity so that the curvature of space (and also time) is responsible for gravitational force experienced by an observer who is actually moving in “straight” lines but feels gravitational force (or equivalently, acceleration) because space-time itself is curved. The curvature of space-time is given by the distribution of mass, energy and also, what was new idea, the distribution of pressure. Consequently, once the concept of higher dimensions of space emerged, there were many who were grokking that this realm was full of spirits, ghosts and sometimes even the home of God (A. T. Schofield). Not even scientists were completely immune to such ideas. For example German physicist Johann Karl Friedrich Zöllner (1834–1882) and British physicist William Crookes (1832–1919) studied a work of American medium Henry Slade (1835–1905). Under their watchful eyes, Slade performed a number of seemingly impossible tricks. Both of them were amazed by his abilities and concluded that the only possible explanation was that Slade had found a way how to move things through a higher dimensions (see Hereward Carrington’s and Harry Houdini’s book The Physical Phenomena of Spiritualism, 1907 [Carrington and Hou- Figure 1.2: The cover to Flatland, dini, 1907]). The public suspicion of a connection be- first edition. tween higher dimensions and mysticism was strengthened when Zöllner openly defended Slade against accusations of fraud during a trial in London in 1877. Apparently, during the trial he was calling for respected authorities like William Weber, William Thomson and Lord Rayleigh. Another very interesting look at higher dimensions was introduced by Charles Howard Hinton (1853–1907) in his illuminating book Scientific Romances [Hinton, 1976]. He theo-

1.2. THE QUEST FOR UNIFICATION

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rized that four-dimensional fluid (water in his example) spreading into three-dimensional subspace of four-dimensional universe would seem to inhabitants of that subspace as a gas, a substance which is spreading in all directions. Edwin Abbott Abbott (1838–1926) was also interested in higher dimensions. As a teacher and priest who believed in the importance of education he offered his particular and pedagogical way how to envision higher dimensions in his book Flattland: A Romance of Many Dimensions, 1884. Protagonist, Mr. Square (Abbott2 ) lives in a two dimensional world together with other polygons. As in our world, symmetry is very important, not only to attract sexual partners but in the case of flattlanders also to possess a good position in the social hierarchy. By suppressing one spacial dimension Abbott was able to imagine properties of three-dimensional being seen from the perspective of two dimensional creature like Mr. Square, who was visited by Lord Sphere from higher dimensions. In the end Mr. Square was convinced that obscure and invisible third dimension indeed exists, but was unable to make his case with another inhabitants of Flattland and was imprisoned for his blasphemy. We can also mention Salvador Dali’s (1904–1989) canvas Corpus Hypercubus where he uses a net of four-dimensional cube (tesseract) instead of cross as a torturing tool for the body of Christ.

1.2

The quest for unification

In this section we will very briefly outline the main motivation for introducing a higher dimensions to physical theories. As we have already mentioned, these theories were found to have the property of unifying two distinct theories of nature into one4 . Unification has always been a bonanza in physics. The Isaac Newton (1642–1727) showed that mechanic of celestial bodies is governed by the same forces which makes apples fall to the ground. James Clerk Maxwell (1831–1879) united electrical and magnetic forces into one and also showed that light is an electromagnetic wave and all of optics is actually a part of the electromagnetic theory. Such examples show that the mathematical elegance and convenience would be a truly compelling reason for adding extra dimensions. A first good example of unification with the aid of EDs is Einstein’s special theory of relativity (STR) where time and space is put on an equal footing. Mathematically, Einstein proposed that physical background of the universe is not three-dimensional fibered (by time parameter) manifold endowed with a simple Euclidean (flat) geometry, as it is in classical mechanics, but it is a simple four-dimensional manifold endowed with Minkowski (also flat, but time dimension is treated differently) geometry. The word “simple” is present in both views, but in different positions. Same principle was lately used in GR. Experimentally, it is now proved, that Einstein’s theories 4 Naturally,

unification do not apply to physics only. An interesting example of grand unification in mathematics is the fundamental theorem of calculus.

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(STR, GR)5 are more precise for describing our visible universe than classical mechanics and Newton’s law of universal gravity. Ehrenfest started to communicate some of his ideas about the dimensionality of the universe with Lorentz in 1917. The result was a paper “In What Way Does It Become Manifest in the Fundamental Laws of Physics that Space Has Three Dimensions?” [Ehrenfest, 1918] where he discuses stability of circular and elliptical orbits of planets orbiting around some central body. In his paper he supposes that orbital motion of planets in any dimensionality of space d > 2 is carry out in a plane and shows that only case with d = 3 admits existence of any stable orbital motions of planets (Bertrand’s theorem). Other physicists like Whitrow and lately Tangherlini continued in his steps. They made use of higher dimensions to demonstrate the validity of Kant’s observation that laws of nature are unique in three spacial dimensions only (see [Tangherlini, 1963]). The fact, that Nature (or God) has chosen simpler mathematical structure with four dimensions over more complicated one with just three dimensions is, roughly speaking, what Einstein called “principle of elegance” and it is a main motivation of higher dimensional physics. If we accept the logic of extra dimensions, the next step would be add another, fifth dimension.

1.3

First tries

In the times of advent of the GR, there were just two so called field theories in physics: gravity and electromagnetism. Weak and strong interactions were not yet known. In addition, since STR, it was suggested that unification of electricity and magnetism has something to do with higher dimensions, with the idea that the universe is four-dimensional spacetime instead of three dimensional space with time as a parameter. In hindsight, it is not surprising that some physicists started to look for a theory in five dimensions in order to unify gravity and electromagnetism. In this section we will state first serious attempts of incorporating one ED into physical theories.

1.3.1

Gunnar Nordström scalar theory of gravity

The Finnish physicist Gunnar Nordström (1881–1923) first consider modification of Newton’s theory of gravity by adding another spacial dimension. Unfortunately, in his theory the gravity was described as a scalar field which was eventually superseded by Einstein’s tensor theory. So Nordström’s contribution was virtually lost to history before it was excavated in the 1980’s by A. Pais’s biography of Einstein (Subtle is the Lord [Pais, 1982]) and by translation of his original paper (for modern version see for example [Ravndal, 2004]). 5 Technically

there is only one Einstein’s theory because STR ⊂ GR.

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Figure 1.3: First who introduced a fifth dimension into physics (from left): Gunnar Nordström, Theodor Kaluza and Oskar Klein.

Nevertheless, his idea represented an important milestone in the investigation of gravity and unification of physics. Some modern theories of gravitation based on superstrings or supergravity contain additional spacial dimensions and also additional scalar fields. Their physical meaning is not always clear, but they are useful in cosmology as candidates for inflation fields and possibly Dark energy. Here, we will therefore give a short summary of the main ideas behind Nordström’s contributions in a bit more modern formulation. The first gravitational theories were inspired by electrostatics, because Newton’s law of universal gravity has practically the same form as Coulomb’s law. The crucial difference, is that the gravitational force is always attractive while the electrostatic force can be also repulsive, depending on the relative signs of the charges. Introducing a scalar gravitational field theory is indeed a tempting idea. This can be very easily achieved within Lagrange formalism so the particular Poisson equation would be: ∇2 φ = 4πGρ ,

(1.1)

where G is gravitational constant, the role of the electromagnetic potential is replaced by a scalar gravitational field φ and ρ is mass density. A lack of a negative sign in front of the scalar field provides that gravitational force is always attractive. The Nordström’s first try was simply to generalize this equation to relativistic form: φ = 4πGρ ,

(1.2)

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where d’Alembertian operator = −∂2t +∇2 . Nordström was aware of an earlier Abraham’s theory of gravity and so he knew that straightforward generalization of Newton’s force law: dUµ dτ

= ∂µ φ ,

where τ is a proper time, would not work. Instead he proposed: d φUµ = ∂µ φ , dτ

(1.3)

(1.4)

In his version, all particles would fall in a gravitational field independently of their masses so the weak equivalence principle is satisfied. The immediate problem with this theory was that the mass density ρ should be a scalar. Therefore in his second scalar theory of gravitation he proposed that a mass density ρ is proportional to the trace of the energymomentum tensor Tµν . After figuring out the proportionality constant between inertial and gravitational masses he found a field equation φφ = 4πG Tr Tµν .

(1.5)

Since for electromagnetic field, the trace of corresponding energy-momentum tensor is zero, light would not interact with gravitation field and would not bend. The possibility that spacetime has more than four dimensions was first contemplated by Nordström in order to unify his scalar theory of gravitation with Maxwell’s theory of electromagnetism. He imagined that our four-dimensional spacetime is enriched by an extra, fifth dimension and that both interactions are part of five-dimensional Maxwell theory 1 L = − Fab F ab − ja Aa , (1.6) 4 where Latin indices now take on five values. Electromagnetic vector-potential has the components ! φ a µ A = A ,√ (1.7) 4πG and contains the gravitational potential φ . Similarly, the five-vector current density is √ J a = J µ , ρ 4πG . (1.8) Nordström realized that current conservation ∂a J a = 0 implies that ρ is independent of the fifth coordinate ∂5 ρ = 0 (1.9) and then observed that when the vector-potential is also made independent of the fifth coordinate, the five-dimensional wave equation separates into standard four-dimensional

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Maxwell wave equation and into his wave equation for gravitational field (1.2). This independence is now know as “cylinder condition” and was rediscovered by Kaluza and Klein several years later. As we shall see in the next sections, in Kaluza–Klein theory, the cylinder condition is somewhat arbitrary put in. Meanwhile in Nordström theory, it is direct consequence of charge conservation. For more in depth analysis of Einstein communication with Nordström see [Norton, 1993, 2007]

1.3.2

Weyl field theory

An interesting attempt of unification in physics was proposed by German mathematician Hermann Weyl (1885–1955). He sent to Einstein a copy of his article, entitled Gravitation and Electricity, [Weyl, 1918, Einstein, 1918] which attempted to unify those two interactions by a very interesting idea: “A true infinitesimal geometry should, however, recognize only a principle for transferring the magnitude of a vector to an infinitesimally close point and then, on transfer to an arbitrarily distant point, the integrability of the magnitude of a vector is no more to be expected than the integrability of its direction”. In a nutshell, he suggested generalization of a Riemann geometry so that not only direction but also magnitude of vector could change during parallel transporting from one point to another point on a manifold. This was achieved by a new (conformal) symmetry of a theory: gµν → λ(x)2 gµν , (1.10) where λ(x) is an arbitrary smooth function of a position. Consider parallel transport of two vectors U µ and V µ from point P to infinitesimally close point P 0 . According to Weyl’s hypothesis, the relationship between the scalar products of vectors at each point is given by: gµν + dgµν (U µ + dU µ ) (V ν + dV ν ) = (1 + dφ) gµν U µ V ν , (1.11) where filed φ, distinguishes this geometry from Riemann’s. The technical details are not important6 . Let us just remind here the conformal connection operating in Weyl’s theory: nµ o αβ 6 For

µ

= Γαβ +

1 µ µ δα Aβ + δβ Aα + gαβ g νµ Aν , 2

(1.12)

more detailed and more recent treatment of Weyl’s ideas see nice essay On the Origins of Gauge Theory by Callum Quigley [Quigley, 2003].

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µ

where Γαβ is the usual Christoffel connection, derived from the metric and a new field Aα ≡ φ,α ensures the conformal covariance (1.10) of Weyl’s geometry. Weyl postulated that of all the possible conformal connections in the equivalence class, only Aα being four potential of Electromagnetic (EM) field had any relation to reality. The Einstein’s replay on Weyl’s work was truly remarkable: “If light-rays were the only means by which metrical relationships in the neighborhood of a space-time point could be determined, there would indeed be an indeterminate factor left in the line-element ds (as well as in the gik ). This ambiguity is removed, however, when measurements obtained though (infinitesimally small) rigid bodies and clocks are taken into account. A timelike ds can be measured directly by a standard clock whose world-line is contained in ds. Such a definition of the line-element ds would become illusory only if the assumptions concerning ’standard lengths’ and ’standard clocks’ was not valid in principle; this would be the case if the length of a standard rod (resp. speed of a standard clock) depended on its history. If this were really so in Nature, chemical elements with spectral-lines of definite frequency could not exist and the relative frequency of two neighbouring atoms of the same kind would be different in general. As this is not the case it seems to me that one cannot accept the basic hypothesis of this theory, whose depth and boldness every reader must nevertheless admire.” In Weyl’s response to this criticism of his theory, we can glimpse the difference between two brilliant minds of that era; the difference between genial mathematician and brilliant physicist: “The geometry developed here is, it must be emphasized, the true infinitesimal geometry. It would be remarkable if in Nature there was realized instead an illogical quasi-infinitesimal geometry, with an electromagnetic field attached to it.” In 1929, Weyl published the paper Electron and Gravitation [Weyl, 1929] where he introduced a slight modification of his original proposal and derived electromagnetism from what we now call gauge principle. To this mathematical intrusion into physics, the famous physicist Wolfgang Pauli (1900–1958) wrote in a letter to Weyl: “I admire your courage; since the conclusion is inevitable that you wish to be judged, not for your success in pure mathematics, but for your true but unhappy love for physics.” Many agreed with Einstein’s last words but also with Pauli’s remark. Nevertheless, nowadays, Weyl’s proposal forms the foundation of gauge theory (see for example [Quigley, 2003]). The success of Weyl’s paper from 1929 can be seen in Pauli’s response after reading it.

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0 “Here I must admit your ability in Physics. Your earlier theory with gik = λgik was pure mathematics and unphysical. Einstein was justified in criticizing and scolding. Now the hour of your revenge has arrived.”

The gauge theory, nowadays a bedrock of particle physics is also important in theories with more spacial dimensions like is string theory. Consequently, the braneworld scenario, which will be the main topic of the next chapter, has something to do with the gauge principle (for interesting review see [Giveon and Kutasov, 1999]).

1.3.3

Kaluza–Klein theory

In the present subsection we shall consider several key ideas concerning what is now know as Kaluza–Klein theory. Going back to 1919 we found ourself in a situation when Maxwell electromagnetism was well established theory and Albert Einstein recently formulated GR. Theodor Kaluza (1885–1954) a brilliant German mathematician and physicist was able to unite these two theories by postulating another spacial dimension [Kaluza, 1921]. The reason why he chose spatial character of extra dimension can be seen in the next chapter. Kaluza moved into pure five-dimensional Einstein gravity defined by field equations: Gˆ AB = 0 ⇐⇒ Rˆ AB = 0 ,

(1.13)

where we have used a hat symbol to distinguish five-dimensional quantities from ordinary four-dimensional7 . Also indexes in capital Latin letters runs over a set (0, 1, 2, 3, 4) . The number four is sometimes omitted for historical reasons and number five is used instead (0, 1, 2, 3, 5) . Also the additional coordinate xˆ4 is sometimes denoted as y so xÂ = (xµ , y) . Eqs. (1.13) can be derived by varying the five-dimensional Einstein–Hilbert action Z p 1 S =− (1.14) Rˆ −gˆ d5 x , 16πGˆ with respect to metric. Here Gˆ is five-dimensional gravitation constant and gˆ = detgÂB . Kaluza split a general five-dimensional metric as: ! gµν + κ42 φ2 Aµ Aν κ4 φ2 Aµ , (1.15) gÂB = κ4 φ2 Aν φ2 where κ4 is a constant, Aµ some arbitrary vector field and φ an unknown scalar field. In modern view the quantities would be described as the spin 2 graviton, the spin 1 photon 7 This

sort of notation is usually adopt in literature. In this chapter we will also, purely out of nostalgia, use this notation. However, in following chapters we will use more recent kinds of notation to distinguish higher-dimensional quantities.

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and the spin 0 dilaton field. Dilaton was especially problematic back in 1919 and therefore set to equal constant. Nowadays the dilaton plays significant role in superstring theory which makes Kaluza’s idea more appealing. The five-dimensional quantities, Einstein tensor, Ricci tensor, Ricci scalar and Christoffel symbols resp. are formally defined exactly as in four dimensions: 1 Gˆ AB = Rˆ AB − Rˆ gÂB , (1.16) 2 C C C ˆD C ˆD Rˆ AB = ∂C ΓÂB − ∂B ΓÂC + ΓÂB ΓCD − ΓÂD ΓBC , (1.17) AB Rˆ = Rˆ AB gˆ , (1.18) 1 C gˆ CD (∂A gˆDB + ∂B gˆDA − ∂D gÂB ) . (1.19) ΓÂB = 2 In order to justify a separation of metric, vector and scalar field in (1.15) and also ensure their proper transformation properties, Kaluza proposed, as we have already mentioned in previous section, rather artificial restriction, the so called cylinder condition: ∂gÂB = 0. ∂y

(1.20)

This condition provides that coordinate xˆ4 would have no direct physical observations. It was Oskar Klein [Klein, 1926] who lately y '$ showed that this condition had a physical '$ '$ '$ 6 meaning and made Kaluza’s theory less arR tificial. He assumed that ED have circu@ R @ &% &% &% &% lar topology. The coordinate y has peri- xµ odic identification, 0 ≤ y ≤ 2πR , where R would be the radius of the circle S 1 . The Figure 1.4: Illustration of Klein idea. space has therefore topology R4 ×S 1 . Illustrative picture of this idea is given in Fig. 1.4. The cylinder condition is simply the answer to the problematic question: “If the fifth dimension exists, why we can not see it?”. Both Kaluza and Nordström just proposed that physics takes place only on four-dimensional hypersurface of five-dimensional universe. They did not provide explanation for the ED nor physical justification of cylinder condition and both issues were viewed as an useful mathematical trick. With the metric (1.15) the action reads: ! Z R 1 2 1 ∂α φ∂α φ 4 √ µν . (1.21) S = − d x −gφ + φ Fµν F + 16πG 4 6πG φ2 where Fµν ≡ Aν,µ − Aµ,ν and we have integrated with respect to the coordinate y and identified Gˆ G≡ , (1.22) L

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R where L = dy is a length of ED. The unification of general relativity and electromagnetism is now obvious, if we suppose that φ = 1. The interpretation of unusual scalar field (dilaton) φ was for Kaluza problematic. As we can see in the action it satisfies Klein–Gordon equation, developed lately by Oskar Klein8 . The fact that the action (1.14) leads to (1.21) is sometimes referred as Kaluza–Klein miracle. Four-dimensional matter (electromagnetic radiation) has been shown to arise purely from the geometry of empty five-dimensional spacetime. The goal of all subsequent Kaluza– Klein theories has been to extend this success to other kinds of matter. In retrospective view of the gauge theory, the split: xµ → xµ ;

x5 → x5 + f (xµ ) ,

(1.23)

is actually no miracle at all, because no other U (1) gauge invariant quantities can be formed. The remarkable fact that electromagnetism is contained in five-dimensional GR now receives a clear explanation: the internal U (1) gauge invariance of electromagnetism is a manifestation of the local coordinate invariance of the small circle in y. The absence of matter source in Eq. (1.13) is Kaluza’s first key assumption that the universe in higher dimensions is empty. This idea is inspired by Einstein’s dream to explain matter by pure geometry. Indeed, modern Kaluza–Klein theories have three main features: 1. They treat nature as pure geometry. 2. They are minimal extensions of general relativity. 3. They are cylindrical. This means that the gravitational and electromagnetic fields are completely contained in a five-dimensional Einstein tensor, that there are no modification to Einstein’s theory other than the assumption of one more spacial dimension and that physics do not depends on it. In fact, any modern unified field theory to be called Kaluza–Klein-like has to satisfied these three conditions. A complete Kaluza–Klein theory would achieve not only a unification of gravity and electromagnetism, but also of matter and geometry. However, to unify more than just gravity and electromagnetism, it seems that one has to abandon Einstein’s goal of geometrization of physics, at least in the sense of point 1 above. Rather than explaining the “base wood” of four-dimensional matter and forces as manifestations of the “pure marble” 8 Kaluza’s

identification of the electromagnetic potential is not quite the right one, because he chooses it equal to gµ5 (up to a constant), instead of taking the quotient gµ5 /g55 . This does not matter in his further analysis, because he considers only the linearized approximation of the field equations.

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of geometry in higher dimensions, one has essentially been driven to invent new kinds of wood. It is useful to keep in mind that the new coordinates y need not necessarily be measured in meters, or even be spacelike (in regard to their metric signature). An example which violates both of these expectations was introduced by Minkowski, who showed that the successes of Maxwell’s electromagnetic theory and Einstein’s special relativity could be understood geometrically if time, along with space, were considered part of a fourdimensional spacetime manifold where x0 = ict. In original Kaluza–Klein theory, the extra dimension is real, space-like and compact. However, if one gives up realness of the additional dimensions, one can consider projective theories or PUFT in abbreviation [Schmutzer, 1995]. Or one can ignore compactness of additional dimensions and study noncompactified theories (see for example Space, Time, Matter theory [Wesson et al., 1996]). The Cylinder Condition The third feature of Kaluza–Klein theory, the cylinder condition, means that physics do not depend on EDs. Mathematically, we have to drop any derivations with respect to them. Kaluza and also Nordstöm were well aware of it, but they did not provide any physical interpretation of cylinder condition or even additional dimensions. Both viewed them as a useful mathematical tool to unify gravity and electromagnetism. Oscar Klein, the author of the name “cylinder condition”, recognize that it can be interpreted as a compactification of an additional dimension to a circle of some very small radius R. This idea introduced new features into the theory (quantization of charge) and also some justification of higher dimension. We cannot see them because they are too small. It was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980’s to the possible theory of everything ten-dimensional superstrings. Several modifications of Kaluza’s idea were suggested by Jordan, Einstein, Bergmann over the following years. Naturally, it was not extended to more than five dimensions until theories of the strong and weak interactions were developed. An alternative to Klein’s compactification mechanism was introduced by Veblen and Hoffman in 1931. Assuming that the cylinder is homogeneous and that the metric is flat, one writes a complete set of wave functions of a free massless particle on this cylinder (e.g., solutions to five-dimensional Klein–Gordon equation), µ

φn = eipµ x , einz/R .

(1.24)

ˆ = 0. φ

(1.25)

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pµ p µ =

n2 . R2

(1.26)

Hence, inhomogeneous modes with n > 0 carry energy of order 1/R, and they cannot be excited in low energy processes. Below the energy scale 1/R, only homogeneous modes with n = 0 are relevant, and low energy physics is effectively four-dimensional. As a bonus, the expansion of the electromagnetic field into Fourier modes explains the quantization of electric charge. This aspect of the theory has had to be abandoned, however, as the charge-to-mass ratio of the higher modes did not match that of any known particles. Nowadays, elementary charges are identified with the ground state Fourier modes only, and their small mass is attributed to spontaneous symmetry-breaking. The early 1980’s lead to a revitalization of these ideas partly due to the realization that a consistent string theory will necessarily include extra dimensions. Besides already mentioned deviation of electron mass and electric charge ratio from experimental data, Edward Witten and Steven Weinberg have proved the so-called Weinberg– Witten no-go theorem. Therefore, for Kaluza–Klein-like theories, it is difficult to obtain massless fermions chirally coupled to the Kaluza–Klein gauge fields in (1 + 3) dimension. KKtheories are therefore usually not in agreement with the standard model of particles. Despite all of its drawbacks, the theory was never completely abandoned. Over many decades physicists were trying to improve the Kaluza’s and Klein’s idea, resulting in many new theories, like string theory [Blagojevic, 2001] For instance, in 1963 De Witt suggested to incorporating the non-Abelian SU (2) gauge group of Yang and Mills into a Kaluza–Klein theory of (4 + d) dimensions. A minimum of three extra dimensions were required. This problem was picked up by others and solved completely by the time of Cho and Freund in 1975 [Overduin and Wesson, 1997].

1.3.4

Simple KK black holes

Regarding the topic of thesis, we will here provide small introductions to some ideas concerning KK black holes, because even though the original five-dimensional KK-theory is not a realistic theory of nature, it gives insight into more sophisticated theories such as string theory and supergravity [Rasheed, 1995]. Kaluza–Klein theory has also been of interest recently in connection with noncommutative differential geometry [Landi et al., 1994] which may be viewed as Kaluza–Klein theory in which the ED is taken to be a discrete set of points rather than a continuum. The simplest black hole solution in KK-theory is the product of the Schwarzschild solution with a circle of circumference L. 2M 2M −1 2 2 2 ds = − 1 − dt + 1 − dr + r dΩ2 + dy 2 . r r 2

(1.27)

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This gives a black string solution for which cross-sections of the event horizon have topology S 2 × S 1 . In the decompactified limit L → ∞ it gives a black string of infinite length. Another simple solution describes a black hole of topology S 2 localized on the KK circle. Such solutions are not known explicitly. Solutions describing black holes with radius much smaller than L have been constructed perturbatively [Harmark, 2004]. Larger black hole solutions, with some interesting limitations on mass, have been constructed numerically [Kudoh and Wiseman, 2005]. Even though the pure, higher-dimensional Schwarzschild solution is stable against linearized gravitational perturbations, black strings suffer from the Gregory–Laflamme (GL) instability [Gregory and Laflamme, 1993]. The relation (1.22) between the four and five dimensional Newton’s constant helps explain a puzzle about black hole entropy. In five dimensions, the entropy is proportional to the three dimensional “area” of the event horizon. Since this clearly increases with the length of the circle, one might think that the five dimensional black string can have many more microstates than the four dimensional black hole. But in fact the five dimensional entropy is the same as the four dimensional entropy: A5 A4 L A4 = = = S4 . (1.28) 4G 4Gˆ 4Gˆ In fact, the same area law for black hole entropy is again and again discovered in any dimensions with by the means of several different methods. This is an example of the universality principle, similar to e.q. critical points in phase transitions. S5 =

1.3.5

Nonrotating charged black holes

We have seen that neutral black holes can be obtained by simply taking a product of the Schwarzschild solution and a circle. The same is not true for charged black holes. These are not simply related to the Reissner–Nordström solution, since if Fµν , 0, it acts as a source for φ and φ cannot be constant. However one can use a trick. From the five dimensional standpoint, one can generate a Maxwell field by a simple boost in the y direction with rapidity parameter α. If a five dimensional spacetime is invariant under translations around a small circle, it can be viewed as an effective four dimensional spacetime coupled to certain matter fields. As an example of such trick we provide[Horowitz and Wiseman, 2011]: 1 gµν dxµ dxν = −f dt 2 + dr 2 + R2 dΩ2 , f where f =p

r − 2M r 2 + r(q − 2M)

,

q R = r r 2 + r(q − 2M) , 2

(1.29)

(1.30)

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where a “charge” q is identified with rapidity α by expression: q = 2M cosh2 α .

(1.31)

This is a black hole with event horizon at r = 2M and singularity at r = 0. The dimensional reduction and boost has not changed the location of the horizon and singularity. Unlike Reissner–Nordström, these black holes do not have a regular inner horizon. The singularity is spacelike, like the Schwarzschild solution. Magnetically-charged solutions can be obtained by a duality rotation, because the field equations are invariant under F →∗ F, φ → −φ. The Eqs. (1.16) admit electrically charged black hole solutions with two horizons r+ ≥ r− (see for details [Duff, 1995]). Interestingly the electric charge q and ADM (Richard Arnowitt, Stanley Deser and Charles W. Misner) mass m are related to r± . The extreme r+ = r− case represents nonsingular magnetic monopole. It is remarkable that Kaluza–Klein theory has a nonsingular magnetic monopole solution since e.g. Einstein–Maxwell theory does not (Of course the five dimensional metric is still a vacuum solution to Einstein’s equation. The magnetic charge arises in the reduction to four dimensions.) This is interesting especially because Kaluza–Klein monopoles have been shown to correspond to exact solutions in string theory and N = 8 supersymmetric theory in five-dimensions [Rasheed, 1995].

1.3.6

General Kaluza–Klein black holes

The advantage of Kaluza–Klein theory is that it admits Kerr–Newman family of black hole solutions without requirement of source terms on the right hand side of Einstein Equations. Pure five-dimensional gravity couplet with U (1) Maxwell field reproduces most general axi-symmetric black hole solutions to Einstein–Maxwell theory. To obtain a rotating, electrically charged KK solution, one can repeat the construction in the previous section starting with the Kerr metric. In other words, one takes the product of Kerr and a line, boosts along the line and then compactifies the extra dimension. These solutions are all stationary, axisymmetric, and invariant under translations in y. This solutions were found by a solution generating technique which uses hidden symmetries of Einstein’s equations with Killing fields. It turns out that black holes with both electric and magnetic charge (sometimes called dyonic black holes) have a nonsingular extremal limit with nonzero horizon area. An excellent review concentrating on the general relativity, rather than particle physics side of the KK theories is represented in [Overduin and Wesson, 1997].

1.3.7

Induced-matter theory

An interesting follow up to Kaluza–Klein theories is Wessons’s Space-Time-Matter theory, better know as induced-matter theory (IMT) [Wesson et al., 1996]. The so-called fundamental constants like c , G and h have as their main purpose the transformation of physical

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dimensions. Thus, a mass can be in GR regarded as a length and so on. One of the famous Einstein goals was to give to all quantities a geometrical interpretation and so transmute the “base wood” of physics to the “marble” of geometry. An early attempt to achieve this goal was made by Kaluza and Klein, but via applying restriction to geometry (cylindricity and compactification). The induced-matter theory is another attempt to achieve Einsteins dream through higher dimensions, but without assumptions of the topology of additional dimensions. A related idea due originally to Ernst Mach (1838–1916) is that the local properties of matter should be connected with the properties and distribution of other material in the universe The science of Mechanics [Mach and McCormack, 2009]. The central thesis of IMT is that four-dimensional Einstein equations: Gµν = κ42 Tµν ,

(1.32)

are a subset of their five-dimensional version Rˆ AB = 0 ,

(1.33)

with an effective or induced 4D energy-momentum tensor. A question that naturally arises is why it is possible to provide a geometrical description of matter using only one extra dimension, whereas other theories of extended spacetime involve several and even then run into well-know problems. The answer lies in generality of IMT approach. It regards macroscopic matter as being geometrically “induced” by a mechanism that locally embeds our four-dimensional spacetime in a Ricci-flat five-dimensional manifold. Moreover, Wesson asserted that only one extra dimension should be sufficient to explain all the phenomenological properties of matter. One interesting point is that the matter created by this process is of a very general kind. Due to Campbell–Magaard theorems [Seahra and Wesson, 2003] any energy-momentum tensor can be produced by choosing the appropriate embedding.

1.3.8

Brans–Dicke theory

Setting φ to a constant gives some nice mathematical results that correspond to our common view of our four-dimensional world. But why do this? If one does not set equal to a constant, one can arrive at a special case of Brans–Dicke theory which is a competing theory with general relativity which says that the gravitational interaction is mediated by a scalar field as well as a tensor field. Kaluza set φ to exactly one. But he could choose any constant and then conformally transformed its influence out of the action. In literature we often find five-dimensional

1.4. THESIS OUTLINE

21

metric in form after this procedure: gÂB = φ

−1/3

gµν + κ42 φAµ Aν κ4 φAµ κ4 φAν φ

! .

This metric leads to conformally rescaled action ! Z R 1 1 ∂α φ∂α φ 4 √ µν S = − d x −g + φF F + . 16πG 4 µν 6πG φ2

(1.34)

(1.35)

where the gravitational part of the action has desired conventional form. But if one does not set φ = constant, then Kaluza’s five-dimensional theory contains besides electromagnetic effects a Brans–Dicke-like scalar field theory, as becomes clear when one considers the case in which the electromagnetic potentials vanish, Aµ = 0 [Brans and Dicke, 1961, Brans, 1962]. Notice, that setting Aµ = 0 has influence on physics only because we impose cylinder condition. Without this condition, the identification Aµ = 0 would be just setting a particular set of coordinates (gauge).

1.4

Thesis outline

The first chapter dealt with some historical introductions to ED. The topic of this thesis, multidimensional black holes, is too large to be even tackled. Therefore the main focus will be concentrated around braneworld black hole of Kerr–Newman structure, found in [Aliev ˇ 2005, Dadhich et al., 2000b] and its subsequent development in work and Gümrükçüoglu, we have done [Stuchlík et al., 2011a, Blaschke and Stuchlík, 2016]. Our focus lies in five-dimensional black holes for several reasons. Five-dimensional braneworld physics can be important as a low energy limit for M theory. Also, fivedimensional black hole spacetimes have particular significance in gauge-gravity correspondence [Witten, 1998]. Finally, five-dimensional black hole is the first, and therefore simplest, example of the spacetime allowing far richer mathematical structures [Emparan and Reall, 2008]. Many results from four-dimensional GR turn out to be specific to this dimensions. In second chapter we will review some basic points about higher-dimensional black holes in braneworld scenario inspired by sting theory. In third chapter We will follow work [Shiromizu et al., 2000] to find induced Einstein equations on the brane via Gauss–Godazzi equations. Then finally, in chapter four, we will show several solution of induced metric for one additional dimension in bulk and set focus on braneworld Kerr–Newman black hole.

1.5

Summary

Nordström and Kaluza–Klein theory were the first significant attempts to go to more dimensions than are observed. They succeeded in unifying general relativity and electromag-

1.5. SUMMARY

22

netism into a single theory. Klein introduced compactification as a explanation of cylinder condition. An ED is a so small circle that observable physics would almost certainly never see any effects of it. Many modifications were made that included a projective approach and noncompactified approach. Therefore, it is so far impossible to say which one really describes reality, as all three are consistent with experiment. The mathematical trick of throwing on the electromagnetic potential (and a scalar field) onto the metric is far away from a complete theory describing the universe. Kaluza–Klein theory become a precursor to theories with more fundamental ambitions, like string theory. It would not be surprise, if a theory of everything is ever found, that it lives in more than four dimensions. The historical and philosophical implication of extra dimensions, especially as the notion matured in the minds of our culture, is a very interesting and vast subject. For its vastness we will just provide a small sample of interesting books, reviews and lecture notes, that we would like to pay homage to by this thesis: [Greene, 1999] [Halpern, 2004] [O’Raifeartaigh, 1997] [Wesson, 2006] [Chru´sciel et al., 2010]

Chapter

2

Braneworld models What do undead string theorists absolutely crave? Branes! Anonymous author

I

n this chapter, we will provide some elemental motivations for implementing extra dimensions (EDs) into the physical theories. We will proceed with some examples of asymptomatically flat higher-dimensional black holes. These solutions are important, because if we are indeed living in some sort of braneworld scenario, then the properties of very small black holes would be influenced by the existence of EDs. This influence could be in principle measurable in particle colliders in the near future. Then we are going to review some basic braneworld models and discuss their possible black hole solutions.

2.1

Motivation

In the previous chapter, we have seen that unification of physics is a main motivation of introducing EDs. A major obstacle in the amalgamation of gravity and particle physics is that general relativity is a classical theory whereas the Standard Model (SM) is quantum in nature. Direct efforts to quantize gravity have met with results that are incalculable or without direct physical interpretation [Carlip, 2001]. One of the candidates for such an unified theory is string theory [Polchinski, 1998]. It is a quantum theory where particles are represented as vibrational modes of elementary one-dimensional strings. Matter (fermions) and gauge bosons are represented as vibrations of open strings, while vibration modes of a closed string represents graviton, particle of gravity. Another fundamental aspects of string theory is that it needs EDs to work and in particular, it dictates in how many dimensions wants to live (10,11 and 26). 23

2.1. MOTIVATION

24

Since its inception, five distinct superstring (strings + supersymmetry) theories had been discovered and it is believed that they all arise as different limits of a single theory, M-theory [Witten, 1995]. The original bosonic string theory is constructed in 26 dimension. Superstring theories usually live in 10 dimensions and enigmatic M-theory is enriched by additional one spacial dimension, which is “large” in the sense we will discus shortly. Any higher-dimensional theory have to hide the EDs from everyday physical experience via some mechanism, like e.g. already discussed Kaluza–Klein (KK) compactification. String theory uses several strategies, one of which we are going to discuss in more details. Braneworld models arise as a direct inspiration from dynamical solutions in string theory, particularly, soliton interpretation of two-dimensional strings endowed with Dirichlet boundary conditions (D-branes). First work, which truly launched interest in braneworld models was done by the Russian physicists V. A. Rubakov and M. E. Shaposnikov in 1983. In their aptly named article Do we live inside a domain wall? [Rubakov and Shaposhnikov, 1983], they used topological defects to model the braneworld. In their scenario all particles are trapped inside a potential well, sufficiently narrow in the direction of EDs and flat in ordinary space directions. This means that propagation in EDs is energetically unfavored resulting in effective three-dimensional appearance of our universe. This kind of matter localization on the brane is called dynamical. Generally speaking, in braneworld models, the universe corresponds to a (3+1)-dimensional hypersurface, a 3-brane, living in some higher-dimensional bulk spacetime with “large” EDs. The meaning of “large” differs in various versions and can go from a size as large as a millimeter to infinity. We will refer to the surface along which some of the particles are localized as branes, which is short for a membrane that could have more spatial dimensions than the usual two. The electromagnetic, weak and strong forces are sensitive to the existence of EDs. If, for example, gauge bosons were allowed to propagate in the EDs, their interactions would be modified beyond any acceptable phenomenological limits unless the size of the EDs was smaller than 10−16 cm. In the framework of braneworld this problem is solved by several methods. We have already mentioned, compactiffication in KK theory and dynamical approach by Rubakov and Shaposnikov. Another, quite popular solution is the assumption that all SM particles are restricted to live on the brane and that only gravity can escape from the brane. This rather straightforward idea, similar to dimension reduction depicted in Abbot’s Flatland is directly inspired by string theory (see Fig. 2.1). The width of the 3-brane along the EDs is considered to be effectively zero or of the order 10−16 cm. The 3-brane, playing the role of our four-dimensional universe, is then embedded in the higher-dimensional spacetime (Bulk), where only gravity can freely propagate. Before we dive into some particular examples of braneworld models, let us contemplate several basic ideas concerning EDs. At high enough energies, Einstein’s GR breaks down, and will be superseded by a

2.1. MOTIVATION

25

Bulk

rane b D closed string

ing r t s open

Figure 2.1: The SM model particles are represented by open strings and are attached to the D-brane, while the gravitons represented by the closed strings move freely in the bulk.

quantum theory of gravity, or more generally, by a theory of quantum gravity. The classical singularities infesting GR in e.g. gravitational collapse and in the hot big bang will hopefully be removed by this next theory. Most analyses of EDs are within the context of quantum field theory. Following ideas suggested in Wesson’s Induced-Matter theory [Wesson et al., 1996], let us consider a massless particle moving in five-dimensional space. Assuming that five-dimensional Lorentz invariance holds, the momentum of the particle is 0 = p2 = gab pa pb .

(2.1)

This relation can be recast into the form pµ pµ = ±p52 ,

(2.2)

where the ± depends of the nature of fifth dimension and our choice of metric signature. This expression is very similar to well-known normalization condition for particles satisfying 4D Lorentz invariance: pµ pµ = m2 , (2.3) where m is mass of the particle. To make this interpretation, we have to choose right sign in 5D metric to avoid tachyons. Tachyons are well-known to be detrimental in most, even classical models, because they can cause problems with causality [Recami and Giannetto, 1985]. This simple model suggest that we should choose the space-like solution. Generally, it turns out that to avoid tachyons we must always choose additional dimensions to be

2.1. MOTIVATION

26

space-like. Indeed, considerations of just two times lead to many difficulties [Dvali et al., 2000c]. Assuming one space-like ED, consider a massless scalar field in 5D flat spacetime. The field is a solution to 5D Klein–Gordon (KG) equation: ∂µ ∂µ − ∂2y φ(xµ , y) = 0 ,

(2.4)

where we have again introduced y for label the ED. By the virtue of compactification of extra dimension or other explanation of cylinder condition, the separation of variables is: X φ= φn (xµ )χn (y) . (2.5) n

Puting this back to the 5D KG equation, we find a set of equations X χn ∂µ ∂µ + m2n φn = 0 ,

(2.6)

n

where we have used identification (inspired by orthonormality of χn ) ∂2n χn = −m2n χn .

(2.7)

The final result looks like an infinite set of equations for a distinct 4D scalar fields φn with masses mn . This is called a KK tower. The fields χn can be thought of as the wave functions of the various KK states in the 5th dimension. Because in most circumstances we assume that EDs are compact, for a flat 5th dimension of length L, the analysis of boundary conditions tells us that the KK masses are given by mn = n/L (see Eq. (1.26)). The KK masses are therefore large if the size of the ED is small. This leads no natural explanation why we have not seen any EDs in nature, or colliders. The collider energies are very small and the corresponding KK states are then too massive to be produced in low energy phenomena. Note, that it is useful to be mindful of the most frequently used notations for constant appearing in RHS of Einstein equations or in the action. In natural units (c = ~ = 1) it reads: 8π κd2 ≡ 8πGd+1 ≡ d−1 , (2.8) Md+1 where Gd+1 is Newton’s gravitational constant and Md+1 is a Planck mass in (d + 1)dimensions. Here (d − 1) is just power of Md+1 . Armed with notation and elemental ideas about braneworld models, let as continue to focus on main thesis topic – black holes.

2.2. HIGHER-DIMENSIONAL BLACK HOLES

2.2

27

Higher-dimensional black holes

In string theories, a variety of classical solutions corresponding to black holes exists, to name just a few [Garfinkle et al., 1991, Callan et al., 1989, Horowitz and Strominger, 1991]. However, for the purpose of this thesis we will concentrate on braneworld black holes, because these models has evolved and far outgrown initial string theory motivations and has proved as a better testing ground for new possibilities in cosmology, astrophysics, and quantum gravity. In braneworld scenarios, sufficiently small black holes (with respect to bulk dimensions) can be well-approximated by asymptotically flat black holes in higher dimensions. Possibility of creation of such “mini” black holes in colliders were discussed in [Giddings and Thomas, 2002, Dimopoulos and Landsberg, 2001]. For these reasons, we are going to provide a small review of higher-dimensional black hole spacetimes that are solutions to the Einstein equations. But first, we are going to start with 4-dimensional Kerr–Newman black hole and write down Carter’s equations, because this particular equations will play an important role in the next chapter and the rest of the thesis. Also, let us note that we are going to focus on simple thin brane models. For more information on thick braneworld scenarios see [Dewolfe et al., 2000, Emparan et al., 2001]

2.2.1

The Kerr–Newman metric

It is assumed that a sufficiently massive star will, at the end of its life, undergo gravitational collapse to form a singularity hidden under event horizon (cosmic censorship hypothesis). The end point of this collapse is thought to be described by the Kerr solution [Kerr, 1963], which is an asymptotically flat, stationary solution to the vacuum Einstein equations. Electrically charged version of the collapse is described by Kerr–Newman (KN) metric. It is a solution to the Einstein–Maxwell equations [Newman et al., 1965]. Using geometric units (c = G = 1) and Boyer–Lindquist coordinates, the exterior region of this spacetime can be described by the metric: ! 2Mr − Q2 ds = − 1 − dt 2 − Σ 2

+

2a(2Mr − Q2 ) Σ sin2 θ dt dϕ + dr 2 + Σ dθ 2 Σ ∆ ! 2 2Mr − Q 2 2 r 2 + a2 + a sin θ sin2 θ dϕ 2 , Σ

(2.9)

where ∆ = r 2 − 2Mr + a2 + Q2 , 2

2

2

Σ = r + a cos θ .

(2.10) (2.11)

M is the mass parameter of the spacetime, a = J/M is the specific angular momentum of the spacetime with internal angular momentum J and Q parameter representing charge.


28

Continuous isometries of KN spacetimes are generated by Killing vector fields K a , satisfying Lg K a = 0 where Lg is a Lie derivative with respect to Kerr metric gab . The Kerr and Kerr–Newman metric admits two such vector fields, an asymptotically timelike ∂t which generates time translations, and an asymptotically spacelike ∂ϕ which has closed orbits and generates rotations around the axis of symmetry. Existence of Killing vectors provides two constant of motion. Remarkably, in [Carter, 1968, 1973b] was found another constant of motion K, quadratic in particle momentum. Lately in [Walker and Penrose, 1970] was traced the existence of K to the existence of an additional (hidden) symmetry, described by a rank-2 Killing tensor, satisfying the generalized Killing equation ∇(a Kbc) = 0 .

(2.12)

This was generalized into arbitrary dimensions. So if a (d + 1)-dimensional metric has at least d commuting Killing vectors, corresponding to d Noether symmetries then its associated Hamilton–Jacobi (HJ) equation has separable solutions. If it has fewer Killing vectors, but its HJ equation is still separable, then this separability can be linked to a hidden phase space symmetry, related to the existence of a higher-rank Killing tensor K satisfying the generalized Killing equation ∇(a Kb1 ...bp ) = 0 .

(2.13)

The direct generalization of Killing tensor, especially useful in higher-dimensional GR, is Killing–Yano form and so called Principal tensor. For extensive review see [Frolov et al., 2017]. Carter’s separated, first order differential equations of the geodesic motion, for neutral test particles take form:

dr dw dθ Σ dw dϕ Σ dw dt Σ dw Σ

where

p = ± R(r),

(2.14)

p = ± W (θ),

(2.15)

PW aP + R, 2 ∆ sin θ 2 (r + a2 )PR = −aPW + , ∆

(2.16)

= −

(2.17)


29

˜ R(r) = PR2 − ∆(m2 r 2 + K), W (θ) = (K − a2 m2 cos2 θ) −

(2.18)

Pw sin θ

2 ,

PR (r) = E(r 2 + a2 ) − aΦ,

(2.19) (2.20)

PW (θ) = aE sin2 θ − Φ.

(2.21)

Along with the conservative rest energy m, three constants of motion related to the spacetime symmetries has been introduced: E is the energy (related to the time Killing vector field ∂t ), Φ is the axial angular momentum (related to the axial Killing vector field ∂ϕ ) and K is the constant of motion related to total angular momentum. For the constant K related to the Killing tensor field we have an expression: K = Kab pa pb ,

(2.22)

where pa is a four-momentum of the test particle. This constant is in the literature usually replaced by the constant Q˜ = K − (E − aΦ)2 ,

(2.23)

since for the motion in the equatorial plane (θ = π/2) there is Q˜ = 0. Generally, these equations can be integrated and expressed in terms of the hyper-elliptic integrals [Misner et al., 1973, Kraniotis, 2005, 2007]. The Carter equations can be also generalized to the motion in the Kerr–Newman-de Sitter spacetimes [Carter, 1973b, Stuchlík, 1983, Stuchlík and Hledík, 1999, Kraniotis, 2005, 2014].

2.2.2

Schwarzschild–Tangherlini black holes

There are now many known, exact, black hole solutions to the higher-dimensional vacuum Einstein equations. The first such solution was found by Frank R. Tangherlini [Tangherlini, 1963]. He generalized the Schwarzschild solution ((2.9) with a = Q = 0) to arbitrary dimension d, finding that the metric appears very similar to the 4-dimensional version: r ds = − 1 − 0 r 2

d−3 !

r dt + 1 − 0 r 2

d−3 !−1

dr 2 + r 2 dΩ2d−2 ,

(2.24)

where dΩ2d is the metric on a unit d-sphere given by 2 2 dΩ2d = dθd−1 + sin2 θd−1 dθd−2 + sin2 θd−2 · · · + sin2 θ2 dθ12 + sin2 θ1 dϕ 2 . . . (2.25) and r0 > 0 some arbitrary parameter.


30

By applying Gauss law in the d-dimensional spacetime, we obtain the following relation between the horizon radius rh and the mass M of the black hole [Argyres et al., 1998]

rh = √

1 M πM4 M4

!

1 d−3

1  d−1  d−3  8Γ 2     d − 2  .

(2.26)

For d > 4, the relation between rh and M is not linear, and it the fundamental Planck scale M4 appears in the expression of the horizon radius. Let us also note that although the metric (2.24) is very similar in four and higher dimensions, for d > 4 there are no bounded timelike geodesics, corresponding to orbits of massive bodies about the central mass.

2.2.3

Myers–Perry black holes

In a d > 4 dimensional spacetime there are more independent planes of rotation. Therefore, one might expect a higher-dimensional generalization of the Kerr black hole to be specified by this number of independent angular momenta Ji . Such a direct generalization was derived in [Myers and Perry, 1986]. When only one angular momentum is nonzero, the solution can be described by the following line-element ds = 1 − 2

µ 2aµ sin2 θ Σ 2 dt + dtdϕ − dr 2 − Σdθ 2 d−5 d−5 ∆ Σr Σr 2 ! 2 a µ sin θ − r 2 + a2 + dϕ 2 − r 2 cos2 θdΩ2d−4 , Σr d−5

where ∆ = r 2 + a2 −

µ r d−5

,

Σ = r 2 + a2 cos2 θ ,

(2.27)

(2.28)

In case of d = 4 the last term dΩ20 is understood as zero, and metric reduces to Kerr solution. The mass and angular momentum (transverse to the rϕ plane) of the black hole are given by (d − 2)Ad−2 2 µ, J = Ma , (2.29) M= 16πGd+1 d −2 with Gd+1 being the (d + 1)-dimensional Newton’s constant, and Ad the area of a ddimensional unit sphere given by 2πd/2 . (2.30) Ad = Γ [d/2] When more than one angular momentum is turned on, the form of the solutions is more complicated [Emparan and Reall, 2008, Vasudevan et al., 2005].


31

Hawking, Hunter and Taylor-Robinson [Hawking et al., 1999] have given a generalization of the five-dimensional MP solution to include a cosmological constant. What form should cosmological constant take in higher dimensions is contemplated in [Gibbons et al., 2005].

2.2.4

Other solutions

Naturally, the Myers–Perry solution is not the only black hole solution of the five dimensional vacuum Einstein equations. Beyond the MP black hole there are the black rings with one more phase of rotating black holes, if one restricts to phases with a single angular momentum that are in thermal equilibrium. This is the black Saturn phase consisting of a central MP black and one black ring around it, having equal temperature and angular velocity. If one abandons the condition of thermal equilibrium there are many more black Saturn phases with multiple rings as well as multi-black ring solutions. For example see [Reall, 2003] Black rings The Black rings, discovered by Emparan and Reall [Emparan and Reall, 2002], are asymptotically flat black hole solutions of the vacuum Einstein equations, with an event horizon of spatial topology S 1 × S 2 . They represent a qualitatively different solution to the Myers– Perry regular, asymptotically flat, solutions of the vacuum Einstein equations. It represents spinning black ring with only one non-zero angular momentum (it rotates around the S 1 direction only). The properties and structure of this spacetime are described in detail in the review article [Emparan and Reall, 2006] It was interesting to wonder whether these new black holes have the same angular momentum and mass as MP solutions and therefore might violate higher-dimensional equivalents to uniqueness theorems. It was not obvious whether or not this was true, but for a small range of angular momentum per unit mass J1 /M , there were both MP and two different black ring solutions. The black hole uniqueness problem was studied in [Emparan and Reall, 2008]. For MP and black rings, the black hole uniqueness, at least in the familiar sense, fails. Pomeransky–Sen’kov black hole The Pomeransky–Sen’kov black hole [Pomeransky and Sen’kov, 2006] is a doubly-spinning generalization of the black ring to include rotation around the S 2 as well as around the S 1 . It was found using solution generating techniques for higher-dimensional Weyl solutions.


32

This two parameter family of black rings does admit an extremal limit. It is more complicated than black ring, but reduces to particular version of that solution in a limit. Various authors have studied this solution [Kunduri et al., 2007, Elvang and Rodriguez, 2008] and there is a more general version, with conical singularities [Morisawa et al., 2008]. Black Saturn Another interesting feature of higher dimensional GR is the existence of a variety of asymptotically flat solutions to the vacuum Einstein equations with multiple horizons. Here we only mention interesting, five-dimensional black Saturn [Elvang and Figueras, 2007]. It represents planet-like composition of an S 3 black hole horizon as a “planet” and a S 1 × S 2 black ring as a “ring”. In five dimensions there are also black di-rings [Izumi, 2008, Evslin and Krishnan, 2009] and bicycling black rings [Elvang and Rodriguez, 2008]. In even higher dimensions, few solutions are known exactly, but it is generally believed that there exists a progressively richer and richer family of black hole solutions as the number of dimension grows.

2.2.5

ADD braneworlds

Brane

A basic requirement of any higher-dimensional theory of gravity is that it reproduces Newton’s law of gravity in the appropriate limit. At first glance, braneworld gravity in d spacial dimensions fails at this first requirement as we might expect the nature of gravity to be ∼ 1/r d−1 4D force law, rather than the familiar ∼ 1/r 2 behavior. There are several different strategies how to solve this problem. In 1998 the Arkani-Hamed, Dimopoulos and Dvali (ADD) proposed braneworld model [Arkani-Hamed et al., 1998] for solving the hierarchy problem without supersymmetry or techED. nicolor. The hierarchy problem can be simply put: Why there are at least two seemingly fundamental energy scales in nature? Why they differ by Figure 2.2: Illustration of ADD 16 orders in magnitude?. The solution to hierarchy braneworld model with n = 1. problem in ADD model originated the interest of the physics of TeV-scale EDs. The key idea is that the gravitational and Standard Model (gauge) interactions are united at the electroweak scale, Mew , which is considered as the only fundamental short


33

distance scale in the Universe [Antoniadis et al., 1998, Arkani-Hamed et al., 1999]. This unification is possible because model contains number of EDs, large in comparison to Plank scale, M4 ∼ 1019 GeV, which is now just an effective scale whose value is a consequence of our four-dimensional viewpoint. To see how this can work, a simple Gauss’ law calculation and dimensional argument analysis is sufficient. They approach is directly inspired by Kaluza–Klein theory. Suppose that the n EDs are compactified with the same radii R. We again see the creation of “image masses” as in the KK case. The force on a test particle at a “small” distance will be higher-dimensional in nature. At “very large” distances the resolution between discrete image masses disappears and they will appear as continuous strings of uniform mass density. By demanding that R is chosen to reproduce the observed M4 , we have R ∼ 10

30 n−17

1TeV cm Mew

!1+2/n .

(2.31)

The static weak field limit of the field equations leads to the d-dimensional Poisson’ equation, whose solution is the gravitational potential: V (r) ∼

κd2 r d−2

.

(2.32)

If the length scale of the EDs is L, then on scales r L, the potential is d + 1-dimensional. By contrast, on scales large relative to L, where the extra dimensions do not contribute to variations in the potential, V behaves like a 4-dimensional potential, i.e., V ∼ L−d r −1 . This means that the usual Planck scale becomes an effective coupling constant, describing gravity on scales much larger than the EDs, and related to the fundamental scale via the volume of the extra dimensions: d−1 d−3 M42 = Md+1 L .

(2.33)

If the ED volume is Planck scale, i.e., L ∼ M4−1 , then Md ∼ M4 . But if the ED volume is significantly above Planck scale, then the true fundamental scale Md+1 can be much less than the effective scale M4 ∼ 1019 GeV. In this case, we understand the weakness of gravity as due to the fact that it “spreads” into EDs. The case d = 1 gives R ∼ 1013 cm implying deviation from Newtonian gravity over the size of the Solar system. Such a result is excluded by direct observation. The case d = 2 gives R ∼ 0.1 − 1 mm. Recent torsion balance experiments [Kapner et al., 2007] have shown the accuracy of Newton’s law down to ∼ 50 µm and therefore suggests that n = 2 case is excluded also. ADD approach to hierarchy problem is justified by observation that electroweak scale Mew is experimentally verified, meanwhile the Plank scale is not. Indeed, our interpretation of MP l as a fundamental energy scale (where gravitational interactions should become


34

strong) is based on the assumption that gravity is unmodified over the 29 orders of magnitude between where it is measured at ∼ 50 µm down to the Planck length ∼ 10−33 cm! The non-trivial task in any explicit realization of ADD framework is localization of the SM fields. A number of ideas for such localizations have been proposed in the literature, both in the context of trapping zero modes on topological defects[Rubakov and Shaposhnikov, 1983] and within string theory. It has been understood for some time that a quantum field theory can contain topological defects of various types and dimensionality, which can have low-energy particle-like modes trapped on them. It therefore seems plausible that our 3-brane is a 4-dimensional defect in a higher dimensional field theory, and the SM particles are some of the light modes trapped on the defect. The idea that the Standard Model gauge and matter fields propagate in the bulk has also been considered [Davoudiasl et al., 2001]. Experimental searches for EDs have been carried for example on at the TEVATRON [Abazov et al., 2005]

2.2.6

Randall–Sundrum braneworlds

In the 1998 the ADD model offered an explanation to hierarchy problem using large EDs that eliminated one disparity in scales by creating another, namely that between the weak scale and the compactification scale Mew 1/L . To avoid this unpleasant feature a new approach to the problem within the braneworld framework was introduced in 1999 by Randall and Sundrum [Randall and Sundrum, 1999a,b]. Randall and Sundrum (RS) models has two types, but both have some common features. Our 4-dimensional universe is again confined to a brane. But bulk space has only one ED and is endowed with cosmological constant Λ5 . Therefore bulk is no longer flat but has a curvature. Motivation to this model can be traced to heterotic M-theory [Hoˇrava and Witten, 1996a,b] and domain wall [Lukas et al., 1999a,b]. Some parallel with type IIB string theory can also be seen, where the RS model can be roughly associated with the near horizon limit of a stack of D3-branes. This led to testing the idea of ADS/CFT (see [Maldacena, 1999, Duff and Liu, 2000, Gubser, 2001])

2.2.7

Randall–Sundrum I

The first of the model version [Randall and Sundrum, 1999a] utilizes a non-factorisable fivedimensional geometry, in which the four-dimensional metric of our universe is dependent on position of the brane φ in the bulk. The ED is compactified with Z2 symmetry on two branes siting in an orbifold S1 /Z2 fixed points. Our brane has negative tension λvis and the second (“hidden”) brane has positive tension λhid (see Fig. 2.3). Bulk has negative cosmological constant Λ5 and so it is an anti-de Sitter space.


35

Z2

Z2

2

Λ5 = − 6k κ5

6

6

t

t

φ=0 λhid = + 6k κ5

φ=π λvis = − 6k κ5

Figure 2.3: Illustration of RS braneworld model I.

If it is assumed that a non-factorisable solution to the above configuration exists that respects four-dimensional Poincare invariance in the coordinates on the branes, xa , then the bulk metric in Gaussian normal coordinates, (xa , φ) based on the brane at φ = 0 is given by ds2 = e−2krc |φ| ηab dxa dxb + rc2 dφ2 , (2.34) where rc is radius of compactification, k is the adS curvature of the bulk and ηab is Minkowski flat metric. There is necessarily a fine tuning between the brane tensions and the bulk cosmological constant: Λ5 = −

k2 , κ5

λvis = −λhid =

6k , κ5

(2.35)

where κ5 is defined in relation (2.8). In the coordinates of Eq. (2.34) all slicing of constant φ have metrics proportional to the Minkowski vacuum. They are related to one another by a “warp” factor. This warp factor is also illustrated in Fig. 2.3. Essentially, the warp factor has effect of “squeezing” or “localizing” gravity around the brane so that it spreads mostly in the four directions parallel to the brane and usual Newton’s law of gravity is recovered, despite that gravity can escape to an ED. Unfortunately, separation of two branes has to be stabilized by some special process, otherwise the branes have tendency to escape into infinity (see [Charmousis et al., 2000, Pilo et al., 2000, Goldberger and Wise, 1999]).


36

Z2

6

t

2

Λ5 = − 6k κ5

y

-

λbrane = 6k κ5 Figure 2.4: Illustration RS braneworld model II.

2.2.8

Randall–Sundrum II

In their second braneworld model [Randall and Sundrum, 1999b] Randall and Sundrum discarded the second brane. Model is therefore stable but do not solve the hierarchy problem. Roles of the branes is now reversed. The visible brane has positive tension λvis ≡ λbrane = 6k/κ5 . Disregarding of second brane can be considered as sending compactification radius to infinity (rc → ∞). Stability issue is therefore solved, because tendency of two branes in RSI model to escape to infinity is here advantage. This leaves a single, positive tension brane residing in a five-dimensional adS bulk which is Z2 symmetric in the brane, see Fig. 2.4. The metric for this model is given by ds2 = e−2k|y| ηµν dxµ dxν + dy 2 ,

(2.36)

where y = rc φ. Brane is located at y = 0 and 0 ≤ y < ∞. However, since the extra dimension is now infinite in extent, it is no longer possible to neglect the massive Kaluza–Klein modes as there will be no energy gap between the zero mode and the lowest mode with mass. Randall and Sundrum demonstrated that if the extra dimension is warped then it may be infinite in extent and still consistent with known gravitational law. An analysis of the Newtonian potential experience on the brane due to a particle of


37

mass M reveals it is of the form G4 M 2 V (r) = 1+ 2 2 , r 3k r

(2.37)

where the short scale correction is due to the combined effect of the continuum of suppressed massive Kaluza–Klein modes [Garriga and Tanaka, 2000]. Since it accurately reproduces both the Newtonian force law and correct four-dimensional structure of the graviton propagator in the low-energy limit, the Randall–Sundrum single brane model has become something of a paradigm of extra-dimensional model building. The fine tuning between the bulk and brane cosmological constants is still present and exactly cancels the effect of the bulk negative energy density on the brane, so that the induced metric is Minkowski. No mechanism by which this fine tuning may be realized is suggested in the model either.

2.2.9

Black holes in RSII

It was soon realized that the form of the Randall–Sundrum metric (2.36) allows a convenient generalization [Brecher and Perry, 2000]. It is possible to replace the Minkowski metric with any four-dimensional metric that is Ricci flat (R = 0) and the resulting fivedimensional metric is also a solution to the five-dimensional Einstein equations with negative cosmological constant. This presents an obvious candidate for a braneworld black hole. The first attempt to find a black hole on an RS brane was that of Chamblin, Hawking, and Reall [Chamblin et al., 2000]. They replaced ηab with metric of 4-dimensional Schwarzschild black hole and obtained “black string” solution: 2 ds2 = e−2k|y| dsSch + dy 2 ,

(2.38)

2 where dsSch just represents Schwarzschild metric. By taking this brane-based approach they guaranteed that the correct description of a four-dimensional black hole is found on the brane. However, from a five-dimensional bulk perspective this metric can be considered as the foliation of the five-dimensional spacetime by Schwarzschild slices with magnitude scaled by the warp factor. From the higher-dimensional viewpoint it is clear that the Schwarzschild singularity is not localized on the brane, but projects out along the extra dimension. This braneworld black hole is a slice of an infinite five-dimensional black string. Collapse to a black hole in the Randall–Sundrum brane-world scenario was studied by [Garriga and Sasaki, 2000] [Casadio and Harms, 2000]


2.2.10

38

DGP braneworld

Shortly after it was clear that RS models could be compatible with Newtonian gravity in a warped spacetime, Dvali, Gadabaze and Porrati (DGP) demonstrated that such a situation was also possible with infinite flat EDs [Dvali et al., 2000a]. They examined the competition between the bulk five-dimensional curvature scalar, R5 and the corresponding intrinsic curvature scalar on the brane, R, in an action of the form Z Z √ √ 1 1 5 S= −g5 R5 d x + −gR d4 x , (2.39) κ5 κ4 where gµν (xα ) is induced by gab (xα , 0) . This model correctly reproduces the 1/r form of Newtonian potential at small distances, with corrections corresponding to logarithmic repulsions. However, the tensor structure of the graviton propagator is found to correspond to massive 4-dimensional graviton. The difference is in incorrect coefficient in front of graviton propagator 1/3 instead of correct 1/2. This may seem small, but massive graviton would have another one polarization state which would manifest as an additional four-dimensional scalar field. This scalar field would have to been included in the four-dimensional effective theory and would add another attractive force. This scalar-tensor gravity causes that DGP model predict anomalous results for experimental tests of general relativity. For example, bending of light by the Sun would be just 3/4 of its measured value [Dvali et al., 2000b] No dedicated investigations of possible extensions of the black hole properties in the framework of the DGP model have been done. Partly because even in the Schwarzschild case, which should translate into an axially symmetric metric in the DGP model, the result would be much more complicated than the five-dimensional Schwarzschild metrics.

2.2.11

Summary

In this chapter we have discus several braneworld scenarios and some higher-dimensional black holes. The motivation to study properties of black hole solutions of higher-dimensional vacuum Einstein equations are many, but within the framework of braneworld scenario, the main reason can be found in ability of microscopic black holes to feel influence of the bulk dimensions. Therefore the knowledge of precise properties of microscopical black holes can be crucial in a search for higher dimensions. The idea of localization of SM gauge physics to hypersurfaces embedded in higherdimen-sional bulk is in particular important in string theory where open strings (SM particles) are attached at each end to some D-brane. Braneworld models represent an interesting realization of string theory ideas, especially in domain of black hole physics and cosmology, where these ideas might become testable in foreseeable future. Given the possible lowering of the fundamental scale of gravity down to levels achievable in the LHC, there is a chance of production of microscopic black holes in ground-based experiment. The chance of finding quantum gravity effects has never been so close.


39

It it important to stress that we have outlined just a few, most known, examples of braneworld models. The numerous other brane configurations have been explored in the literature [Lykken and Randall, 2000, Gregory et al., 2000, Kogan et al., 2001, Karch and Randall, 2001] and many extensive review articles have been written [Rubakov, 2001, Dick, 2001, Maartens and Koyama, 2010, Kanti, 2004, Rizzo, 2004].

Chapter

3

A General Covariant Approach I can say without a doubt that there are an infinite number of universes. Some are just like our own...but for one or two significant events, exactly the same. Lex Luthor

W

ith the limitations of the simplest methods, outlined in the previous chapter, a more general brane-based method to find the black hole solutions is needed. Because not all gravitational phenomena can be treated in the weak field limit, we require a concrete understanding of the true non-perturbative nature of gravity on the brane. In this chapter we will follow the first and very simple, non-perturbative approach to braneworld gravity provided by [Shiromizu et al., 2000]. The basic idea is to project the five-dimensional Einstein equations onto the brane to obtain 4-dimensional effective (induced) Einstein equations. These equations, which we are going to derive, resemble the standard Einstein equations, but with correction terms arising from the non-local bulk gravitational effects.

3.1

Gauss–Godazzi equations

The induced metric hab (the first fundamental form) is the metric tensor defined on a hypersurface Σ (submanifold) which is described by the metric tensor on a larger manifold into which the hypersurface is embedded. Here we are using abstract index notation and follow notation used in Wald’s book [Wald, 1984]. In abstract index notation, the indexes serves mainly as a remainder of the tensorial structure and should not be understand as components of object in question. The larger manifold is in the framework of braneworld 41

3.1. GAUSS–GODAZZI EQUATIONS

42

models usually called bulk. The induced metric is defined by following relation: hab =

∂y c ∂y d g , ∂xa ∂xb cd

(3.1)

where y a are coordinates on hypersurface and xa are coordinates on bulk spacetime. The functions y a (x) define the embedding into the higher-dimensional manifold. Similarly, if we are interested in hypersurface with only (n − 1) dimension, we can use relation: hab = gab ± na nb , (3.2) where na is a unit normal vector to hypersurface and sign ± depends on the type of the vector na , whether it is time-like or space-like vector. The last equation can be difficult to understand in abstract index notation. The left hand side has to be understand as induced metric projected back onto bulk space-time, e.i. hab ≡

∂xc ∂xd h . ∂y a ∂y b cd

(3.3)

With just an affine structure on the manifold one would not be able to get an induced connection. In differential geometry there is lemma that says: Let (M, gab ) be a spacetime and let Σ be a smooth spacelike hypersurface in M. Let hab , denote the induced metric on Σ and let Da denote the derivative operator associated with hab . Then Da is given by the formula a ...a a e d ...d a e f Dc Tb 1...b k = hd1 . . . hdk hb1 . . . hbl hc ∇f Te11...el k , (3.4) 1

l

1

k

1

l

where ∇a is the derivative operator associated with gab and hab ≡ ∂y a /∂xb is a projection operator. The expression (3.3) can be recast with the use of projective operator into the new form: hab = hca hdb gcd .

(3.5)

Here we leave up to the reader the span of upper and lower indexes. However, in concrete coordinate system it is useful to somehow distinguish indexes on bulk space and indexes on hyperspace to avoid confusion. As an example we provide calculation of induced metric tensor on S 2 sphere embedded into R3 flat manifold. As a projective operators we use transformation (see Fig. 3.1) from Cartesian coordinates to spher- Figure 3.1: system.

Spherical coordinate


43

ical coordinates: x = r sin θ cos ϕ ,

(3.6)

y = r sin θ sin ϕ ,

(3.7)

z = r cos θ ,

(3.8)

where the radius of sphere r > 0 is a constant. By (3.1) we find that only two non-zero components of metric hab are: hθθ = r 2 ,

hϕϕ = r 2 sin2 θ .

(3.9)

We can also easily convince ourselves that Eq. (3.4) provides correct form of derivation operator defined by induced metric: f

c Da ωb ≡ ∂a ωb − Γab ωc = hdb ha ∂f ωd .

(3.10)

The Riemann tensor with no torsion is formally defined by commutation relation: [∇a ∇b ] ωc = Rdabc ωd ,

(3.11)

where ωd is arbitrary one form (dual vector) and ∇a is the connection associated with metric gab of the spacetime. It represents the difference between vector parallely transformed along the infinitesimally small closed loop and the original vector, at some point, Fig. 3.2. The definition (3.11) can be naturally used for defining Riemann tensor on hypersurfaces: [Da Db ] ωc =

(m) d

R

abc ωd ,

(3.12)

where Da is the connection associated to the induced metric hab on the hypersurface with dimension m. Furthermore, if we are interested in case where hyperspace is spacelike with Figure 3.2: Illustration of parallel transport on m = d − 1, it is possible to do following the sphere. decomposition. According to relation (3.4) we have: f g Da Db ωc = Da hdb hec ∇d ωe = ha hb hkc ∇f hdg hek ∇d ωe f

= ha hdb hec ∇f ∇d ωe + hec Kab nd ∇d ωe + hdb Kac ne ∇d ωe ,

(3.13)


44

where we have used extrinsic curvature tensor Kab ≡ hca ∇c nb with na being unit normal vector to the hypersurface. From the definition of extrinsic curvature tensor it is clear that its geometric meaning is a measure of difference between parallelly transported normal vector nb along some curve on the hypersuface and vector nb . The Eq. (3.2) with the definition of Kab shows to as a property hba hdc ∇b hed = Kab ne ,

(3.14)

which we have also used in Eq. (3.13). Now, the middle term on the right-hand side of Eq. (3.13) is symmetric with respect to a and b so in final relation for induced Riemann tensor it cancels. Furthermore, we have hdd ne ∇d ωe = −Kbe ωe .

(3.15)

Putting together all these results, we obtain Gauss equation: (n−1)

d

Rabc =

(n)

j f g

Rf gk ha hb hkc hdj − Kac Kbd + Kbc Kad ,

(3.16)

where we have given explicitly the dimensions of the Riemann tensors. Similarly, we can show that f g (n) d Da Kbc − Dc Kba = R ef g nd heb ha hc . (3.17) This equation is called Codazzi equation.

3.1.1

Israel’s formalism

Before we continue to develop braneworld energy-momentum tensor, let us do small caveat. Let us contemplate how do we deal with the question of discontinuous geometries. In the framework of GR, this jumps in geometry are directly translated into jumps in energy momentum tensor. A classical example is the connection between curvature of Schwarzschild exterior and interior solution, where along the boundary of the star surface there is a jump in energy density. W. Israel [Israel, 1966] developed a mathematical framework now called Israel formalism to handle these kinds of questions. Let us consider situation illustrated in Fig. 3.3, i.e., a (d + 1)−dimensional spacetime M separated into two different regions M + and M − with a common boundary Σ : Σ = ∂M + ∩ ∂M − .

(3.18)

In the interior of each regions M ± , Einstein equations are satisfied. Thus ± ± Gab = κd2 Tab .

(3.19)


45

Σ M−

M+ n

Figure 3.3: The hyperspace Σ divides the spacetime M into two regions M + and M −

The surface Σ can be space-like or time-like. The null surfaces are usually more complicated (see, e.g. ([Poisson, 2004])). Therefore the norm of unit normal vector n is: n · n = gab na nb ≡ ,

(3.20)

where values are ±1 and are determined by the signature of the metric gab . Using the extrinsic curvature of hypersurface Σ induced by the two different regions we can compare the two different geometries. It is clear that for hypersurface to make any sense, the metric induced by spaces M + and M − has to agree. In fact, we usually demand that metric tensor is continuous throughout the whole spacetime. However, the embeddings and so extrinsic ± curvature tensors Kab do not need to be the same. Using Gauss (3.16) and Coddazzi equations (3.17) and thin shall approximation, we find Lanczos equation [Grøn and Hervik, 2007]: [Kab ] − hab [K] = κd2 Sab ,

(3.21)

where K is trace of Kab and Sab represents energy-momentum tensor on the hypersurface Σ. We have also used the bracket operation defined as [X] ≡ X + − X − . By contracting the Lanczos equation and substituting the result into the same equation, we get: 1 2 h S , (3.22) [Kab ] = κd Sab − n − 1 ab where n stands for dimensionality of a hypersurface Σ . Together with the condition on induced metric (3.23) [hab = 0] , we have the so called Israel junction conditions.

3.2. MODEL WITH ONE EXTRA DIMENSION

3.2

46

Model with one extra dimension

Let us now concentrate on the simple model under braneworld framework, developed in [Shiromizu et al., 2000]. In this influential work our 4-dimensional universe is represented as a domain wall (3-brane) (M, hab ) embedded into five-dimensional bulk spacetime (V , gab ). The unit vector na is normal to M and the induced metric on M is defined by Eq. (3.2). They have chosen minus sign − because the normal vector na is considered to be space-like. Let us consider the Gauss Eq. (3.16) in the form suited for the braneworld model: (4) a R bcd

=

(5) g

R

a n c s nrs hg hb hr hd

+ 2K[ca Kbd] ,

(3.24)

and the Codazzi equation in contracted form: Da Kba − Db K =

(5)

Rrc nr hcb .

By contraction of Gauss equation we find: 1 (5) (4) (5) Gab = Rrs − grs R hra hsb + (5) Rrs nr ns hab + KKab − Kar Krb 2 1 − hab K 2 − K cd Kcd − Eãb , 2 where

g Eãb ≡ (5) Rcdrg nc nr hda hb .

(3.25)

(3.26) (3.27)

Using the five-dimensional Einstein equations (5)

1 Gab ≡ (5) Rab − gab (5) R = κ52 Tab 2

(3.28)

together with the decomposition of the Riemann tensor in n-dimensions into the Weyl curvature tensor 2 (n) ga[c (n)Rd]b − gb[c (n)Rd]a Cabcd =(n) Rabcd − n−2 2 (n) (3.29) R ga[c gd]b , + (n − 1)(n − 2) we obtain the 4-dimensional induced equations in the form: (4)

where

Gab

2κ52 r s 1 r r s = Trs ha hb + Trs n n − Tr hab + KKab − Kar Krb 3 4 1 − h K 2 − K cd Kcd − Eab , 2 ab g

c Eab ≡ (5) Cdrg nc nr hca hb .

(3.30) (3.31)


47

Finally, putting (3.28) into Codazzi Eq. (3.25) we find: g

Da Kba − Db K = κ52 (5) Trg nr hb ,

(3.32)

because it can be easily verified that gab na hbd = 0 .

(3.33)

So far we have not considered any particular symmetry nor particular form of the energy momentum tensor. For convenience, let us abandon abstract index notation and choose a coordinate ξ, such that the hypersurface ξ = 0 coincides with the brane world and nA dxA = dξ . This implies the five-dimensional metric in the form: ds2 = dξ 2 + hµν dxµ dxν .

(3.34)

Bearing the particular braneworld model and the thin shell approximation in mind, let us assume that the five-dimensional energy-momentum tensor has the form Tµν = −Λgµν + Sµν δ (ξ) ,

(3.35)

Sµν = −λhµν + τµν ,

(3.36)

where with nµ τνµ = 0 . The delta function δ (ξ) is here to ensure that ordinary matter exists solely on the brane. The Λ is the cosmological constant of the bulk spacetime and λ and τµν are the braneworld vacuum energy and the energy-momentum tensor, respectively. Note that λ is the tension of the brane in five-dimensions. Properly speaking τµν should be evaluated by the variational principle of the fourdimensional Lagrangian for matter fields because the normal matter except for gravity is assumed to be living only in the ξ = 0 brane, but let us continue in this simplified way. The singular behavior in the energy-momentum tensor is usually handled by the Israel’s junction conditions (3.22). For this particular model they read: h i hµν = 0 , h i 1 2 Kµν = −κ5 Sµν − hµν S . (3.37) 3 Now it is very useful to impose the Z2 -symmetry on this spacetime, with the brane as the fixed point. The particular usefulness can be seen in that, interestingly, the Z2 symmetry uniquely determines the extrinsic curvature Kµν of the brane in terms of the energy-momentum tensor 1 , κ2 1 Kµν = − 5 Sµν − hµν S . (3.38) 2 3 1 More

precisely, there should be another ± sign determining which side of the brane we are having in mind, but due to Z2 -symmetry imposed on the brane it can be omit.


48

Substituting Eq. (3.38) into Eq. (3.30), we obtain the gravitational equations on the 3-brane in the form: (4) Gµν = −Λ4 hµν + 8πG4 τµν + κ54 πµν − Eµν , (3.39) where Λ4 G4 πµν

1 2 1 2 2 = κ Λ + κ5 λ , 2 5 6 4 κ5 λ , = 48π 1 1 1 1 = ττµν − τµα τνα + hµν ταβ τ αβ − hµν τ 2 . 12 4 8 24

(3.40) (3.41) (3.42)

It should be noted that Eµν in the above is the limiting value at but not the value exactly on the brane. Induced equations resembles the conventional Einstein equations in 4 dimensions. In fact, the Einstein equations can be recovered by taking the limit κ5 → 0 while keeping G4 finite. Nevertheless, there are some important differences. As can be easily seen, the existence of Newton’s gravitational constant G4 strongly relies on the presence of the vacuum energy λ. In other words, it becomes impossible to define Newton’s gravitational constant during an era when the distinction between the vacuum energy and the normal matter energy is ambiguous. Furthermore, we would have the wrong sign of G4 if λ < 0. The πµν term, which is quadratic in τµν could play a very important role, especially in the early universe when the matter energy scale is high. In addition to these features that have been pointed out previously, Eq. (3.39) contains a new term, Eµν . It is a part of the five-dimensional Weyl tensor and carries information of the gravitational field outside the brane. It is non-vanishing if the bulk spacetime is not purely adS. At the same time, it is not freely specifiable but is constrained by the motion of the matter on the brane. Let us show this feature now. Together with Eq. (3.38), Eq. (3.32) implies the conservation law for the matter, 1 µ Dµ Kν − Dν K = − κ52 Dν τµν = 0 . 2

(3.43)

Therefore the contracted Bianchi identities D µ (4)Gµν = 0 imply the relation between Eµν and τµν as D µ Eµν = K αβ Dν Kαβ − Dβ Kνα 1 1 4 αβ µ = κ τ Dν ταβ − Dβ τνα + τµν − hµν τ D τ . (3.44) 4 5 3 Thus Eµν is not freely specifiable but its divergence is constrained by the matter term. If one TT , and the longitudinal part, further decomposes Eµν into the transverse-traceless part, Eµν L , the latter is determined completely by the matter. Hence if the E TT part is absent, the Eµν µν equations will be closed solely with quantities that reside in the brane. However, as usually

3.3. CLASSES OF VACUUM SOLUTIONS ON THE BRANE

49

L part corresponds to gravitational waves or the case in the conventional gravity, the Eµν gravitons in five-dimensions, and they will be inevitably excited by matter motions and their excitations affect matter motions in return. This implies the effective gravitational equations on the brane are not closed but one must solve the gravitational field in the bulk L at the same time in general. The derivation of equations that govern the evolution of Eµν can be found in [Shiromizu et al., 2000]. In pure vacuum (τµν = 0 = Tµν ) we can choose the bulk cosmological constant to satisfy

1 Λ = − κ52 λ2 . 6

(3.45)

With this restrictions we can simplify Eq. (3.39) to Rµν = −Eµν .

(3.46)

Here Eµν carries the influence of nonlocal gravitational degrees of freedom in the bulk onto the brane, including the tidal and transverse traceless aspects of the free gravitational field. This tensor satisfies the divergence constraint (3.44) in form D µ Eµν = 0 .

(3.47)

For static solutions, Eqs. (3.46) and (3.47) form a closed system of equations on the brane. This leads to a very powerful result for using the 4-dimensional GR solutions to brane-world solutions in five-dimensional gravity [Dadhich et al., 2000b]: A stationary general solution with tracefree energy-momentum tensor gives rise to a vacuum brane-world solution in five-dimensional gravity . This result will be especially important in the following section.

3.3

Classes of vacuum solutions on the brane

The symmetry properties of Eµν imply that in general we can decompose it irreducibly with respect to a chosen 4-velocity field U µ as in [Maartens, 2001]: 1 4 (3.48) Eµν = −κ U Uµ Uν + hµν + Pµν + 2Q(µ Uν) , 3 where κ = κ5 /κ4 , the “dark radiation” term U = −κ4 Eµν U µ U ν is a scalar, Qµ = κ4 hαµ Eαβ a spacial vector and 1 β Pµν = −κ4 hα(µ hν) − hµν hαβ Eαβ (3.49) 3 a spacial, symmetric and trace-free tensor.


50

In a static vacuum, with U µ along the static Killing direction, we have Qµ = 0 and the conservation constrain for Eµν takes the form: 1 4 Dµ U + U Aµ + D ν Pµν = 0 , (3.50) 3 3 where Aµ = U ν Dν Uµ is the 4-acceleration. In the static, spherically symmetric case we choose 1 Aµ = A(r)rµ and Pµν = P (r) rµ rν − hµν , (3.51) 3 where A(r) and P (r) (the “dark pressure”) are some scalar functions of radial (areal) distance r , and rµ is a unit radial vector. We chose the static spherically symmetric metric on the brane in the form: ds2 = −eν(r) dt 2 + eµ(r) dr 2 + r 2 (dθ 2 + sin2 θdϕ 2 ) . Then the gravitational field equations and the vation equation in the vacuum take the form ! µ0 1 −µ 1 − + 2 −e 2 r r r ! 0 1 ν 1 + 2 − 2 e−µ r r r ! ν 0 2 ν 0 − µ0 ν 0 µ0 −µ 00 e ν + + − 2 r 2 ν0

(3.52)

effective energy-momentum tensor conser48πG4 U, κ4 λ 16πG4 (U + 2P ) , = κ4 λ 32πG4 (U − P ) , = κ4 λ U 0 + 2P 0 6P = − − , 2U + P r (2U + P ) =

(3.53)

where 0 ≡ d/dr . The system of structure equations (3.53) is not closed until a further condition is imposed on the functions U and P . By choosing some particular forms of these functions, several classes of static vacuum solutions can be generated in the framework of the braneworld model. As a first case we consider that the dark radiation U and the dark pressure P satisfy the constraint 2U + P = 0 . (3.54) Therefore the metric on the brane is 2G4 M + U0 24πG4 P0 − , (3.55) r κ4 λr 2 where U0 and P0 are integration constants. This form of the metric has been first obtained in [Dadhich et al., 2000b] with identifications: eν = e−µ = 1 −

U0 = 0 , 24πG4 P0 b = . κ4 λ

(3.56)


51

Here b is a dimensionless tidal charge parameter. This parameter will be important in the next chapter, where we will show how precisely can the value of this parameter influence some astrophysical properties of black holes. For b > 0 there is a direct analogy to the Reissner–Nordström metric (b ↔ Q2 , where Q2 is a dimensionless charge parameter). The intriguing new possibility that b < 0, which is impossible in the GR Reissner–Nordström case, leads to only one horizon. In the b < 0 case, the (single) horizon has a greater area than its Schwarzschild counterpart, so that bulk effects act to increase the entropy and decrease the temperature of the black hole. In general relativity, the electric field in the Reissner–Nordström solutions acts to weaken the gravitational field, and the same is true for the braneworld black hole with b > 0. By contrast, the b < 0 case corresponds to the opposite effect, i.e., bulk effects tend to strengthen the gravitational field. A second class of solutions of the system of Eqs. (3.53) can be obtained [Harko and Mak, 2004] by assuming that U + 2P = 0 . (3.57) For this class of solutions the projections of the Weyl bulk tensor are given by U = −2P =

κ4 λ . 72πG4 r 2

(3.58)

In the case of a vanishing dark radiation, U = 0, which also implies a vanishing dark mass Q = 0, the dark pressure P satisfies a Bernoulli type equation, given by 2 4r P P G4 M + 16πG κ4 λ dP 3P = 0, (3.59) + + dr r r 2 1 − 2Gr4 M with the general solution P= r3

1 !, q 2G4 M 16π C1 1 − r − κ4 λM

(3.60)

where C1 is an arbitrary constant of integration. Hence for U = 0 the metric tensor components are given by e

µ

eν

2G4 M −1 = 1− , r #2 " 2G4 M 2G4 M −1/2 16π 1− , = C2 1 − C1 − 4 r r κ λM

(3.61) (3.62)

where C2 is another arbitrary constant of integration. The deviations from the Schwarzschild geometry are very small. The standard GR results are recovered for κ4 λ → ∞, which gives C12 C2 = 1.


3.3.1

52

Braneworld Kerr–Newmann solution

Very similar approach was adopted by A. N. Aliev and Gümrükçüoglu [Aliev and Gümˇ 2005] to derive braneworld equivalent to Kerr metric. Formally this solution is rükçüoglu, also endowed with tidal charge parameter b. Again for b > 0 there is direct analogy to 4dimensional GR Kerr–Newman solution (2.9). Therefore it is sometimes called, braneworld Kerr–Newman solution or sometimes even tidal charged Kerr solution. The derivation of branewrold Kerr–Newmann metric is in the scope of methods we have already outlined. They used Gaussian normal coordinates and the lapse function and the shift vector in the spirit of ADM [Arnowitt et al., 2008]. The resulting line element, in the standard Boyer–Lindquist coordinates (t, r, θ, ϕ), is: ! 2Mr − b ds = − 1 − dt 2 − Σ 2

2a(2Mr − b) Σ sin2 θ dt dϕ + dr 2 Σ ∆

! 2Mr − b 2 2 + Σ dθ + r + a + a sin θ sin2 θ dϕ 2 , Σ 2

2

2

(3.63)

where ∆ = r 2 − 2Mr + a2 + b , 2

2

2

Σ = r + a cos θ .

(3.64) (3.65)

M is the mass parameter of the spacetime, a = J/M is the specific angular momentum of the spacetime with internal angular momentum J. The form of the metric (3.63) is again the same as that of the standard Kerr–Newman solution of the 4D Einstein–Maxwell equations, with squared electric charge Q2 being replaced by the tidal charge b [Misner et al., 1973]. Overall, we can separate the discussion for the metric (3.63) into three cases: a) The zero tidal parameter b = 0. In this case we are dealing with the standard Kerr metric. b) The positive tidal parameter b > 0. Where we are dealing with the standard Kerr– Newman metric. c) The negative tidal parameter b < 0. Where we are dealing with the non-standard Kerr–Newman metric. This case can be also considered, rather artificially, as the Kerr–Newman spacetime with imaginary charge parameter Q, so that its square is equal to tidal charge parameter b. Notice that in the braneworld Kerr–Newman spacetimes, the geodesic structure is relevant also for the motion of electrically charged particles, as there is no electric charge related to these spacetimes. On the other hand, the case (b) can be equally considered for the analysis of the uncharged particle motion in the standard electrically charged Kerr–Newman spacetime.


53

For simplicity we put in the following considerations M = 1. Then the spacetime parameters a and b, and the time t and radial r coordinates become dimensionless. This is equivalent to the redefinition when we express all the quantities in units of M: a/M → a, b/M 2 → b, t/M → t and r/M → r. Separation between the black hole and naked singularity spacetimes is given by the relation of the spin and tidal charge parameters in the form a2 + b = 1

(3.66)

determining the so called extreme black hole with coinciding horizons. The condition 0 < a2 +b < 1 governs black hole spacetimes with two distinct event horizons, while the condition a2 +b < 0 governs black hole spacetimes with only one distinct event horizon at r > 0. For a2 + b > 1, the spacetime describes a naked singularity. For positive tidal charges the black hole spin has to be a2 < 1, as in the standard Kerr– Newman spacetimes, but for negative tidal charges there can exist black holes violating the well know Kerr limit, having a2 > 1 [Stuchlík and Kotrlová, 2009]. The original derivation was done in Kerr–Schild coordinates. With substitutions ! 2 + a2 r dt = dx0 + − 1 dr , (3.67) ∆ a dϕ = dϕ˜ + dr , (3.68) ∆ ˜ + a sin(ϕ)) ˜ sin θ , x = (r cos(ϕ) (3.69) ˜ − a cos(ϕ)) ˜ sin θ , y = (r sin(ϕ)

(3.70)

z = r cos θ ,

(3.71)

the braneworld Kerr–Newman geometry can be transformed back into the Kerr–Schild form using the Carterian coordinates: ds2 = − (dx0 )2 + (dx)2 + (dy)2 + (dz)2

+

2 (2Mr − b)r 2 1 1 0 r(xdx + ydy) + a(xdy − ydx) − dx − zdz , r r 2 + a2 r 4 + a2 z 2

(3.72)

where r is defined, implicitly, by r 4 − r 2 (x2 + y 2 + z2 − a2 ) − a2 z2 = 0 .

(3.73)

This particular solution to effective braneworld Einstein equations, plays important role in the following chapter. There, we are going to study some effects of non-local gravity expressed by parameter b on the “classical” Kerr black hole and its singularity. This is therefore a natural stop for review, even thou there has been plenty of subsequent work developing the outlined braneworld model (see e.g. [Maartens and Koyama, 2010] and reference within).

3.4. SUMMARY

3.4

54

Summary

In this chapter we have outlined derivation of Einstein’s equations on a hypersurface via analysis of induced metric and extrinsic curvature. It turned out that closed form of these equations cannot be find just by consideration of mater and energy on the brane. Nevertheless, the effective form of vacuum Einstein equations was found to be of such simple form that for static case the equations are closed. We have used these equation to find some examples of braneworld black holes. Especially we have explicitly shown the braneworld Kerr–Newman metric, which properties we will develop in the next chapter.

Chapter

4

Braneworld Kerr–Newman spacetime What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school... It is my task to convince you not to turn away because you don’t understand it. You see my physics students don’t understand it... That is because I don’t understand it. Nobody does. Richard Feynman

I

n this chapter we follow our original work to show, in a coherent and unified way, some properties of Braneworld Kerr–Newman (KN) black hole and its singularity. We will use mainly our already published results [Stuchlík et al., 2011a, Blaschke and Stuchlík, 2016] and occasionally, partial unpublished ones, if we deemed them relevant to points in discussion. The individual papers are listed in chapter 6 and 7 as attachment. The information content of this chapter is superset of the published articles. The braneworld KN spacetime is determined by the metric (3.63). It is instructive to look at properties of general spacetime endowed with same symmetries. Therefore, we will start our discussion with circular geodetics of arbitrary axially symmetric and stationary spacetime. 55

4.1. GENERAL CIRCULAR GEODESIC MOTION OF AXIALLY SYMMETRIC ...

4.1

56

General circular geodesic motion of axially symmetric spacetime

In general stationary and axially symmetric spacetime with the Boyer–Lindquist coordinate system (t, r, θ, ϕ) and (− + ++) signature of the metric tensor, the line element is given by ds2 = gtt dt 2 + 2gtϕ dtdϕ + grr dr 2 + gθθ dθ 2 + gϕϕ dϕ 2 .

(4.1)

The metric (4.1) is adapted to the symmetries of the spacetime, endowed with the Killing vectors (∂/∂t) and (∂/∂ϕ) for time translations and spatial rotations, respectively. For geodesic motion in the equatorial plane (θ = π/2), the metric functions gtt , gtϕ , grr , gθθ and gϕϕ depend only on the radial coordinate r. So except the rest energy m, two integrals of the motion are relevant as Q˜ = 0: (4.2)

Ut = − E , U ϕ = L ,

where the 4-velocity Uα = gαν dxν /dτ , with τ being the affine parameter. In the case of ˜ asymptotically flat spacetime, we can identify at infinity the motion constant E = E/m as ˜ the specific energy, i.e., energy related to the rest energy, and the motion constant L = Φ/m as the specific angular momentum. The geodesic equations of the equatorial motion take the form (see, e.g., [Kovács and Harko, 2010]) Egϕϕ + Lgtϕ Egtϕ + Lgtt dϕ dt = 2 =− 2 , , (4.3) dτ dτ gtϕ − gtt gϕϕ gtϕ − gtt gϕϕ and grr

dr dτ

!2 (4.4)

= R(r),

where the radial function R(r) is defined by R(r) ≡ −1 +

E 2 gϕϕ + 2ELgtϕ + L2 gtt 2 gtϕ − gtt gϕϕ

.

(4.5)

4.1.1 E , L and Ω of circular geodesics For circular geodesics in the equatorial plane, the conditions R(r) = 0

and

∂r R(r) = 0

(4.6)

must be satisfied simultaneously. These conditions determine the specific energy E, the specific angular momentum L and the angular velocity Ω = dϕ/dt related to distant observers,

4.1. GENERAL CIRCULAR GEODESIC MOTION OF AXIALLY SYMMETRIC ...

57

for test particles following the circular geodesics, as functions of the radius and the spacetime parameters in the form gtt + gtϕ Ω E = ±q , 2 − gtt + 2gtϕ Ω + gϕϕ Ω gtϕ + gϕϕ Ω L = ∓q , 2 − gtt + 2gtϕ Ω + gϕϕ Ω q −gtϕ,r ± (gtϕ,r )2 − gtt,r gϕϕ,r Ω = , gϕϕ,r

(4.7) (4.8)

(4.9)

where the upper and lower signs refer to two families of solutions. To avoid any misunderstanding, we will refer to these two families as the upper sign family, and the lower sign family. At large distances in the asymptotically flat spacetimes, the upper family orbits are corotating, while the lower family orbits are counterrotating with respect to rotation of the spacetime. This separation holds in the whole region above the event horizon of the Kerr–Newman black hole spacetimes, but it is not necessarily so in all the Kerr– Newman naked singularity spacetimes – in some of them the upper family orbits become counterrotating close to the naked singularity as demonstrated in [Stuchlík, 1980]. Using the spacetime line element of the braneworld rotating spacetimes given by (3.63), with the assumption of M = 1, we obtain the radial profiles of the specific energy, specific axial angular momentum and the angular velocity related to infinity of the circular geodesics in the form [Dadhich and Kale, 1977]: √ r 2 − 2r + b ± a r − b E = p , (4.10) √ r r 2 − 3r + 2b ± 2a r − b √ √ r − b r 2 + a2 ∓ 2a r − b ∓ ab L = ± , (4.11) p √ r r 2 − 3r + 2b ± 2a r − b 1 . (4.12) Ω = ± 2 r √ ±a r−b

From Eqs. (4.10)–(4.12) we immediately see that two restrictions on the existence of circular geodesics have to be satisfied: √ r 2 − 3r + 2b ± 2a r − b ≥ 0 , (4.13) r ≥ b.

(4.14)

The equality in the first condition determines the photon circular geodesics – this demonstrates that positions of circular orbits of test particles are limited by the circular geodesics

4.2. EFFECTIVE POTENTIAL

58

of massless particles. The second, reality condition is relevant in the Kerr–Newman spacetimes with positive tidal charge b only, if we restrict attention to the region of positive radii.

4.2

Effective potential

Next to the radial function R(r, a, b, E, L) and Carter’s equations (2.14)-(2.17), the equatorial motion of test particles can be conveniently treated by using the so called effective potential VEff (r, a, b, L). This function is related to the particle specific energy and depends on the specific angular momentum of the motion and the spacetime parameters. The equation E = VEff determines the turning points of the radial motion of the test particle. The notion of the effective potential is useful in treating the Keplerian (quasigeodesic) accretion onto the central object that is directly related to the circular geodesic motion [Novikov and Thorne, 1973, Page and Thorne, 1974]. The circular geodesics are governed by the local extrema of the effective potential; the accretion process is possible in the regions of stable circular geodesics corresponding to the local minima of the effective potential. The effective potential can be easily derived using the normalization condition for the test particle motion Uα U α = −1 (4.15) that implies via Eqs. (4.3) dr dτ

grr

!2 (4.16)

= (E − VEff+ )(E − VEff− ) .

In the general, stationary and axisymmetric spacetimes, the effective potential is in the form p β ± β 2 − αγ , (4.17) VEff± (r, a, b, L) = α where α = γ =

gϕϕ 2 gϕt − gϕϕ gtt

,

β=

−Lgtϕ 2 gϕt − gϕϕ gtt

(4.18)

,

L2 gtt − 1. 2 gϕt − gϕϕ gtt

(4.19)

This form can be simplified into

VEff =

q 2 −Lgtϕ ± L2 + gϕϕ gtϕ − gϕϕ gtt gϕϕ

.

(4.20)

4.2. EFFECTIVE POTENTIAL

59

We have to choose the upper (plus) sign of the general expression of the effective potential, as this case represents the boundary of the motion of particles in the so called positive root states having positive locally measured energy and future-oriented time component of 4-velocity. The lower (minus) sign expression of the effective potential is irrelevant here, as it determines in the regions of interest particles in the so called negative-root states having negative locally measured energy and past-oriented time component of the 4-velocity, being thus related to the Dirac particles – for details see [Misner et al., 1973, Biˇcák et al., 1989]. The physically relevant condition of the test particle motion reads E ≥ VEff .

(4.21)

For particles with the non-zero rest mass, m > 0, the explicit form of the effective potential in the braneworld Kerr–Newman spacetimes reads √ p aL(2r − b) + r ∆ L2 r 2 + r 4 + a2 (r 2 + 2r − b) VEff (r, a, b, L) = . r 4 + a2 (r 2 + 2r − b)

(4.22)

For massless particles, m = 0, we formally obtain VEffp (r, a, b) L

√ a(2r − b) ± r 2 ∆ . = 4 2 2 r + a (r + 2r − b)

(4.23)

Here the + sign is valid if L > 0 and the − sign is valid if L < 0. Of course, we know that the photon geodesic motion is independent of the photon energy, being dependent of the impact parameter λ = L/E for the equatorial motion [Bardeen, 1973, Stuchlík and Schee, 2010]. The effective potential is symmetric under transformation a → −a, L → −L, therefore, we will only study the Kerr–Newman braneworld spacetimes with non-negative values of the spin parameter a. The effective potential has a discontinuity (divergence) at radii determined by the conditions: 2 gϕt − gϕϕ gtt = 0 , 4

2

2

r + a (r + 2r − b) = 0 .

(4.24) (4.25)

2 At the equatorial plane, the quantity gϕt − gϕϕ gtt ≡ ∆, and the condition (4.24) implies that the effective potential diverges at the event horizons. The second condition (4.25) for possible divergence of the effective potential can be transformed to the relation r 2a2 + a2 r + r 3 b = bs ≡ . (4.26) a2

4.3. CAUSALITY VIOLATION REGION In the limit of b → bs , the numerator of (4.22) reads r − (L − |L|) a2 + r 3 . a

60

(4.27)

Thus if L ≥ 0, both numerator and denominator of Eq. (4.22) are zero and we have to use the L’Hopital rule to obtain  4 4 2 2 2 22 r +a +2a (L +r )+L r   L≥0  2aL(a2 +r 2 ) limb→bS VEff =  , (4.28)   ∞ L , (4.29) a2 therefore the effective potential can be undefined for small values of r. However, this could happen only in the causality violation region where the effective potential looses its relevance because of the modified meaning of the axial coordinate that has time-like character in this region.

4.3

Causality violation region

In the causality violation region (sometimes called time-machine region) the axial coordinate ϕ takes time-like character implying possible existence of closed time-like curves. The causality violation region is defined by the condition gϕϕ < 0.

(4.30)

In the equatorial plane the boundary of the causality violation region is determined by the condition r 4 + a2 (r 2 + 2r − b) = 0 . (4.31) The boundary of the causality violation region can be expressed by the relation r 2a2 + a2 r + r 3 . b = bCV ≡ a2

(4.32)

Notice that the functions bCV (r, a) and bs (r, a) defined in Eq. (4.26) are equivalent – therefore, the divergence could occur just at the boundary of the causality violation region. At

4.3. CAUSALITY VIOLATION REGION

61

r(a,b) - radius of causality violation region

3

0.9 2.5

2

b

0.7 1.5

0.5 1

0.3

black holes

0.5

0.1

0 0

0.5

1

1.5

2

2.5

3

a Figure 4.1: Contour plot for radii of boundary of the causality violation region in the equatorial plane.

Fig. 4.1 we give some examples of the extension of the causality violation region. We see that for this region to exist above the ring singularity, the tidal charge has to be positive. With increasing values of the parameters b > 0 and a, the causality violation region expands. The Eq. (4.31) gives us maximal possible extension of causal violation region located at √ rMax = 1 + b − 1 . (4.33) For positive b the value of rMax is less than b and therefore, as we shall see later, the causality violation region cannot reach the region where the circular geodesics exist. In the Kerr–Schild coordinates the boundary of the causality violation region is given by the relations 3 a2 + r 2 x2 = , (4.34) a2 ∆ r 2 (a2 (b − 2r − r 2 ) − r 4 ) z2 = (4.35) a2 ∆

4.4. SINGULARITY

62

0.8

a=1

0.6 0.4

z

0.2

1

0

0.4

b=4 -0.2 -0.4 -0.6 -0.8 0

0.2

0.4

0.6

0.8

1

1.2

1.4

x Figure 4.2: Examples of causality violation region.

In the following, we consider geodesic motion only in the regions above the causality violation region. For astrophysical phenomena occurring in the naked singularity spacetimes, it is usually assumed that above the boundary of the causality violation region the Kerr or Kerr–Newman spacetime is removed and substituted by different solution that could be inspired by the String theory – such objects are called superspinars [Gimon and Hoˇrava, 2009, Stuchlík and Schee, 2010, 2013]. Therefore, it is quite natural to assume that in the braneworld model framework, the inner boundary of the superspinars is located at radii larger than those related to the boundary of the causality violation region.

4.4

Singularity

The Kerr–Schild form of the metric, given in Eq. (3.72), is analytical everywhere except at points satisfying the condition x2 + y 2 + z2 = a2

and

z = 0.

(4.36)

4.4. SINGULARITY

K

63

400 300 200 100 0 -100 -200 -300 -400

3 2.5 0

2 0.2

0.4

1.5 0.6

r

0.8

1 1

1.2

θ

0.5 1.4

0

Figure 4.3: Example of the behaviour of Kretschmann’s scalar K for a = 0.8 and b = −0.8 to illustrate it’s similarity to Kerr–Newman case.

This condition is the same as in the case of the Kerr black holes or naked singularities, so we clearly see that the braneworld parameter b has no influence on the position of the physical singularity of the space-time. The physical “ring” singularity of the braneworld rotating black holes (and naked singularities) is located at r = 0 and θ = π/2, as in the Kerr spacetimes. We describe the influence of the braneworld tidal charge parameter b on the Kerrlike ring singularity at r = 0 , θ = π/2 using the Kretschmann scalar K = Rαβγδ Rαβγδ representing an appropriate tool to probe the structure of spacetimes singularities. Using (3.63) we obtain

K=

8 4 2 2 2 4 4 6 2 6 r A − 2a r Bf + a Cf − 6a M f , (r 2 + a2 f 2 )6

(4.37)

4.5. ERGOSPHERE

64

where f

(4.38)

= cos θ , 2

2 2

A = (7b − 12bMr + 6M r ) , 2

(4.39)

2 2

B = (17b − 60bMr + 45M r ) ,

(4.40)

C = (7b2 − 60bMr + 90M 2 r 2 ) .

(4.41)

The Kretschmann scalar is formally the same as in the case of the Kerr–Newman metric with Q2 → b [Henry, 2000]. Naturally, the negative values of brane parameter would have some effect onto K, but as we can see from denominator of (4.37), it does not influence the location of the singularity. As an example we demonstrate behaviour of the scalar K for a = 0.8 and b = −0.8 near the ring singularity in the Fig. 4.3. For completeness we give also the Ricci tensor whose components take the form: Rtt = 4b Rtϕ = Rϕt = Rθθ = Rϕϕ = −

a2 + 2∆ − a2 cos(2θ)

, (a2 + 2r 2 + a2 cos(2θ))3 (a2 + ∆) sin2 θ , −8ab (a2 + 2r 2 + a2 cos(2θ))3 R Rtϕ , Rrr = − θθ , ∆ 2b , 2 2 a + 2r + a2 cos(2θ) 3a4 + 2r 4 + a2 b − 2Mr + 5r 2 4b sin2 (θ) (a2 + 2r 2 + a2 cos(2θ))3 a2 ∆ cos(2θ) . (a2 + 2r 2 + a2 cos(2θ))3

(4.42) (4.43) (4.44) (4.45) (4.46) (4.47)

Ricci scalar is zero automatically by construction of the braneworld Kerr–Newman solution [Dadhich et al., 2000a].

4.5

Ergosphere

Here we demonstrate influence of the braneworld tidal charge parameter b on the ergosphere whose boundary is defined by the condition: gtt = r 2 − 2r + a2 cos2 θ + b = 0 .

(4.48)

Extension of the ergosphere in the latitudinal coordinate θ is determined by the maximal latitude given by the relation 1−b cos2 θmax = 2 . (4.49) a

4.5. ERGOSPHERE

65

1.6

a=1

1.4 1.2

b=0

z

1 0.8

b=0.3

0.6

b=0.7 ax

an g

le

0.4

M

0.2

b=0.95

singularity 0 0

0.5

1

1.5

2

2.5

x Figure 4.4: Polar slice through the braneworld Kerr–Newman spacetime in the Cartesian Kerr–Schild coordinates. Dimensionless spin parameter a is fixed to value 1 and the braneworld parameter b is appropriately chosen to demonstrate its influence on the ergosphere.

We can see that existence of the ergosphere is limited by the condition b < 1.

(4.50)

We can infer that the ergosphere extension increases as the tidal charge parameter b decreases. It is convenient to represent location of the ergosphere in the Kerr–Schild coordinates (3.72). Using the spacetime symmetry we can focus only on the polar slices with y = 0. In this case the condition for the static limit surface governing the border of the ergosphere is simply given by [Carter, 1973b] a2 + r 2 ∆ x2 = , a2 (2r − b)r 2 − r 4 z2 = . (4.51) a2

4.5. ERGOSPHERE

66

1.2

b=0.9 1

a=0.32

z

0.8

0.6

a=0.6

0.4

0.2 singularity at x = a

a=1.2 a=2.5

0 0

0.5

1

1.5

2

2.5

3

x Figure 4.5: Polar slice through the braneworld Kerr–Newman spacetime in the Cartesian Kerr–Schild coordinates. The braneworld parameter b is fixed to value 0.9 and the spin parameter a is appropriately chosen to demonstrate its influence on the ergosphere.

In Fig. 4.4 and Fig. 4.5 we illustrate influence of the braneworld tidal charge b on the ergosphere extension. As we can see the ergosphere does not always completely surrounds the ring singularity. For b < 1, the ergosphere exists for each dimensionless spin a > 0, covering all values of the latitudinal angle for the Kerr–Newman black holes. However, as the spin a increases for the Kerr–Newman naked singularities, the ergosphere extension shrinks – the maximal angle α decreases. It can be proved that the causality violation region never overlaps with ergosphere and its extension is influenced by the braneworld parameter b in an opposite way. While causality violation region increases with increasing b, the ergosphere extension is getting smaller. Example of both regions is depicted in Fig. 4.6.

4.6. CIRCULAR GEODESICS OF PHOTONS

67

0.6

a=0.5, b=0.98 0.4

ergosphere causality violation region

0.2

z

0

-0.2

-0.4

-0.6 0

0.5

1

1.5

x Figure 4.6: Ergosphere and causality violation region.

4.6

Circular geodesics of photons

Here we study motion of photons, as the photon circular orbits represent a natural boundary for existence of the circular geodesic motion [Bardeen, 1973, Bálek et al., 1989]. The general photon motion in the braneworld KN black hole spacetimes was studied in [Schee and Stuchlík, 2009]. Here we concentrate on the equatorial photon motion and especially on the existence of the photon circular orbits. In case of the equatorial photon orbits, the radial function R(r) is determined by the Eq. (4.5) with removed term −1 (as the rest energy m = 0) that can be transformed into the form [Schee and Stuchlík, 2009]: h i2 r 2 − a(λ − a) − ∆(λ − a)2 R = , (4.52) E2 r 2∆ where the impact parameter λ is defined by the relation λ=

L . E

Notice that photon orbits depend only on the impact parameter λ.

(4.53)

4.6. CIRCULAR GEODESICS OF PHOTONS r

\

1 r1 , r 2 4 3b

aphMin

b

(−∞; 0)

0; 34

max min

max min

− −b

1

1; 89

min min

− min

− min

− −

min

max

max

max

min

√ √b (8b − 9) 3 3

∞

∞

0

∞

∞

−

3 4;1

68

9 8;∞

aph (r, b) √ 1−b 0

Table 4.1: All kinds of extrema of function aph (r, b). We show the (+) sign part only because of the symmetry corresponding to interchangeability between signs (±) and nature of local minima (max/min). Function aphMin is value of function aph (r, b) at lowest possible r, which is r = 0 for non-positive values of b and r = b otherwise.

Applying conditions for the circular motion (4.6), we find that the equatorial photon circular orbits are given by the equations i2 r 2 − a(λ − a) − ∆(λ − a)2 = 0 ,

(4.54)

2r(r 2 + a2 − aλ) − (r − 1)(λ − a)2 = 0 .

(4.55)

h

These two conditions imply that the radii of circular photon orbits are determined by equation √ r 2 − 3r + 2a2 + 2b ± 2a ∆ = 0 , (4.56) and the impact parameter λ is given by the equation λ = −a

r 2 + 3r − 2b . r 2 − 3r + 2b

Furthermore, the equation (4.56) can by transformed into the form √ r 2 − 3r + 2b ± 2a r − b = 0

(4.57)

(4.58)

that implies the same reality condition on the radius of the photon orbit rph as the one that follows from Eqs. (4.10)-(4.12): rph ≥ b . (4.59) Due to the reality condition the numerator in Eq. (4.57) is positive, while the (±) sign of the denominator is determined by the sign in Eq. (4.58). Thus we obtain corotating orbits (λ > 0) for the upper sign in (4.58) and counterrotating orbits (λ < 0) in the other case. The solution of the Eq. (4.58) can be expressed in the form 3r − r 2 − 2b a = aph (r, b) ≡ ± . (4.60) √ 2 r −b


69

2

V(0,0) 1.5

IV(1,1)

VII(2,2)

1

b

VI(2,2)

III(1,1)

na

I(0,2)

0.5

ked

sin g ck h ularit ies ole s

bla

II(1,3)

0

IX(0,3)

-0.5

VIII(0,2)

X(0,1)

-1 0

0.5

1

1.5

a Figure 4.7: Braneworld Kerr–Newman black holes and naked singularities can be divided into ten distinguish classes according to the properties of circular photon geodesics. Curve aph (4b/3, b) (full line), given by (4.63), plays the main role in the classification. The corresponding regions of b–a plane are denoted by I–X; the numbers in brackets denote the number of the circular photon orbits in the respective class. The first number determines number of the stable circular photon geodesics, the second number determines number of the unstable circular photon geodesics.

For given a and b the points of a line a = const crossing the function aph (r, b) determine radius rph of the photon circular orbits. We restrict our discussion to the solutions corresponding to a > 0, giving both corotating and counterrotating orbits. The zeros of the function aph (r, b) are located at rph± =

1 √ 3 ± 9 − 8b . 2

(4.61)

Note that these solutions represent radii of photon circular orbits in the Reissner–Nordström spacetimes [Stuchlík and Hledík, 2002, Pugliese et al., 2011]. Since ∂aph ∂r

=±

(r − 1)(3r − 4b) 4 (r − b)3/2

,

(4.62)


70

Class

Horiz.

Ergo.

Orbits

Class

Horiz.

Ergo.

Orbits

I II III IV V

yes yes no no no

yes yes yes no no

0,2 1,3 1,1 1,1 0,0

VI VII VIII IX X

no no yes yes no

yes no yes yes yes

2,2 2,2 0,2 0,3 0,1

Table 4.2: Ten possible divisions of braneworld KN spacetimes with respect to existence of the horizon, existence of the ergosphere and the number of stable and unstable circular photon orbits. The first number in the column orbits corresponds to amount of stable circular photon orbits, the second corresponds to amount of unstable circular photon orbits.

the extrema of the curves aph (r, b) are located at r = 1 and at r = 4b/3 . The value of the function aph (r, b) at the point r = 1 reads (recall that we consider the positive values of spin) √ aph−ex (r = 1, b) = 1 − b , (4.63) corresponding to the extreme Kerr–Newman black holes, while at r = 4b/3 it reads √ b aph−ex (r = 4b/3, b) = ± √ (8b − 9) . 3 3

(4.64)

The position, value and kind of the extrema of the function aph (r, b) are listed in Table (4.1). The results are summarized in Fig. 4.7. We see that all curves drawn there – the curve aph−ex (r = 4b/3, b), the line a2 + b = 1 (corresponding to the extremal black holes), the line a2 + b = 0, and the line b = 1 (separating the braneworld Kerr–Newman naked singularities with the ergosphere from those without it) – divide the b – a plane into ten regions. In this sense, the braneworld Kerr–Newman spacetimes can be divided into ten different classes, characterized by: 1. existence of the horizon 2. existence of the ergosphere 3. the number of stable and unstable circular photon orbits. The situation is summarized in Table (4.2) and is also visualized in Fig. 4.7, in accord with the analysis of circular photon orbits in the standard KN spacetimes [Bálek et al., 1989]. In the case of the braneworld KN black holes new regions VIII,IX and X corresponding to negative values of the tidal charge, (b < 0), occur in addition to the standard Kerr–Newman spacetimes.

4.7. EQUATORIAL GEODESICS WITH ZERO ANGULAR MOMENTUM

71

2

r=1

1.5

aSol+ aH

a

1

0.5

,,

V Eff

aSol0 r = 3/4

-0.5

-1 0

0.5

1

1.5

2

r Figure 4.8: Functions aSol± and aH defining location of horizons. Double prime represents second derivations with respect to r.

4.7

Equatorial geodesics with zero angular momentum

The equatorial motion with zero angular momentum requires automatically Q˜ = 0 and vanishing axial angular momentum, L = 0. Then the motion is governed by the specific energy E having the allowed values restricted by an effective potential giving turning points of the motion (where dr/dτ = 0) due to the condition E = Veff . We shall thus discuss properties of the effective potential Veff (r; a, b, Q˜ = 0, L = 0) = Veff (r; a, b).

4.7.1

Kerr spacetimes

It is convenient to remind first as a reference the case of the Kerr geometry when b = 0. Then the effective potential takes the form √ r ∆

VEff (r, a, b = 0) = p . r 4 + a2 (r 2 − 2r)

(4.65)


72

0.6

a > 1.2

99

0.4

a = 1.299 0.3

1.299 > a > 1

>0

0.2

a=

1

0.1

1>a

VEff (r,a,b=0,L=0)

0.5

a=0

0 0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

r Figure 4.9: Six different types behavior for function VEff (r, a) with parameters b = 0 and L = 0 according to spin parameter a.

The local extrema of the effective potential are given by the solutions of the relation dVEff (r, a, b = 0) = 0, dr

(4.66)

p a2sol± = 2r − r 2 ± 2r (r − 1) .

(4.67)

and can be expressed in the form

The local extrema of the effective potential determine the special class of equatorial circular geodesics with L = 0. The character of these local extrema of the effective potential is determined by the sign of the second derivative of Veff . As demonstrated in Fig. 4.8, for r > 3/4 all the local extrema are minima corresponding to stable circular geodesics, while at r < 3/4 maxima of the effective potential occur, corresponding to unstable circular geodesics. We can see that the function asol− (the minus part of the solution) exists always between horizons, defined by √ (4.68) aH = 2r − r 2 , and has no physical relevance. The inflexion point of the effective potential occurs at √ 3 3 3 r = , a= ∼ 1.299 . (4.69) 4 4


73

3.0

2.5

aext(ex)

a2

2.0

a0

1.5

G 3 4 3 the effective potential is monotonically increasing with increasing radius. In the black hole spacetimes, only unstable circular geodesic with L = 0 exist under the inner horizon.

4.7.2

Braneworld Kerr–Newman spacetimes

For non-zero values of the tidal charge parameter b, the effective potential takes the form VEff (r; a, b) = p

√ r ∆ r 4 + a2 (r 2 − 2r − b)

.

(4.70)

Assuming the spin a > 0, the effective potential vanishes at r = 0 and at the inner and the horizons in the black hole spacetimes. Radius of divergence of the effective potential is governed by vanishing of denominator in the VEff (r; a, b) and can be determined by the

4.7. EQUATORIAL GEODESICS WITH ZERO ANGULAR MOMENTUM 2

2

a = 0.7

causality violation

a = 0.98

causality violation

1.5

1.5

VII

bex

bdiv

bdiv

1

b

1

b

74

0.5

bex

VII

0.5

IV V

0

VI V

0

II

inside event horizons


-0.5

-0.5

II

I -1

-1 0

0.5

1

1.5

0

2

0.5

2

1.5

2

2

a = 1.2

causality violation

a = 1.299

causality violation

bdiv

1.5

1.5

bex

0.5

bex

VII

0.5 VI

VI

0

bdiv

1

VII

b

1

b

1

r

r

0

III -0.5

III

-0.5


II


-1

II

-1 0

0.5

1

1.5

2

0

0.5

r

1

1.5

2

r 2

2

a = 1.35

causality violation

a = 1.39754

causality violation 1.5

1.5 VII

bdiv

bex

bdiv

1

b

b

1

0.5

bex

VII

0.5

VI

VII 0

0 VIII

VI

-0.5

-0.5 III

0

0.5

1

III

II


-1

1.5

II

-1

2

0

r

0.5

1

1.5

2

r

Figure 4.11: Functions bex and bdiv . On the right edge of each figure there is indicated the classification according to number of extrema in effective potential, see Fig. 4.12.

relation b = bdiv (r, a) ≡

r 4 + a2 (r 2 − 2r − b) . a2

(4.71)

Notice that the divergence radius of the effective potential corresponds to the boundary of the causality violation region in the equatorial plane (4.31); the corresponding test particles have thus turning points before reaching this region. The local extrema of the effective potential, determined by the relation dVEff (r; a, b) = 0, dr

(4.72)


75

0.4 P2

VII(1) ) b 1(a

P

b2 (a)

IV(0)

0.2

VI(3)

V(2) 0.0

III(2)

b

-0.2

2

a

1 2 b= a+

-0.4 =0

+b

I(0)

-0.6

VIII(0)

II(1)

-0.8

-1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

a Figure 4.12: Parameter space (a − b) divided into eight separate regions I-VIII. The number in brackets correspond to number of extrema in effective potential wit zero angular momentum. Point P = (0.938756 , 0.118485) and point P2 = (1.39771 , 0.312654). Also point where function b2 (a) intersects zero axis is a = 1.299 .

are for a > 0 governed by the condition b = bext± =

p −G(r; a) ± a2 + r 2 G(r; a) 2a2

,

(4.73)

where G(r; a) = r 4 + a4 + 2a2 (r − 2),

(4.74)

while for the Reissner–Nordström (RN) spacetimes with a = 0 we obtain simple relation b = bext(RN) = r .

(4.75)

In the RN case the local extrema are located at r = b and their character is given by the second derivative of the effective potential that takes the form d2 VEff (r = b, a = 0, b) 1 = p . 2 2 dr b (b − 1)b

(4.76)


76

0.7

b=-0.4

VEff (r,a,b,L=0)

0.6

0.5

0.4

VIII(0) 0.3

0.2

III(2)

0.1

II(1)

I(0)

II(1) 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

r Figure 4.13: Behavior of effective potential with zero angular momentum VEff (r, a, b, L = 0) in regions I(0),II(1),III(2) and VIII(0) for brane parameter b = −0.4 and corresponding values of parameter a.

That is the reason why for b ∈ (1; ∞) there is minimum of the effective potential at r = b, while for b ∈ (0; 1) the second derivative is not well defined and there is no local extremum of the effective potential outside the black hole horizons [Stuchlík and Hledík, 2002, Pugliese et al., 2011]. The behavior of the effective potential VEff (r, a, b) in the KN spacetimes is thus determined by the functions bdiv (r; a) and bextr± (r; a) where the condition ∆ > 0 guaranteeing positions outside the dynamic regions of the spacetime has to be satisfied. For the braneworld KN black holes the equation bextr = b , has solutions only under the inner horizon. For naked singularities the situation is more complex. The limit of the reality of the function bextr± (r; a) is governed by the by the function that is identical with the function governing its zero points. The limits of reality and the zero points are given by the relation √ a2 = a2extr(r)± (r) = a2extr(z) (r) ≡ r(2 − r) ± 2r 1 − r.

(4.77)

The function a2extr(r)± (r) is represented in Fig. 4.10. In order to fully understand the behaviour of the effective potential (4.70), we have to determine the second derivative of the

4.7. EQUATORIAL GEODESICS WITH ZERO ANGULAR MOMENTUM IV(0)

V(2) 0.7

0.2

a=0.1

a=0.3

a=0.7

a=0

.79 1.5

a=0.8

1.0

a=0.63

a=0.5

0.5

0.1

2.0

2.5

3.0

3.5

0.0 0.0

4.0

0.5

1.0

a=0.7

0.2

0.3

49 1/2

0.3

0.4

0.9

VEff (r,a,b,L=0)

0.4

0.5

9 1/2

VEff (r,a,b,L=0)

0.5

0.0 0.0

b=0.05

0.6

a=

b=0.2

0.6

a=0.9

0.7

0.1

1.5

r VI(3)

b=0.1

3.0

3.5

4.0

VEff (r,a,b,L=0)

0.5

a=1.1

0.2

a=1

0.5

1.0

2.5

3.0

3.5

4.0

8368

0.3 a=1 0.2

0.1

a=

0.1

VI(3) a=1.1

.80

0.3

a=1.5 a=1.33052

0.4

1 1/2

a=1.3 a=1.2

0.5

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.0

a=0

0.4

5

a=

0.6

0.9 1/ 2

VEff (r,a,b,L=0)

2.5

0.7

b=0.2

0.0 0.0

2.0

r VII(1)

0.7

0.6

77

0.5

r

1.0

1.5

2.0

r

Figure 4.14: Behavior of effective potential with zero angular momentum VEff (r, a, b, L = 0) in regions IV(0),V(2),VI(3) and VII(1).

effective potential at the extrema points, ∂2 VEff (r, a, bextr± ) , ∂r 2

(4.78)

that determine the character of the extremal points of the functions bextr± (r; a). The local extrema of the function bextr± (r; a) are given by the function a2extr(ex) (r) that is implicitly determined by the relation p 2(r 2 + a2 )[(r − 1)a2 + r 3 ] 4a2 (1 − r) ± 2r G(r; a) ± = 0. p G(r; a)

(4.79)

The resulting function a2extr(ex) (r) is determined numerically and is also illustrated in Fig. 4.10. The zeros of the function bextr± given in Fig. 4.10 determine a branching point at a = 1.299 , r0 = 0.755; i.e., the highest value of the spin parameter a where the functions


78

bextr± (r; a) intersect the b = 0 line. For a > 1.299, there is bextr+ (r; a) > 0 and bextr− (r; a) < 0. The characteristic functions of the effective potential, i.e., the functions bdiv and bextr± , are illustrated for six typical situations in Fig. 4.11 for six typical values of the spin parameter a. We give also explicitly the character of the extremal points of the effective potential. In the black hole spacetimes, there is no extremum above the outer horizon, while under the inner horizon there can be no, one or two extrema if b > 0, while only one local extremum exist for b < 0. In the naked singularity spacetimes, one two or three local extrema exist if b > 0, while no, one or two local extrema exist, if b < 0. Let rsd± (a, b) be the positions of the inflexion points of the effective potential, i.e., the solution of the equation ∂2 VEff (r, a, bextr± ) = 0. (4.80) ∂r 2 Then we define the functions giving the limiting behavior of the effective potential in the form b1 (a) ≡ bextr+ (r = rsd+ ) ,

b2 (a) ≡ bextr− (r = rsd− ).

(4.81)

The functions b1 (a) and b2 (a) govern the classification of the braneworld KN spacetimes according to the equatorial motion with vanishing angular momentum, giving the values of the spacetime parameters a and b for whom the effective potential changes its character, namely, the effective potential changes the number and/or kind of its local extrema. These two functions are illustrated in Fig. 4.12 giving the classification of the braneworld KN spacetimes. In the classification we consider three criteria: number of the extrema of the effective potential, sign of the braneworld parameter b, and type of the spacetime (black hole or naked singularity). The spacetime parameter space (a − b) can be then separated into eight areas (I-VIII) with differing number of solutions of the equation bextr± = b . The results can be seen in Fig. 4.12. The arabic number in the brackets corresponds to the number of solutions for each region. The functions b1 (a) and b2 (a) were obtained by numerical calculations. They intersect at the point P2 with the spacetime parameters taking the values a = 1.39771 , b = 0.312654; the function b2 (a) intersects the zero axes at a = 1.299 . The graph of the function b2 (a) is always above the curve a2 + b = 1 corresponding to the extreme black holes and separates black holes from naked singularities. Point P in Fig. 4.12 where function b1 (a) intersects the curve a2 + b = 1 is a = 0.938756 , b = 0.118485. Now we can give for the established classes of the braneworld KN spacetimes typical sequences of the effective potential, taking fixed braneworld parameter b and correspondingly varying spin parameter a. We separate the braneworld spacetime with negative and positive parameter b, and use only a > 0 values as the equatorial motion with vanishing angular momentum is independent of the sign of the spin. We also give distribution of the

4.8. RADIAL FALL FROM INFINITY

79

local maxima (minima) of the effective potential giving unstable (stable) circular geodesics with L = 0. Typical effective potentials corresponding to the classes I(0), II(1), III(2), VIII(0) of the braneworld KN spacetimes with negative values of brane parameter, b < 0, are represented in Fig. 4.13). The local extrema of the effective potential correspond to the function bextr− (r; a). The behaviour of the effective potential for the b < 0 spacetimes is quite simple as suggested by Fig. 4.13. In the black hole region I(0), the effective potential is simply descending to zero value at the horizon, having no local extremum. In the black hole region II(1), one local maximumm exists under the inner horizon. If we increase sufficiently value of the spin parameter a, we enter the naked singularity region III(2) where the effective potential demonstrates one local maximum and one local minimum. Further increase of the spin a parameter will eventually cause that these two extrema coalesce into an inflexion point, and we then enter the naked singularity region VIII(0) where no extrema points of the effective potential exist. For the braneworld spacetimes with b > 0, the behaviour of the effective potential is more complex in comparison with the b < 0 spacetimes, as shown in Fig. 4.14. In the black hole region IV(0) there are no local extrema of the effective potential that not defined between the horizons, being decreasing (increasing) above the outer (under the inner) black hole horizon. As the spin parameter a increases and the brane parameter is kept at b < 0.118485, we enter the black hole region V(2). In this region there are two extrema points (one minimum and one maximum) under the inner horizon. Further increase of the parameter a causes transition from the black holes region to the naked singularities region VI(3) where the effective potential demonstrates existence of two local minima and one local maximum. Finally, the region VII(1) corresponds to naked singularity spacetimes demonstrating one local minimum of the effective potential; notice that such in Fig. 4.14 the corresponding case illustrated for b = 2 is mixed with the case VI(3) as can be seen from the classification map in Fig. 4.12.

4.8

Radial fall from infinity

As demonstrated in [Bicak and Stuchlik, 1976, Stuchlík et al., 1999], there is a special class of test particle orbits corresponding to the purely “radial” trajectories, i.e., trajectories keeping constant latitude θ = const., but with varying azimuthal coordinate ϕ. Such particles fall freely from infinity, having zero angular momentum, L = 0, Q˜ = 0, and energy equal to the rest energy, E = m. Such test particles move purely radially relative to the family of the locally non-rotating frames introduced in [Bardeen et al., 1972a]. For the purely radial geodesics with the motion constants L = 0, Q˜ = 0, E = m, the function governing the radial motion takes in the braneworld KN spacetimes the following

4.9. STABLE CIRCULAR GEODESICS

80

simple form R(r; a, b) = m2 (r 2 + a2 )(2r − b).

(4.82)

We can see that the particles falling from infinity with L = 0, Q˜ = 0, E = m have turning point of their radial motion at the radius r = b/2. Clearly, in the equatorial plane the turning point is located outside the causality violating region, as follows from behavior of the effective potential Veff (r; a, b). Notice that the turning point of the equatorial radial geodesics is given by solution of equation Veff (r; a, b) = 1.

4.9

Stable circular geodesics

It is well known that character of the test particle (geodesic) circular motion governs structure of the Keplerian (geometrically thin) accretion disks orbiting a black hole [Novikov and Thorne, 1973, Page and Thorne, 1974] or a naked singularity (superspinar) [Stuchlík et al., 2011b, Stuchlík and Schee, 2012]; similarly, it can govern also motion of a satellite orbiting the black hole or the naked singularity (superspinar) along a quasicircular orbit slowly descending due to gravitational radiation of the orbiting satelite [Ruffini, 1973]. The Keplerian accretion, starting at large distances from the attractor, is possible in the regions of the black hole or naked singularity spacetimes where local minima of the effective potential exist and the energy corresponding to these minima decreases with decreasing angular momentum [Misner et al., 1973]. In other words, in terms of the radial profiles of the quantities characterizing circular geodesics, the Keplerian accretion is possible where both specific angular momentum and the specific energy of circular geodesics decrease with decreasing radius. In the standard model of the black hole accretion disks, the inner edge of the accretion disk is located in the so called marginally stable circular geodesic where the effective potential has an inflexion point [Novikov and Thorne, 1973], but the situation can be more complex in the naked singularity spacetimes [Stuchlík and Hledík, 2002, Stuchlík and Schee, 2014]. We study stability of the circular geodesic motion of test particles relative to the radial perturbations in the braneworld Kerr–Newman spacetimes. Note that the equatorial circular motion is then always stable relative to the latitudinal perturbations perpendicular to the equatorial plane [Biˇcák and Stuchlík, 1976]. We show that the most interesting and, in fact, unexpected result occurs for test particles orbiting the special class of the braneworld mining-unstable Kerr–Newman naked singularities demonstrating an infinitely deep gravitational well enabling (formally) unlimited energy mining from the naked singularity spacetime. Of course, such a mining must be limited by violation of the assumption of the test particle motion.


4.9.1

81

Marginally stable circular geodesics

The loci of the stable circular orbits are given by the condition related to the radial motion R(r) function ∂2 R(r, a, b, E, L) ≤ 0, (4.83) ∂r 2 or the relation ∂2 VEff (r, a, b, L) ≤ 0, (4.84) ∂r 2 related to the effective potential VEff (r), where the case of equality corresponds to the marginally stable circular orbits at rms with L = Lms , corresponding to the inflexion point of the effective potential – for lower values of the specific angular momentum L the particle cannot follow a circular orbit. Such marginally stable circular orbit represents the innermost stable circular orbit and the inner edge of the Keplerian disks in the Kerr black hole and naked singularity spacetimes. Using the relations (4.10) and (4.11), we obtain for the braneworld KN spacetimes [Stuchlík and Kotrlová, 2009]1 r(6r − r 2 − 9b + 3a2 ) + 4b(b − a2 ) ∓ 8a(r − b)3/2 = 0 .

(4.85)

In the previous studies, only the braneworld black hole spacetimes were usually considered [Kotrlová et al., 2008a, Stuchlík and Kotrlová, 2009]. Standard KN naked singularity spacetimes were discussed in [Biˇcák et al., 1989, Pugliese et al., 2013]. Here we consider whole family of the braneworld KN spacetimes, with both positive and negative tidal charges. The solution of Eq. (4.85) can be express in the form p 4 (r − b)3/2 ± 3br 2 − (2 + 4b)r 3 + 3r 4 ams = ∓ , 4b − 3r

(4.86)

where the ∓ signs correspond to the upper and lower family of the circular geodetics. The ± signs correspond to the two possible solutions of Eq. (4.85). The local extrema of the function ams (r, b) are given by the relation √ p ams(extr) ≡ ∓ 2 b ± b (4b − 1) . (4.87) Thus it can be shown that there is no solution for Eq. (4.85) related to the lower family of circular geodesics when b> 1 Formally

5 4

∧

√ p a < −2 b + b (4b − 1)

same results relevant for KN spacetime can be found in [Aliev and Galtsov, 1981]

(4.88)


3.0

82

ms. is not defined

2.0

b

ms. is not defined for upper family 1.0

0.0

black holes -1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

a Figure 4.15: Mapping of existence of the marginally stable circular geodesics in the parameter space of the braneworld Kerr–Newman spacetimes.

and there is no solution for the upper sign family of circular geodesics when 1 > b > 1/4 and √ p √ p 2 b − b (4b − 1) < a < 2 b + b (4b − 1) .

(4.89)

The existence of the marginally stable circular geodesics in dependence on the dimensionless parameters of the braneworld KN spacetimes is represented in Fig. 4.15. This figure will be crucial for construction of the classification of the KN spacetimes according to the Keplerian accretion, but it is not sufficient, as the classification of the photon circular geodesics plays a crucial role too. The function ams (r, b) determines in a given KN spacetime location of the marginally stable circular geodesics that are usually considered as the boundary of Keplerian accretion discs determined by the quasi-geodesic motion.

4.10. ANALYTICAL PROOF

4.9.2

83

Innermost stable circular geodesics

The standard treatment when the inner edge of the Keplerian accretion discs is located at the marginally stable orbits defined by the inflexion point of the effective potential (this point is also the innermost stable circular orbit (ISCO) [Novikov and Thorne, 1973]) works perfectly in the braneworld KN black hole spacetimes, but the situation is more complex in the braneworld naked singularity spacetimes, as the innermost stable circular orbits (ISCO) do not always correspond to the marginally stable orbit defined by Eq. (4.85)2 . The ISCO can be determined by the function rex (L, a, b) defined implicitly by Eq. (4.11). There are two relevant extrema of this function given by the conditions √ a) 0 = r 2 − 3r + 2b ± 2a r − b , (4.90) b)

r = b.

(4.91)

The case a) tells us that the innermost circular geodesics corresponds to the photon circular geodesics that can be also stable with respect to radial perturbations so that this condition is also applicable as a limit on the stable circular orbits of test particles, as demonstrated in [Stuchlík and Hledík, 2002]. Then the specific energy and the specific angular momentum tend asymptotically to E → ±∞ and L → ±∞ but the impact parameter λ = L/E remains finite. The condition r > b could restrict the condition implied by the photon circular geodesics. The case b) can be relevant for the braneworld spacetimes with positive braneworld parameter b, as demonstrated in [Stuchlík and Hledík, 2002] where the effective potential VEff (r, b, L) for the Reissner–Nordström naked singularity spacetimes clearly demonstrates that the inner edge of the Keplerian disc is located at r = b, having L = 0, while no marginally stable circular orbit, corresponding to an inflexion point of the effective potential, exist.

4.10

Analytical proof

The Keplerian accretion works if there exist a continuous sequence of local minima of the effective potential with decreasing values of angular momentum L. In terms of the effective potential (4.22), the conditions for existence of Keplerian accretion disks can be express in the form ∂VEff (r, a, b, L) = 0, ∂r ∂2 VEff (r, a, b, L) ≤ 0, ∂r 2 ∂VEff (r, a, b, L) < 0. ∂L 2 See

for example [Stuchlík and Hledík, 2002, Favata, 2011]

(4.92)

4.10. ANALYTICAL PROOF

84

6

Vc

Va

5

(10)

Vb

4

b

3

IVb

2

IVa (9)

1

IIIa

I

(8)

IIIb

II

0

IX

X

-1 VIII

0

5

10

15

20

a Figure 4.16: Classification of the braneworld KN spacetimes according to the properties of circular geodesics relevant for the Keplerian accretion. The parameter space b − a is separated by curves governing the extrema of the functions determining the √ photon circular orbits (thick lines) and the marginally stable orbits (dashed lines). Point ( 0.5,0.5) is an intersection of dashed line and curve separating black holes from naked singularities (b = 1 − a2 ). This two curves are tangent at the common point.

Except the possibility to stop this procedure by the inflexion point of the effective potential, there exist other possible way to stop validity of these conditions: ∂VEff (r, a, b, L) =0 ∂L that will be satisfied at a turning point where ±a(2r − b) L = LT ≡ √ . r r 2 − 2r + b

(4.93)

(4.94)

At this turning point the minimum of effective potential, given by ∂VEff (r, a, b, LT )/∂r = 0, is located where: r −b = 0 ⇒ r = b. (4.95) √ r 2 r 2 − 2r + b

4.10. ANALYTICAL PROOF 1.3

Va (1)

85

Vb

1.2

Vc

1.125 1.1

(2)

(3)

IVa

IVb

VII (4) VI

1

(5) IIIb

b

0.9

IIIa

0.8

(6)

I

0.7

0.6

(7)

0.5

II 0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a Figure 4.17: Classification of the braneworld KN spacetimes according to the properties of circular geodesics relevant for the Keplerian accretion. The parameter space b − a is separated by curves governing the extrema of the functions determining the photon circular orbits (thick lines) and the marginally stable orbits (dashed lines). Detailed structure for small values of spin a and b ∼ 1.

In such situations, the inner edge of the Keplerian accretion disk is located at r = b. Putting this result into the definition of the function LT , we find ±ab LT (r = b) = p ⇒ b > 1. b b(b − 1)

(4.96)

We see that the effect of existence of the lowest possible value of angular momentum L associated with local minima of effective potential occurs only for values of the tidal charge b > 1 when Eq. (4.95) is well defined at r > b. The second possible way the conditions (4.92) are not well matched is related to the situation when the local minima of the effective potential turn into an inflexion point at r defined by (4.85). So the inflexion point of the effective potential is not always defined, if the tidal param-

4.11. CLASSIFICATION OF BRANEWORLD KN SPACETIMES

86

eter b > 5/4 for the lower family solution of Eq. (4.85). In this case the Keplerian accretion stops at the point r = b where the minimum of the effective potential starts to increase in energy level with decreasing L. On the other hand, for the upper family solution of Eq. (4.85) situation becomes extraordinary and much more interesting for the tidal charge in the interval 1 > b > 1/4, and appropriately tuned spin a; then the inflexion point of the effective potential does not exist – for this reason, the Keplerian accretion starting at large values of the angular momentum of accreting matter cannot be stopped, and it continues with no limit to unlimitedly large negative values of the angular momentum and unlimitedly large negative values of the energy. Therefore, in this case, unrestricted mining of the energy due to the Keplerian accretion could occur. Of course, this mining has to be stopped at least when the energy of the accreting matter starts to be comparable to the mass parameter of the Kerr–Newman naked singularity, and the approximation of the test particle motion of matter in the disk is no longer valid.

4.11

Classification of braneworld KN spacetimes

In order to create classification of the braneworld KN black hole and naked singularity spacetimes according to possible regimes of the Keplerian accretion, we consider existence of the event horizons, existence of the ergosphere, and we use the characteristics of the circular geodesics: existence of the circular photon geodesics and their stability, existence of the marginally stable circular geodesics related to inflexion points of the effective potential, and relevance of the limiting radius r = b. We use the classification of the braneworld Kerr– Newman spacetimes introduced for the characterization of the photon circular geodesics, and generate a subdivision of the introduced classes according to the criteria related to the marginally stable orbits. The individual classes of the KN spacetimes will be represented by typical radial profiles of the specific angular momentum L, specific energy E, and the effective potential VEff that enable understanding of the Keplerian accretion and calculation of its efficiency. We first briefly summarize results of the two special cases – Kerr and Reissner–Nordström spacetimes. In the following classification of the braneworld KN spacetimes the characteristic types of the behavior of the circular geodesics in the special Kerr and RN spacetimes occur, but also some quite new and extraordinary situations arise. The results of the circular geodesic analysis in the braneworld Kerr–Newman spacetimes can be directly applied also for the circular geodesics in the standard Kerr–Newman spacetimes, if we make transformation b → Q2 where Q2 represents the squared electric charge parameter of the KN background.


4.11.1 2

Case b=0: Kerr black hole and naked singularity spacetimes b=0

upper family

5

1.5

b=0

upper family

4

1

3

0.5

2

L

E

87

0

1 -0.5

-1.5 -2

3

0

a=0 a=0.9 a=1 a=1.1

-1

1

2

3

r

4

5

-1

6

lower family

-2

7 b=0

0

a=0 a=0.9 a=1 a=1.1

2.5

a=0 a=0.9 a=1 a=1.1 1

2

3

r

4

5

6

lower family

7 b=0

a=0 a=0.9 a=1 a=1.1

-2

2

L

E

-4 1.5

-6 1 -8

0.5 0

1

2

3

4

5

r

6

7

8

9

10

-10

1

2

3

4

5

r

6

7

8

9

10

Figure 4.18: E and L for Kerr black hole and naked singularities. The limiting case of the well known results of the test particle circular orbits in the Kerr spacetimes that were studied in detail in [Bardeen et al., 1972a, Stuchlík, 1980] demonstrates clearly the necessity of very careful treating of the families of circular orbits in the naked singularity spacetimes where the simple decomposition of the circular orbits to corotating and counterrotating (retrograde) is not always possible. Namely, in the spacetimes with 1 < a < ac = 1.3 the circular orbits that are corotating at large distances from the ring singularity become retrograde near the ring singularity, at the ergosphere; moreover, in the spacetimes with 1 < a < a0 = 1.089 the covariant energy of such orbits can be negative. The specific energy and specific angular momentum of the circular geodesics of the Kerr black hole and naked singularity spacetimes are illustrated in Fig. 4.18. Notice that the unstable circular geodesics approach the radius r = 0 with unlimitedly increasing covariant energy and axial angular momentum, however, the photon circular geodesic cannot exist at the ring singularity. The Kerr naked singularities are classically unstable as Keplerian accretion from both the corotating and counterrotating disks inverts the naked singularity


88

into an extreme Kerr black hole – the transition is discontinuous (continuous) for corotating (counterrotating) Keplerian disk [de Felice, 1974, Stuchlík, 1980, 1981b, Stuchlík et al., 2011b]. We shall see latter that the Keplerian accretion cannot be generally treated simply in the Kerr–Newman naked singularity spacetimes due to the complexities discussed in detail in [Stuchlík and Schee, 2014]. We expect addressing this issue in future work.

4.11.2

Case a=0: RN black hole and naked singularity spacetimes

The other limiting case of the Reissner–Nordström and Reissner–Nordtröm-(anti-)de Sitter black hole and naked singularity spacetimes has been treated in [Pugliese et al., 2011, Stuchlík and Hledík, 2002]. It has been demonstrated that in the RN naked singularity spacetimes even two separated regions of circular geodesics could exist. The doubled regions of the stable circular motion occur in the RN naked singularity spacetimes with the charge parameter 1 < Q2 < 5/4, or even only stable circular geodesics could exist, if the charge parameter Q2 > 5/4. In the RN naked singularity spacetimes also doubled photon circular geodesics can occur, with the inner one being stable relative to radial perturbations, if the charge parameter is in the interval of 1 < Q2 < 9/8 [Stuchlík and Hledík, 2002, Pugliese et al., 2013]. The same phenomena occur in the naked singularity Kehagias– Sfetsos spacetimes of the Hoˇrava quantum gravity [Stuchlík and Schee, 2014, Stuchlík et al., 2014] or in the no-horizon regular Bardeen or Ayon–Beato–Garcia spacetimes [Stuchlík and Schee, 2015, Schee and Stuchlík, 2015]. We shall see that in the braneworld Kerr–Newman naked singularity spacetimes the special naked singularity effects of the Kerr and RN case are mixed in an extraordinary way leading to existence of an infinitely deep gravitational well implying the new effect we call mining instability.

4.11.3

Characteristic points of the KN spacetime classification

The classification of the braneworld Kerr–Newman spacetimes according to the character of the circular geodesics and related effective potential are determined by the functions governing the local extrema of the functions giving the photon circular geodesics and the marginally stable circular geodesics corresponding to the inflexion points of the effective potential. In the space of the spacetime parameters b − a then exist fourteen regions corresponding to classes of the braneworld Kerr–Newman spacetimes demonstrating different behavior of the circular geodesics and Keplerian accretion as demonstrated in Fig. 4.16 and in Fig. 4.17 giving details of the regions of low values of the dimensionless parameters a and b. These regions are governed by intersection points of the curves (4.63) and (4.87) that give thirteen characteristic points in the parameter space that are summarized in the following way: the pairs (a, b) are ordered gradually from the top to the bottom and

4.11. CLASSIFICATION OF BRANEWORLD KN SPACETIMES Class I II IIIa IIIb IVa IVb Va Vb Vc VI VII VIII IX X

ISCO =MSO =MSO =Photon =MSO at r=b at r=b at r=b at r=b at r=b =MSO at r=b =MSO =MSO =MSO

MSO(u) classic classic − classic − classic − − classic classic classic classic classic classic

MSO(l) classic classic classic classic classic classic − classic classic classic classic classic classic classic

Hor./Erg. yes/yes yes/yes no/yes no/yes no/no no/no no/no no/no no/no no/yes no/no yes/yes yes/yes no/yes

89 SP 0 1 1 1 1 1 0 0 0 2 2 0 0 0

UP 2 3 1 1 1 1 0 0 0 2 2 2 3 1

Table 4.3: Classification of parameter space b−a with respect to: ISCO - radius of innermost stable circular orbit; MSO(u) - radius of Marginally Stable Orbit for upper sign family; MSO(l) - radius of Marginally Stable Orbit for lower sigh family; SP - number of stable photon circular orbit; UP - number of unstable photon circular orbit. ISCO has only two possible outcomes. It can be identical with MSO or lies at r = b. Word “classic” in this context means that MSO is defined by Eq. (4.85). from the left to the right, (1)

→ (0, 1.25) ,

(2)

→ (0, 1.125) , ! A− (12 + A− ) 3 (12 + A− ) = (0.0831, 1.1748) , , → √ 32 16 2 ! 1 → √ , 1 = (0.19245, 1) , 3 3 √ → 2 − 3, 1 = (0.268, 1) , 3 → 0.5, , √ 4 → 0.5, 0.5 ,

(3) (4) (5) (6) (7) (8) (9)

→ (1, 0.25) , √ → 2 + 3, 1 = (3.732, 1) ,

! A+ (12 + A+ ) 3 (12 + A+ ) = (15.0992, 5.361) , (10) → , √ 32 16 2

4.11. CLASSIFICATION OF BRANEWORLD KN SPACETIMES where

√

q √ A± = 9 + 8 3 ± 3 83 + 48 3

90

(4.97)

The point (6) is the crossing point of the function of the extrema of the photon circular orbit function aph−ex , and the curve separating black hole and naked singularity spacetimes, b = 1 − a2 . The point (7) is the common point of the dotted curve, given by the function ams(extr) and limiting the spacetimes allowing for existence of the marginally stable orbits, and the curve separating black holes from naked singularities (b = 1−a2 ). These two curves are tangent at the common point.

4.11.4

Character of circular geodesics in the KN spacetimes

The parameter space of the braneworld KN spacetimes b − a is divided into fourteen regions due to the criteria reflecting basic properties of the spacetimes and properties of their circular geodesics: • Existence of event horizons and ergosphere • Existence of unstable and stable circular photon geodesics • Existence of the marginally stable geodesics or the innermost stable circular geodesics (ISCO) The classification is summarized in Table (4.3). Basically, we combine Fig. 4.15 with Fig. 4.7 to obtain Figs. 4.16-4.17 where properties of the photon circular geodesics and properties of marginally stable geodesics or ISCO’s are reflected. We will show that the most surprising properties of the Keplerian accretion arise in the spacetimes of Class IIIa. Now we give properties of the circular geodesics in all the fourteen classes of the braneworld KN spacetimes, presenting and discussing typical radial profiles of their specific energy and specific angular momentum, complemented by sequences of the effective potential. Classification of the standard KN spacetimes according to properties of the circular geodesics contains all the classes except those related to b < 0 – therefore, classes VIII, IX and X are excluded. Class I Class of the black hole spacetimes with two horizons, two unstable photon circular orbits and ergosphere. Class border is given by line b = aph−ex (r = 4b/3, b) , b = 1 − a2 with intersection at point (0.5, 3/4) (point number (6) in Fig. 4.16) and line a = 0 . Marginally stable orbits for test massive particles are given by the inflexion point of the effective potential are defined by Eq. (4.85) and coincide with the ISCO’s (this is the standard scenario of Keplerian accretion: shortly – classic).

4.11. CLASSIFICATION OF BRANEWORLD KN SPACETIMES 6 5

2

2 1

VEff (r,a,b,L)

Inner horizon

E

Outer horizon

Inner horizon

-20

Outer horizon

1.6

3 0

a=0.2, b=0.5 L=2.95432 L=4 L=6 L=8

Class: I

1.8

4

20

L

a=0.2, b=0.5 upper sign lower sign

Class: I

0

1.4 Horizon


Class: I 40

91

1.2 1 0.8

-1

-40

0.6

0.1

1

r

10

-2 0.1

100

1

r

10

100

1

2.48

13.32

33.52 61.55 100

r

Figure 4.19: L, E and effective potential for class I. Class II Class of the black hole spacetimes with two horizons, one stable and three unstable photon circular orbits and ergosphere. Notice that the stable and unstable photon circular geodesic are located under the inner horizon, being thus irrelevant for the Keplerian accretion. Border is given by line b = aph−ex (r = 4b/3, b) and b = 1 − a2 with intersection at point (0.5, 3/4) (point number (6) in Fig 4.16) and line b = 0 . Marginally stable orbits for test massive particles are given by the inflexion point of the effective potential, coincide with ISCO’s (classic). a=0.8, b=0.1 upper sign lower sign

Class: II 40

6

a=0.8, b=0.1

Class: II

L=-0.216085 L=2.25431 L=0 L=4 L=12 L=-0.23

0.1

Outer horizon

1

Inner horizon

VEff (r,a,b,L)

Outer horizon

1 0 -1

-40 0.1

2

Inner horizon

E

Outer horizon

Inner horizon

L

3

-20

10

4

20

0


Class: II

5

1

r

10

100

-2 0.1

1

r

10

100

0.01 0.17

1.77

13.91

141.44

r

Figure 4.20: L, E and effective potential for class II. Class IIIa Class of the naked singularity spacetimes with one stable and one unstable photon circular geodesics and ergosphere. The border of the Class IIIa region is given by line b = ams(extr) √ and line b = 1 with intersection points at 2 − 3, 1 = (0.268, 1) (point number (5)) and √ 2 + 3, 1 = (3.732, 1) (point number (9)). We have also marked the point number (7) √ with coordinates ( 0.5, 0.5) where the line b = ams(extr) and line b = 1 − a2 touch and are tangent to each other. This theoretically means that effect of mining instability can be √ achieved for extremal Kerr–Newman black holes with spin parameter a = 0.5 and charge or braneworld tidal charge parameter b = 0.5. But it occurs under the event horizon. We have also marked the point number (8) with coordinates (1, 0.25) giving information on minimal amount of electric charge or braneworld tidal charge parameter b.


92

Marginally stable orbit of massive test particles is in case of the lower family circular geodesics given by inflexion point of the effective potential and coincides with ISCO (classic).


Class: IIIa 40

6


Class: IIIa

5

4

VEff (r,a,b,L)

E

L

3 0

2 1

-20

-40 1

r

10

100

3 2 1

0

0

-1

-1

-2 0.1

a=1.5, b=0.8 L=10 L=3 L=0 L=-10 L=-20

Class: IIIa

5

4

20

0.1

6

1

r

10

100

-2 0.1

0.804

9.3

98.6

r

Figure 4.21: L, E and effective potential for class IIIa.

In case of the upper family circular geodesics, the inflexion point of the effective potential is not defined and the sequence of minima of the effective potential continues with decreasing specific energy and specific angular momentum of the accreting matter down to the stable photon orbit. This orbit can be therefore considered as ISCO of the massive test particles. We have thus found an infinitely deep gravitational well enabling theoretically an unlimited mining of energy from the naked singularity. In fact, such a mining instability could work only up to the energy contained in the naked singularity spacetime. We can expect that the energy mining could work also in more realistic situations when the naked singularity is removed and an astrophysically more plausible superspinar is created by joining a regular (e.g. stringy) solution to the Kerr–Newman spacetime at a radius overcoming the outer radius of the causality violation region [Gimon and Hoˇrava, 2009, Stuchlík and Schee, 2012]. The mining could work if the matching radius of the internal stringy spacetime and the outer Kerr–Newman spacetime is smaller than the radius of the stable photon orbit related to the mining instability of the Class IIIa spacetimes. For completeness, we give in this case also the locally measured (LNRF) specific energy of the upper family circular geodesics. As shown in Fig. 4.22, the specific energy ELNRF diverges, along with the covariant specific energy E as the orbit approaches the limiting photon circular orbit. On the other hand, Fig. 4.23 clearly demonstrates that the ratio |E|/ELNRF remains finite while the orbits approach the location corresponding to the stable photon circular orbit.

4.11. CLASSIFICATION OF BRANEWORLD KN SPACETIMES 2

upper sign, b=0.7

Class: IIIa

2

upper sign, a=1

Class: IIIa

b=0.25 b=0.5 b=0.9

1.8

1.8

1.6

1.6

ELNRF

ELNRF

a=0.6 a=0.8 a=1.2

93

1.4

1.4

1.2

1.2

1

1

0.8

0.8 0

0.5

1

1.5

2

2.5

3

3.5

0

4

0.5

1

1.5

2

2.5

3

3.5

4

r

r

Figure 4.22: Energy measured by the LNRF observers (upper sign family orbits only). The energy diverges at the radius of the stable photon circular orbit. Of course, the mining instability could work only if the assumption of the test particle motion of the accreting matter is satisfied. Therefore, the assumption requires validity of the relation ˜ M, |E| (4.98) the covariant energy of the particle (accreting matter) has to be much smaller than the naked singularity mass parameter M. Of course, the issue of the mining instability and the related interaction of the mining unstable Kerr–Newman naked singularity (Kerr–Newman superspinar) and the accreting mass is much more complex and deserves a more detailed study. 0.8

upper sign, b=0.7 a=0.6 a=0.8 a=1.2

Class: IIIa

0.7

2

1.5

0.5

|E|/ELNRF

|E|/ELNRF

0.6

upper sign, a=1 b=0.25 b=0.5 b=0.9

Class: IIIa

0.4 0.3 0.2

1

0.5

0.1 0

0 0.6

0.7

0.8

0.9

1

1.1

r

1.2

1.3

1.4

1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

r

Figure 4.23: Absolute value of covariant energy /E/ divided by LNRF energy (for the upper sign family orbits). For class IIIa, the fraction is defined up to radius of stable circular photon orbit and at this point it has finite value. Class IIIb Class of naked singularity spacetimes with one stable and one unstable photon circular orbit and ergosphere. In the parameter space b − a the area related to this class is not


94

compact and disintegrates into two separated areas. The first area is infinitely large, its border is given by the line b = ams(extr) , line b = 1 − a2 , line b = 1 and b = 0 with in √ √ tersections at the point ( 0.5, 0.5) (point number (7)) and 2 + 3, 1 (point number (9)). The second area is compact and finite. Its border is given by the line b = ams(extr) , line b = aph−ex (r = 4b/3, b), line b = 1 − a2 and line b = 1 with intersection points number (4), (5) and (7). It is not obvious from figures, but b = ams(extr) and b = aph−ex (r = 4b/3, b) do not intersect. Marginally stable orbits of both the lower and upper family of circular geodesics are given by the inflexion point of the effective potential and coincide with ISCO’s (classic). Notice that in this case the sequence of the upper family orbits with descending specific energy E and specific angular momentum L is interrupted by a sequence where both E and L increase with decreasing radius, corresponding thus to the unstable geodesics. In this case, the infinitely deep gravitational well still exists, but the Keplerian accretion sequence is interrupted and this gravitational well cannot be applied in an astrophysically natural accretion process. Nevertheless, it is still possible to use this gravitational well, if matter with appropriate initial conditions (values of the motion constants), enabling starting of the mining instability, could appear close to the naked singularity. a=3.5, b=0.5 upper sign lower sign

Class: IIIb 40

6


Class: IIIb

5

VEff (r,a,b,L)

4

E

L

3 0

2 1

-20

-40 1

r

10

100

3 2 1

0

0

-1

-1

-2 0.1

a=3.5, b=0.5 L=20 L=10 L=3 L=1.34083 L=-10 L=-20

Class: IIIb

5

4

20

0.1

6

1

r

10

100

-2 0.1

0.5

10.6

99.3

398.5 1000

r

Figure 4.24: L, E and effective potential for class IIIb. Class IVa Class of naked singularity spacetimes with one stable and one unstable photon circular orbit. Class is without ergosphere. Border of this class in the spacetime parameter space is given by line b = aph−ex (r = 4b/3, b) and b = ams(extr) with intersection points (3),(5),(9) and (10). For the lower family circular geodesics, the marginally stable orbits defined by the inflexion point of the effective potential occur (classic). For the upper family circular geodesics marginally stable orbit is not defined, and the ISCO is located at r = b as it is always for all classes with b > 1. A sequence of stable circular geodesics with sharply increasing specific energy occurs near (above) the radius r = b, approaching the stable photon circular orbit. (Such sequences of stable circular orbits were discussed in [Stuchlík and Schee, 2014].) Note that probability we are actually living in a spacetime with the braneworld tidal


95

charge parameter greater than one is very small [Kotrlová et al., 2008a, Böhmer et al., 2008]. a=2, b=1.5 upper sign lower sign

Class: IVa 40

6

a=2, b=1.5 upper sign lower sign

Class: IVa

5

6 5

VEff (r,a,b,L)

4

20

E

3

L

a=2, b=1.5 L=10 L=5 L=0 L=-5 L=-10

Class: IVa

0

2 1

-20

4 3 2

0

1

-1

-40 0.1

1

r

10

-2 0.1

100

1

r

10

0 0.1

100

1.5

4.2

25.6

99.3

r

Figure 4.25: L, E and effective potential for class IVa. Class IVb Class of naked singularity spacetimes with one stable and one unstable photon circular orbit. Class is without ergosphere. In the parameter space b − a this class is not compact and disintegrates into two separated areas. The first area is infinitely large, the border is given by the line b = aph (4b/3, b), line b = ams(extr) and line b = 1 with intersection points (9) and (10). The second area is finite and its border is given by same lines and intersection points (2), (3), (4) and (5). For both upper and lower families of circular geodesics, the marginally stable orbits of test massive particles are given by the inflexion point of the effective potential, governing the sequence of geodesics related to the standard Keplerian accretion (classic). There is additional internal sequence of stable circular geodesic, with ISCO located at r = b as it is always for all classes with b > 1. This sequence approaches the stable photon circular geodesic. a=10, b=2 upper sign lower sign

Class: IVb 40

6

2

upper sign lower sign

5

a=10, b=2 L=15 L=10 L=4.69702 L=0 L=-7.07107 L=-12

Class: IVb

1.8

VEff (r,a,b,L)

1.6

20

4

E

L

a=10, b=2

Class: IVb

0

3 2

-20

1.4 1.2 1 0.8

1 -40

0.6 1

10

r

100

0

1

10

r

100

0.73 1

2

10

100

r

Figure 4.26: L, E and effective potential for class IVb. Class Va Class of naked singularity spacetimes with no stable or unstable photon circular orbit. Class is also without ergosphere. The border of the Class Va region in the parameter space is given by the line b = ams(extr) and line a = 0, with intersection point (1).


96

For both lower and upper family circular geodesics, the marginally stable orbits are not defined. The circular geodesics are only stable, ISCO’s are located at r = b, as it is always for all classes with b > 1. a=2, b=3 upper sign lower sign

Class: Va 40

a=2, b=3 upper sign lower sign

Class: Va

1.4 1.3

VEff (r,a,b,L)

E

1.1 0

1 0.9

-20

a=2, b=3 L=2 L=-0.816497 L=-5 L=-8 L=-12 L=20

Class: Va

1.8

1.2

20

L

2

0.8

1.6 1.4 1.2 1 0.8

0.7

-40

0.6

0.1

1

r

10

0.6

100

1

10

1

100

r

3

7.5

22.2

65.5

159.4 400.6 1000

r

Figure 4.27: L, E and effective potential for class Va. Class Vb Class of naked singularity spacetime with no stable or unstable photon circular orbit. Class is also without ergosphere. The border of the Class Vb region in the parameter space is given by the line b = ams(extr) , and line b = aph (4b/3, b), with intersection points (1), (3) and (10). The class is infinitely large. The upper family circular geodesics are stable only, finishing at ISCO at r = b. The marginally stable orbit exists for the lower family orbits giving the limit of standard Keplerian accretion. The lower family orbits continue downwards by sequence of unstable orbits and finally stable orbits finishing at r = b. a=2, b=2.1 upper sign lower sign

Class: Vb 40

6


Class: Vb

5

6 5

VEff (r,a,b,L)

4

20

E

L

3 0

2 1

-20

4 3 2

0

1

-1

-40 0.1

a=2, b=2.1 L=10 L=5 L=-1.3159 L=-5 L=-10

Class: Vb

1

r

10

100

-2 0.1

1

r

10

100

0

1.5

4.2

25.6

99.3

r

Figure 4.28: L, E and effective potential for class Vb. Class Vc Class of naked singularity spacetimes with no stable or unstable photon circular orbit. This class is also without ergosphere. In the parameter space b − a this class disintegrates into two separated areas. The first one is infinitely large, its border is given by the line b = aph−ex (r = 4b/3, b) and line b = ams(extr) , with the intersection point (10). The second area is finite and its border is given by same line b = aph−ex (r = 4b/3, b), line b = ams(extr) and line a = 0, with intersection points (1), (2) and (3).


97

The marginally stable orbit exists for both the lower and upper family circular geodesics, giving thus the standard limit of the Keplerian accretion (classic). Both the lower and upper family orbits continue downwards by sequence of unstable orbits and finally stable orbits finishing at r = b. a=0.01, b=1.15 upper sign lower sign

Class: Vc 40

2

1.6

a=0.01, b=1.15 L=3 L=2.6462 L=2 L=-0.02408 L=-1

Class: Vc

1.4

E

0

-20

VEff (r,a,b,L)

1.5

20

L


Class: Vc

1

0.5

1.2 1 0.8 0.6

-40

0.4

0.1

1

r

10

0 0.1

100

1

r

10

100

0.1

0.31

1

2.2

6.5 10

100

r

Figure 4.29: L, E and effective potential for class Vc. Class VI Class of naked singularity spacetimes with two stable and two unstable photon circular orbits and ergosphere. In the parameter space the area of the Class VI spacetimes has the boundary is given by the line b = aph−ex (r = 4b/3, b), line b = 1 − a2 and the line b = 1, with the intersection points (4) and (6). For both lower and upper family of circular geodesics the marginally stable orbit exists, giving thus the inner edge of the standard Keplerian accretion. The upper family orbits have also a very narrow region of stable circular orbits near the radius r = b, finishing at the stable circular photon orbit. a=0.15, b=0.99 upper sign lower sign

Class: VI 40

6


Class: VI

5

2

a=0.15, b=0.99 L=5 L=2 L=0 L=-2 L=-5

Class: VI

1.5

VEff (r,a,b,L)

4

20

E

L

3 0

2 1

-20

1

0.5

0

0 -1

-40 0.1

1

r

10

100

-2 0.01

0.1

1

r

10

100

-0.5 0.3

1

1.78 2.56

10

r

Figure 4.30: L, E and effective potential for class VI. Class VII Class of naked singularity spacetimes with two stable and two unstable photon circular orbits. Class is without the ergosphere. In the parameter space the area of the Class VII spacetimes has the boundary given by the line b = aph−ex (r = 4b/3, b), line a = 0 and the line b = 1, with the intersection points (2) and (4).


98

For both lower and upper family circular geodesics, the marginally stable orbit exists giving thus the edge of the standard Keplerian accretion. Further, both lower and upper family orbits have an ISCO at r = b where sequence of stable orbits starts, finishing for each family at the related photon circular geodesic. a=0.15, b=1.01 upper sign lower sign

Class: VII 40

6


Class: VII

5

2

a=0.15, b=1.01 L=5 L=2 L=0 L=-2 L=-5

Class: VII

1.5

VEff (r,a,b,L)

4

20

E

L

3 0

2 1

-20

1

0.5

0

0 -1

-40 0.1

1

r

10

-2 0.1

100

1

r

10

-0.5 0.3

100

1

1.78 2.56

10

r

Figure 4.31: L, E and effective potential for class VII. Class VIII Class of black hole spacetimes with negative braneworld tidal charge parameter b that has only one horizon located at r > 0, two unstable photon circular orbits and ergosphere. In the parameter space b − a, boundary of the region related to this class is given by the line b = −a2 , and the line a = 0. For both lower and upper family circular geodesics the marginally stable orbit exist, determining the inner edge of the standard Keplerian disk. We obtained thus the standard situation typical for Kerr black holes, but no geodesic structure occurs at r > 0 under the event horizon. 6

a=0.5, b=-1 upper sign lower sign

Class: VIII

5

E

Outer horizon

L

-20

1

-40 1

2

r

10

100

VEff (r,a,b,L)

Outer horizon

4

3 0

3 2 1

0

0

-1

-1

-2 0.1

a=0.5, b=-1 L=-10 L=0 L=3.44726 L=10 L=20

Class: VIII

5

4

20

0.1

6

1

r

10

100

Horizon


Class: VIII 40

-2 0.1

1

4.34

95.52

396.1 1000

r

Figure 4.32: L, E and effective potential for class VIII. Class IX Class of black hole spacetimes with negative braneworld parameter b having two horizons, three unstable photon circular orbits and ergosphere. Border of the related region of the spacetime parameter space is given by the line b = 1 − a2 , line b = −a2 , and the line b = 0. For both lower and upper circular orbits the marginally stable orbit exists, giving in standard way the inner edge of the Keplerian accretion. Unstable orbits exist under the inner horizon.

4.12. EFFICIENCY OF THE KEPLERIAN ACCRETION a=0.9, b=-0.3 upper sign lower sign

Class: IX

5

1

-40 0.01

0.1

2

1

10

r

3 2 1

0

0

-1

-1

-2 0.01

100

VEff (r,a,b,L)

Inner horizon

E

Outer horizon

Inner horizon

-20

Outer horizon

4

3 0

a=0.9, b=-0.3 L=-10 L=0 L=2.29179 L=4 L=10 L=18

Class: IX

5

4

20

L

6

0.1

1

r

10

-2 0.01

100

Outer horizon

6

Inner horizon

a=0.9, b=-0.3 upper sign lower sign

Class: IX 40

99

0.053 0.17

1 2.05

13.55

97.1 320.4 1000

r

Figure 4.33: L, E and effective potential for class IX. Class X Class of naked singularity spacetimes with negative brane parameter b having one unstable photon circular orbit and ergosphere. Border of the related region of the parameter space is given by the line b = 1 − a2 , and the line b = 0. In these naked singularity spacetimes the marginally stable orbit exists for both lower and upper family, representing thus in both cases the inner edge of the Keplerian accretion disks. Under the marginally stable orbits only unstable orbits exist for both families. From the point of view of the geodesic structure, the naked singularity spacetimes of Class X resemble the standard Kerr naked singularity spacetimes. a=2, b=-0.01 upper sign lower sign

Class: X 40

6

a=2, b=-0.01 upper sign lower sign

Class: X

5

VEff (r,a,b,L)

1.6

E

3 0

2 1

-20

a=2, b=-0.01 L=-4 L=-2 L=0.647934 L=2 L=3

Class: X

1.8

4

20

L

2

1.4 1.2 1

0 0.8 -1

-40 0.1

0.6 1

r

10

100

-2 0.01

0.1

1

r

10

100

0.1

0.3

1

4.8

10

100

r

Figure 4.34: L, E and effective potential for class X.

4.12

Efficiency of the Keplerian accretion

Now we are able to determine the energetic efficiency of the Keplerian accretion. From the astrophysical point of view, the standard Keplerian accretion is relevant in the regions enabling starting of accretion at large distance (infinity) and its finishing at the first inner edge that can be approached by a continuous accretion process. We determine efficiency of the Keplerian accretion for all the classes of the braneworld Kerr–Newman spacetimes for the standard Keplerian accretion. In some of these spacetimes there exists also an inner region where the Keplerian accretion could work due to the decline of both energy and angular momentum with decreasing radius. However, these regions are not related to the standard notion of Keplerian accretion and will not be considered here for calculations

4.12. EFFICIENCY OF THE KEPLERIAN ACCRETION

100

of the accretion efficiency. Moreover, there could exist also complexities of the Keplerian accretion process related to the behaviour of the angular velocity that are described in detail in [Stuchlík and Schee, 2014] – we shall not discuss these subtleties in the present paper. We concentrate our attention in determining the efficiency for the Keplerian accretion following the upper family circular geodesics, when the efficiency can be very high, being in some cases even unlimitedly high (formally). In the case of the upper family Keplerian accretion, the efficiency is discontinuous when transition between the naked singularity with sufficiently high dimensionless spin and the related extreme black hole state is considered. The critical value of the spin, and the related critical tidal charge reads 1 acr = √ , 2

bcr =

1 . 2

(4.99)

We have to stress that the efficiency of the Keplerian accretion in the near-extreme naked singularity spacetimes exceeds significantly efficiency in the extreme black hole spacetimes. On the other hand, the efficiency of the Keplerian accretion in the upper family regime is fully continuous in the case of transition of the naked singularity to extreme black hole spacetime with sufficiently low spin, a < acr , and for all the braneworld KN spacetimes in the case of the Keplerian accretion in the lower family accretion regime. Generally, the efficiency of the Keplerian accretion is substantially smaller in comparison to the upper family regime in a given Kerr–Newman spacetime. The efficiency of the accretion for the geometrically thin Keplerian disks governed by the circular geodesics is defined by the relation η(a, b) = 1 − E(redge , a, b) ,

(4.100)

where redge denotes location of the inner edge of the standard Keplerian accretion disks. For the Keplerian disks following the lower family circular geodesics, the inner edge of the disk is always located at the marginally stable geodesic giving thus always the scenario of the Keplerian accretion in the Kerr spacetimes. On the other hand, for the upper family Keplerian disks, the situation is more complex, as follows from the classification of the braneworld Kerr–Newman spacetimes. There can occur three qualitatively different cases in dependence on the combinations of dimensionless spacetime parameters a and b. In the first family of classes of the KN spacetimes, the redge is simply located at the marginally stable geodesic, giving thus the scenario of the Keplerian accretion onto Kerr black holes – this case includes all the braneworld Kerr–Newman black hole spacetimes. In the second family of the Kerr–Newman classes, the inner edge of the Keplerian disk is located at the radius r = b, giving thus the special case discovered at first for the Reissner–Nordström naked singularity spacetimes [Stuchlík and Hledík, 2002, Pugliese et al., 2011]. In all classes with b > 1 (IV,V) the efficiency of the Keplerian accretion along


η (a,b)

η

101

∞

∞ 1.5 1 0.5 0

1 0.8 0.6

0

0.5

1

1.5

2

2.5

a

0.2

3

3.5

4 0

1.8

1.7071

1.6

13 1.5774 1.60

1.5345

1.5222

a=3 4

1.4

0.3 0.1

10

a=√3

1.2

η (a,b)

b

0.4

0.4 0.6 0.8

a=√2

η

1.0

1

∞

0.8 a= 2

0.6 0.4778

0.4655

0.42270.3987

0.2 0.0 -2.5

a=1

a=3.732

-2

-1.5

-1

-0.5

a=

0.

9

0.4

0

0.2929

/2 √2 a= a=0.268 a=0

0.5

0.134 0.081

1

1.5

2

b Figure 4.35: Energetic efficiency η(a, b) of the Keplerian accretion following the upper family circular geodesics is given in dependence on the spacetime dimensionless tidal charge parameter b and the spin parameter a. 3D diagram is reflecting the position of the special class of the mining unstable Kerr–Newman spacetimes of Class IIIa in the plane of the spacetime parameters. Due to a complex character of the efficiency function we give also the characteristic a = const sections in the η − b plane.


102

0.08

η (a,b)

0.07

a=0.000 a=0.707 a=1.000 a=1.411 a=1.730 a=2.000

0.06

0.05

0.04

0.03 -4.0

-3.0

-2.0

-1.0

0.0

1.0

b Figure 4.36: Energy efficiency of the Keplerian accretion following the lower family circular geodesics is given in dependence on the spacetime dimensionless tidal charge parameter b for characteristic values of the spin parameter a = const.

the upper family of circular geodesics is independent of the spin parameter a being defined by the simple relation 3 r 1 η(b) = 1 − 1 − . (4.101) b The efficiency goes slowly to 0% for b → ∞ – see Fig. 4.35. In the third and most interesting family of the Kerr–Newman spacetime classes, redge corresponds to the radius of the stable photon orbit approached by particles with specific energy E → −∞ and specific angular momentum L → −∞ – notice that the limiting photon circular geodesic is a corotating one as the impact parameter λ = L/E > 0. In the third case, the Keplerian accretion efficiency approaches (theoretically) infinity. This effect occurs explicitly in the Class IIIa KN spacetimes as clearly demonstrated in Fig. 4.21. For this class the tidal charge parameter b ∈ (1/4, 1) and the dimensionless spin 3 Interestingly,

the Keplerian efficiency relation for the lower family of geodetics is much more complex and depends on the spin parameter a.


103

√ p √ p a ∈ 2 b − b(4b − 1), 2 b + b(4b − 1) . The Keplerian accretion efficiency is given for the upper family of circular geodesics in Fig. 4.35, and for the lower family circular orbits in Fig. 4.36. Because of its complexity, we represent the case of the upper family accretion regime by a 3D figure with addition of the figure representing the relevant sections a = const. In the case of the lower family accretion regime, the representative a = const sections are sufficient to clearly demonstrate the character of the efficiency of the Keplerian accretion. For the Keplerian accretion along the lower family circular geodesics the situation is quite simple and the efficiency is always continuously matched between the naked singularity and the extreme black hole states. The efficiency of the lower family regime accretion for fixed dimensionless spin a of the braneworld KN spacetimes always decreases with decreasing tidal charge parameter b. Moreover, for fixed tidal charge b, the efficiency decreases with increasing spin a. In order to understand the upper family Keplerian accretion regime and its efficiency, the dependences η (b, a = const) are most instructive. They are governed by two crucial families of curves. First, the efficiency of the Keplerian accretion in the extreme braneworld KN black hole spacetimes, and the related near-extreme braneworld KN naked singularity spacetimes is given by the relation 1 ηjump (b) = 1 ± q , 1 4 − 1−b

(4.102)

where the + sign corresponds to the efficiency in the near-extreme naked singularity spacetimes, while the − sign corresponds to the related extreme black holes. Naturally, this formula is relevant in the interval of the tidal charge b ∈ (−∞, 0.5), i.e., up to the critical value of the tidal charge. Second crucial curve is given by the efficiency of the accretion in the limiting spacetimes governed by the boundary of Class IIIa spacetimes, ηmining (b, amining± (b)), √ p where b ∈ (1/4, 1), and amining (b) = 2 b ± b(4b − 1). The results can be summarized in the following way. For whole the braneworld spacetimes with the negative tidal charge parameter b < 0, and for those with positive charge parameter 0 ≤ b ≤ 1/2, the large jump of the efficiency in transition between the+naked singularity to the related extreme black hole state occurs. Such a jump was observed for the first time in the case of transition between the Kerr naked singularity and the extreme Kerr black hole (b = 0) where η ∼ 1.57 goes down to η ∼ 0.43 [Stuchlík, 1980]. For the braneworld KN extreme black holes (related near-extreme naked singularities), the efficiency slightly increases (decreases) with negatively valued tidal charge increasing in its magnitude, so the efficiency jump slightly decreases from its maximal Kerr value. On the other hand, for b ∈ (0, 1/2), the efficiency for the extreme black holes (near-extreme naked singularities) decreases (increases), and the jump fastly increases – for b = 1/2, the efficiency jumps from η ∼ 1.707 down to η ∼ 0.293.


104

For the naked singularities with thetidal in the interval √ charge √ p1/4 < b a > 1/ 2, the curves η(b, a = 1) start at the extreme state√and finish at the state with 0 < b < 1/2 and efficiency η > 1. For spin approaching a = 1/ 2, the curve η(b, a = const) degenerates at the point with √ b = 1/2, and efficiency approaching η ∼ 1.707. For higher values of spin, a ∈ (1, 2 + 3), the efficiency curves η(b, a = const) decrease to the curve ηmining (b, amining± (b)) with b increasing in the interval √ b ∈ (1/4, 1) and the efficiency decreasing down to the limiting value of η(b = 1, a = 2 + 3) ∼ 0.134. For tidal charge b ∈ (1/2, 1), the efficiency of the Keplerian accretion at the transition between the extreme black hole and the related near-extreme naked singularity is continuously matched. The efficiency of η ∼ 0.134 is reached for the Kerr–Newman √spacetime √ √ with b = 1 and a = 2 − 3. For values of the spin in the interval of a ∈ 2 − 3, 1/ 2 , the transition of the function η(b, a = const) between the black hole and naked singularity states, obtained due to increasing tidal charge b, is still continuous, and the curve ηmining (b, amining± (b)) is reached at values η < 0.293. With increasing spin a, the efficiency of the Keplerian accretion decreases. It is interesting that for naked singularities having spin a higher than ∼ 4.97 and appropriately valued tidal charge b, the efficiency reaches values smaller than those corresponding to the Schwarzschild black holes (η ∼ 0.057). Note that the results of the Keplerian accretion analysis for the braneworld Kerr– Newman spacetimes can be directly applied also for the Keplerian accretion in the standard Kerr–Newman spacetimes, if we make transformation b → Q2 where Q2 represent the electric charge parameter of the Kerr–Newman background.

4.12.1

Locally non-rotating frames and orbital motion

The orbital velocity of matter orbiting a braneworld Kerr black hole along circular orbits is given by appropriate projections of its 4-velocity U = (U t , 0, 0, U ϕ ) onto the tetrad of a locally non-rotating frame (LNRF) [Bardeen et al., 1972b] 1 e(t) = ω2 gϕϕ − gtt 2 dt , (4.103) 1

e(ϕ) = (gϕϕ ) 2 (dϕ − ωdt) ,

(4.104)


e

(r)

Σ = ∆

105

! 21

(4.105)

dr , 1

e(θ) = Σ 2 dθ ,

(4.106)

where ω is the angular velocity of the LNRF relative to distant observers and reads ω=−

gtϕ gϕϕ

=

a(2r − b) . Σ(r 2 + a2 ) + (2r − b) a2 sin2 θ

(4.107)

For the circular motion, the only non-zero component of the 3-velocity measured locally in the LNRF is the azimuthal component that is given by 2 r 2 + a2 − a2 ∆ sin2 θ sin θ (Ω − ω) [Ω − ω] (ϕ) VLNRF = r , √ = Σ ∆ g tt ω2 − gϕϕ where Ω=

lgtt + gtϕ Uϕ =− t U lgtϕ + gϕϕ

(4.108)

is the angular velocity of the orbiting matter relative to distant observers and l=−

Uϕ

(4.109)

Ut

is its specific angular momentum; Ut , Uϕ are the covariant components of the 4-velocity field of the orbiting matter. Using (3.63) we arrive to the formula a(2r−b) l + Σ sin2 θ 1 − 2r−b Σ . (4.110) Ω= 2 sin2 θ sin2 θ − l a(2r−b) sin2 θ r 2 + a2 + 2r−b a Σ Σ

4.12.2

Energy measured in LNRF

It is useful to determine for particles on the circular geodesics the locally measured energy, related to some properly defined family of observers. The specific energy related to the LNRF (ELNRF ) is given by the projection of the 4-velocity on the time-like vector of the frame:

ELNRF = U

(t)

=

! dt (t) = e dτ t √ r2 ± a r − b

(t) U µ eµ

= p

r 4 + a2 (r 2 + 2r

(4.111) √ ∆

. √ − b) r 2 − 3r + 2b ± 2a r − b p

4.13. ASCHENBACH EFFECT FOR BRANEWORLD KN SPACETIMES

106

The locally measured particle energy must be always positive for the particles in the positive-root states assumed here – while it is negative for the negative-root states that are physically irrelevant in the context of our study [Biˇcák et al., 1989]. The LNRF energy of particle following the circular geodesics diverges on the photon circular orbit as well as the covariant energy E. It also diverges for circular orbits approaching the boundary of the causality violation region given by Eq. (4.25).

4.12.3

Future-oriented particle motion

For the positive-root states the time evolution vector has to be oriented to future, i.e., dt/dτ > 0. On the other hand, the negative-root states have past oriented time vectors, dt/dτ < 0, being thus physically irrelevant for our study. To be sure that we are using the solutions related to the proper effective potential Veff with the correct upper sign, we have to check that the considered geodesics have proper orientation dt/dτ > 0. Using the metric (3.63) and relations for the specific energy (4.10) and specific angular momentum (4.11) in Eq. (4.3), we obtain the time component of the 4-velocity for both the upper and lower family circular geodesics in the form √ dt r2 ± a r − b = p . dτ r r 2 − 3Mr + 2b ± 2a√Mr − b

(4.112)

We see from this equation that the time component is always positive for both the orbits of the upper and lower family, so we always have the positive-root states and no mixing with the negative-roots states occurs.

4.13

Aschenbach effect for braneworld KN spacetimes

Properties of accretion discs can be appropriately represented by circular orbits of test particles or fluid elements orbiting black holes (superspinars). The local properties can be efficiently expressed when related to the locally non-rotating frames (LNRF’s), since these frames corotate with the spacetime in a way that enables us to cancel the frame-dragging effects as much as possible [Bardeen et al., 1972b]. A new phenomenon related to the LNRF-velocity profiles of matter orbiting near-extreme Kerr black holes has been found by Aschenbach [Aschenbach, 2004, Stuchlík et al., 2005] namely a non-monotonicity in the velocity profile of the Keplerian motion in the field of Kerr black holes with dimensionless spin a > 0.9953. Such a hump in the LNRF-velocity profile of the corotating orbits is a typical and relatively strong feature in the case of Keplerian motion in the field of Kerr naked singularities, but in the case of Kerr black holes it is a very small effect appearing for near-extreme black holes only (see Fig. 4.37). In the naked singularity case, we call the orbits to be of the first family rather than corotating, since these can be retrograde relative

4.13. ASCHENBACH EFFECT FOR BRANEWORLD KN SPACETIMES 1.0

1.0

a=0.999 a=0.777 a=0.555 a=0.000 Newton

0.9

0.8 0.6

)

0.7

(ϕ)

(ϕ)

V K (r;a)

0.8

V K (r;a)

107

0.4 0.2 0.0

0.6

Newton a=1.0 a=1.3 a=1.8 a=5.0

-0.2 0.5 -0.4

( 0.4 1.0

1.5

2.0

2.5

3.0

r

3.5

4.0

4.5

5.0

-0.6 0.0

1.0

2.0

3.0

4.0

5.0

r

Figure 4.37: Keplerian velocity profiles related to the LNRF. Left: Kerr black holes-the velocity profiles presented for some values of the black hole spin. The Aschenbach effect appears for near-extreme black holes and is weak. Right: Kerr naked singularities-the velocity profiles are given for some values of the spin, demonstrating existence of the Aschenbach effect for orbits with negative-valued velocity. For completeness, the velocity profile is also given for the extreme black hole, demonstrating a velocity jump at r = 1.

to the LNRF in vicinity of the ring singularity for small values of spin (a < 5/3), while they are corotating for larger values of the spin [Stuchlík, 1980]; the humpy character of the LNRF-velocity profile ceases for naked singularities with a > 4.0014 as demonstrated in Fig. 4.37. A study of non-Keplerian distribution of specific angular momentum (` = const), related to geometrically thick discs of perfect fluid, has shown that the “humpy” LNRFvelocity profile appears for near-extreme Kerr black holes with a > 0.9998 [Stuchlík et al., 2005]. The humpy LNRF-velocity profile emerges in the ergosphere of near-extreme Kerr black holes, at the vicinity of the marginally stable circular orbit. The maximal velocity difference between the local minimum and maximum of the humpy Keplerian velocity profiles is ∆v ≈ 0.07c and takes place for a = 1. Motion of test particles in the field of braneworld rotating black holes is given by the geodesic structure of the Kerr–Newman spacetimes with the tidal charge b. The braneworld parameter reflects the tidal effects of the bulk space and has no influence on the motion of charged particles. The geodesic structure given by the Carter equations [Carter, 1973a] is relevant for both uncharged and charged test particles. The circular test particle orbits of the braneworld Kerr black holes are identical to the circular geodesics of the Kerr–Newman spacetime with properly chosen charge parameter. We shall study the Aschenbach effect, i.e., we look for the non-monotonicity (humps) in the LNRF-velocity profiles of Keplerian discs orbiting near-extreme braneworld Kerr black holes or naked singularities.

4.14. VELOCITY PROFILES FOR THE KEPLERIAN DISTRIBUTION OF `

108

0.6

BH Aschenbach ef.

NS Aschenbach ef.

0.2

0.0

(ϕ)

V K+ (r;a,b)

0.4

NS-retrograde Aschenbach ef. -0.2

-0.4

-0.6 0.0

b=-2

a=1.73 a=1.76 a=1.83 1.0

2.0

3.0

4.0

5.0

6.0

r Figure 4.38: Non-monotonic plus-family orbital velocity profiles in braneworld KN black hole and naked singularity spacetimes with the tidal charge b = −2, given for three appropriately chosen values of spin a reflecting the whole variety of possible behavior related to the Aschenbach effect.

4.14

Velocity profiles for the Keplerian distribution of `

Motion of test particles following circular geodetical orbits in the equatorial plane (θ = π/2) is described by the Keplerian distribution of the specific angular momentum, which in the braneworld Kerr backgrounds takes the form [Stuchlík and Kotrlová, 2009] √ (r 2 + a2 ) r − b ∓ a(2r − b) `K± (r; a, b) = ± ; (4.113) √ r 2 − 2r + b ± a r − b the signs ± refer to two distinct families of orbits in the Kerr braneworld spacetimes. From the LNRF point of view the 2nd family, or minus-family (given by the lower sign), represents retrograde orbits, while the 1st family, or plus-family (given by the upper sign), represents direct orbits in the black hole spacetimes [Bardeen et al., 1972b], but can represent both direct and retrograde orbits in the naked singularity spacetimes. We can find a similar situation for Kerr spacetime in [Stuchlik, 1980, Stuchlík, 1981a]. For both families, we find a formal limit of the Keplerian motion to be located at r = b. In the black hole spacetimes, it is located under the inner horizon and is irrelevant from the astrophysical point of view. In the naked singularity spacetimes, it is relevant for b > 0, while for b < 0, it is located under the ring singularity at r = 0 which represents the limit on the location of circular orbits.


109

0.8 0.6

(ϕ)

V K (r;a,b)

0.4

b

0.2 0.0 -0.2

b=0.3

-0.4 -0.6

a=0.8366 a=0.8410 a=3.0000 a=8.0000

-0.8 -1.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

r Figure 4.39: Non-monotonic plus-family orbital velocity profiles in braneworld KN black hole and naked singularity spacetimes with the tidal charge b = 0.3 .

The corresponding Keplerian angular velocity related to distant observers is given by the relation 1 ΩK± (r; a, b) = ± 2 , (4.114) √r ±a r−b

and the Keplerian orbital velocity related to the LNRF in braneworld Kerr backgrounds is thus given by the relation √ r − b(r 2 + a2 ) ∓ a(2r − b) (ϕ) √ . (4.115) VK± (r; a, b) = ± √ r2 ± a r − b ∆ Clearly, there is a limit on existence of circular equatorial geodesics at r = b that is relevant (and located above the ring singularity) for spacetimes with positive tidal charge. Like in the Kerr backgrounds (b = 0), also in the braneworld Kerr backgrounds only the plus-family orbits exhibit the “minimum-maximum” humpy structure of the LNRF(ϕ) related orbital velocity profiles VK (r) . On the other hand, in braneworld Kerr nakedsingularity spacetimes with positive tidal charge (b > 0), there is always at least one local


110

(ϕ)

V K- (r;a,b) 0.0

-0.5

-1.0

b=-0.50, a=1 b=0.000, a=0 b=1.125, a=0 b=1.020, a=0 b=2.000, a=1 b=4.000, a=3

-1.5

-2.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

r 7.0

8.0

Figure 4.40: Minus-family orbital velocity profiles in braneworld Kerr black hole and naked singularity spacetimes.

(ϕ)

(ϕ)

extreme in the orbital velocity profiles (maximum in VK+ and minimum in VK− profiles) where the orbital velocity gradient changes its sign. This non-monotonicity, however, does not correspond to the Aschenbach effect. (ϕ) We illustrate the typical character of VK+ (r; a, b) velocity profiles in the braneworld rotating black hole and naked singularity spacetimes with a negative tidal charge in Fig. 4.38 in this case, the profiles are extended down to r = 0, but here we do not consider the region of r < 0. For special values of parameters of naked singularity spacetimes, the Aschenbach effect can be related to the retrograde region of first family orbits (the humpy (ϕ) structure contains orbits with VK+ < 0). (ϕ)

For positive tidal charge the characteristic profiles of VK+ are presented in the Fig. 4.39 – in this case the profiles finish their validity at r = b. There is also depicted velocity profile under the inner horizon of the black hole. Numerical calculations indicate that for those kinds of velocity profiles there is no Aschenbach effect, so for the black hole spacetimes (ϕ) we shall focus our attention on the behavior of the Keplerian profiles VK+ (r; a, b) above the outer horizon. (ϕ) For both positive and negative tidal charges the profiles of VK− in black hole and naked singularity spacetimes are presented in Fig. 4.40. These profiles do not exhibit any “minimum-maximum” structures. Therefore, in the following we shall focus attention to

4.14. VELOCITY PROFILES FOR THE KEPLERIAN DISTRIBUTION OF ` 1.0

111

1.4

b=0

b=0.5

aR-

aR+

1.2

0.8 1.0

0.6

a(r;b)

a(r;b)

ams ahah+

0.4

aph

0.8

ams

0.6

aph 0.4

ah-

ah+

aR-

0.2 0.2

0.0 0.0

1.0

2.0

3.0

4.0

0.0 0.0

5.0

1.0

2.0

3.0

4.0

5.0

6.0

r

r 2.0

1.6

aR+

aR+

b=-1

1.8

1.4

b=-2

1.6 1.2

1.4

ams

ah-

ams

1.2

a(r;b)

a(r;b)

1.0

ah-

0.8

aph 0.8

aph

0.6

1.0

0.6

ah+

0.4

ah+

0.4 0.2 0.0 0.0

aR-

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.2

aR-

0.0 0.0

1.0

r

2.0

3.0

4.0

5.0

6.0

7.0

8.0

r

Figure 4.41: The loci of the photon circular orbits aph (r, b); the marginally stable orbits ams (r, b); function aR± (r, b) and function ah± (r, b) defined implicitly by position of event horizon for appropriately chosen fixed values of b.

(ϕ)

(ϕ)

the plus-family orbits only and use notation VK (r; a, b) instead of VK+ (r; a, b). (ϕ)

We have to consider the function VK (r; a, b) (and also the functions lK (r; a, b), ΩK (r; a, b)) within the range of definition of the Keplerian motion [Stuchlík and Kotrlová, 2009]. The range is governed by the radii of the photon circular geodesics rph given implicitly by the relation r(3 − r) − 2b , (4.116) √ 2 r −b and by the radii of the marginally stable circular geodesics rms , implicitly given by a = aph (r; b) ≡

p 4(r − b)3/2 ∓ r 3r 2 − 2r(1 + 2b) + 3b a = ams (r; b) ≡ . (4.117) 3r − 4b The functions aph (r; b) and ams (r; b) are illustrated in Fig. 4.41 – for detailed discussion of the properties of photon and marginally stable orbits see [Stuchlík and Kotrlová, 2009, Kotrlová et al., 2008b]. Above the black hole outer horizon, the stable orbits are located in


112

4.0

b=r 2.0

0.0

b

-2.0

-4.0

-6.0

-8.0

-10.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

r Figure 4.42: Naked singularity regions with potentially retrograde plus-family orbits.

the interval of rms < r < ∞, while unstable orbits are located in the interval rph < r < rms . Note that for Kerr naked singularity the situation is generally more complicated, see for (ϕ) example [Stuchlík and Kotrlová, 2009]. For r → rph there is VK (r; a, b) → 1. The stable circular orbits are relevant for Keplerian accretion discs, therefore, it is reasonable to put the limits on the physical relevance of the Aschenbach effect to the region of stable circular geodesics. It is well known that for standard 4D Kerr black holes the Aschenbach effect, i.e., the non-monotonic LNRF-related orbital velocity profile appears for strongly limited class of near-extreme black holes [Aschenbach, 2004, Stuchlík et al., 2005]. It appears in the regions where the LNRF-related velocity of Keplerian motion reaches relatively large magni(ϕ) tude VK ∼ 0.5 − 0.6, but the velocity difference of the minimum-maximum hump is much (ϕ)

smaller (∆VK ∼ 0.01). These are the reasons why the effect was overlooked for a relatively long time. On the other hand, the Aschenbach effect is much stronger for Kerr naked singularities and is manifested for a large range of spin 1 < a < 4.0005. Moreover, for Kerr naked singularities with spin close to the extreme black hole state (a = 1), the Aschenbach effect is connected to another interesting effect related to circular geodesics – namely the retrograde


113

4

Q

2

P 0

-2

b

BH NS NS retrograde

-4

-6

P=(0.768082, 0.41005) Q=(5.99, 2.14)

-8

-10 0

1

2

3

4

5

6

7

8

9

10

a Figure 4.43: Classification of the braneworld Kerr spacetimes according to existence of the Aschenbach effect. The Aschenbach effect is allowed in the black region representing black holes, blue region representing naked singularities with corotating orbits only, and green region representing naked singularities with retrograde motion in the LNRF-velocity (φ) profile (corresponding to negative values of the function VK ).

character of the 1st family circular geodesics related to the LNRF. The counter-rotating orbits of the 1st family can constitute a part of the non-monotonic Keplerian profile (see Fig. 4.3). The “retrograde” Kerr naked singularities manifesting (strongly) the Aschenbach (ϕ) effect, can be determined by the relation (implied by the condition VK = 0) √ √ a = aR± (r) ≡ r 1 ± 1 − r .

(4.118)

It is illustrated in Fig. 4.41 (the case b = 0) – we see that √ the retrograde 1st-family orbits exist at radii 0 < r < 1, for spin parameters 1 < aR+ < 3 3/4; the Keplerian LNRF-related (ϕ) velocity profile touches VK = 0 at r = 1 for a = 1. For braneworld Kerr naked singularities, the retrograde motion of plus-family circular geodesics appears for spin determined by the condition aR− < a < aR+ , where


114

0.09 0.08

∆ a(b)

0.07

∆ a(b)

0.06 0.05 0.04

∆a(b)/aex

0.03 0.02 0.01 0.00 -10

-8

-6

-4

-2

0

b Figure 4.44: Spin range of braneworld Kerr black holes allowing for the Aschenbach effect given as a function of the tidal charge b. It is illustrated by the function ∆a(b) and the relative spin range determined by the function ∆a(b)/aex . The lowest area represents most interesting case (from astrophysical point of view) when the radius of marginally stable orbit rms is less than the local minimum of the Keplerian velocity profile.

√ (2r − b) ± D , aR± (r; b) ≡ √ 2 r −b

(4.119)

D = b2 − 4r(1 − r)b + 4r 2 (1 − r) .

(4.120)

where Functions aR± (r; b) are illustrated in Fig. 4.41. The conditions D ≥ 0 and r > b put limit on the radii where for given tidal charge b the retrograde plus-family orbits can exist, as illustrated in Fig. 4.42.

4.14.1

Aschenbach effect in braneworld spacetimes

Character of the LNRF-related velocity profile of the Keplerian (equatorial) circular motion is determined by the behavior of the velocity gradient that can be expressed in the form


115

0.622

b=0.3

0.620

b=0

0.585

0.618

V K (r;a,b)

0.995

(ϕ)

0.575

(ϕ)

V K (r;a,b)

0.8362 0.580

0.99616 0.570

0.8363

0.616

0.8364

0.614 0.612

0.8365 aex=0.8367..

0.610

0.997 0.608

0.565

0.8366

aex=1

0.606

0.998 0.560 1.2

1.4

1.6

1.8

2.0

2.2

1.1

1.2

1.3

r

1.4

1.5

1.6

r

0.560

0.580

b=-2

0.560

b=-1

0.550

0.540

V K (r;a,b)

1.39 0.530

1.395

(ϕ)

(ϕ)

V K (r;a,b)

0.540

0.520

0.510

aex=1.4143...

1.4

1.689

0.520

1.7 0.500

1.71

0.480 0.460 0.440

aex=1.7321...

1.72

0.500 0.420

1.405 0.490 1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.400 1.2

r

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

r

Figure 4.45: Non-monotonic LNRF-related velocity profiles for braneworld Kerr black hole backgrounds given for some values of the tidal charge b and appropriately chosen values of the a. The black points denote loci of rms . (ϕ)

∂VK A A A = 1 − 324 , ∂r A2 A2 where

(4.121)

√

! r 2 + a2 A1 = r − b + 2r − 2a , (4.122) 2(r − b) √ √ A2 = a r − b + r 2 ∆ , (4.123) √ A3 = r − b r 2 + a2 − a(2r − b) , (4.124) " ! # √ a r −1 √ 2 A4 = ∆ + 2r + a r −b+r . (4.125) √ ∆ 2 r −b Considering the braneworld Kerr black √ holes, we restrict our attention to the region above the event horizon at r > r+ = 1 + 1 − a2 − b. The rotation (spin) parameter of black hole spacetimes is limited by √ aex = 1 − b

(4.126)


0.618

0.57

0.616

116

0.614

0.56

V K,ex

(ϕ)

(ϕ)

V K,ex

0.612

0.55

b=0

0.54

b=0.3

0.610 0.608 0.606

0.53 0.604

0.52

0.51 0.995

0.602

0.996

0.997

0.998

0.999

0.600 0.8362

1

0.8363

0.8364

0.8365

0.8366

0.8367

a

a 0.55

0.55

0.50 0.50

(ϕ)

V K,ex

(ϕ)

V K,ex

0.45

b=-1

0.45

(ϕ)

b=-2

0.40

∆V K,ex

0.35 0.40

0.30 0.35 1.385

1.390

1.395

1.400

1.405

1.410

1.415

0.25 1.68

1.69

1.70

1.71

a

1.72

1.73

1.74

a

(ϕ)

Figure 4.46: Function VK,ex which determines values of the local minimum and maximum (ϕ)

of the function VK . The upper line represents the local maximum, while the lower line represents the local minimum. These curves demonstrate that maximum of the velocity difference is reached for the extreme black hole states. In (d), there is an indication for the definition of extremal velocity difference (occurring for the extreme black hole states) (ϕ) as a function of the tidal charge b, function ∆VK,ex giving maximal difference between (ϕ)

value of the local maximum and local minimum of the function VK,ex , considered with fixed b, is in Fig. 4.48.

for a given tidal charge b. When braneworld Kerr naked singularities are considered, the region r > b has to be studied for the existence of the humpy LNRF-velocity profiles when b > 0, while for b < 0 we have to analyze the whole region of r > 0 above the ring singularity. (ϕ)

Local extrema of VK (r) profiles (giving their “minimum-maximum” humpy parts) are (ϕ)

determined by the condition ∂VK /∂r = 0 , i.e., by zero points of the function (4.121) that are identical to the roots of the polynomial √ g(Z) : r h1 Z 4 + h2 Z 3 + h3 Z 2 + h4 Z + h5 = 0 ,

(4.127)


V K ( r=rms;a,b)-V K ( r=rmin;a,b)

0.0035

b=0

0.0030

0.0020 0.0015 0.0010 0.0005 0.0000 0.995

b=0.3

0.0025

0.0020

(ϕ)

0.0025

(ϕ)

(ϕ)

(ϕ)


0.0040

0.996

0.997

0.998

0.999

0.0015

0.0010

0.0005

0.0000 0.8362

1.000

0.8363

0.8364

0.8366

0.8367

0.0020


0.0014 0.0012

(ϕ)

b=-1

0.0010

b=-2

0.0015

0.0008 0.0006 0.0004

0.0010

0.0005

(ϕ)

(ϕ)

(ϕ)


0.8365

a

a 0.0016

0.0002 0.0000 1.385

117

1.390

1.395

1.400

1.405

1.410

0.0000 1.68

1.415

1.69

1.70

1.71

a

1.72

1.73

1.74

a

(ϕ)

(ϕ)

Figure 4.47: Function VK (r = rms , a, b)−VK (r = rmin,a,b ) which defines difference between (ϕ)

VK with radius for marginally stable orbit and with radius for local minimum. The dark grey region represent more astrophysically interesting cases, when radius of marginally stable orbit rms is lower than the radius for the local minimum rmin , see Fig. 4.44. where h i h1 = r 3r 2 + 2(1 − 2b)r − 3b ,

(4.128)

h2 = −2r 2 (1 − b/r)3/2 (3r + 1) , 4

3

2

(4.129) 2

h3 = 4r − 2(5 + 3b)r + 16br + 2br(1 − 3b) − b , n h i √ h4 = 1 − b/r 2r 3 4 (1 − b/r)2 + 4 (1 − b/r) + 1 − o − 2r 4 [1 + 4 (1 − b/r)] ,

(4.130)

h5 = r 3 (r 2 − 2br + b) .

(4.132)

(4.131)

The occurrence of the Aschenbach effect can be analyzed by numerical study of the roots of Eq. (4.127). The numerical approach enables to determine the region in the parameter space (a − b) where this effect is admitted, see Fig. 4.43. In naked singularity region, there is a subregion with retrograde 1st-family orbits. We can see that for black holes the region is highly sensitive on the choice of the tidal charge b. For positive values of b increasing, it becomes narrowed as compared with the case of b = 0 and disappears when b > 0.41005.


118

0.25

(ϕ)

∆V K,ex

0.20

0.15

0.10

0.05

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

b (ϕ)

Figure 4.48: Function ∆VK,ex giving maximal difference between values of the local maxi(ϕ)

mum and local minimum of the function VK,ex , for a fixed b. In the limit value of b = 0.41005, the corresponding value of the spin reads a = 0.76808. On the other hand, the interval of the black hole spin allowing for the Aschenbach effect enlarges significantly with descending of negative tidal charge b. We demonstrate this dependence in Fig. 4.44 presenting the function ∆a(b), which denotes spin interval of the black holes (with tidal charge b) allowing for the Aschenbach effect. Since the maximal black hole spin aex also depends on b, we give the dependence of the ratio ∆a/aex on b for completeness. Notice that for tidal charge b = −1, the spin interval ∆a (and even its relative magnitude given by ∆a/aex ) increases by almost one order as compared to the case of b = 0. In the naked singularity spacetimes, the Aschenbach effect appears in wide region of a, b parameters – both for positive and negative tidal charges. The allowed naked singularity region is limited by b = 2.14 and a = 5.99 – the point Q in Fig. 4.44. The behaviour of humpy LNRF-velocity profiles of Keplerian orbits in the field of braneworld Kerr black holes is represented by a series of figures for both positive and negative tidal charges. In order to clearly illustrate all the aspects of the Aschenbach effect in dependence on the black hole parameters, we give figures with both fixed value of the tidal charge b and fixed value of the rotation parameter a, see Fig. 4.45. (The boundary

4.15. SUMMARY

119

(ϕ)

0.6

b= 0.03 b= 0.02 b= 0.01 b= 0.00 b=-0.01 b=-0.02

0.580

) 0.4

V K (r;a,b)

0.578

0.576

0.2

0.0

(ϕ)

V K (r;a =0.996 aex,b)

0.582

-0.2

0.574 -0.4 0.572 1.2

( 1.4

1.6

1.8

r

2.0

2.2

-0.6 0.0

0.5

1.0

a=1.00,b= 0.0 a=1.05,b= 0.0 a=1.66,b=-1.5 a=2.10,b=-3.0 a=2.57,b=-5.0 1.5

2.0

r

Figure 4.49: LEft: a sequence of non-monotonic LNRF-related velocity profiles for near extreme black hole background. In the sequence, there is a = 0.996 aex and we change the parameter b. Right: a sequence of non-monotonic LNRF-related velocity profiles √ for naked singularity background close to the extreme black hole state given by aex = 1 − b. In the sequence, there is a = 1.05 aex , and we can see the transform between the retrograde and purely corotating velocity profiles. For comparison, the velocity profile constructed for the extreme Kerr black hole is included. of stable orbits given by the marginally stable orbit is represented by a point on all the profiles.) Moreover, we include the dependences of the LNRF velocities in the minimum and maximum of the humpy profile on the black hole parameters (see Figs. 4.46-4.47). These extremal LNRF-related velocities decrease with the tidal charge descending, but their difference increases significantly – for √fixed b, the depth of the humpy profile grows with black holes spin approaching aex = 1 − b . Finally, in Fig. 4.48, there is the dependence of the velocity difference at the minimum-maximum part of the humpy profile on the tidal charge b for extreme braneworld Kerr black holes. We can se that for b = −1 the extremal difference increases by a factor ∼ 2 as compared to the case of b = 0. We also demonstrate there, how the Aschenbach effect disappears about the point Q for positive tidal charges, and how the LNRF-related Keplerian profiles in the naked singularity spacetime can be reduced into those for the corresponding extreme black hole spacetime, when the retrograde naked singularity profile is transformed into a discontinuity occurring at the radius r = 1, and the negatively valued part under the horizon becomes physically irrelevant. Behavior of Keplerian velocity profiles in both black hole and naked singularity backgrounds corresponding to near-extreme black hole cases is shown in the Fig. 4.49.

4.15

Summary

In this chapter we have discussed several properties of braneworld KN black hole and naked singularity spacetimes. Namely, the influence of tidal charge parameter b on Aschenbach

4.15. SUMMARY

120

effect and effectiveness of Keplerian accretion. The main motivation for studying five-dimensional black hole models lie in the interesting possibility that five-dimensional braneworld models, i.e., models with one “large” bulk dimension, could be regarded as being effective description of the M theory. We have demonstrated, for the first time according to our knowledge, an interesting new “mining” phenomenon, i.e., that some classes of braneworld KN naked singularity spacetime show unstable behavior with respect to test particles. We have demonstrated that even one test particle can destabilize this solution via accretion onto its central body. Here accretion of only one particle has to be understand as accretion in gravitational mode, when the energy and momentum is carried out by gravitational waves.

Chapter

5

Conclusion Today’s science fiction is tomorrow’s science fact. Isaac Asimov

I

n this thesis, we have studied black holes (and naked singularities) in the multidimensional spacetimes. After introductory chapters we concentrated out attention to particular kind of braneworld black holes and naked singularities of Kerr–Newman type in five-dimensions. The main motivation of studying this dimensional case is its potential relation in future M-theory, where the five-dimensional spacetimes can play role in the effective low-energy limit. We have shown that the Aschenbach effect is a typical feature of the circular geodetical motion in the field of both standard and braneworld Kerr naked singularities with a relatively large interval of spins above the extreme black hole limit. For naked-singularity spin sufficiently close to the extreme black hole state, the Aschenbach effect is manifested by the retrograde plus-family circular orbits. For black hole spacetimes, such retrograde orbits can appear under the inner horizon, thus being irrelevant from the astrophysical point of view. In the field of near-extreme rotating black holes, the Aschenbach effect located above the outer black hole horizon can be considered as a small remnant of the typical naked singularity phenomenon. The circular geodesics of the braneworld Kerr–Newman black hole and naked singularity spacetimes have been studied and classification of these spacetimes according to character of the circular geodesic structure has been presented. The circular geodesics have been separated into two families – the lower family containing only the counterrotating circular geodesics, and the upper family with corotating geodesics at large distance, but possible transformation to counterrotating geodesics in vicinity of the naked singularity. It has been demonstrated that fourteen different classes of the Kerr–Newman spacetimes 121

5.1. OUTLOOK

122

can exist, mainly due to the properties of the upper family of circular geodesics. Implications of the geodesic structure to the Keplerian accretion have been given, and efficiency of the Keplerian accretion have been determined. The accretion efficiency is continuously matched between the naked singularity and extreme black hole spacetimes for the Keplerian accretion along the lower family circular geodesics. On the other hand, there is a strong discontinuity occurring in the transition between the naked singularities and the extreme black holes for the Keplerian accretion along the upper family circular geodesic, √ if the dimensionless spin of the Kerr–Newman spacetime is sufficiently high (a > 1/ 2) – the energy efficiency of the Keplerian accretion is then substantially higher for the naked singularity spacetimes. The accretion efficiency could then go up to the value of η ∼ 1.707 √ for Kerr–Newman near-extreme naked singularity spacetimes with b ∼ 1/2 and a ∼ 1/ 2. For the Keplerian accretion along the lower family circular geodesics, the inner edge of the disk has to be always located at the marginally stable circular geodesic corresponding to an inflexion point of the effective potential of the motion in accord with the scenario of Keplerian accretion onto Kerr black holes and naked singularities. It has been shown that the Keplerian accretion along the upper family geodesics can give three different scenarios. It can finish at the inner edge located at the marginally stable circular geodesic – this is the standard accretion scenario present in the black hole spacetimes. However, two other scenarios could occur in the naked singularity spacetimes. The inner edge of the Keplerian accretion could occur at r = b that is the special limit on existence of the circular geodesics. For b > 1 the efficiency of the upper family Keplerian accretion is independent of the naked singularity spin. The most interesting is the third scenario, related to the Kerr–Newman naked singularity spacetimes of Class IIIa having an infinitely deep gravitational potential of the upper family Keplerian accretion. Then the inner edge of the Keplerian accretion could occur even at the stable photon circular geodesic, and the accretion efficiency could be formally unlimited, making such naked singularity spacetimes unstable relative to “mining” accretion.

5.1

Outlook

Though some ground has been covered in this thesis, we have only tackled upon a small fraction of the implications of extra dimensions in physics. There are many outstanding problems, and the possible directions of future work. We have some ideas how to further develop properties of mining spacetimes. Especially, the mining Kerr–Newman naked singularities demonstrate extraordinary behavior in the case of the so called Banados–Sith–West effect. This effect regards particle collisions with ultra-high center-of-mass energy (see [Bañados et al., 2009]). Another interest of ours is optical phenomena relates to observers orbiting in the so called “mining regime”. That is, in closed vicinity of the stable photon geodesic representing

5.1. OUTLOOK

123

limit on Keplerian accretion in the mining Kerr–Newman naked singularity spacetimes. These two ideas will be presented in future work [Stuchlík et al., 2017].

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Z. Stuchlík and J. Schee. Optical effects related to Keplerian discs orbiting KehagiasSfetsos naked singularities. Classical and Quantum Gravity, 31(19):195013, October 2014. doi: 10.1088/0264-9381/31/19/195013. 80, 88, 94, 100 Z. Stuchlík and J. Schee. Circular geodesic of Bardeen and Ayon-Beato-Garcia regular black-hole and no-horizon spacetimes. International Journal of Modern Physics D, 24: 1550020-289, December 2015. doi: 10.1142/S0218271815500200. 88 Z. Stuchlík, J. Biˇcák, and V. Bálek. The Shell of Incoherent Charged Matter Falling onto a Charged Rotating Black Hole. General Relativity and Gravitation, 31:53–71, January 1999. doi: 10.1023/A:1018863304224. 79 Z. Stuchlík, P. Slaný, G. Török, and M. A. Abramowicz. Aschenbach effect: Unexpected topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr black holes. Physical Review D, 71(2):024037, January 2005. doi: 10.1103/PhysRevD.71.024037. 106, 107, 112 Z. Stuchlík, M. Blaschke, and P. Slaný. Non-monotonic Keplerian velocity profiles around near-extreme braneworld Kerr black holes. Classical and Quantum Gravity, 28(17):175002, September 2011a. doi: 10.1088/0264-9381/28/17/175002. 21, 55 Z. Stuchlík, S. Hledík, and K. Truparová. Evolution of Kerr superspinars due to accretion counterrotating thin discs. Classical and Quantum Gravity, 28(15):155017, August 2011b. doi: 10.1088/0264-9381/28/15/155017. 80, 88 Z. Stuchlík, J. Schee, and A. Abdujabbarov. Ultra-high-energy collisions of particles in the field of near-extreme Kehagias-Sfetsos naked singularities and their appearance to distant observers. Physical Review D, 89(10):104048, May 2014. doi: 10.1103/PhysRevD. 89.104048. 88 Z. Stuchlík, M. Blaschke, and J. Schee. Particle collisions and optical effects in the mining Kerr–Newman spacetimes. in preparation, 2017. 123 F. R. Tangherlini. Schwarzschild Field in n Dimensions and the Dimensionality of Space Problem. Nuovo Cimento, 27:636–651, 1963. 8, 29 M. Tegmark. LETTER TO THE EDITOR: On the dimensionality of spacetime. Classical and Quantum Gravity, 14:L69–L75, April 1997. doi: 10.1088/0264-9381/14/4/002. 4 M. Tegmark. Is “the Theory of Everything” Merely the Ultimate Ensemble Theory? Annals of Physics, 270:1–51, November 1998. doi: 10.1006/aphy.1998.5855. 4 M. Vasudevan, K. A. Stevens, and D. N. Page. Particle motion and scalar field propagation in Myers Perry black-hole spacetimes in all dimensions. Classical and Quantum Gravity, 22:1469–1482, April 2005. doi: 10.1088/0264-9381/22/7/017. 30

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Part II Individual papers

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6

Paper 1: Non-monotonic Keplerian velocity profiles around near-extreme braneworld Kerr black holes Not only is the Universe stranger than we think, it is stranger than we can think. Werner Heisenberg

T

he first paper is dedicated to the study of Aschenbach effect in braneworld Kerr– Newman spacetime. Back in 2011 we though that Aschenbach effect can be a hint of some interesting new effects around very rapidly rotating black holes. Even though, the effect itself depends on the specific frame of reference (LNRF), we hope that it is actually an indication of some frame free phenomena. We can see some similarity, e.g., with the retrograde motion of planets on the night sky. This motion is also solely caused by the frame of reference, when we look at elliptic motion from a point under elliptic motion. But, the retrograde aspect of the planetary motion was indication that Earth might not be in the center of all motion.

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IOP PUBLISHING

CLASSICAL AND QUANTUM GRAVITY

Class. Quantum Grav. 28 (2011) 175002 (18pp)

doi:10.1088/0264-9381/28/17/175002

Non-monotonic Keplerian velocity profiles around near-extreme braneworld Kerr black holes Zdenˇek Stuchl´ık, Martin Blaschke and Petr Slaný Institute of Physics, Faculty of Philosophy and Science, Silesian University, Bezruˇcovo nám. 13, CZ-746 01 Opava, Czech Republic E-mail: [email protected], [email protected] and [email protected]

Received 22 April 2011, in final form 24 June 2011 Published 26 July 2011 Online at stacks.iop.org/CQG/28/175002 Abstract We study the non-monotonic Keplerian velocity profiles related to locally nonrotating frames (LNRF) in the field of near-extreme braneworld Kerr black holes and naked singularities in which the non-local gravitational effects of the bulk are represented by a braneworld tidal charge b and the 4D geometry of the spacetime structure is governed by the Kerr–Newman geometry. We show that positive tidal charge has a tendency to restrict the values of the black hole dimensionless spin a admitting the existence of the non-monotonic Keplerian LNRF-velocity profiles; the non-monotonic profiles exist in the black hole spacetimes with tidal charge smaller than b = 0.410 05 (and spin larger than a = 0.768 08). With decreasing value of the tidal charge (which need not be only positive), both the region of spin allowing the non-monotonicity in the LNRF-velocity profile around braneworld Kerr black hole and the velocity difference in the minimum–maximum parts of the velocity profile increase implying growing astrophysical relevance of this phenomenon. PACS numbers: 04.70.−s, 04.20.−q, 98.80.−k, 04.50.−h

1. Introduction Fast rotating black holes play a crucial role in understanding processes observed in quasars and active galactic nuclei (AGN) or in microquasars. It has been shown that supermassive black holes in AGN evolve into states with dimensionless spin a ∼ 1 due to accretion from thin discs [1, 2]. This statement is supported by the analysis of profiled x-ray (Fe56) lines observed in some AGN (e.g. in MCG-6-30-15) [3–5] and in some microquasars (e.g. GRS 1915+105) [6]. Evidence for the existence of near-extreme Kerr black holes comes from high-frequency quasi-periodic oscillations (QPOs) of the observed x-ray flux in some microquasars [7, 8]. A fast rotating black hole could also be located in the galaxy centre source Sqr A∗ [9–11]. 0264-9381/11/175002+18$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA

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(ϕ)

(ϕ)

V K (r;a)

V K (r;a)

1.0

1.0

a=0.998 a=0.700 a=0.500 a=0.000

0.9 0.8

0.8 0.6 0.4

0.7

0.2 0.0

0.6

a=1.0000 a=1.2000 a=1.5000 a=4.0014

-0.2

0.5 0.4 1.0

)

-0.4 1.5

2.0

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3.0

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3.5

4.0

4.5

5.0

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-0.6 0.0

( 1.0

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5.0

r

(b)

Figure 1. Keplerian velocity profiles related to the LNRF. (a) Kerr black holes—the velocity profiles presented for some values of the black hole spin. The Aschenbach effect appears for near-extreme black holes and is weak. (b) Kerr naked singularities—the velocity profiles are given for some values of the spin, demonstrating existence of the Aschenbach effect for orbits with negative-valued velocity. For completeness, the velocity profile is also given for the extreme black hole, demonstrating a velocity jump at r = 1.

It is widely accepted that the phenomena observed in AGN and microquasars are related to accretion discs orbiting Kerr black holes. However, we can also consider the possibility of explaining these phenomena by Kerr superspinars with the external field described by the geometry of Kerr naked singularity spacetime [12]. Then both accretion and related optical effects and the QPO effects enable us to find clear signature of the Kerr superspinar’s presence [13–18]. Properties of accretion discs can be appropriately represented by circular orbits of test particles or fluid elements orbiting black holes (superspinars). The local properties can be efficiently expressed when related to the locally non-rotating frames (LNRFs), since these frames corotate with the spacetime in a way that enables us to cancel the frame-dragging effects as much as possible [19]. A new phenomenon related to the LNRF-velocity profiles of matter orbiting near-extreme Kerr black holes has been found by Aschenbach [9, 20, 21], namely a non-monotonicity in the velocity profile of the Keplerian motion in the field of Kerr black holes with dimensionless spin a > 0.9953. Such a hump in the LNRF-velocity profile of the corotating orbits is a typical and relatively strong feature in the case of Keplerian motion in the field of Kerr naked singularities, but in the case of Kerr black holes it is a very small effect appearing for near-extreme black holes only—see figure 1. In the naked singularity case, we call the orbits to be of the first family rather than corotating, since these can be retrograde relative to the LNRF in vicinity of the ring singularity for small values of spin (a < 5/3), while they are corotating for larger values of the spin [15]; the humpy character of the LNRFvelocity profile ceases for naked singularities with a > 4.0014—as demonstrated in figure 1. A study of non-Keplerian distribution of specific angular momentum (l = const), related to geometrically thick discs of perfect fluid, has shown that the ‘humpy’ LNRF-velocity profile appears for near-extreme Kerr black holes with a > 0.9998 [21]. The humpy LNRF-velocity profile emerges in the ergosphere of near-extreme Kerr black holes, at the vicinity of the marginally stable circular orbit. The maximal velocity difference between the local minimum and maximum of the humpy Keplerian velocity profiles is v ≈ 0.07 c and takes place for a = 1 [22]. Using the idea of ‘hump-induced’ oscillations related to humpy LNRF-velocity profiles, the extended orbital resonance model (EXORM) was developed and applied to explain complex 2

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QPO patterns observed in some black hole sources [22–24]. In the EXORM, the resonant phenomena between oscillations with orbital frequencies (Keplerian and epicyclic—radial or vertical) and the so-called humpy frequency, given by the maximal slope of the LNRF-velocity profile in its humpy part, are assumed to appear at the radius where the ‘humpy’ oscillations are expected to be generated by the non-monotonicity of the LNRF-velocity profile [23]. The EXORM is able to explain all five high-frequency QPOs observed in the microquasar GRS 1915+105 by the humpy frequency, the radial (and vertical) epicyclic frequencies and their simple combinations taken at the common ‘humpy’ radius, implying the black hole parameters of the source M = 14.8 M , a = 0.9998 [22], in good agreement with estimates given by different methods [6, 20]. This model can give interesting results also in the case of the x-ray binary system XTE J1650-500 [24], and an ULX candidate system NGC 5408 X-1. On the other hand, QPOs observed in Sqr A∗ [9] cannot be explained by the EXORM [25]. The humpy LNRF-velocity profiles for both Keplerian and l = const specific angular momentum distributions of orbiting matter were studied in the Kerr–de Sitter and Kerr–antide Sitter black hole spacetimes and values of the black hole spin a allowing for the existence of the humpy profiles were found in dependence on the value of the cosmological constant [26, 27]. The last decade gave rise to a plenty of models modifying the 4D Einstein general relativity due to hidden dimensions; therefore, it is interesting to investigate the Aschenbach effect in rotating black hole (naked singularity) spacetimes allowed in alternative gravitational theories. The string theories, describing gravity as the higher-dimensional interaction, appearing to be effectively 4D at low energies, inspired braneworld models assuming the observable universe to be a 3-brane, i.e. the ‘domain wall’, to which the non-gravitational matter fields are confined, while gravity enters the extra spatial dimensions that could be much larger than lp ∼ 10−33 cm. The model of Randall and Sundrum (RS model) [28] allows gravity localized near the brane with an infinite size extra dimension while the warped spacetime satisfies the 5D Einstein equations with a negative cosmological constant. An arbitrary energy–momentum tensor could then be allowed on the brane and effective 4D Einstein equations have to be satisfied on the brane. The RS model implies standard 4D Einstein equations in the low energy limit, but significant deviations occur at high energies, near black holes or compact stars. The combination of high-energy (local) and bulk stress (non-local) effects alters the matching problem on the brane in comparison with the standard 4D gravity [29]. The bulk gravity stresses imply that the matching conditions do not have a unique solution on the brane and the 5D Weyl tensor is needed as a minimum condition for uniqueness. No exact solution of the 5D braneworld Einstein equations is known at present, but a numerical solution has been found quite recently [30]. On the other hand, the 4D stationary and axisymmetric vacuum solution describing a braneworld rotating black hole has been found by solving the braneworld constrained equations under an assumption of the specialized form of the metric (namely of the Kerr–Schild form) [31]. Of course, it is not an exact solution satisfying the full system of 5D equations, but in the framework of the constrained equations it represents a consistent rotating black hole solution reflecting the influence of the extra dimension through a single braneworld parameter. The braneworld rotating black holes are described by the metric tensor of the Kerr–Newman form with the braneworld tidal charge b determining the 5D non-local gravitational coupling between the brane and the bulk [31]. For non-rotating braneworld black holes, the metric is reduced to the Reissner– Nordström form containing the tidal charge [32]. This spacetime can also represent the external field of braneworld neutron stars described by the uniform density internal spacetime [29]. The influence of the braneworld tidal charge on physical processes has been extensively investigated for both the black holes [33–38] and neutron stars [39–43], or in the weak field 3

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limit [44, 45]. In the case of microscopic black holes, experimental evidence is assumed in the LHC [46]. The braneworld tidal charge can be, in principle, both positive and negative, but the negative values are probably more relevant [32]. Note that for b > 0, the braneworld spacetime can be identified with the Kerr–Newman spacetime by b → Q2 , where Q2 is the squared electric charge; however, it is not the Kerr–Newman background since the electromagnetic part of this background is missing. Some astrophysically relevant restrictions on the value of the tidal charge b were obtained both in the weak-field limit [44] and in the strong-field limit. Here, we will study the existence of the humpy LNRF-velocity profiles in the field of braneworld rotating black holes considering both negative and positive values of the braneworld tidal charge. Our results related to b > 0 are relevant also in the case of the standard Kerr– Newman spacetimes (with b → Q2 ) for uncharged particles. We restrict our attention to the Keplerian LNRF-velocity profiles postponing the study of perfect fluid configurations to future work. 2. Effective gravitational equations in braneworld models In the 5D warped space models of Randall and Sundrum, the gravitational field equations in the bulk can be expressed in the form [32, 47] ˜ AB = k˜ 2 [− ˜ g˜ AB + δ(χ )(−λgAB + TAB )], G

(1)

˜ P enters via k˜ 2 = 8π/M ˜ 3 , λ is the brane tension, where the fundamental 5D Planck mass M P ˜ and is the negative bulk cosmological constant; gAB = g˜ AB − nA nB is the induced metric on the brane, with nA being the unit vector normal to the brane. The effective gravitational field equations induced on the brane are determined by the bulk field equations (1), the Gauss–Codazzi equations and the generalized matching Israel conditions. They can be expressed in the form of modified Einstein’s equations containing additional terms reflecting bulk effects onto the brane [32, 47] Gμν = −gμν + k 2 Tμν + k˜ 2 Sμν − Eμν ,

(2)

where k 2 = 8π/MP2 , with MP being the braneworld Planck mass. The relations of the energy scales and cosmological constants are given in the form ˜ 2 3 M 4π ˜ 4π ˜ P; λ2 . (3) = 3 MP = + √P M ˜ ˜3 4π M 3 M λ P P Local bulk effects on the matter are determined by the ‘squared energy–momentum’ tensor Sμν , while the non-local bulk effects are given by the tensor Eμν . Assuming zero cosmological constant on the brane ( = 0), we arrive to the condition 2

˜ = − 4π λ . ˜3 3M P

(4)

In the vacuum case, Tμν = 0 = Sμν , the effective gravitational field equations on the brane reduce to the form [47] Rμν = −Eμν ,

Rμ μ = 0 = Eμ μ

(5)

implying divergence constraint [47] ∇ μ Eμν = 0,

where ∇μ denotes the covariant derivative on the brane. 4

(6)

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Equation (6) represents Bianchi identities on the brane, i.e. an integrability condition for the field equations Rμν = −Eμν [31]. For stationary and axisymmetric (or static, spherically symmetric) solutions, (5) and (6) form a closed system of equations on the brane. The 4D general relativity energy–momentum tensor Tμν (with Tμ μ = 0) can be formally identified to the bulk Weyl term on the brane due to the correspondence k 2 Tμν

↔

−Eμν .

(7)

The general relativity conservation law ∇ Tμν = 0 then corresponds to the constraint equation on the brane (6). This behaviour indicates that the Einstein–Maxwell solutions in standard general relativity should correspond to constrained braneworld vacuum solutions. This was indeed shown in the case of braneworld (Reissner–Nordström and Kerr–Newman) black hole solutions [31]. In both of these solutions, the influence of the non-local gravitational effects of the bulk on the brane is represented by a single ‘braneworld’ parameter b. The 1/r 2 behaviour of the second term in the Newtonian potential μ

=−

b M + 2 2 MP r 2r

(8)

inspired the name ‘tidal charge’ for the parameter b [32]. 3. Orbital motion in the braneworld Kerr spacetimes The motion of test particles in the field of braneworld rotating black holes is given by the geodesic structure of the Kerr–Newman spacetimes with the tidal charge b. The braneworld parameter reflects the tidal effects of the bulk space and has no influence on the motion of charged particles. The geodesic structure given by the Carter equations [48] is relevant for both uncharged and charged test particles. The circular test particle orbits of the braneworld Kerr black holes are identical to the circular geodesics of the Kerr–Newman spacetime with properly chosen charge parameter. We will study the Aschenbach effect, i.e. we look for the non-monotonicity (humps) in the LNRF-velocity profiles of Keplerian discs orbiting near-extreme braneworld Kerr black holes or naked singularities. 3.1. Geometry Using standard Boyer–Lindquist coordinates (t, r, θ, ϕ) and geometric units (c = G = 1), we can write the line element of rotating (Kerr) black hole on the 3D brane in the form 2Mr − b 2a(2Mr − b) 2 dt 2 − sin θ dt dϕ ds 2 = − 1 − 2Mr − b 2 2 (9) + dr 2 + dθ 2 + r 2 + a 2 + a sin θ sin2 θ dϕ 2 , where = r 2 − 2Mr + a 2 + b,

(10)

= r 2 + a 2 cos2 θ,

(11)

M and a = J /M are the mass parameter and the specific angular momentum of the background, while the braneworld parameter b, called the ‘tidal charge’, represents the imprint of non-local (tidal) gravitational effects of the bulk space [31]. The physical ‘ring’ singularity of the 5

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braneworld rotating black holes (and naked singularities) is located at r = 0 and θ = π/2, as in the Kerr spacetimes. The form of metric (9) is the same as that of the standard Kerr–Newman solution of the 4D Einstein–Maxwell equations with tidal charge b being replaced by the squared electric charge Q2 [49]. The stress tensor on the brane Eμν takes the form b Ett = −Eϕϕ = − 3 [ − 2(r 2 + a 2 )], (12) Err = −Eθθ = −

b , 2

(13)

2ab 2 (r + a 2 ) sin2 θ, (14) 3 that is, fully analogical (b → Q2 ) to components of the electromagnetic energy–momentum tensor of the Kerr–Newmann solution in Einstein’s general relativity [31]. For negative values of the tidal charge (b < 0), the values of the black hole spin a > M are allowed. Such a situation is forbidden for the standard 4D Kerr black holes. In the following, we put M = 1 in order to work with completely dimensionless formulae. ϕ

Eϕt = −(r 2 + a 2 ) sin2 Et = −

3.2. LNRFs and orbital motion The orbital velocity of matter orbiting a braneworld Kerr black hole along circular orbits is given by appropriate projections of its 4-velocity U = (U t , 0, 0, U ϕ ) onto the tetrad of a LNRF [19]: 1

e(t) = (ω2 gϕϕ − gtt ) 2 dt, 1

e(ϕ) = (gϕϕ ) 2 (dϕ − ω dt), (r)

e

=

(15) (16)

12 dr,

1

e(θ) = 2 dθ,

(17) (18)

where ω is the angular velocity of the LNRF relative to distant observers and reads a(2r − b) gtϕ . (19) = ω=− gϕϕ (r 2 + a 2 ) + (2r − b) a 2 sin2 θ For the circular motion, the only non-zero component of the 3-velocity measured locally in the LNRF is the azimuthal component that is given by [ − ω] (ϕ) VLNRF = ω2 − ggϕϕtt =

[(r 2 + a 2 )2 − a 2 sin2 θ ] sin θ ( − ω) , √

where lgtt + gtϕ Uϕ =− t U lgtϕ + gϕϕ is the angular velocity of the orbiting matter relative to distant observers and Uϕ l=− Ut =

6

(20)

(21)

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is its specific angular momentum; Ut , Uϕ are the covariant components of the 4-velocity field of the orbiting matter. Using (9) we arrive to the formula 1 − 2r−b sin2 θ l + a(2r−b) = . (22) r 2 + a 2 + 2r−b a 2 sin2 θ sin2 θ − l a(2r−b) sin2 θ 4. Velocity profiles for the Keplerian distribution of the specific angular momentum 4.1. Circular geodesics The motion of test particles following circular geodetical orbits in the equatorial plane (θ = π/2) is described by the Keplerian distribution of the specific angular momentum, which in the braneworld Kerr backgrounds take the form [33] √ (r 2 + a 2 ) r − b ∓ a(2r − b) ; (23) lK± (r; a, b) = ± √ r 2 − 2r + b ± a r − b the signs ± refer to two distinct families of orbits in the Kerr braneworld spacetimes. From the LNRF point of view, the second family, or minus-family (given by the lower sign), represents retrograde orbits, while the first family, or plus-family (given by the upper sign), represents direct orbits in the black hole spacetimes [19], but can represent both direct and retrograde orbits in the naked singularity spacetimes. We can find a similar situation for Kerr spacetime in [15, 16]. For both families, we find a formal limit of the Keplerian motion to be located at r = b. In the black hole spacetimes, it is located under the inner horizon and is irrelevant from the astrophysical point of view. In the naked singularity spacetimes, it is relevant for b > 0, while for b < 0, it is located under the ring singularity at r = 0 which represents the limit on the location of circular orbits. The corresponding Keplerian angular velocity related to distant observers is given by the relation 1 , (24) K± (r; a, b) = ± √ r 2/ r − b ± a and the Keplerian orbital velocity related to the LNRF in braneworld Kerr backgrounds is thus given by the relation √ r − b(r 2 + a 2 ) ∓ a(2r − b) (ϕ) VK± (r; a, b) = ± . (25) √ √ (r 2 ± a r − b) Clearly, there is a limit on the existence of circular equatorial geodesics at r = b that is relevant (and located above the ring singularity) for spacetimes with a positive tidal charge. Like in the Kerr backgrounds (b = 0), also in the braneworld Kerr backgrounds only the plus-family orbits exhibit the ‘minimum–maximum’ humpy structure of the LNRF-related orbital velocity profiles VK(ϕ) (r). On the other hand, in braneworld Kerr naked-singularity spacetimes with a positive tidal charge (b > 0), there is always at least one local extreme in (ϕ) (ϕ) the orbital velocity profiles (maximum in VK+ and minimum in VK− profiles) where the orbital velocity gradient changes its sign. This non-monotonicity, however, does not correspond to the Aschenbach effect. (ϕ) We illustrate the typical character of VK+ (r; a, b) velocity profiles in the braneworld rotating black hole and naked singularity spacetimes with a negative tidal charge in figure 2— in this case, the profiles are extended down to r = 0, but here we do not consider the region of r < 0. For special values of parameters of naked singularity spacetimes, the Aschenbach 7

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(ϕ)

V K+ (r;a,b) 0.6

BH Aschenbach ef.

0.4 0.2

NS Aschenbach ef.

0.0

NS-retrograde Aschenbach ef. -0.2 -0.4

b=-2

-0.6 0.0

a=1.73 a=1.76 a=1.83 1.0

2.0

3.0

4.0

5.0

6.0

r

Figure 2. Non-monotonic plus-family orbital velocity profiles in braneworld Kerr black hole and naked singularity spacetimes with the tidal charge b = −2, given for three appropriately chosen values of spin a reflecting the whole variety of possible behaviour related to the Aschenbach effect.

(ϕ)

V K+ (r;a,b) 0.8 0.6 0.4

b

0.2 0.0 -0.2 -0.4

b=0.3

a=0.8366 a=0.8410 a=3.0000 a=8.0000

-0.6 0.0

2.0

4.0

6.0

8.0

r

10.0

Figure 3. Non-monotonic plus-family orbital velocity profiles in braneworld Kerr black hole and naked singularity spacetimes with the tidal charge b = 0.3.

effect can be related to the retrograde region of first family orbits (the humpy structure contains (ϕ) orbits with VK+ < 0). (ϕ) For the positive tidal charge, the characteristic profiles of VK+ are presented in figure 3—in this case, the profiles finish their validity at r = b. The velocity profile under the inner horizon of the black hole is also depicted. Numerical calculations indicate that for those kinds of velocity profiles there is no Aschenbach effect, so for the black hole spacetimes (ϕ) (r; a, b) above the we will focus our attention on the behaviour of the Keplerian profiles VK+ outer horizon. (ϕ) For both positive and negative tidal charges, the profiles of VK− in black hole and naked singularity spacetimes are presented in figure 4. These profiles do not exhibit any ‘minimum– maximum’ structures. Therefore, in the following, we will focus our attention on the plus(ϕ) (ϕ) family orbits only and use the notation VK (r; a, b) instead of VK+ (r; a, b). 8

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(ϕ)

V K- (r;a,b) 0.0

-0.5

-1.0

b= -1.00, a=1 b= -0.50, a=1 b=0.000, a=0 b=1.125, a=0 b=1.020, a=0 b=2.000, a=1

-1.5

-2.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

r

Figure 4. Minus-family orbital velocity profiles in braneworld Kerr black hole and naked singularity spacetimes.

(ϕ)

We have to consider the function VK (r; a, b) (and also the functions lK (r; a, b), K (r; a, b)) within the range of definition of the Keplerian motion [33]. The range is governed by the radii of the photon circular geodesics rph given implicitly by the relation a = aph (r; b) ≡

r(3 − r) − 2b , √ 2 r −b

(26)

and by the radii of the marginally stable circular geodesics rms , implicitly given by

4(r − b)3/2 ∓ r 3r 2 − 2r(1 + 2b) + 3b . (27) a = ams (r; b) ≡ 3r − 4b The functions aph (r; b) and ams (r; b) are illustrated in figure 5—for a detailed discussion of the properties of photon and marginally stable orbits, see [33, 39]. Above the black hole outer horizon, the stable orbits are located in the interval of rms < r < ∞, while unstable orbits are located in the interval rph < r < rms . Note that for Kerr naked singularity, the situation is (ϕ) generally more complicated; see, for example, [33]. For r → rph , there is VK (r; a, b) → 1. The stable circular orbits are relevant for Keplerian accretion discs; therefore, it is reasonable to put the limits on the physical relevance of the Aschenbach effect to the region of stable circular geodesics. It is well known that for standard 4D Kerr black holes, the Aschenbach effect, i.e. the non-monotonic LNRF-related orbital velocity profile, appears for the strongly limited class of near-extreme black holes [9, 21]. It appears in the regions where the LNRF-related velocity (ϕ) of Keplerian motion reaches relatively large magnitude VK ∼ 0.5–0.6, but the velocity (ϕ) difference of the minimum–maximum hump is much smaller VK ∼ 0.01 . These are the reasons why the effect was overlooked for a relatively long time. On the other hand, the Aschenbach effect is much stronger for Kerr naked singularities and is manifested for a large range of spin 1 < a < 4.0005. Moreover, for Kerr naked singularities with spin close to the extreme black hole state (a = 1), the Aschenbach effect is connected to another interesting effect related to circular geodesics—namely the retrograde character of the first family circular geodesics related to the LNRF. The counter-rotating orbits of the first family can constitute a part of the non-monotonic Keplerian profile (see figures 1 and 2). The ‘retrograde’ Kerr naked 9

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a(r;b)

a(r;b)

1.0

1.4

a R0.8

1.2

b=0.5

0.6

1.0

ams

ah0.4

a R+

b=0

0.8 0.6

ah+ aph

0.4

0.2

ah-

aR-

ah+

0.2

0.0 0.0

1.0

2.0

3.0

4.0

5.0

r

0.0 0.0

1.0

2.0

3.0

(a)

1.4

a R+

b=-1 ams

1.0 ah0.8 0.6

aph

0.4 0.0 0.0

5.0

6.0

r

a(r;b)

1.2

0.2

4.0

(b)

a(r;b) 1.6

ams

aph

ah+ aR1.0

2.0

3.0

4.0

5.0

6.0

7.0

r

2.0 a 1.8 R+ 1.6 1.4 ah1.2 1.0 0.8 0.6 0.4 aR0.2 0.0 0.0 1.0

(c)

b=-2 ams aph ah+ 2.0

3.0

4.0

5.0

6.0

7.0

8.0

r

(d)

Figure 5. The loci of the photon circular orbits aph (r, b); the marginally stable orbits ams (r, b); function aR± (r, b) and function ah± (r, b) defined implicitly by the position of the event horizon for appropriately chosen fixed values of b.

singularities manifesting (strongly) the Aschenbach effect can be determined by the relation (ϕ) (implied by the condition VK = 0) a = aR± (r) ≡

√ √ r(1 ± 1 − r).

(28)

It is illustrated in figure 5 (the case b = 0)—we see that √ the retrograde first-family orbits exist at radii 0 < r < 1 for spin parameters 1 < aR+ < 3 3/4; the Keplerian LNRF-related (ϕ) velocity profile touches VK = 0 at r = 1 for a = 1. For braneworld Kerr naked singularities, the retrograde motion of plus-family circular geodesics appears for spin determined by the condition aR− < a < aR+ , where √ (2r − b) ± D aR± (r; b) ≡ , √ 2 r −b

(29)

D = b2 − 4r(1 − r)b + 4r 2 (1 − r).

(30)

where

Functions aR± (r; b) are illustrated in figure 5. The conditions D 0 and r > b put a limit on the radii where for the given tidal charge b the retrograde plus-family orbits can exist, as illustrated in figure 6. 10

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b 4.00

b=r 2.00 0.00 -2.00 -4.00 -6.00 -8.00 -10.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

r

Figure 6. Naked singularity regions with potentially retrograde plus-family orbits.

4.2. The Aschenbach effect in braneworld spacetimes The character of the LNRF-related velocity profile of the Keplerian (equatorial) circular motion is determined by the behaviour of the velocity gradient that can be expressed in the form (ϕ)

∂ VK A1 A3 A4 = − , ∂r A2 A22 where

(31)

r 2 + a2 + 2r − 2a, (32) 2(r − b) √ √ A2 = (a r − b + r 2 ) , (33) √ A3 = r − b(r 2 + a 2 ) − a(2r − b), (34) √ r −1 √ a (a r − b + r 2 ) . A4 = + 2r + (35) √ 2 r −b Considering the braneworld Kerr √ black holes, we restrict our attention to the region above the event horizon at r > r+ = 1 + 1 − a 2 − b. The rotation (spin) parameter of black hole spacetimes is limited by √ (36) aex = 1 − b A1 =

√

r −b

for a given tidal charge b. When braneworld Kerr naked singularities are considered, the region r > b has to be studied for the existence of the humpy LNRF-velocity profiles when b > 0, while for b < 0 we have to analyse the whole region of r > 0 above the ring singularity. (ϕ) The local extrema of VK (r) profiles (giving their ‘minimum–maximum’ humpy parts) are determined by the condition ∂ VK(ϕ) /∂r = 0 , i.e. by zero points of the function (31) that are identical to the roots of the polynomial √ (37) g(Z) : r(h1 Z 4 + h2 Z 3 + h3 Z 2 + h4 Z + h5 ) = 0, where h1 = r[3r 2 + 2(1 − 2b)r − 3b],

(38) 11

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4 0.4

Q

2

0.3

P

0.2

0

0.1 -2

0.0

b

0.75

0.85

0.95

BH NS NS retrograde

-4

-6

P=(0.768082, 0.41005) Q=(5.99, 2.14)

-8

-10

0

1

2

3

4

5

6

7

8

9

10

a Figure 7. Classification of the braneworld Kerr spacetimes according to the existence of the Aschenbach effect. The Aschenbach effect is allowed in the black region representing black holes, dark grey region representing naked singularities with corotating orbits only and the light grey region representing naked singularities with retrograde motion in the LNRF-velocity profile (ϕ) (corresponding to negative values of the function VK ).

h2 = −2r 2 (1 − b/r)3/2 (3r + 1),

(39)

h3 = 4r 4 − 2(5 + 3b)r 3 + 16br 2 + 2br(1 − 3b) − b2 ,

h4 = 1 − b/r{2r 3 [4(1 − b/r)2 + 4(1 − b/r) + 1] − 2r 4 [1 + 4(1 − b/r)]},

(40)

h5 = r 3 (r 2 − 2br + b).

(42)

(41)

The occurrence of the Aschenbach effect can be analysed by the numerical study of the roots of equation (37). The numerical approach enables us to determine the region in the parameter space (a − b) where this effect is admitted, see figure 7. In the naked singularity region, there is a subregion with retrograde first-family orbits. We can see that for black holes the region is highly sensitive on the choice of the tidal charge b. For positive values of b increasing, it becomes narrow as compared to the case of b = 0 and disappears when b > 0.410 05. In the limit value of b = 0.410 05, the corresponding value of the spin reads a = 0.768 08. On the other hand, the interval of the black hole spin allowing for the Aschenbach effect enlarges significantly with descending of the negative tidal charge b. We demonstrate this dependence in figure 8 presenting the function a(b), which denotes the spin interval of the black holes (with tidal charge b) allowing for the Aschenbach effect. Since the maximal black hole spin aex also depends on b, we give the dependence of the ratio a/aex on b for completeness. Note that for tidal charge b = −1, the spin interval a (and even its relative magnitude given by a/aex ) increases by almost one order as compared to the case of b = 0. In the naked singularity spacetimes, the Aschenbach effect appears in the wide region of a, b parameters—both for positive and negative tidal charges. The allowed naked singularity region is limited by b = 2.14 and a = 5.99—the point Q in figure 8. The behaviour of humpy LNRF-velocity profiles of Keplerian orbits in the field of braneworld Kerr black holes is represented by a series of figures for both positive and negative tidal charges. In order to clearly illustrate all the aspects of the Aschenbach effect in dependence on the black hole parameters, we give figures with both fixed value of the tidal 12

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Δ a(b) 0.09 0.08

Δ a(b)

0.07 0.06 0.05 0.04

Δa(b)/aex

0.03 0.02 0.01

b

0.00 -10

-8

-6

-4

-2

0

Figure 8. Spin range of braneworld Kerr black holes allowing for the Aschenbach effect given as a function of the tidal charge b. It is illustrated by the function a(b) and the relative spin range determined by the function a(b)/aex . The darkest area represents the most interesting case (from astrophysical point of view) when the radius of marginally stable orbit rms is less than the local minimum of the Keplerian velocity profile.

(ϕ)

(ϕ)

V K (r;a,b)

V K (r;a,b)

0.622

0.590

b=0.3

0.620 0.618

0.585

0.8362 0.8363

0.616

0.580

0.995

0.8364

0.614

b=0

0.575

0.99616

0.612

0.8365

0.610

0.570

aex=0.8367..

0.997 0.608

0.565

0.8366 0.606 1.1

1.2

1.3

1.4

1.5

1.6

r

0.560

aex=1

0.998 1.2

1.4

1.6

1.8

2.0

2.2

r

(b)

(a) (ϕ)

(ϕ)

V K (r;a,b)

V K (r;a,b)

0.560

0.580

b=-2

0.560

b=-1

0.550

0.540 0.540

0.480

1.71

0.460

aex=1.4143...

1.4

0.440

0.500

1.4

aex=1.7321...

1.72

0.420

1.405 0.490 1.2

1.7

0.500

1.395 0.520 0.510

1.689

0.520

1.39 0.530

1.6

1.8

2.0

(c)

2.2

2.4

2.6

r

0.400 1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

r

(d )

Figure 9. Non-monotonic LNRF-related velocity profiles for braneworld Kerr black hole backgrounds given for some values of the tidal charge b and appropriately chosen values of the a. The black points denote the loci of rms .

charge b and fixed value of the rotation parameter a, see figure 9. (The boundary of stable orbits given by the marginally stable orbit is represented by a point on all the profiles.) Moreover, we include the dependences of the LNRF velocities in the minimum and maximum of the humpy 13

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(ϕ)

(ϕ)

V K,ex

(ϕ)

V K (r=rms;a,b)-V K (r=rmin;a,b)

0.618

0.0030

0.616

0.0025

0.614 0.612

b=0.3

0.0020

0.610

b=0.3

0.0015

0.608 0.606

0.0010

0.604

0.0005

0.602 0.600 0.8362

a 0.8363

0.8364

0.8365

(a)

0.8366

0.0000 0.8362

0.8367

(ϕ)

a 0.8363

0.8364

(e)

(ϕ)

V K,ex

0.8365

0.8366

0.8367

(ϕ)


0.58

0.0040

0.57

0.0035 0.0030

0.56

b=0

0.0025

0.55 0.54

0.0020

b=0

0.0015

0.53

0.0010

0.52

0.0005

0.51 0.995

0.996

0.997

0.998

0.999

a

1

0.0000 0.995

(b) (ϕ)

a 0.996

0.997

(f)

(ϕ)

V K,ex

0.998

0.999

1.000

(ϕ)


0.55

0.0016 0.0014

0.50

0.0012 0.0010

0.45

b=-1

b=-1

0.0008 0.0006

0.40

0.0004 0.0002

0.35

a

1.385

1.390

1.395

1.400

(c)

1.405

1.410

0.0000 1.385

1.415

(ϕ)

1.395

(ϕ)

V K,ex

1.400

(g)

1.405

1.410

1.415

(ϕ)


0.55

0.0020

0.50

0.0015

0.45 0.40

(ϕ)

ΔV K,ex

b=-2

b=-2

0.0010

0.35

0.0005

0.30 0.25 1.68

a 1.390

a 1.69

1.70

1.71

1.72

1.73

0.0000 1.68

1.74

(d)

a 1.69

1.70

1.71

1.72

1.73

1.74

(h) (ϕ)

Figure 10. (a)–(d) Function VK,ex which determines values of the local minimum and maximum (ϕ)

of the function VK . The upper line represents the local maximum, while the lower line represents the local minimum. These curves demonstrate that maximum of the velocity difference is reached for the extreme black hole states. (d) An indication for the definition of extremal velocity difference (occurring for the extreme black hole states) as a function of the tidal charge (ϕ) b, function VK,ex giving maximal difference between value of the local maximum and local (ϕ)

minimum of the function VK,ex , considered with fixed b, as in figure 11. (e)–(h) Function (ϕ)

(ϕ)

(ϕ)

VK (r = rms , a, b) − VK (r = rmin,a,b ) which defines the difference between VK with radius for the marginally stable orbit and with radius for the local minimum. The dark grey region represents more astrophysically interesting cases, when the radius of the marginally stable orbit rms is lower than the radius for the local minimum rmin , see figure 8.

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(ϕ)

ΔV K,ex 0.25

0.20

0.15

0.10

0.05

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

b

0.0

0.5

(ϕ)

Figure 11. Function VK,ex giving maximal difference between values of the local maximum and (ϕ)

local minimum of the function VK,ex for a fixed b.

(ϕ)

(ϕ)

V K (r;a =0.996 aex,b)

V K (r;a,b)

0.582

0.6

b= 0.03 b= 0.02 b= 0.01 b= 0.00 b=-0.01 b=-0.02

0.580 0.578

) 0.4 0.2 0.0

0.576

-0.2

0.574 0.572 1.2

a=1.00,b= 0.0 a=1.05,b= 0.0 a=1.66,b=-1.5 a=2.10,b=-3.0 ( a=2.57,b=-5.0

-0.4 1.4

1.6

1.8

(a)

2.0

2.2

r

-0.6 0.0

0.5

1.0

1.5

2.0

r

(b)

Figure 12. (a) Sequence of non-monotonic LNRF-related velocity profiles for the near-extreme black hole background. In the sequence, there is a = 0.996 aex and we change the parameter b. (b) Sequence of non-monotonic LNRF-related velocity profiles √ for the naked singularity background close to the extreme black hole state given by aex = 1 − b. In the sequence, there is a = 1.05 aex , and we can see the transform between the retrograde and purely co-rotating velocity profiles. For comparison, the velocity profile constructed for the extreme Kerr black hole is included.

profile on the black hole parameters (see figure 10). These extremal LNRF-related velocities decrease with the tidal charge descending, but their difference increases significantly—for √ fixed b, the depth of the humpy profile grows with black holes spin approaching aex = 1 − b . Finally, in figure 11, there is the dependence of the velocity difference at the minimum– maximum part of the humpy profile on the tidal charge b for extreme braneworld Kerr black holes. We can see that for b = −1 the extremal difference increases by a factor ∼ 2 as compared to the case of b = 0. We also demonstrate there, how the Aschenbach effect disappears about the point Q for positive tidal charges, and how the LNRF-related Keplerian profiles in the naked singularity spacetime can be reduced into those for the corresponding extreme black hole spacetime, 15

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when the retrograde naked singularity profile is transformed into a discontinuity occurring at the radius r = 1, and the negatively valued part under the horizon becomes physically irrelevant. The behaviour of Keplerian velocity profiles in both black hole and naked singularity backgrounds corresponding to near-extreme black hole cases is shown in figure 12. 5. Conclusions We have shown that the Aschenbach effect is a typical feature of the circular geodetical motion in the field of both standard and braneworld Kerr naked singularities with a relatively large interval of spins above the extreme black hole limit. For naked-singularity spin sufficiently close to the extreme black hole state, the Aschenbach effect is manifested by the retrograde plus-family circular orbits. For black hole spacetimes, such retrograde orbits can appear under the inner horizon, thus being irrelevant from the astrophysical point of view. In the field of near-extreme rotating black holes, the Aschenbach effect located above the outer black hole horizon can thus be considered as a small remnant of the typical naked singularity phenomenon. We have demonstrated that in the braneworld near-extreme Kerr black hole spacetimes, the non-monotonic LNRF-related orbital velocity profiles of the Keplerian motion are suppressed for positive tidal charges as compared with the standard Kerr black holes, and disappear for the tidal charge b > 0.410 05, while they are strongly enlarged for decreasing negative tidal charge. The spin range of black holes allowing for the Aschenbach effect increases significantly with decreasing tidal charge, strengthening possible relevance of the Aschenbach effect in astrophysical phenomena (see, e.g., [50]). This possibility is further supported by increasing the magnitude of the velocity difference in the minimum–maximum part of the LNRF-related Keplerian profile with decreasing negative tidal charge. Recent observations of high-frequency quasiperiodic oscillations observed in the x-ray spectra of some neutron-star binaries [51–54] that could be used to test braneworld models in the strong-field limits imply negatively valued dimensionless tidal charges as high as b = −2 when the Aschenbach effect could be quite significant and well observed [50]. We conclude that the Aschenbach effect can play a significant role in explaining a variety of physical phenomena (optical effects and related line profiles, explanation of a variety of high-frequency quasi-periodic oscillations observed in some microquasars and active galactic nuclei, etc) expected to appear in the strong gravitational field of braneworld Kerr black holes, especially in the case of negative tidal charge. Acknowledgment ˇ This work was supported by the Czech grants MSM 4781305903, LC 06014, GACR 205/09/H033 and the internal grant SGS/2/2010. References [1] Volonteri M, Madau P, Quataert E and Rees M 2005 The distribution and cosmic evolution of massive black hole spins Astrophys. J. 620 69–77 [2] Shapiro S 2005 Spin, accretion, and the cosmological growth of supermassive black holes Astrophys. J. 620 59–68 [3] Tanaka Y et al 1995 Gravitationally redshifted emission implying an accretion disk and massive black-hole in the active galaxy MCG-6-30-15 Nature 375 659 [4] Miyakawa T, Ebisawa K, Terashima Y, Tsuchihashi F, Inoue H and Zycki P 2009 Spectral variation of the Seyfert 1 galaxy MCG 6-30-15 observed with Suzaku Publ. Astron. Soc. Japan 61 1355 16

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[5] Reynolds C S, Fabian A C, Brenneman L W, Miniutti G, Uttley P and Gallo L C 2009 Constraints on the absorption-dominated model for the x-ray spectrum of MCG-6-30-15 Mon. Not. R. Astron. Soc. 397 L21–5 [6] McClintock J E, Shafee R, Narayan R, Remillard R A, Davis S W and Li L-X 2006 The spin of the near-extreme Kerr black hole GRS 1915+105 Astrophys. J. 652 518–39 [7] Török G, Abramowicz M A, Klu´zniak W and Stuchl´ık Z 2005 The orbital resonance model for twin peak kHz quasi-periodic oscillations in microquasars Astron. Astrophys. 436 1–8 [8] Steiner J F, McClintock J F, Orosz J A, Narayan R, Torres M A and Remillard R A 2008 Estimating the spin of the stellar-mass black hole XTE J1550-564 AAS/High Energy Astrophysics Division Meeting vol 10 [9] Aschenbach B 2004 Measuring mass and angular momentum of black holes with high-frequency quasi-periodic oscillations Astron. Astrophys. 425 1075–82 [10] Török G 2005 QPOs in microquasars and Sgr A∗ : measuring the black hole spin Astron. Nachr. 326 856–60 [11] Meyer L, Eckart A, Schödel R, Duschl W J, Mu˙zić K, Dovèiak M and Karas V 2006 Near-infrared polarimetry setting constraints on the orbiting spot model for Sgr A∗ flares Astron. Astrophys. 460 15 [12] Gimon E G and Hoˇrava P 2009 Astrophysical violations of the Kerr bound as a possible signature of string theory Phys. Lett. B 672 299–302 [13] de Felice F 1974 Repulsive phenomena and energy emission in the field of a naked singularity Astron. Astrophys. 34 15 [14] de Felice F 1978 Classical instability of a naked singularity Nature 273 429–31 [15] Stuchl´ık Z 1980 Equatorial circular orbits and the motion of the shell of dust in the field of a rotating naked singularity Bull. Astron. Inst. Czech. 31 129–44 [16] Stuchl´ık Z 1981 Evolution of Kerr naked singularities Bull. Astron. Inst. Czech. 32 68–72 [17] Stuchl´ık Z and Schee J 2010 Appearance of Keplerian discs orbiting Kerr superspinars Class. Quantum Grav. 27 215017 [18] Stuchl´ık Z, Hled´ık S and Truparová K 2011 Evolution of Kerr superspinars due to accretion counter-rotating thin discs Class. Quantum Grav. 28 155017 [19] Bardeen J M, Press W H and Teukolsky S A 1972 Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation Astrophys. J. 178 347–69 [20] Aschenbach B 2008 Measurement of mass and spin of black holes with QPOs Chin. J. Astron. Astrophys. Suppl. 8 291–6 [21] Stuchl´ık Z, Slaný P, Török G and Abramowicz M A 2005 Aschenbach effect: unexpected topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr black holes Phys. Rev. D 71 024037 [22] Stuchl´ık Z, Slaný P and Török G 2007 LNRF-velocity hump-induced oscillations of a Keplerian disc orbiting near-extreme Kerr black hole: a possible explanation of high-frequency QPOs in GRS 1915+105 Astron. Astrophys. 470 401–4 [23] Stuchl´ık Z, Slaný P and Török G 2007 Humpy LNRF-velocity profiles in accretion discs orbiting almost extreme Kerr black holes: a possible relation to quasi-periodic oscillations Astron. Astrophys. 463 807–16 [24] Slaný P and Stuchl´ık Z 2008 Mass estimate of the XTE J1650-500 black hole from the extended orbital resonance model for high-frequency QPOs Astron. Astrophys. 492 319–22 [25] Stuchl´ık Z, Slaný P and Török G 2008 Humpy LNRF-velocity profiles in accretion discs orbiting rapidly rotating Kerr black holes Proc. 11th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity: Gravitation and Relativistic Field Theories (Berlin, 23–29 July 2006) ed H Kleinert and R T Jantzen (Singapore: World Scientific) (series editor R Ruffini) pp 1060–2 [26] Müller A and Aschenbach B 2007 Non-monotonic orbital velocity profiles around rapidly rotating Kerr–(anti)de Sitter black holes Class. Quantum Grav. 24 2637–44 [27] Slaný P and Stuchl´ık Z 2008 On non-monotonic orbital velocity profiles around rapidly rotating Kerr-(anti)de Sitter black holes: a comment to the recently published results Class. Quantum Grav. 25 038001 [28] Randall L and Sundrum R 1999 A large mass hierarchy from a small extra dimension Phys. Rev. D 83 3370–3 [29] Germani C and Maartens R 2001 Stars in the braneworld Phys. Rev. D 64 124010 [30] Figueras P and Wiseman T 2011 Gravity and large black holes in Randall–Sundrum II braneworlds arXiv:1105.2558 [31] Aliev A N and Gümrük¸cu¨ o˘glu A E 2005 Charged rotating black holes on a 3-brane Phys. Rev. D 71 104027 [32] Dadhich N, Maartens R, Papadopoulos P and Rezania V 2000 Black holes on the brane Phys. Lett. B 487 1–6 [33] Stuchl´ık Z and Kotrlová A 2009 Orbital resonances in discs around braneworld Kerr black holes Gen. Rel. Grav. 41 1305–43 [34] Aliev A N and Talazan P 2009 Gravitational effects of rotating braneworld black holes Phys. Rev. D 80 044023 [35] Schee J and Stuchl´ık Z 2009 Profiles of emission lines generated by rings orbiting braneworld Kerr black holes Gen. Rel. Grav. 41 1795–818

17

Chapter

7

Paper 2: Efficiency of the Keplerian accretion in braneworld Kerr–Newman spacetimes and mining instability of some naked singularity spacetimes We feel free because we lack the very language to articulate our unfreedom. Slavoj Žižek

T

he second paper is dedicated to study of effectiveness of the accretion onto braneworld KN black hole and naked singularity. We have discovered new kind of classical instability for special class of naked singularity spacetimes. We call this instability “mining” because the effective potential looks like a shaft of an infinitely deep mine. This mining instability is so strong that theoretically a single particle in the vicinity of the singularity can cause its destruction or “mined out” all its energy.

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PHYSICAL REVIEW D 94, 086006 (2016)

Efficiency of the Keplerian accretion in braneworld Kerr-Newman spacetimes and mining instability of some naked singularity spacetimes Martin Blaschke* and Zdeněk Stuchlík† Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo náměstí 13, CZ-746 01 Opava, Czech Republic (Received 1 August 2016; published 18 October 2016) We show that the braneworld rotating Kerr-Newman black hole and naked singularity spacetimes with both positive and negative braneworld tidal charge parameters can be separated into 14 classes according to the properties of circular geodesics governing the Keplerian accretion. We determine the efficiency of the Keplerian accretion disks for all braneworld Kerr-Newman spacetimes. We demonstrate the occurrence of an infinitely deep gravitational potential in Kerr-Newman naked singularity spacetimes having the pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi braneworld dimensionless tidal charge b ∈ ð1=4; 1Þ and the dimensionless spin a ∈ ð2 b − bð4b − 1Þ, pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b þ bð4b − 1ÞÞ, implying unbound efficiency of the Keplerian accretion and the possibility of extracting the whole naked singularity mass. Therefore, we call them braneworld “mining-unstable” KerrNewman naked singularity spacetimes. Fundamental restriction on the relevance of the extraordinary—but fully classical—phenomenon of the mining instability is given by validity of the assumption of geodesic motion of the accreting matter. DOI: 10.1103/PhysRevD.94.086006

I. INTRODUCTION In recent years, one of the most interesting and promising approaches to force-unification theory is represented by the higher-dimensional string theory and, in particular, M-theory [1,2]. In string theory and M-theory, gravity is described as a truly higher-dimensional interaction becoming effectively 4D at low enough energies. These theories inspired the so-called braneworld models, in which the observable Universe is a 3D-brane to which the standard particle-model fields are confined, while gravity enters the extra spatial dimensions [3]. The braneworld models provide an elegant solution to the hierarchy problem of the electroweak and quantum gravity scales, as these scales could become of the same order (in TeV) due to the large scale extra dimensions [3]. In fact, gravity can be localized near the D3-brane in the bulk space with a noncompact, infinite size extra dimension, with the warped spacetime satisfying the 5D Einstein equations [4]—the noncompact dimension can be related to M-theory. Future collider experiments can test the braneworld models quite well, including even the hypothetical mini–black hole production [5]. The 5D Einstein equations at the bulk space can be constrained to the 3D-brane, thus implying modified 4D Einstein equations [6]. Solution of these constrained 4D Einstein equations is quite complex in the presence of the matter stress-energy tensor, e.g., in the case of models of neutron stars [7–9]. However, it can be relatively simple in * †

[email protected] [email protected]

2470-0010=2016=94(8)=086006(28)

the case of vacuum solutions related to braneworld black holes. For both the spherically symmetric and static black holes that can be described by the Reissner-Nordström geometry [10] and the axially symmetric and stationary rotating black holes that can be described by the KerrNewman geometry [11], the influence due to the tidal effects from the bulk is simply represented by a single parameter. This parameter is called tidal charge because of the similarity of the effective stress-energy tensor of the tidal effects of the bulk space and the stress-energy tensor of the electromagnetic field [10]. The rotating braneworld black hole spacetimes and the related naked singularity spacetimes are thus represented by the Kerr-Newman geometry, but without the associated electromagnetic field occurring in standard general relativity [12]. The tidal charge parameter can be either positive or negative [10,11], while, in standard general relativity, only a positive parameter corresponding to the square of the electric charge occurs. The standard studies of the Reissner-Nordström or KerrNewman black hole and naked singularity geodesic motion [13–17] can thus be directly applied for the braneworld black holes and naked singularities with positive tidal charge. The astrophysically relevant implications of the geodesic motion were extensively studied for the braneworld black holes (with both positive and negative tidal charges) in a number of papers related to the optical effects [18–25], or the test particle motion [26–33]. Here, we study the circular motion of test particles and photons in the braneworld Kerr-Newman spacetimes and give classification of the braneworld black hole and naked

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MARTIN BLASCHKE and ZDENĚK STUCHLÍK


singularity spacetimes according to the properties of the radial profiles of specific angular momentum and specific energy of sequences of corotating and retrograde circular orbits. We give the classification for both the positive and negative values of the dimensionless tidal charge parameter. We also determine the efficiency of the Keplerian accretion disks that is related to the astrophysically relevant accretion from infinity (large distance) downwards to the first limit on existence of stable circular geodesics. A very detailed analysis of the circular motion of electrically neutral test particles in the standard Kerr-Newman spacetimes was presented in [17], where both black hole and naked singularity spacetimes were discussed. The results of this study are relevant in the braneworld spacetimes with positive tidal charges; we do not repeat them, focusing our study on the phenomena related to the Keplerian accretion, its efficiency, and the phenomenon of a new special instability of the naked singularity spacetimes that were not considered in [17]. Along with the standard classical instability due to the Keplerian accretion occurring in the Kerr naked singularity spacetimes, leading to their conversion to a Kerr black hole [34–37], we have found a special class of classical instability, called here “mining” instability, as this instability is related to an unlimitedly deep gravitational potential well, occurring in the class of the KerrNewman naked singularity spacetimes with appropriately restricted values of their dimensionless spin a and dimensionless tidal charge b. We briefly discuss the limits on the applicability of the Keplerian accretion in relation to the mining instability.

angular momentum J, and the braneworld parameter b, called the “tidal charge,” represents an imprint of the nonlocal (tidal) gravitational effects of the bulk space [11]. The form of the metric (1) is the same as that of the standard Kerr-Newman solution of the 4D EinsteinMaxwell equations, with the squared electric charge Q2 being replaced by the tidal charge b [12]. We can separate out three cases: (a) b ¼ 0, in which we are dealing with the standard Kerr metric. (b) b > 0, in which we are dealing with the standard KerrNewman metric. (c) b < 0, in which we are dealing with the nonstandard Kerr-Newman metric. Notice that, in the braneworld Kerr-Newman spacetimes, the geodesic structure is relevant also for the motion of electrically charged particles, as there is no electric charge related to these spacetimes. On the other hand, case (b) can be equally considered for the analysis of the uncharged particle motion in the standard electrically charged KerrNewman spacetime. For simplicity, we put in the following considerations M ¼ 1. Then the spacetime parameters a and b and the time t and radial r coordinates become dimensionless. This is equivalent to the redefinition when we express all the quantities in units of M: a=M → a, b=M2 → b, t=M → t, and r=M → r. Separation between the black hole and naked singularity spacetimes is given by the relation of the spin and tidal charge parameters in the form

II. BRANEWORLD KERR-NEWMAN GEOMETRY

a2 þ b ¼ 1;

Using the standard Boyer-Lindquist coordinates ðt; r; θ; φÞ and the geometric units (c ¼ G ¼ 1), we can write the line element of a rotating (Kerr-Newman) black hole or naked singularity, representing a solution of the Einstein equations constrained to the 3D-brane, in the form [10,11] 2Mr − b 2að2Mr − bÞ 2 ds2 ¼ − 1 − dt2 − sin θdtdφ Σ Σ Σ þ dr2 þ Σdθ2 Δ 2Mr − b 2 2 a sin θ sin2 θdφ2 ; þ r2 þ a2 þ ð1Þ Σ

ð4Þ

determining the so-called extreme black hole with coinciding horizons. The condition 0 < a2 þ b < 1 governs black hole spacetimes with two distinct event horizons, while the condition a2 þ b < 0 governs black hole spacetimes with only one distinct event horizon at r > 0. For a2 þ b > 1, the spacetime describes a naked singularity. For positive tidal charges the black hole spin has to be a2 < 1, as in the standard Kerr-Newman spacetimes, but, for negative tidal charges, there can exist black holes violating the well-known Kerr limit, having a2 > 1 [28]. Using the substitutions

where

dt ¼ dx0 þ

2 r þ a2 − 1 dr; Δ

ð5Þ

a dr; Δ

ð6Þ

dφ ¼ dφ~ þ

Δ ¼ r2 − 2Mr þ a2 þ b;

ð2Þ

Σ ¼ r2 þ a2 cos2 θ:

ð3Þ

~ þ a sinðφÞÞ ~ sin θ; x ¼ ðr cosðφÞ

ð7Þ

M is the mass parameter of the spacetime, a ¼ J=M is the specific angular momentum of the spacetime with internal

~ − a cosðφÞÞ ~ sin θ; y ¼ ðr sinðφÞ

ð8Þ

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EFFICIENCY OF THE KEPLERIAN ACCRETION IN …

z ¼ r cos θ;


ð9Þ

the braneworld Kerr-Newman geometry can be transformed into the so-called Kerr-Schild form using the Cartesian coordinates:

K

ds2 ¼ −ðdx0 Þ2 þ ðdxÞ2 þ ðdyÞ2 þ ðdzÞ2 ð2Mr − bÞr2 1 0 dx rðxdx þ ydyÞ þ 4 − r2 þ a2 r þ a2 z2 2 1 þ aðxdy − ydxÞ − zdz ; ð10Þ r

400 300 200 100 0 -100 -200 -300 -400

3 2.5 0

2 0.2

0.4

1.5 0.6

0.8

r

where r is defined, implicitly, by

1 1

1.2

θ

0.5 1.4

0

FIG. 1. Example of the behavior of the Kretschmann scalar K for a ¼ 0.8 and b ¼ −0.8 to illustrate its similarity to the KerrNewman case.

r4 − r2 ðx2 þ y2 þ z2 − a2 Þ − a2 z2 ¼ 0: A. Singularity The metric (10) is analytical everywhere except at points satisfying the condition x2 þ y2 þ z2 ¼ a2

and z ¼ 0:

ð11Þ

This condition is the same as in the case of the Kerr black holes or naked singularities, so we clearly see that the braneworld parameter b has no influence on the position of the physical singularity of the spacetime. The physical “ring” singularity of the braneworld rotating black holes (and naked singularities) is located at r ¼ 0 and θ ¼ π=2, as in the Kerr spacetimes. We describe the influence of the braneworld tidal charge parameter b on the Kerr-like ring singularity at r ¼ 0, θ ¼ π=2 using the Kretschmann scalar K ¼ Rαβγδ Rαβγδ, representing an appropriate tool to probe the structure of spacetime singularities. Using (1) we obtain 8 K¼ 2 ðr4 A − 2a2 r2 By2 þ a4 Cy4 − 6a6 M2 y6 Þ; ðr þ a2 y2 Þ6 ð12Þ

denominator of (12), it does not influence the location of the singularity. As an example we demonstrate the behavior of the scalar K for a ¼ 0.8 and b ¼ −0.8 near the ring singularity in Fig. 1. For completeness we also give the Ricci tensor whose components take the form Rtt ¼ 4b

Rtφ ¼ −8ab

ð13Þ

A ¼ ð7b2 − 12bMr þ 6M 2 r2 Þ;

ð14Þ

B ¼ ð17b2 − 60bMr þ 45M2 r2 Þ;

ð15Þ

C ¼ ð7b2 − 60bMr þ 90M 2 r2 Þ:

ð16Þ

The Kretschmann scalar is formally the same as in the case of the Kerr-Newman metric with Q2 → b [38]. Naturally, the negative values of the brane parameter would have some effect on K, but, as we can see from the

ða2 þ ΔÞsin2 θ ; ða2 þ 2r2 þ a2 cosð2θÞÞ3

Rθθ ¼ Rφφ ¼ 4bsin2 ðθÞ −

ð17Þ ð18Þ

Rθθ ; Δ

ð19Þ

2b ; a þ 2r þ a2 cosð2θÞ

ð20Þ

Rφt ¼ Rtφ ;

where y ¼ cos θ;

a2 þ 2Δ − a2 cosð2θÞ ; ða2 þ 2r2 þ a2 cosð2θÞÞ3

2

Rrr ¼ −

2

3a4 þ 2r4 þ a2 ðb − 2Mr þ 5r2 Þ ða2 þ 2r2 þ a2 cosð2θÞÞ3

ð21Þ

a2 Δ cosð2θÞ : ða þ 2r2 þ a2 cosð2θÞÞ3

ð22Þ

2

Ricci scalar is automatically zero by construction of the braneworld Kerr-Newman solution [10]. B. Ergosphere Here, we demonstrate the influence of the braneworld tidal charge parameter b on the ergosphere whose boundary is defined by the condition

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gtt ¼ r2 − 2Mr þ a2 cos2 θ þ b ¼ 0:

ð23Þ

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ða2 þ r2 ÞΔ ; a2 ð2r − bÞr2 − r4 z2 ¼ : a2

Extension of the ergosphere in the latitudinal coordinate θ is determined by the maximal latitude given by the relation cos2 θmax ¼

1−b : a2

x2 ¼

ð24Þ

We can see that the existence of the ergosphere is limited by the condition b < 1:

ð25Þ

We can infer that the ergosphere extension increases as the tidal charge parameter b decreases. It is convenient to represent location of the ergosphere in the Kerr-Schild coordinates (10). Using the spacetime symmetry, we can focus only on the polar slices with y ¼ 0. In this case the condition for the static limit surface governing the border of the ergosphere is simply given by [39]

In Fig. 2 we illustrate the influence of the braneworld tidal charge b on the ergosphere extension. The ergosphere does not always completely surround the ring singularity. To illustrate this phenomenon, we also give the dependence of the maximal allowed latitudinal angle of the ergosphere on the dimensionless spin and dimensionless tidal charge. For b < 1, the ergosphere exists for each dimensionless spin a > 0, covering all values of the latitudinal angle for the Kerr-Newman black holes. However, as the spin a increases for the Kerr-Newman naked singularities, the ergosphere extension shrinks—the maximal angle α decreases. 1.2

1.6 1.

a=1

1.4

b=0.9

1

a=0.32

1.2

0.8

0.8

z

b=0

1

z

ð26Þ

b=0.3

0.6

0.6

a=0.6

0.4

b=0.7

an g

le

0.4

0.2

M ax

0.2

singularity at x = a

b=0.95

singularity 0 0

0.5

1

1.5

2

a=1.2 a=2.5

0

2.5

0

0.5

1

x

1.5

2

2.5

3

x 0.6

0.8

a=0.5, b=0.98

a=1

0.6

0.4

ergosphere causality violation region

0.4 0.2

1

0

0.4

z

z

0.2 0

b=4 -0.2 -0.2

-0.4 -0.4

-0.6 -0.8

-0.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.5

1

1.5

x

x

FIG. 2. (Upper left panel) Polar slice through the braneworld Kerr-Newman spacetime in the Cartesian Kerr-Schild coordinates. The dimensionless spin parameter a is fixed at 1 and the braneworld parameter b is appropriately chosen to demonstrate its influence on the ergosphere. (Upper right panel) Polar slice through the braneworld Kerr-Newman spacetime in the Cartesian Kerr-Schild coordinates. The braneworld parameter b is fixed at 0.9 and the spin parameter a is appropriately chosen to demonstrate its influence on the ergosphere. (Lower left panel) Causality violation region. (Lower right panel) Ergosphere and causality violation region.

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C. Causality violation region In the “causality violation region” (sometimes called the time-machine region), the axial coordinate φ takes timelike character, implying the possible existence of closed timelike curves. The causality violation region is defined by the condition gφφ < 0:

ð27Þ

In the equatorial plane, the boundary of the causality violation region is determined by the condition r4 þ a2 ðr2 þ 2r − bÞ ¼ 0:

ð28Þ

The boundary of the causality violation region can be expressed by the relation b ¼ bCV ≡

rð2a2 þ a2 r þ r3 Þ : a2

ð29Þ

In Fig. 3 we give some examples of the extension of the causality violation region. We see that, for this region to exist above the ring singularity, the tidal charge has to be positive. With increasing values of the parameters b > 0 and a, the causality violation region expands. Equation (28) gives us maximal possible extension of the causal violation region located at rMax

pffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ b − 1:

ð30Þ

For a positive b, the value of rMax is less than b and therefore, as we shall see later, the causality violation region cannot reach the region where the circular geodesics exist. In the Kerr-Schild coordinates, the boundary of the causality violation region is given by the relations

z2 ¼

1

b

ð34Þ

1 Σ 2 dr; Δ 1

ð35Þ

eðθÞ ¼ Σ2 dθ;

0.5

where ω is the angular velocity of the LNRFs relative to distant observers and reads

0.3

ω¼−

0.1 1

1

0.7

1

0.5

ð33Þ

eðrÞ ¼

1.5

0

eðtÞ ¼ ðω2 gφφ − gtt Þ2 dt; eðφÞ ¼ ðgφφ Þ2 ðdφ − ωdtÞ;

2

0

ð32Þ

In the rotating Kerr-Newman spacetimes, physical processes can be most conveniently expressed in the family of locally nonrotating frames (LNRFs), corresponding to zero angular momentum observers, with tetrad vectors given by the relations [43]

2.5

black holes

r2 ða2 ðb − 2r − r2 Þ − r4 Þ : a2 Δ

D. Locally nonrotating frames

0.9

0.5

ð31Þ

It can be proved that the causality violation region never overlaps with the ergosphere, and its extension is influenced by the braneworld parameter b in the opposite way. While causality violation region increases with an increasing b, the ergosphere extension gets smaller. This phenomenon is illustrated in Fig. 2, where the Kerr-Schild coordinates are used. In the following, we consider geodesic motion only in the regions above the causality violation region. For astrophysical phenomena occurring in the naked singularity spacetimes, it is usually assumed that, above the boundary of the causality violation region, the Kerr or Kerr-Newman spacetime is removed and substituted for a different solution that could be inspired by string theory—such objects are called superspinars [40–42]. Therefore, it is quite natural to assume that, in the braneworld model framework, the inner boundary of the superspinars is located at radii larger than those related to the boundary of the causality violation region.

r(a,b) - radius of causality violation region

3

ða2 þ r2 Þ3 ; a2 Δ

x2 ¼

1.5

2

2.5

3

a FIG. 3. Contour plot for radii of the boundary of the causality violation region in the equatorial plane.

ð36Þ

gtφ að2r − bÞ ¼ : ð37Þ 2 2 gφφ Σðr þ a Þ þ ð2r − bÞa2 sin2 θ

Convenience of the LNRFs can be demonstrated, e.g., in the case of the free fall of particles from infinity, which is purely radial only if related to the family of the LNRFs [44].

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E. Geodesic motion and Carter’s equations Using the Hamilton-Jacobi method, Carter found separated first order differential equations of the geodesic motion [39,45], which in the case of the braneworld Kerr spacetimes take the form Σ

pffiffiffiffiffiffiffiffiffi dr ¼ RðrÞ; dw

ð38Þ

Σ

pffiffiffiffiffiffiffiffiffiffiffi dθ ¼ WðθÞ; dw

ð39Þ

dφ P aP ¼ − W2 þ R ; dw Δ sin θ

ð40Þ

dt ðr2 þ a2 ÞPR ¼ −aPW þ ; dw Δ

ð41Þ

III. CIRCULAR GEODESIC MOTION

ð42Þ

In general stationary and axially symmetric spacetime with the Boyer-Lindquist coordinate system ðt; r; θ; φÞ and the ð− þ þþÞ signature of the metric tensor, the line element is given by

Σ Σ

in [19]; we use the results of these works in the following discussions. We have to construct classification of the braneworld Kerr-Newman spacetimes according to the properties of circular geodesics governing the Keplerian accretion, which can be related not only to the standard accretion disks but also to the quasicircular motion of gravitationally radiating particles. We give classifications according to the properties of the circular null geodesics and the stability of the circular geodesics that become the critical attribute of the Keplerian accretion. Finally, we combine the effects given by these two classifications. Of course, we have to also include in the classification as relevant criteria the existence of the event horizons and the existence of the ergosphere.

where ~ RðrÞ ¼ P2R − Δðm2 r2 þ KÞ; WðθÞ ¼ ðK~ − a2 m2 cos2 θÞ −

Pw sin θ

2 ;

ð43Þ

~ ~ 2 þ a2 Þ − aΦ; PR ðrÞ ¼ Eðr

ð44Þ

~ ~ 2 θ − Φ: PW ðθÞ ¼ aEsin

ð45Þ

Along with the conservative rest energy m, three constants of motion related to the spacetime symmetries have been introduced: E~ is the energy (related to the time Killing ~ is the axial angular momentum (related to vector field), Φ the axial Killing vector field), and K~ is the constant of motion related to the total angular momentum (related to the Killing tensor field) that is usually replaced by the ~ ¼ K~ − ðaE~ − ΦÞ~ 2 since, for the motion in the constant Q ~ ¼ 0. equatorial plane (θ ¼ π=2), we have Q Note that the separable Eqs. (38)–(41) are quaranteed in the Petrov type D spacetimes, particularly when the metric in the Boyer-Lindquist coordinates can be expressed in the Kerr-like form by replacing the mass parameter M by a function MðrÞ independent of latitude θ. In the braneworld b rotating black hole spacetimes, we have MðrÞ ¼ M − 2r . Generally, these equations can be integrated and expressed in terms of the hyperelliptic integrals [12,46,47]. The Carter equations can also be generalized to the motion in the KerrNewman–de Sitter spacetimes [39,46,48–50]. For the geodesic motion of photons, we put m ¼ 0 in the Carter equations. Analysis of the photon motion in the standard Kerr-Newman spacetimes [17,51,52] can be directly applied to the case of photon motion in the braneworld Kerr-Newman spacetimes. This has been done

ds2 ¼ gtt dt2 þ 2gtφ dtdφ þ grr dr2 þ gθθ dθ2 þ gφφ dφ2 : ð46Þ The metric (46) is adapted to the symmetries of the spacetime, endowed with the Killing vectors (∂=∂t) and (∂=∂φ) for time translations and spatial rotations, respectively. For geodesic motion in the equatorial plane (θ ¼ π=2), the metric functions gtt , gtφ , grr , gθθ , and gφφ in Eq. (46) depend only on the radial coordinate r. Thus, omitting the rest energy m, two integrals of the motion are ~ ¼ 0: relevant when Q Ut ¼ −E;

Uφ ¼ L;

ð47Þ

where the 4-velocity Uα ¼ gαν dxν =dτ, with τ being the affine parameter. In the case with an asymptotically flat spacetime, we can identify at infinity the motion constant ~ E ¼ E=m as the specific energy, i.e., energy related to the ~ rest energy, and the motion constant L ¼ Φ=m as the specific angular momentum. The geodesic equations of the equatorial motion take the form (see, e.g., [53]) dt Egφφ þ Lgtφ ¼ 2 ; dτ gtφ − gtt gφφ

Egtφ þ Lgtt dφ ¼− 2 ; dτ gtφ − gtt gφφ

ð48Þ

and grr

2 dr ¼ RðrÞ; dτ

where the radial function RðrÞ is defined by

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ð49Þ

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RðrÞ ≡ −1 þ

2

2

E gφφ þ 2ELgtφ þ L gtt : g2tφ − gtt gφφ


Ω¼

ð50Þ

1 2 prffiffiffiffiffiffi r−b

a

:

ð57Þ

A. Energy, angular momentum, and angular velocity of circular geodesics

From Eqs. (55)–(57) we immediately see that two restrictions on the existence of circular geodesics have to be satisfied:

For circular geodesics in the equatorial plane, the conditions

pffiffiffiffiffiffiffiffiffiffiffi r2 − 3r þ 2b 2a r − b ≥ 0;

ð58Þ

r ≥ b:

ð59Þ

RðrÞ ¼ 0

and ∂ r RðrÞ ¼ 0

ð51Þ

must be satisfied simultaneously. These conditions determine the specific energy E, the specific angular momentum L and the angular velocity Ω ¼ dφ=dt related to distant observers, for test particles following the circular geodesics, as functions of the radius and the spacetime parameters in the form gtt þ gtφ Ω E ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; −ðgtt þ 2gtφ Ω þ gφφ Ω2 Þ gtφ þ gφφ Ω L ¼∓ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; −ðgtt þ 2gtφ Ω þ gφφ Ω2 Þ

Ω¼

−gtφ;r

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðgtφ;r Þ2 − gtt;r gφφ;r gφφ;r

;

ð52Þ

ð53Þ

ð54Þ

where the upper and lower signs refer to two families of solutions. To avoid any misunderstanding, we will refer to these two families as the upper sign family and the lower sign family. At large distances in the asymptotically flat spacetimes, the upper family orbits are corotating, while the lower family orbits are counterrotating with respect to the rotation of the spacetime. This separation holds in the whole region above the event horizon of the Kerr-Newman black hole spacetimes, but this is not necessarily so in all of the Kerr-Newman naked singularity spacetimes—in some of them the upper family orbits become counterrotating close to the naked singularity, as demonstrated in [35]. Using the spacetime line element of the braneworld rotating spacetimes given by (1) [11,54] and assuming that M ¼ 1, we obtain the radial profiles of the specific energy, the specific axial angular momentum, and the angular velocity related to infinity of the circular geodesics in the form pffiffiffiffiffiffiffiffiffiffiffi r2 − 2r þ b a r − b E ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ; r r2 − 3r þ 2b 2a r − b

The equality in the first condition determines the photon circular geodesics—this demonstrates that positions of circular orbits of test particles are limited by the circular geodesics of massless particles. The second reality condition is relevant in the Kerr-Newman spacetimes with the positive tidal charge b only if we restrict our attention to the region of positive radii. B. Effective potential Instead of the radial function Rðr; a; b; E; LÞ, the equatorial motion of test particles can be conveniently treated by using the so-called effective potential V Eff ðr; a; b; LÞ, which is related to the particle specific energy and depends on the specific angular momentum of the motion and the spacetime parameters. The equation E ¼ V Eff determines the turning points of the radial motion of the test particle. The notion of the effective potential is useful in treating the Keplerian (quasigeodesic) accretion onto the central object that is directly related to the circular geodesic motion [55,56]. The circular geodesics are governed by the local extrema of the effective potential; the accretion process is possible in the regions of stable circular geodesics corresponding to the local minima of the effective potential. The effective potential can be easily derived using the normalization condition for the test particle motion Uα Uα ¼ −1; which implies, for the equatorial motion relation, 2 dr grr ¼ ðE − V Effþ ÞðE − V Eff− Þ; dτ

ð61Þ

and, in the general stationary and axisymmetric spacetimes, the effective potential can be expressed in the form V Eff ðr; a; b; LÞ ¼

ð55Þ

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 r − bðr þ a2 ∓ 2a r − bÞ ∓ ab pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð56Þ L¼ pffiffiffiffiffiffiffiffiffiffiffi r r2 − 3r þ 2b 2a r − b

ð60Þ

β

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2 − αγ ; α

ð62Þ

where

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α¼

gφφ ; g2φt − gφφ gtt

β¼

−Lgtφ ; g2φt − gφφ gtt

ð63Þ

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γ¼

L2 gtt − 1: g2φt − gφφ gtt

This form can be simplified to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −Lgtφ ðL2 þ gφφ Þðg2tφ − gφφ gtt Þ : V Eff ¼ gφφ


ð64Þ

b ¼ bs ≡ ð65Þ

We have to choose the upper (plus) sign of the general expression of the effective potential, as this case represents the boundary of the motion of particles in the so-called positive-root states having a positive locally measured energy and a future-oriented time component of the 4velocity. The lower (minus) sign expression of the effective potential is irrelevant here, as it determines in the regions of interest particles in the so-called negative-root states having a negative locally measured energy and a past-oriented time component of the 4-velocity, thus being related to the Dirac particles—for details, see [12,57]. The physically relevant condition of the test particle motion reads E ≥ V Effþ :

diverges at the event horizons. The second condition (70) for a possible divergence of the effective potential can be transformed to the relation

ð66Þ

For particles with the nonzero rest mass, m > 0, the explicit form of the effective potential in the braneworld KerrNewman spacetimes reads V Eff ðr; a; b; LÞ

pffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aLð2r − bÞ þ r Δ L2 r2 þ r4 þ a2 ðr2 þ 2r − bÞ : ¼ r4 þ a2 ðr2 þ 2r − bÞ ð67Þ

For massless particles, m ¼ 0, we formally obtain pffiffiffiffi V Effp ðr; a; bÞ að2r − bÞ r2 Δ ¼ 4 : ð68Þ L r þ a2 ðr2 þ 2r − bÞ Here, the plus sign is valid if L > 0 and the minus sign is valid if L < 0. Of course, we know that the photon geodesic motion is independent of the photon energy, being dependent on the impact parameter l ¼ L=E for the equatorial motion [41,58]. The effective potential is symmetric under the transformation a → −a, L → −L; therefore, we will only study the Kerr-Newman braneworld spacetimes with nonnegative values of the spin parameter a. The effective potential has a discontinuity (divergence) at radii determined by the conditions

rð2a2 þ a2 r þ r3 Þ : a2

ð71Þ

Notice that the functions bs ðr; aÞ and bCV ðr; aÞ are equivalent—therefore, the divergence could occur just at the boundary of the causality violation region. In the limit of b → bS , the numerator of (67) reads r − ðL − jLjÞða2 þ r3 Þ: a

ð72Þ

Thus, if L ≥ 0, both the numerator and the denominator of (67) are zero and we have to use the L’Hôpital rule to obtain limb→bS V Eff ¼

r4 þa4 þ2a2 ðL2 þr2 ÞþL2 r2 2aLða2 þr2 Þ

L≥0

∞

L

rð2a2 þ a2 r þ L2 r þ r3 Þ : a2

ð74Þ

Therefore, the effective potential can be undefined for small values of r. However, this could happen only in the causality violation region, where the effective potential looses its relevance because of the modified meaning of the axial coordinate that has a timelike character in this region. C. Energy measured in LNRFs

ð70Þ

It is useful to determine, for particles on the circular geodesics, the locally measured energy related to some properly defined family of observers. The specific energy related to the LNRF (ELNRF ) is given by the projection of the 4-velocity on the timelike vector of the frame: dt ðtÞ ðtÞ ELNRF ¼ U ðtÞ ¼ Uμ eμ ¼ e dτ t pffiffiffiffiffiffiffiffiffiffiffi r2 a r − b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r4 þ a2 ðr2 þ 2r − bÞ pffiffiffiffi Δ × pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi : ð75Þ 2 r − 3r þ 2b 2a r − b

At the equatorial plane, the quantity g2φt − gφφ gtt ≡ Δ, and the condition (69) implies that the effective potential

The locally measured particle energy must always be positive for the particles in the positive-root states assumed

g2φt − gφφ gtt ¼ 0; 4

2

2

r þ a ðr þ 2r − bÞ ¼ 0:

ð69Þ

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here, while it is negative for the negative-root states that are physically irrelevant in the context of our study [57]. The LNRF energy of the particle following the circular geodesics diverges on the photon circular orbit as well as the covariant energy E. It also diverges for circular orbits approaching the boundary of the causality violation region given by Eq. (70). D. Future-oriented particle motion For the positive-root states, the time evolution vector has to be oriented to the future, i.e., dt=dτ > 0. On the other hand, the negative-root states have past-oriented time vectors, dt=dτ < 0, and thus are physically irrelevant for our study. To be sure that we are using the solutions related to the proper effective potential V eff with the correct upper sign, we have to ensure that the considered geodesics have the proper orientation, dt=dτ > 0. Using the metric (1) and relations for the specific energy (55) and specific angular momentum (56) in Eq. (48), we obtain the time component of the 4-velocity for both the upper and lower family circular geodesics in the form pffiffiffiffiffiffiffiffiffiffiffi dt r2 a r − b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : dτ r r2 − 3Mr þ 2b 2a Mr − b

ð76Þ

We see from this equation that the time component is always positive for the orbits of both the upper and the lower family, so we always have the positive-root states, and no mixing with the negative-root states occurs.

Applying conditions for the circular motion (51), we find that the equatorial photon circular orbits are given by the equations ½r2 − aðλ − aÞ2 − Δðλ − aÞ2 ¼ 0; 2

L : E

ð78Þ

ð80Þ

and the impact parameter λ is given by the equation λ ¼ −a

r2 þ 3r − 2b : r2 − 3r þ 2b

ð82Þ

Furthermore, Eq. (81) can by transformed into the form pffiffiffiffiffiffiffiffiffiffiffi ð83Þ r2 − 3r þ 2b 2a r − b ¼ 0; which implies the same reality condition on the radius of the photon orbit rph as the one that follows from Eqs. (55)–(57): rph ≥ b:

ð84Þ

Because of the reality condition, the numerator in Eq. (82) is positive, while the () in the denominator is determined by the sign in Eq. (83). Thus, we obtain corotating orbits (λ > 0) for the upper sign in (83), and counterrotating orbits (λ < 0) in the other case. The solution of Eq. (83) can be expressed in the form ð3r − r2 − 2bÞ pffiffiffiffiffiffiffiffiffiffiffi : 2 r−b

ð85Þ

For a given a and b, the points of a line a ¼ const crossing the function aph ðr; bÞ determine the radius, rph , of the photon circular orbits. We restrict our discussion to the solutions corresponding to a > 0, giving both corotating and counterrotating orbits. The zeros of the function aph ðr; bÞ are located at pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 rph ¼ ð3 9 − 8bÞ: 2

ð86Þ

Note that these solutions represent radii of photon circular orbits in the Reissner-Nordström spacetimes [15,16]. Since ∂aph ðr − 1Þð3r − 4bÞ ; ¼ ∂r 4ðr − bÞ3=2

ð77Þ

where the impact parameter λ is defined by the relation

ð79Þ

These two conditions imply that the radii of the circular photon orbits are determined by the equation pffiffiffiffi r2 − 3r þ 2a2 þ 2b 2a Δ ¼ 0; ð81Þ

a ¼ aph ðr; bÞ ≡

We first study motion of photons, as the photon circular orbits represent a natural boundary for the existence of circular geodesic motion [52,58]. The general photon motion in the braneworld KerrNewman black hole spacetimes was studied in [19]. Here, we concentrate on the equatorial photon motion and, especially, on the existence of the photon circular orbits. In the case of the equatorial photon orbits, the radial function RðrÞ is determined by Eq. (50) with the removed term −1 (with the rest energy m ¼ 0), which can be transformed into the form [19]

λ¼

2

2rðr þ a − aλÞ − ðr − 1Þðλ − aÞ ¼ 0:

IV. CIRCULAR GEODESICS OF PHOTONS

R ½r2 − aðλ − aÞ2 − Δðλ − aÞ2 ¼ ; 2 E r2 Δ

2

ð87Þ

the extrema of the curves aph ðr; bÞ are located at r ¼ 1 and at r ¼ 4b=3. The value of the function aph ðr; bÞ at the point r ¼ 1 reads (recall that we consider the positive values of the spin)

Notice that the photon orbits depend only on the impact parameter λ.

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aph-ex ðr ¼ 1; bÞ ¼

pffiffiffiffiffiffiffiffiffiffiffi 1 − b;

ð88Þ

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PHYSICAL REVIEW D 94, 086006 (2016) 2

TABLE I. All kinds of extrema of the function aph ðr; bÞ. We show the plus sign part only because of the symmetry corresponding to the interchangeability between signs () and the nature of the local minima (max/min). The function aphMin is the value of the function aph ðr; bÞ at the lowest possible r, which is r ¼ 0 for nonpositive values of b and r ¼ b otherwise.

V(0,0)

1.5

VI(2,2)

0.5

b

r 1 r1 , r2 4 3b aphMin

ð−∞; 0Þ ð0; 34Þ ð34 ; 1Þ max min −b

max min min ∞

b 1

IV(1,1)

VII(2,2)

1

bla

ð1; 98Þ ð98 ; ∞Þ

min min min min max max max ∞ 0 ∞

min ∞

aph ðr; bÞ pffiffiffiffiffiffiffiffiffiffiffi 1−b 0 pffiffi pbffiffi ð8b − 9Þ 3 3

ed s

ing ck h ularit ies ole s

II(1,3)

0

-0.5

-1

III(1,1)

nak

I(0,2)

IX(0,3)

VIII(0,2)

0

0.5

X(0,1)

1

1.5

a

corresponding to the extreme Kerr-Newman black holes, while, at r ¼ 4b=3, it reads pffiffiffi b aph-ex ðr ¼ 4b=3; bÞ ¼ pffiffiffi ð8b − 9Þ: 3 3

ð89Þ

The position, value, and kind of the extrema of the function aph ðr; bÞ are listed in Table I. The results are summarized in Fig. 4. We see that all curves drawn there— the curve aph-ex ðr ¼ 4b=3; bÞ, the line a2 þ b ¼ 1 (corresponding to the extremal black holes), the line a2 þ b ¼ 0, and the line b ¼ 1 (separating the braneworld KerrNewman naked singularities with the ergosphere from those without it)—divide the b-a plane into ten regions. In this sense, the braneworld Kerr-Newman spacetimes can be divided into ten different classes, characterized by (1) existence of the horizon, (2) existence of the ergosphere, and (3) the number of stable and unstable circular photon orbits. The situation is summarized in Table II and is also depicted in Fig. 4, in accord with the analysis of circular photon orbits in the standard Kerr-Newman spacetimes [52]. In the case of the braneworld Kerr-Newman black holes, the new regions VIII, IX, and X corresponding to the negative TABLE II. Ten possible divisions of braneworld Kerr-Newman spacetimes with respect to the existence of the horizon, the existence of the ergosphere, and the number of stable and unstable circular photon orbits. The first number in the column orbits corresponds to the amount of stable circular photon orbits, while the second corresponds to the amount of unstable circular photon orbits. Class I II III IV V

Horiz.

Ergo.

yes yes no no no

yes yes yes no no

Orbits 0, 1, 1, 1, 0,

2 3 1 1 0

Class

Horiz.

Ergo.

VI VII VIII IX X

no no yes yes no

yes no yes yes yes

Orbits 2, 2, 0, 0, 0,

2 2 2 3 1

FIG. 4. Braneworld Kerr-Newman black holes and naked singularities can be divided into ten distinguish classes according to the properties of the circular photon geodesics. Curve aph ð4b=3; bÞ (the solid line), given by (88), plays the main role in the classification. The corresponding regions of the b-a plane are denoted by I–X; the numbers in brackets denote the number of circular photon orbits in the respective classes. The first number determines the number of stable circular photon geodesics, while the second number determines the number of unstable circular photon geodesics.

values of the tidal charge, (b < 0), occur in addition to the standard Kerr-Newman spacetimes. V. STABLE CIRCULAR GEODESICS It is well known that the character of the test particle (geodesic) circular motion governs the structure of the Keplerian (geometrically thin) accretion disks orbiting a black hole [55,56] or a naked singularity (superspinar) [36,37]; similarly, it can also govern the motion of a satellite orbiting the black hole or the naked singularity (superspinar) along a quasicircular orbit slowly descending due to the gravitational radiation of the orbiting satellite [59]. The Keplerian accretion, starting at large distances from the attractor, is possible in the regions of the black hole or naked singularity spacetimes where local minima of the effective potential exist, and the energy corresponding to these minima decreases with decreasing angular momentum [12]. In other words, in terms of the radial profiles of the quantities characterizing circular geodesics, the Keplerian accretion is possible where both specific angular momentum and the specific energy of the circular geodesics decrease with a decreasing radius. In the standard model of the black hole accretion disks, the inner edge of the accretion disk is located in the so-called marginally stable circular geodesic where the effective potential has an inflection point [55], but the situation can be more complex in the naked singularity spacetimes [15,60]. We study the stability of the circular geodesic motion of the test particles relative to the radial perturbations in the braneworld Kerr-Newman spacetimes. Note that the

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equatorial circular motion is then always stable relative to the latitudinal perturbations perpendicular to the equatorial plane [61]. We show that the most interesting and, in fact, an unexpected result occurs for test particles orbiting the special class of the braneworld mining-unstable KerrNewman naked singularities, demonstrating an infinitely deep gravitational well enabling (formally) unlimited energy mining from the naked singularity spacetime. Of course, such a mining must be limited by a violation of the assumption of the test particle motion.

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amsðextrÞ ≡ ∓ ð2 b bð4b − 1ÞÞ:

ð94Þ

Thus, it can be shown that there is no solution for Eq. (92) related to the lower family of circular geodesics when b>

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 ∧ a < −2 b þ bð4b − 1Þ; 4

ð95Þ

and there is no solution for the upper sign family of circular geodesics when 1 > b > 1=4 and pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b − bð4b − 1Þ < a < 2 b þ bð4b − 1Þ:

A. Marginally stable circular geodesics The loci of the stable circular orbits are given by the condition related to the radial motion RðrÞ function ∂ 2 Rðr; a; b; E; LÞ ≤ 0; ∂r2

ð90Þ

∂ 2 V Eff ðr; a; b; LÞ ≤ 0; ∂r2

ð91Þ

or the relation

related to the effective potential V Eff ðrÞ, where the case of equality corresponds to the marginally stable circular orbits at rms with L ¼ Lms , corresponding to the inflection point of the effective potential—for lower values of the specific angular momentum L, the particle cannot follow a circular orbit. Such a marginally stable circular orbit represents the innermost stable circular orbit and the inner edge of the Keplerian disks in the Kerr black hole and in naked singularity spacetimes. Using the relations (55) and (56), we obtain for the braneworld Kerr-Newman spacetimes [28,62] rð6r − r2 − 9b þ 3a2 Þ þ 4bðb − a2 Þ ∓ 8aðr − bÞ3=2 ¼ 0:

ð96Þ

The existence of the marginally stable circular geodesics dependent on the dimensionless parameters of the braneworld Kerr-Newman spacetimes is represented in Fig. 5. This figure will be crucial for construction of the classification of the Kerr-Newman spacetimes according to the Keplerian accretion, but it is not sufficient, as the classification of the photon circular geodesics plays a crucial role, too. The function ams ðr; bÞ determines, in a given KerrNewman spacetime, the location of the marginally stable circular geodesics that are usually considered to be the boundaries of the Keplerian accretion disks determined by the quasigeodesic motion. B. Innermost stable circular geodesics The standard treatment when the inner edge of the Keplerian accretion disks is located at the marginally stable orbits defined by the inflection point of the effective potential [this point is also the innermost stable circular orbit (ISCO) [55]] works perfectly in the braneworld KerrNewman black hole spacetimes, but the situation is more complex in the braneworld naked singularity spacetimes, as the ISCOs do not always correspond to the marginally

ð92Þ

ams ¼∓

4ðr − bÞ3=2

3.0

ms. is not defined

2.0

ms. is not defined for upper family

b

In the previous studies, only the braneworld black hole spacetimes were typically considered [27,28]. Standard Kerr-Newman naked singularity spacetimes were discussed in [17,57]. Here, we consider the whole family of the braneworld Kerr-Newman spacetimes, with both positive and negative tidal charges. The solution of Eq. (92) can be expressed in the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3br2 − ð2 þ 4bÞr3 þ 3r4 ; 4b − 3r

1.0

0.0

black holes -1.0

ð93Þ 0.0

where ∓ corresponds to the upper and the lower family of the circular geodetics. corresponds to the two possible solutions of Eq. (92). The local extrema of the function ams ðr; bÞ are given by the relation

0.5

1.0

1.5

2.0

2.5

3.0

a

FIG. 5. Mapping of existence of the marginally stable circular geodesics in the parameter space of the braneworld Kerr-Newman spacetimes.

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stable orbit defined by Eq. (92) [64]. The ISCOs that are not coinciding with a marginally stable circular geodesic, related to an inflection point of the effective potential, correspond to the orbits with the lowest radius in the sequence of stable circular geodesics. Contrary to the case of the marginally stable circular orbits from which the particles can move inwards, in the case of ISCOs representing the inner limit of stable circular geodesics, the particle remains captured at this orbit or in its vicinity. Such ISCO orbits were found for the first time in the ReissnerNordstrom(–de Sitter) naked singularity spacetimes when they corresponded to orbits with vanishing angular momentum (particles at static positions) [15,16]. Here, we demonstrate the existence of a new class of this kind of ISCO representing the limit of stable circular geodesics located at the stable photon circular geodesic. In order to allow for the standard accretion with decreasing E’s and L’s with a decreasing radius of the stable orbits, we consider the stable circular geodesics with E → −∞ and L → −∞. The ISCO can be formally determined if we consider the function of the radius of the circular geodesic rc ðLc ; a; bÞ, given implicitly by Eq. (56), or rc ðEc ; a; bÞ, given implicitly by Eq. (55). Then the rISCO can be defined in a given spacetime, with fixed parameters a, b, by the relations drc =dLc ¼ 0, drc =dEc ¼ 0, which can be expressed as dLc =drc → −∞ and dEc =drc → −∞, related to the standard accretion with decreasing energy and angular momentum of accreting matter. Note that the conditions dLc =drc → ∞ and dEc =drc → ∞ can determine the outermost stable circular geodesics from which the accretion could start, but such a situation is not related to a plausible astrophysical situation, as discussed in detail in [60]. There are two relevant cases where the conditions dLc =drc → ∞ and dEc =drc → ∞ can be satisfied: ðaÞ

pffiffiffiffiffiffiffiffiffiffiffi 0 ¼ r2 − 3r þ 2b 2a r − b; ðbÞ r ¼ b:

located at r ¼ b, having L ¼ 0, while no marginally stable circular orbits corresponding to an inflection point of the effective potential exist. Note that, in some spacetimes, the sequence of the stable circular geodesics can start at the outermost stable circular geodesic with dLc =drc → þ∞ and dEc =drc → þ∞ corresponding to the stable circular geodesic. C. Effective potential dependence on specific angular momentum and analytical proof that the Keplerian accretion with infinite efficiency can exist in approximation to geodesic motion The Keplerian accretion works if a continuous sequence of local minima of the effective potential with decreasing values of angular momentum L exists. In terms of the effective potential (67), conditions for the existence of the Keplerian accretion disks can be expressed in the form ∂V Eff ðr; a; b; LÞ ¼ 0; ∂r ∂ 2 V Eff ðr; a; b; LÞ ≤ 0; ∂r2 ∂V Eff ðr; a; b; LÞ < 0: ∂L

Along with the possibility of stopping this procedure by the inflection point of the effective potential, there is another possible way to negate the validity of these conditions: ∂V Eff ðr; a; b; LÞ ¼ 0; ∂L

Case (a) tells us that the innermost circular geodesics correspond to the photon circular geodesics that can also be stable with respect to radial perturbations, so this condition is also applicable as a limit on the stable circular orbits of the test particles, as demonstrated in [15]. Then the specific energy and the specific angular momentum tend asymptotically to E → ∞ and L → ∞, but the impact parameter λ ¼ L=E remains finite. The condition r > b could restrict the condition implied by the photon circular geodesics. Here, we consider the case where dLc =drc → −∞ and dEc =drc → −∞. Case (b) can be relevant to the braneworld spacetimes with the positive braneworld parameter b, as demonstrated in [15], where the effective potential V Eff ðr; b; LÞ for the Reissner-Nordström naked singularity spacetimes clearly demonstrates that the inner edge of the Keplerian disk is

ð100Þ

which will be satisfied at a turning point where að2r − bÞ ffi: L ¼ LT ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r2 − 2r þ b

ð97Þ ð98Þ

ð99Þ

ð101Þ

At this turning point, the minimum of the effective potential, given by ∂V eff ðr; a; b; LT Þ=∂r ¼ 0, is located where r−b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 ⇒ r ¼ b: r2 r2 − 2r þ b

ð102Þ

In such situations, the inner edge of the Keplerian accretion disk is located at r ¼ b. Putting this result into the definition of the function LT , we find ab LT ðr ¼ bÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ⇒ b > 1: b bðb − 1Þ

ð103Þ

We see that the effect of the existence of the lowest possible value of angular momentum L associated with local minima of effective potential occurs only for values of

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the tidal charge b > 1 when Eq. (102) is well defined at r > b. The second possible way the conditions (99) are not well matched is related to the situation when the local minima of the effective potential turn into an inflection point at r defined by (92). Therefore, the inflection point of the effective potential is not always defined, if the tidal parameter b > 5=4, for the lower family solution of Eq. (92). In this case the Keplerian accretion stops at the point r ¼ b, where the minimum of the effective potential starts to increase in energy level with a decreasing L. On the other hand, for the upper family solution of Eq. (92), the situation becomes extraordinary and much more interesting for the tidal charge in the interval 1 > b > 1=4 and the appropriately tuned spin a; then the inflection point of the effective potential does not exist. For this reason, the Keplerian accretion starting at large values of the angular momentum of accreting matter cannot be stopped, and it continues with no limit to unlimitedly large negative values of the angular momentum and unlimitedly large negative values of the energy. Therefore, in this case, unrestricted mining of the energy due to the Keplerian accretion could occur. Of course, this mining has to be stopped—at least when the energy of the accreting matter starts to be comparable to the mass parameter of the Kerr-Newman naked singularity and the approximation of the test particle motion of matter in the disk is no longer valid. VI. CLASSIFICATION OF BRANEWORLD KERR-NEWMAN SPACETIMES ACCORDING TO RADIAL PROFILES OF CIRCULAR GEODESICS In order to create classifications of the braneworld KerrNewman black hole and naked singularity spacetimes according to possible regimes of the Keplerian accretion, we consider the existence of event horizons and the existence of the ergosphere, and we use the characteristics of the circular geodesics: existence of the circular photon geodesics and their stability, existence of the marginally stable circular geodesics related to inflection points of the effective potential, and relevance of the limiting radius r ¼ b. We use the classification of the braneworld KerrNewman spacetimes introduced for the characterization of the photon circular geodesics, and we generate a subdivision of the introduced classes according to the criteria related to the marginally stable orbits. The individual classes of the Kerr-Newman spacetimes will be represented by typical radial profiles of the specific angular momentum L, specific energy E, and effective potential V Eff that enable understanding of the Keplerian accretion and calculation of its efficiency. We first briefly summarize the results of two special cases—Kerr and Reissner-Nordström spacetimes. In the following

classification of the braneworld Kerr-Newman spacetimes, the characteristic types of behavior of the circular geodesics in the special Kerr and Reissner-Nordström spacetimes occur, but some quite new and extraordinary situations also arise. The results of the circular geodesic analysis in the braneworld Kerr-Newman spacetimes can also be directly applied to the circular geodesics in the standard KerrNewman spacetimes if we make the transformation b → Q2 , where Q2 represents the squared electric charge parameter of the Kerr-Newman background. A. Case b = 0: Kerr black hole and naked singularity spacetimes The limiting case of the well-known results of the test particle circular orbits in the Kerr spacetimes that were studied in detail in [35,43] demonstrates clearly the necessity of very carefully treating the families of circular orbits in the naked singularity spacetimes, where the simple decomposition of the circular orbits to corotating and counterrotating (retrograde) is not always possible. Namely, in the spacetimes with 1 < a < ac ¼ 1.3, the circular orbits that are corotating at large distances from the ring singularity become retrograde near the ring singularity, at the ergosphere; moreover, in the spacetimes with 1 < a < a0 ¼ 1.089, the covariant energy of such orbits can be negative. The specific energy and the specific angular momentum of the circular geodesics of the Kerr black hole and naked singularity spacetimes are illustrated in Fig. 8. Notice that the unstable circular geodesics approach the radius r ¼ 0 with unlimitedly increasing covariant energy and axial angular momentum; however, the photon circular geodesic cannot exist at the ring singularity. The Kerr naked singularities are classically unstable, as the Keplerian accretion from both the corotating and counterrotating disks inverts the naked singularity into an extreme Kerr black hole—the transition is discontinuous (continuous) for corotating (counterrotating) Keplerian disks [34–36,65]. We shall see later that the Keplerian accretion cannot be generally treated simply in the Kerr-Newman naked singularity spacetimes due to the complexities that are discussed in detail in [60]. We expect to address this issue in future work. B. Case a = 0: Reissner-Nordström black hole and naked singularity spacetimes The other limiting case of the Reissner-Nordström (RN) and Reissner-Nordtröm–(anti–)de Sitter black hole and naked singularity spacetimes was treated in [15,16]. It has been demonstrated that in the Reissner-Nordström naked singularity spacetimes that even two separated regions of circular geodesics could exist. The doubled regions of stable circular motion could occur in the RN naked singularity spacetimes with the charge parameter 1 < Q2 < 5=4—or, even, only stable circular geodesics

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Q2

could exist—if the charge parameter > 5=4. In the RN naked singularity spacetimes doubled photon circular geodesics can also occur, with the inner one being stable relative to radial perturbations, if the charge parameter is in the interval 1 < Q2 < 9=8 [15,17]. The same phenomena occur in the naked singularity Kehagias-Sfetsos spacetimes of the Hoˇrava quantum gravity [60,66] or in the no-horizon regular Bardeen or Ayon-Beato-Garcia spacetimes [67,68]. We shall see that, in the braneworld Kerr-Newman naked singularity spacetimes, the special naked singularity effects of the Kerr and Reissner-Nordström case are mixed in an extraordinary way, leading to the existence of an infinitely deep gravitational well and implying the new effect we call mining instability.

regions are governed by intersection points of the curves (88) and (94), which give 13 characteristic points in the parameter space that are summarized in the following way: the pairs ða; bÞ are ordered gradually from top to the bottom and from left to the right, ð1Þ → ð0; 1.25Þ; ð2Þ → ð0; 1.125Þ; A− ð12 þ A− Þ 3 pffiffiffi ð3Þ → ; ð12 þ A− Þ 32 16 2 ¼ ð0.0831; 1.1748Þ; 1 pffiffiffi ; 1 ¼ ð0.19245; 1Þ; ð4Þ → 3 3 pffiffiffi ð5Þ → ð2 − 3; 1Þ ¼ ð0.268; 1Þ; 3 ð6Þ → 0.5; ; 4 pffiffiffiffiffiffiffi ð7Þ → ð 0.5; 0.5Þ;

C. Characteristic points of the Kerr-Newman spacetime classification The classification of the braneworld Kerr-Newman spacetimes according to the character of the circular geodesics and the related effective potential are determined by the functions governing the local extrema of the functions giving the photon circular geodesics and the marginally stable circular geodesics corresponding to the inflection points of the effective potential. In the space of the spacetime parameters b-a, 14 regions then exist corresponding to classes of the braneworld Kerr-Newman spacetimes, demonstrating the different behaviors of the circular geodesics and the Keplerian accretion, as demonstrated in Figs. 6 and 7, giving details of the regions of low values of the dimensionless parameters a and b. These

ð8Þ → ð1; 0.25Þ; pffiffiffi ð9Þ → ð2 þ 3; 1Þ ¼ ð3.732; 1Þ; Aþ ð12 þ Aþ Þ 3 pffiffiffi ; ð12 þ Aþ Þ ð10Þ → 32 16 2 ¼ ð15.0992; 5.361Þ; where

6

Vc

Va

5

(10)

Vb

4

b

3

IVb

2

IVa (9)

1 (8)

IX

-1

IIIb

II

0

X

VIII

0

5

10

15

20

a

FIG. 6. Classification of the braneworld Kerr-Newman spacetimes according to the properties of circular geodesics relevant to the Keplerian accretion. The parameter space b-a is separated by curves governing the extrema pffiffiffiffiffiffi of the functions determining the photon circular orbits (the solid lines) and the marginally stable orbits (the dashed lines). Point ( 0.5, 0.5) is an intersection of the dashed line and the curve separating black holes from naked singularities (b ¼ 1 − a2 ). These two curves are tangent at the common point.

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EFFICIENCY OF THE KEPLERIAN ACCRETION IN … 1.3

Va (1)

Vb

1.2 Vc 1.125 1.1


(3)

(2)

IVa

IVb

VII

(4) VI

1

(5) IIIb b

b

0.9

IIIa

0.8 (6)

I

0.7 0.6 0.5 0.4

(7)

II 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

a FIG. 7. Classification of the braneworld Kerr-Newman spacetimes according to the properties of circular geodesics relevant to the Keplerian accretion. The parameter space b-a is separated by curves governing the extrema of the functions determining the photon circular orbits (the solid lines) and the marginally stable orbits (the dashed lines). Detailed structure for small values of spin a and b ∼ 1.

6

upper family

b=0

10

a=0 a=0.9 a=1 a=1.1

5 4

upper family

b=0

5

2

L

E

3 0

1 0

a=0 a=0.9 a=1 a=1.1

-5

-1 -2

1

2

3

4

5

6

7

8

9

10

-10

1

2

3

4

5

r 3

6

7

8

9

10

r

lower family

b=0

0

a=0 a=0.9 a=1 a=1.1

2.5

lower family

b=0 a=0 a=0.9 a=1 a=1.1

-2

2

L

E

-4 1.5

-6 1 -8

0.5 0

1

2

3

4

5

6

7

8

9

10

-10

r

FIG. 8.

1

2

3

4

5

6

r

E and L for Kerr black holes and naked singularities.

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pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi A ¼ ð9 þ 8 3 3ð83 þ 48 3ÞÞ:

ð104Þ

Point (6) is the crossing point of the function of the extrema of the photon circular orbit function aph-ex , and the curve separating the black hole and naked singularity spacetimes, b ¼ 1 − a2 . Point (7) is the common point of the dotted curve, given by the function amsðextrÞ and limiting the spacetimes, allowing for the existence of marginally stable orbits, and the curve separating the black holes from naked singularities (b ¼ 1 − a2 ). These two curves are tangent at the common point. D. Character of circular geodesics in the Kerr-Newman spacetimes The parameter space of the braneworld Kerr-Newman spacetimes b-a is divided into 14 regions due to the criteria reflecting basic properties of the spacetimes and properties of their circular geodesics: (i) Existence of event horizons and an ergosphere. (ii) Existence of unstable and stable circular photon geodesics. (iii) Existence of the marginally stable geodesics or the ISCO. The classification is summarized in Table III. Basically, we combine Figs. 5 and 4 to obtain Figs. 6 and 7, where properties of the photon circular geodesics and properties of the marginally stable geodesics or ISCOs are reflected. We will show that the most surprising properties of the Keplerian accretion arise in the spacetimes of class IIIa. TABLE III. Classification of parameter space b-a with respect to ISCO—radius of the innermost stable circular orbit; MSO(u) —radius of the marginally stable orbit for the upper sign family; MSO(l)—radius of the marginally stable orbit for the lower sign family; SP—number of stable photon circular orbits; UP— number of unstable photon circular orbits. ISCO has only two possible outcomes. It can either be identical to the MSO or lie at r ¼ b. The word “classic” in this context means that MSO is defined by Eq. (92). Class

ISCO

MSO(u)

MSO(l)

Hor./Erg.

SP

UP

I II IIIa IIIb IVa IVb Va Vb Vc VI VII VIII IX X

¼ MSO ¼ MSO ¼ Photon ¼ MSO at r ¼ b at r ¼ b at r ¼ b at r ¼ b at r ¼ b ¼ MSO at r ¼ b ¼ MSO ¼ MSO ¼ MSO

classic classic classic classic classic classic classic classic classic classic

classic classic classic classic classic classic classic classic classic classic classic classic classic

yes/yes yes/yes no/yes no/yes no/no no/no no/no no/no no/no no/yes no/no yes/yes yes/yes no/yes

0 1 1 1 1 1 0 0 0 2 2 0 0 0

2 3 1 1 1 1 0 0 0 2 2 2 3 1

Now we give properties of the circular geodesics in all 14 classes of the braneworld Kerr-Newman spacetimes, presenting and discussing the typical radial profiles of their specific energy and specific angular momentum, complemented by sequences of the effective potential. Classification of the standard Kerr-Newman spacetimes according to properties of the circular geodesics contains all of the classes except those related to b < 0; therefore, classes VIII, IX, and X are excluded. 1. Class I This class (Fig. 9) of black hole spacetimes has two horizons, two unstable photon circular orbits, and an ergosphere. The class border is given by the line b ¼ aph-ex ðr ¼ 4b=3; bÞ, b ¼ 1 − a2 , with the intersection at point (0.5; 3=4) [point (6) in Fig. 6] and line a ¼ 0. Marginally stable orbits for test massive particles are given by the inflection point of the effective potential are defined by Eq. (92) and coincide with the ISCOs (this is the standard scenario of the Keplerian accretion: for short, classic). 2. Class II This class (Fig. 10) of black hole spacetimes has two horizons, one stable and three unstable photon circular orbits, and an ergosphere. Notice that the stable and unstable photon circular geodesics are located under the inner horizon and thus are irrelevant for the Keplerian accretion. The border is given by the line b ¼ aph-ex ðr ¼ 4b=3; bÞ and b ¼ 1 − a2 , with the intersection being at point (0.5; 3=4) [point (6) in Fig. 6] and line b ¼ 0. Marginally stable orbits for test massive particles are given by the inflection point of the effective potential, coinciding with the ISCOs (classic). 3. Class IIIa This class (Fig. 11) of naked singularity spacetimes has one stable and one unstable photon circular geodesic and an ergosphere. The border of the class IIIa region is given by the lines b ¼ amsðextrÞ and b ¼ 1, with intersection points at pffiffiffi pffiffiffi ð2 − 3; 1Þ ¼ ð0.268; 1Þ [point (5)] and ð2 þ 3; 1Þ ¼ ð3.732; 1Þ [point We have also marked point (7) with pffiffiffiffiffiffi(9)]. ffi coordinates ð 0.5; 0.5Þ, where the lines b ¼ amsðextrÞ and b ¼ 1 − a2 touch and are tangent to each other. This theoretically means that an effect of mining instability can be achieved for extremal pffiffiffiffiffiffiffiKerr-Newman black holes with spin parameter a ¼ 0.5 and the charge or braneworld tidal charge parameter b ¼ 0.5. However, it occurs under the event horizon. We have also marked point (8) with coordinates (1, 0.25), giving information on the minimal amount of the electric charge or braneworld tidal charge parameter b.

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~ ≪ M; jEj

ð105Þ

and the covariant energy of the particle (accreting matter) has to be much smaller than the naked singularity mass parameter M. Of course, the issue of the mining instability Class: I


40

6

This class (Fig. 15) of naked singularity spacetimes has one stable and one unstable photon circular orbit. The class is without an ergosphere. The border of this class in the spacetime parameter space is given by the lines a=0.2, b=0.5 upper sign lower sign

2

VEff (r,a,b,L)

1

Class: I

a=0.2, b=0.5 L=2.95432 L=4 L=6 L=8

1.8 1.6

Outer horizon

2

Inner horizon

E

Outer horizon

Inner horizon

L

5. Class IVa

4

0

1.4 1.2 1 0.8

-1

-40 0.1

This class (Fig. 14) of naked singularity spacetimes has one stable and one unstable photon circular orbit and an ergosphere. In the parameter space b-a, the area related to this class is not compact and disintegrates into two separated areas. The first area is infinitely large: its border is given by lines b ¼ amsðextrÞ , b ¼ 1 − a2 , b ¼ 1, and pffiffiffiffiffiffiffi b ¼ 0, with at the point ð 0.5; 0.5Þ [point (7)] pffiffiintersections ffi and ð2 þ 3; 1Þ [point (9)]. The second area is compact and finite. Its border is given by lines b ¼ amsðextrÞ , b ¼ aph-ex ðr ¼ 4b=3; bÞ, b ¼ 1 − a2 , and b ¼ 1, with intersection points (4), (5), and (7). It is not obvious from the figures, but b ¼ amsðextrÞ and b ¼ aph-ex ðr ¼ 4b=3; bÞ do not intersect. Marginally stable orbits of both the lower and the upper family of circular geodesics are given by the inflection point of the effective potential and coincide with the ISCOs (classic). Notice that in this case the sequence of the upper family orbits with descending specific energy E and specific angular momentum L is interrupted by a sequence where both E and L increase with a decreasing radius, thus corresponding to the unstable geodesics. In this case, the infinitely deep gravitational well still exists, but the Keplerian accretion sequence is interrupted and this gravitational well cannot be applied in an astrophysically natural accretion process. Nevertheless, it is still possible to use this gravitational well, if matter with appropriate initial conditions (values of the motion constants), enabling the start of the mining instability, could appear close to the naked singularity.

Class: I

3

-20 -2

4. Class IIIb

5

20 0

and the related interaction of the mining-unstable KerrNewman naked singularity (the Kerr-Newman superspinar) and the accreting mass is much more complex and deserves a more detailed study.

Horizon

The marginally stable orbit of the massive test particles is in the case of the lower family circular geodesics given by the inflection point of the effective potential, and it coincides with the ISCO (classic). In the case of the upper family circular geodesics, the inflection point of the effective potential is not defined and the sequence of minima of the effective potential continues with decreasing specific energy and specific angular momentum of the accreting matter down to the stable photon orbit. This orbit can therefore be considered an ISCO of the massive test particles. We have thus found an infinitely deep gravitational well enabling, theoretically, an unlimited mining of energy from the naked singularity. In fact, such a mining instability could work only up to the energy contained in the naked singularity spacetime. We can expect that the energy mining could also work in more realistic situations where the naked singularity is removed and an astrophysically more plausible superspinar is created by joining a regular (e.g., stringy) solution to the Kerr-Newman spacetime at a radius overcoming the outer radius of the causality violation region [37,40]. The mining could work if the matching radius of the internal stringy spacetime and the outer Kerr-Newman spacetime is smaller than the radius of the stable photon orbit related to the mining instability of the class IIIa spacetimes. For completeness, we also give in this case the locally measured (LNRF) specific energy of the upper family circular geodesics. As shown in Fig. 12, the specific energy ELNRF diverges, along with the covariant specific energy E as the orbit approaches the limiting photon circular orbit. On the other hand, Fig. 13 clearly demonstrates that the ratio jEj=ELNRF remains finite while the orbits approach the location corresponding to the stable photon circular orbit. Of course, the mining instability could work only if the assumption of the test particle motion of the accreting matter is satisfied. Therefore, the assumption requires validity of the relation

0.6 1

r

10

100

-2 0.1

1

10

100

r

FIG. 9.

L, E, and effective potential for class I.

086006-17

1

2.48

r

13.32

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MARTIN BLASCHKE and ZDENĚK STUCHLÍK Class: II


40

6


Class: II


5

Class: II

a=0.8, b=0.1 L=-0.216085 L=2.25431 L=0 L=4 L=12 L=-0.23

0.1

Outer horizon

1

Inner horizon

2

1

VEff (r,a,b,L)

-20

E

Inner horizon

0

Outer horizon

3

Outer horizon

Inner horizon

4

20

L

10

0 -1

-40 0.1

1

r

10

-2 0.1

100

1

10

r

0.01

100

0.17

1.77

13.91

r

141.44

FIG. 10. L, E, and effective potential for class II. Class: IIIa


40

6

Class: IIIa


5

VEff (r,a,b,L)

E

L

2 1

-20 -40 0.1

1

10

r

100

Class: IIIa

0

-2 0.1

1

10

r

2

ELNRF

1.6

1.4

1.2

1

1

0.8 2.5

3

3.5

4

Class: IIIa

upper sign, a=1 b=0.25 b=0.5 b=0.9

1.4

1.2

2

98.6

This class (Fig. 16) of naked singularity spacetimes has one stable and one unstable photon circular orbit. These spacetimes are without an ergosphere. In the parameter space b-a, this class is not compact and disintegrates into two separated areas. The first area is infinitely large: the border is given by the lines b ¼ aph ð4b=3; bÞ, b ¼ amsðextrÞ ,

1.6

1.5

9.3

r

6. Class IVb

upper sign, b=0.7

1

0.804

Note that the probability that we are actually living in a spacetime with the braneworld tidal charge parameter greater than one is very small [26,27].

1.8

0.5

-2 0.1

100

L, E, and effective potential for class IIIa.

a=0.6 a=0.8 a=1.2

0

1

-1

1.8

0.8

2

-1

b ¼ aph-ex ðr ¼ 4b=3; bÞ and b ¼ amsðextrÞ , with intersection points (3), (5), (9), and (10). For the lower family circular geodesics, the marginally stable orbits defined by the inflection point of the effective potential occur (classic). For the upper family, the circular geodesics’ marginally stable orbit is not defined, and the ISCO is located at r ¼ b, as it is for all classes with b > 1. A sequence of stable circular geodesics with sharply increasing specific energy occurs near (slightly above) the radius r ¼ b, approaching the stable photon circular orbit. (Such sequences of stable circular orbits are discussed in [60].) 2

3

0

FIG. 11.

a=1.5, b=0.8 L=10 L=3 L=0 L=-10 L=-20

4

3 0

Class: IIIa

5

4

20

ELNRF

6

0

0.5

1

1.5

2

2.5

3

3.5

4

r

r

FIG. 12. Energy measured by the LNRF observers (upper sign family orbits only) in the mining-unstable Kerr-Newman spacetimes of class IIIa. The energy diverges at the radius of the stable photon circular orbit.

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EFFICIENCY OF THE KEPLERIAN ACCRETION IN … 0.8

Class: IIIa


upper sign, b=0.7 a=0.6 a=0.8 a=1.2

0.7

2

0.6

Class: IIIa

upper sign, a=1 b=0.25 b=0.5 b=0.9

1.5

|E|/ELNRF

|E|/ ELNRF

0.5 0.4

1

0.3 0.2

0.5

0.1 0 0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0

1.5

0

0.2

0.4

0.6

0.8

r

1

1.2

1.4

r

FIG. 13. Absolute value of covariant energy jEj divided by LNRF energy (for the upper sign family orbits) in the mining-unstable Kerr-Newman spacetimes of class IIIa. The fraction is defined down to the radius of the stable circular photon orbit, and at this point it has a finite value.

8. Class Vb

and b ¼ 1, with intersection points (9) and (10). The second area is finite and its border is given by the same lines and intersection points (2), (3), (4), and (5). For both the upper and lower families of the circular geodesics, the marginally stable orbits of the test massive particles are given by the inflection point of the effective potential, governing the sequence of geodesics related to the standard Keplerian accretion (classic). There is an additional internal sequence for a stable circular geodesic, with the ISCO located at r ¼ b, as it is for all classes with b > 1. This sequence approaches the stable photon circular geodesic at the outer edge.

This class (Fig. 18) of naked singularity spacetimes has no stable or unstable photon circular orbits. These spacetimes are also without an ergosphere. The border of the class Vb region in the parameter space is given by the lines b ¼ amsðextrÞ and b ¼ aph ð4b=3; bÞ, with intersection points (1), (3), and (10). The class is infinitely extended in the parameter space. The upper family circular geodesics are stable only, finishing at the ISCO located at r ¼ b. The marginally stable orbit exists for the lower family orbits, giving the limit of the standard Keplerian accretion. The lower family orbits continue downwards by a sequence of unstable orbits and, finally, stable orbits finishing at r ¼ b.

7. Class Va This class (Fig. 17) of naked singularity spacetimes has no stable or unstable photon circular orbits. These spacetimes are also without an ergosphere. The border of the class Va region in the parameter space is given by the lines b ¼ amsðextrÞ and a ¼ 0, with intersection point (1). For both the lower and upper family circular geodesics, the marginally stable orbits are not defined. The circular geodesics are only stable and the ISCOs are located at r ¼ b, as they are for all spacetime classes with b > 1. Class: IIIb


40

6

9. Class Vc This class (Fig. 19) of naked singularity spacetimes has no stable or unstable photon circular orbits. These are again spacetimes without an ergosphere. In the parameter space b-a, this class disintegrates into two separated areas. The first one is infinitely extended: its border is given by the lines b ¼ aph-ex ðr ¼ 4b=3; bÞ and b ¼ amsðextrÞ , with

Class: IIIb


5

VEff (r,a,b,L)

E

L

2 1

-20 -40 1

r

10

100

3 2 1

0

0

-1

-1

-2 0.1

FIG. 14.

a=3.5, b=0.5 L=20 L=10 L=3 L=1.34083 L=-10 L=-20

4

3 0

Class: IIIb

5

4

20

0.1

6

1

r

10

100

-2 0.1

L, E, and effective potential for class IIIb.

086006-19

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10.6

r

99.3

398.5 1000

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MARTIN BLASCHKE and ZDENĚK STUCHLÍK Class: IVa


40

6


Class: IVa


5

6

a=2, b=1.5 L=10 L=5 L=0 L=-5 L=-10

5

VEff (r,a,b,L)

4

20

E

3

L

Class: IVa

0

2 1

-20

4 3 2

0 1

-1

-40 0.1

1

r

10

-2 0.1

100

1

r

10

0 0.1

100

1.5

r

4.2

25.6

99.3

FIG. 15. L, E, and effective potential for class IVa.

Class: IVb


40

6

a=10, b=2

2


5

VEff (r,a,b,L)

4

E

0

3 2

-20

a=10, b=2 L=15 L=10 L=4.69702 L=0 L=-7.07107 L=-12

1.4 1.2 1 0.8

1 -40

0.6 1

10

0

100

r

1

10

Class: Va


1.3

2

VEff (r,a,b,L)

E

1.1 0

1 0.9

-20

0.8

0.1

1

r

10

0.6

100

10

100

Class: Va

a=2, b=3 L=2 L=-0.816497 L=-5 L=-8 L=-12 L=20

1.6 1.4 1.2 1 0.8

0.7

-40

r

1.8

1.2

20

2

L, E, and effective potential for class IVb.

1.4 Class: Va


40

0.73 1

100

r

FIG. 16.

L

Class: IVb

1.8 1.6

20

L

Class: IVb

0.6 1

10

1

100

r

3

7.5

22.2

r

65.5 159.4 400.6 1000

FIG. 17. L, E, and effective potential for class Va.

Class: Vb


40

6

Class: Vb


5

6 5

VEff (r,a,b,L)

E

3

L

a=2, b=2.1 L=10 L=5 L=-1.3159 L=-5 L=-10

4

20 0

2 1

-20

4 3 2

0

1

-1

-40 0.1

Class: Vb

1

r

10

100

-2 0.1

FIG. 18.

1

r

10

100

0

L, E, and effective potential for class Vb.

086006-20

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4.2

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25.6

99.3

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EFFICIENCY OF THE KEPLERIAN ACCRETION IN … Class: Vc


40

2


1.6

a=0.01, b=1.15 L=3 L=2.6462 L=2 L=-0.02408 L=-1

1.4

VEff (r,a,b,L)

E

0

1

-20

0.5

1.2 1 0.8 0.6

-40

0.4

0.1

1

r

10

0 0.1

100

FIG. 19.

Class: VI


40

6

1

r

10

100

0.1

Class: VI


2

VEff (r,a,b,L)

E

3 2 1 -20

1

2.2

r

6.5 10

Class: VI

100

a=0.15, b=0.99 L=5 L=2 L=0 L=-2 L=-5

1.5

4

0

0.31

L, E, and effective potential for class Vc.

5

20

L

Class: Vc

1.5

20

L


Class: Vc

1 0.5

0

0 -1

-40 0.1

1

r

10

100

-2 0.01

0.1

1

10

r

-0.5 0.3

100

1

1.78 2.56

10

r

FIG. 20. L, E, and effective potential for class VI.

Class: VII


40

6

Class: VII


5

VEff (r,a,b,L)

E

3 0

2 1

-20

Class: VII

a=0.15, b=1.01 L=5 L=2 L=0 L=-2 L=-5

1.5

4

20

L

2

1 0.5

0

0 -1 1

r

10

100

-2 0.1

FIG. 21.

Class: VIII


40

6

1

100

-0.5 0.3

Class: VIII


6

1

2 1

-40

r

10

100

a=0.5, b=-1 L=-10 L=0 L=3.44726 L=10 L=20

2 1 0

-1

-1

-2 0.1

Class: VIII

3

0

FIG. 22.

10

r

4

VEff (r,a,b,L)

E

Outer horizon

-20

1.78 2.56

5

4

0

1

L, E, and effective potential for class VII.

3

L

10

5

20

0.1

r

Outer horizon

0.1

1

r

10

100

-2 0.1

L, E, and effective potential for class VIII.

086006-21

Horizon

-40

1

4.34

r

95.52

396.1 1000

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MARTIN BLASCHKE and ZDENĚK STUCHLÍK 40

6


Class: IX


5

0.01

0.1

1

10

r

3 2 1

0

0

-1

-1

-2 0.01

100

VEff (r,a,b,L)

2 1

-40

Outer horizon

Inner horizon

E

Outer horizon

Inner horizon

-20

a=0.9, b=-0.3 L=-10 L=0 L=2.29179 L=4 L=10 L=18

4

3 0

Class: IX

5

4

20

L

6

0.1

1

10

r

-2 0.01

100

Outer horizon


Inner horizon

Class: IX

0.053 0.17

1 2.05

r

13.55

97.1 320.4 1000

FIG. 23. L, E, and effective potential for class IX.

spacetimes has the boundary given by the lines b ¼ aph-ex ðr ¼ 4b=3; bÞ, a ¼ 0, and b ¼ 1, with intersection points (2) and (4). For both the lower and upper family circular geodesics, the marginally stable orbit exists, thus giving the edge of the standard Keplerian accretion. Furthermore, both the lower and upper family orbits have the ISCO at r ¼ b, where the sequence of stable orbits finishes, starting for each family at the related photon circular geodesic.

intersection point (10). The second area is finite and its border is given by lines b ¼ aph-ex ðr ¼ 4b=3; bÞ, b ¼ amsðextrÞ , and a ¼ 0, with intersection points (1), (2), and (3). The marginally stable orbit exists for both the lower and upper family circular geodesics, thus giving the standard limit of the Keplerian accretion (classic). Both the lower and upper family orbits continue downwards by a sequence of unstable orbits and, finally, stable orbits finishing at r ¼ b.

12. Class VIII

10. Class VI

This class (Fig. 22) of black hole spacetimes has a negative braneworld tidal charge parameter b—which has only one horizon, located at r > 0—two unstable photon circular orbits, and an ergosphere. In the parameter space b-a, the boundary of the region related to this class is given by the lines b ¼ −a2 and a ¼ 0. For both the lower and upper family circular geodesics, the marginally stable orbit exists, determining the inner edge of the standard Keplerian disk. Thus, we obtained the standard situation typical for Kerr black holes, but no geodesic structure occurs at r > 0 under the event horizon.

This class (Fig. 20) of naked singularity spacetimes has two stable and two unstable photon circular orbits and an ergosphere. In the parameter space, the area of the class VI spacetimes has the boundary given by the lines b ¼ aph-ex ðr ¼ 4b=3; bÞ, b ¼ 1 − a2 , and b ¼ 1, with intersection points (4) and (6). For both the lower and the upper family of circular geodesics, the marginally stable orbit exists, thus giving the inner edge of the standard Keplerian accretion. The upper family orbits have also a very narrow region of stable circular orbits near the radius r ¼ b, starting with the stable circular photon orbit.

13. Class IX This class (Fig. 23) of black hole spacetimes has a negative braneworld parameter b with two horizons, three unstable photon circular orbits, and an ergosphere. The border of the related region of the spacetime parameter space is given by the lines b ¼ 1 − a2 , b ¼ −a2 , and b ¼ 0.

11. Class VII This class (Fig. 21) of naked singularity spacetimes has two stable and two unstable photon circular orbits, with no ergosphere. In the parameter space, the area of the class VII Class: X

a=2, b=-0.01

6


40

Class: X

a=2, b=-0.01

VEff (r,a,b,L)

E

2 1

-20

a=2, b=-0.01 L=-4 L=-2 L=0.647934 L=2 L=3

1.6

3 0

Class: X

1.8

4

20

L

2


5

1.4 1.2 1

0 0.8 -1

-40 0.1

0.6 1

r

10

100

-2 0.01

0.1

1

r

10

100

FIG. 24. L, E, and effective potential for class X.

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For both the lower and upper circular orbits, the marginally stable orbit exists, giving in a standard way the inner edge of the Keplerian accretion. Unstable orbits exist under the inner horizon. 14. Class X This class (Fig. 24) of naked singularity spacetimes has a negative brane parameter b with one unstable photon circular orbit and an ergosphere. The border of the related region of the parameter space is given by the lines b ¼ 1 − a2 and b ¼ 0. In these naked singularity spacetimes, the marginally stable orbit exists for both the lower and the upper family of circular geodesics, thus representing in both cases the inner edge of the Keplerian accretion disks. Under the marginally stable orbits, only unstable orbits exist for both families. From the point of view of the geodesic structure, the naked singularity spacetimes of class X resemble the standard Kerr naked singularity spacetimes. VII. EFFICIENCY OF THE KEPLERIAN ACCRETION Now we are able to determine the energetic efficiency of the Keplerian accretion. From the astrophysical point of view, the standard Keplerian accretion is relevant in the regions enabling the start of accretion at large distance (infinity) and its completion at the first inner edge that can be approached by a continuous accretion process. We determine the efficiency of the Keplerian accretion for all classes of the braneworld Kerr-Newman spacetimes for the standard Keplerian accretion. In some of these spacetimes, an inner region also exists where the Keplerian accretion could work due to the decline of both energy and angular momentum with a decreasing radius. However, these regions are not related to the standard notion of Keplerian accretion and will not be considered here for calculations of the accretion efficiency. Moreover, complexities of the Keplerian accretion process related to the behavior of the angular velocity could also exist. These complexities are described in detail in [60]—we shall not discuss these subtleties in this paper. We concentrate our attention on determining the efficiency for the Keplerian accretion following the upper family circular geodesics, where the efficiency can be very high, being in some cases even unlimitedly high (formally). In the case of the upper family Keplerian accretion, the efficiency is discontinuous when the transition between the naked singularity with sufficiently high dimensionless spin and the related extreme black hole state is considered. The critical values of the spin and the related critical tidal charge read 1 acr ¼ pffiffiffi ; 2

1 bcr ¼ : 2

ð106Þ

We must stress that the efficiency of the Keplerian accretion in the near-extreme naked singularity spacetimes exceeds significantly the efficiency in the extreme black hole spacetimes. On the other hand, the efficiency of the Keplerian accretion in the upper family regime is fully continuous in the case of the transition of the naked singularity to an extreme black hole spacetime with sufficiently low spin, a < acr , and for all the braneworld Kerr-Newman (KN) spacetimes in the case of the Keplerian accretion in the lower family accretion regime. Generally, the efficiency of the Keplerian accretion is substantially smaller in comparison to the upper family regime in a given Kerr-Newman spacetime. The efficiency of the accretion for the geometrically thin Keplerian disks governed by the circular geodesics is defined by the relation ηða; bÞ ¼ 1 − Eðredge ; a; bÞ;

ð107Þ

where redge denotes the location of the inner edge of the standard Keplerian accretion disks. For the Keplerian disks following the lower family circular geodesics, the inner edge of the disk is always located at the marginally stable geodesic, thus giving always the scenario of the Keplerian accretion in the Kerr spacetimes. On the other hand, for the upper family Keplerian disks, the situation is more complex, as follows from the classification of the braneworld Kerr-Newman spacetimes. Three qualitatively different cases can occur that depend on combinations of the dimensionless spacetime parameters a and b. In the first family of classes of the Kerr-Newman spacetimes, the redge is simply located at the marginally stable geodesic, thus giving the scenario of the Keplerian accretion onto Kerr black holes—this case includes all of the braneworld Kerr-Newman black hole spacetimes. In the second family of the Kerr-Newman classes, the inner edge of the Keplerian disk is located at the radius r ¼ b, thus giving the special case first discovered for the Reissner-Nordström naked singularity spacetimes [15,16]. In all classes with b > 1 (IV, V), the efficiency of the Keplerian accretion along the upper family of circular geodesics is independent of the spin parameter a being defined by the simple relation [69] rffiffiffiffiffiffiffiffiffiffiffi 1 ηðbÞ ¼ 1 − 1 − : b

ð108Þ

The efficiency goes slowly to 0% for b → ∞; see Fig. 25. In the third and most interesting family of the KerrNewman spacetime classes, redge corresponds to the radius of the stable photon orbit approached by particles with the specific energy E → −∞ and the specific angular momentum L → −∞. Notice that the limiting photon circular geodesic is a corotating one, as the impact parameter

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∞

η (a,b)

η ∞ 1.5 1 0.5 0

1 0.8 0

0.6 0.5

1

0.4

1.5

2

2.5

a

b

0.2

3

3.5

4 0

1.8

1.7071

1.6

1.5345

1.5222

1.6013

1.5774

a=3 4

1.4

0.3

0.1

10

a=√3

1.2

a=√2

0.4

η (a,b)

0.8

1

∞

η

1.0

0.6

0.8 a=

2

0.6 0.4778

0.4655

0.4227 0.3987 0.2929

0.2 0.0 -2.5

a=1

a=3.732

-2

-1.5

-1

-0.5

a= 0.

9

0.4

0

√ a=

2/2 a=0.268 a=0

0.5

0.134 0.081

1

1.5

2

b

FIG. 25. The energetic efficiency ηða; bÞ of the Keplerian accretion following the upper family circular geodesics is given depending on the spacetime dimensionless tidal charge parameter b and the spin parameter a. The 3D diagram reflects the position of the special class of the mining-unstable Kerr-Newman spacetimes of class IIIa in the plane of the spacetime parameters. Because of the complex character of the efficiency function, we also give the characteristic a ¼ const sections in the η-b plane.

λ ¼ L=E > 0. In the third case, the Keplerian accretion efficiency (theoretically) approaches infinity. This effect occurs explicitly in the class IIIa Kerr-Newman spacetimes, as clearly demonstrated in Fig. 11. For this class, we have the tidal charge parameter b ∈ ð1=4; dimensionffi 1Þpand pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi the less spin a ∈ ð2 b − bð4b − 1Þ, 2 b þ bð4b − 1ÞÞ. The Keplerian accretion efficiency is given for the upper family of circular geodesics in Fig. 25, and for the lower family circular orbits in Fig. 26. Because of its complexity, we represent the case of the upper family accretion regime by a 3D figure, with the addition of a figure representing the

relevant sections a ¼ const. In the case of the lower family accretion regime, the representative a ¼ const sections are sufficient to clearly demonstrate the character of the efficiency of the Keplerian accretion. For the Keplerian accretion along the lower family circular geodesics, the situation is quite simple and the efficiency is always continuously matched between the naked singularity and the extreme black hole states. The efficiency of the lower family regime accretion for fixed dimensionless spin a of the braneworld KerrNewman spacetimes always decreases with a decreasing

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0.08

0.07

a=0.000 a=1.000 a=0.707 a=1.730 a=1.411 a=2.000

(a,b)

0.06

0.05

0.04

0.03 -4.0

-3.0

-2.0

-1.0

0.0

1.0

b

FIG. 26. Energy efficiency of the Keplerian accretion following the lower family circular geodesics is given depending on the spacetime dimensionless tidal charge parameter b for characteristic values of the spin parameter a ¼ const.

tidal charge parameter b. Moreover, for a fixed tidal charge b, the efficiency decreases with an increasing spin a. For understanding the upper family Keplerian accretion regime and its efficiency, the dependences ηðb; a ¼ constÞ are most instructive. They are governed by two crucial families of curves. First, the efficiency of the Keplerian accretion in the extreme braneworld KN black hole spacetimes and the related near-extreme braneworld KN naked singularity spacetimes are given by the relation 1 ηjump ðbÞ ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 4 − 1−b

ð109Þ

where the plus sign corresponds to the efficiency in the near-extreme naked singularity spacetimes, while the minus sign corresponds to the related extreme black holes. Naturally, this formula is relevant in the interval of the tidal charge b ∈ ð−∞; 0.5Þ, i.e., up to the critical value of the tidal charge. Second, a crucial curve is given by the efficiency of the accretion in the limiting spacetimes governed by the boundary of class IIIa spacetimes, ηmining ðb; amining ðbÞÞ, where b ∈ ð1=4; 1Þ, and pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amining ðbÞ ¼ 2 b bð4b − 1Þ. The results can be summarized in the following way. For all of the braneworld spacetimes with the negative tidal charge parameter b < 0, and for those with positive charge parameter 0 ≤ b ≤ 1=2, the large jump in efficiency in the transition between the naked singularity and the related extreme black hole state occurs. Such a jump was observed for the first time in the case of the transition between the Kerr naked singularity and the extreme Kerr black hole (b ¼ 0), where η ∼ 1.57 goes down to η ∼ 0.43 [35]. For the braneworld KN extreme black holes (the related near-extreme naked singularities), the efficiency slightly increases

(decreases) with negatively valued tidal charge increasing in its magnitude, so the efficiency jump slightly decreases from its maximal Kerr value. On the other hand, for b ∈ ð0; 1=2Þ, the efficiency for the extreme black holes (the near-extreme naked singularities) decreases (increases), and the jump quickly increases—for b ¼ 1=2, the efficiency jumps from η ∼ 1.707 down to η ∼ 0.293. For the naked singularities with the tidal charge in the interval 1=4 < bp< and the dimensionless spin in the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffi 1 p interval a ∈ ð2 b − bð4b − 1Þ, 2 b þ bð4b − 1ÞÞ, the formally defined efficiency of the Keplerian accretion is unlimited. At the boundary of this region, the efficiency is given by the limit governed by the regular Keplerian accretion finished at the marginally stable orbit. For the naked singularities with b < 1=2, the efficiency of the Keplerian accretion, starting at the near-extreme state and keeping spin a ¼ const, decreases with an increasing tidal charge down to the curve ηmining ðb; amining ðbÞÞ. A further increase of b causes an entrance to the region of unlimited efficiency. The curve ηðb; a ¼ 1Þ starts at b ¼ 0 and finishes at b ¼ 1=4, giving the related efficiency η ¼ 1. pffiffiffi For a spin in the interval 1 > a > 1= 2, the curves ηðb; a ¼ 1Þ start at the extreme state and finish at the state with 0 < b < p 1=2 ffiffiffi and efficiency η > 1. For a spin approaching a ¼ 1= 2, the curve ηðb; a ¼ constÞ degenerates at the point with b ¼ 1=2, and efficiency approaches pffiffiffi η ∼ 1.707. For higher values of spin, a ∈ ð1; 2 þ 3Þ, the efficiency curves ηðb; a ¼ constÞ decrease to the curve ηmining ðb; amining ðbÞÞ, with b increasing in the interval b ∈ ð1=4; 1Þ and the efficiency decreasing down to the limiting pffiffiffi value of ηðb ¼ 1; a ¼ 2 þ 3Þ ∼ 0.134. For the tidal charge b ∈ ð1=2; 1Þ, the efficiency of the Keplerian accretion at the transition between the extreme black hole and the related near-extreme naked singularity is continuously matched. The efficiency of η ∼ 0.134 is reached for pffiffiffi the Kerr-Newman spacetime with b ¼ 1 and a ¼ 2 − p3ffiffi.ffi Forpffiffivalues of the spin in the interval of ffi a ∈ ð2 − 3; 1= 2Þ, the transition of the function ηðb; a ¼ constÞ between the black hole and naked singularity states, obtained due to the increasing tidal charge b, is still continuous, and the curve ηmining ðb; amining ðbÞÞ is reached where η < 0.293. With an increasing spin a, the efficiency of the Keplerian accretion decreases. It is interesting that, for naked singularities having a spin a higher than ∼4.97 and an appropriately valued tidal charge b, the efficiency reaches values smaller than those corresponding to the Schwarzschild black holes (η ∼ 0.057). Note that the results of the Keplerian accretion analysis for the braneworld Kerr-Newman spacetimes can also be directly applied to the Keplerian accretion in the standard Kerr-Newman spacetimes, if we make the transformation b → Q2 , where Q2 represents the electric charge parameter of the Kerr-Newman background.

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VIII. CONCLUSIONS In this paper the circular geodesics of the braneworld Kerr-Newman black hole and naked singularity spacetimes have been studied, and classification of these spacetimes according to the character of the circular geodesic structure has been presented. The circular geodesics have been separated into two families—the lower family containing only the counterrotating circular geodesics and the upper family with corotating geodesics at a large distance, but a possible transformation to counterrotating geodesics in the vicinity of the naked singularity. It has been demonstrated that 14 different classes of the Kerr-Newman spacetimes can exist, mainly due to the properties of the upper family of circular geodesics. Implications of the geodesic structure to the Keplerian accretion have been given, and efficiency of the Keplerian accretion has been determined. The accretion efficiency is continuously matched between the naked singularity and extreme black hole spacetimes for the Keplerian accretion along the lower family circular geodesics. On the other hand, there is a strong discontinuity occurring in the transition between the naked singularities and the extreme black holes for the Keplerian accretion along the upper family circular geodesic, if the dimensionless spinpofffiffiffi the Kerr-Newman spacetime is sufficiently high (a > 1= 2)—the energy efficiency of the Keplerian accretion is then substantially higher for the naked singularity spacetimes. The accretion efficiency could then go up to the value of η ∼ 1.707 for Kerr-Newman near-extreme pffiffiffi naked singularity spacetimes with b ∼ 1=2 and a ∼ 1= 2. For the Keplerian accretion along the lower family circular geodesics, the inner edge of the disk always has to be located at the marginally stable circular geodesic corresponding to an inflection point of the effective potential of the motion, in accord with the scenario of the Keplerian accretion onto Kerr black holes and naked singularities. It has been shown that the Keplerian accretion along the upper family geodesics can give three different scenarios. It can finish at the inner edge located at the marginally stable circular geodesic—this is the standard accretion scenario present in the black hole spacetimes. However, two other scenarios could occur in the naked singularity spacetimes. The inner edge of the Keplerian accretion could occur at r ¼ b, which is the special limit on the existence of the circular geodesics. For b > 1 the efficiency of the upper family Keplerian accretion is independent of the naked

singularity spin. The most interesting is the third scenario, related to the Kerr-Newman naked singularity spacetimes of class IIIa having an infinitely deep gravitational potential of the upper family Keplerian accretion. Then the inner edge of the Keplerian accretion could occur even at the stable photon circular geodesic, and the accretion efficiency could be formally unlimited, making such naked singularity spacetimes unstable relative to accretion mining. The mining instability of the Kerr-Newman naked singularity spacetimes (or related superspinars) is a classical instability that could imply interesting consequences, which we plan to study in the future. Nevertheless, it is clear that the mining instability has to at least be restricted by the validity of the test particle approximation used in this paper for the Keplerian accretion. The other classical instability, related to the conversion of Kerr naked singularities to extreme black holes due to the Keplerian accretion [36], has to be treated in future work, but this instability necessarily has a much more complex character in the Kerr-Newman naked singularity spacetimes in comparison to the relatively simple Kerr case, due to the complexities related to the mining instability and the influence of the tidal charge. Interesting phenomena could be expected in the miningunstable Kerr-Newman spacetimes (class IIIa) even if it will represent an astrophysically more acceptable concept of the Kerr-Newman superspinar, with the inner boundary of the Kerr-Newman spacetime located at least at the outer boundary of the causality violation region [37,40,41]. We can expect observations of extremely high energy coming out of collisions in the vicinity of such a superspinar, enabled by the nonexistence of the event horizon and the fast rotation of the superspinar or, inversely, the strong magnification of the incoming radiation [42,66]. We could expect a hot doughnut-shaped configuration of accreting matter surrounding the superspinar, as discussed in [60]. ACKNOWLEDGMENTS Z. S. and M. B. have been supported by the Albert Einstein Centre for Gravitation and Astrophysics, financed by Czech Science Agency Grant No. 14-37086G and by Silesian University at Opava Grant No. SGS/14/2016. M. B. thanks Petr Slaný for several discussions.

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