Black hole parameters estimation from its shadow Rahul Kumara∗ and Sushant G. Ghosha, b, c† a
Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India b
Multidisciplinary Centre for Advanced Research and Studies (MCARS),
arXiv:1811.01260v1 [gr-qc] 3 Nov 2018
Jamia Millia Islamia, New Delhi 110025 India and c
Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag 54001, Durban 4000, South Africa
Abstract Event Horizon Telescope is on the verge to capture the first ever image of the black hole shadow with an impressive resolution up to the event horizon scale. Shadow is believed to provide a potential way to understand the black hole properties and nearby matter dynamics. Considering a shadow of any general shape and size, we present observables which completely depends upon the shadow geometry. Based on these observables we characterize various black hole shadows and also estimate black hole parameters. PACS numbers:
∗
Electronic address:
[email protected]
†
Electronic address:
[email protected],
[email protected]
1
I.
INTRODUCTION
Black holes have always been mysterious objects since their discovery, however, despite ample observational evidence of their existence still, we do not have a real image of these objects. It is generally believed that probing the immediate environment of a black hole will not only provide an image of these object and the dynamics of nearby matters but also assist to learn about the strong gravity effects near the horizon. In this quest, the Event Horizon Telescope (EHT) [1] come as an important and potential prospect, which is a network of radio telescopes across the globe; based on the Very Long Baseline Interferometry (VLBI) technique this network works as an effective Earth-sized telescope and concerning the shadow of black hole candidate Sgr A* at the center of Milky-Way galaxy and also at the center of nearby M87 galaxy. The EHT is likely to reach an impressive angular resolution of 10µas in mm and sub-mm wavelengths, sufficient to resolve the horizon structure of black hole, the result from EHT observations are highly awaited and expected to be announced very soon. In principle, shadow observational results can also be used to predict the exact nature of astrophysical black holes, and in turn, they could also be used to place constraints on modified gravity theories which also admit the black hole solutions. Analyzing the propagation of light around a black hole is phenomenologically very important to unveil the near horizon spacetime properties. The strong gravitational field around black hole forces photons to move in circular orbits. In subsequent studies it is found that these unstable circular orbits have a very important influence on quasinormal modes [2, 3], gravitational lensing [4] and black hole shadow. Black hole shadow is a prominent feature of the strong gravitational field around black hole. A black hole due to its defining property i.e., the event horizon, along with surrounded photon region is expected to cast a dark region over an observer bright sky, which is known as the shadow. Synge [5] and Luminet initiated the study of black hole shadow by discussing the shadow cast by a Schwarzschild black hole, thereafter, Bardeen [6] in his pioneering work studied the shadow of Kerr black hole. Since then shadow has been extensively studied in the literature. In particular, over a past decade a flurry of activities in the analytical investigation, observational studies and numerical simulation of shadows have been reported. Even though the no-hair theorem encapsulate that the Kerr black hole is the unique stationary vacuum solution of Einstein field equation, still the exact Kerr nature of astrophysical 2
black hole has not been confirmed. Having this assertion that the astrophysical black holes may show deviation from the exact Kerr black hole [7, 8], numerous Kerr deviated black hole solutions have been obtained namely, with non-linear electrodynamics as a source, in higher dimension, in modified theories, etc. These black hole cast variety of shadows which have been studied numerically and analytically, e.g., Kerr-Newman black hole [9], black holes in extended Chern-Simons modified gravity [10], Kaluza-Klein rotating dilaton black hole [11], Kerr-Taub-NUT black hole, rotating braneworld black hole, regular black hole [12, 13, 50], black hole in higher dimensions [15–18]. Recently, the study of shadow is also extended to the wormhole spacetime [19–22]. Developing some methodological ways to estimate the parameters of astrophysical black hole candidates is one of the contemporary challenges in astronomy. The possibility of extracting some information about black holes from imaging their shadow, increase the relevance of shadow study. In this commencement, Hioki and Maeda discussed numerical estimation of Kerr black hole spin and inclination angle from the shadow observables [23], which is extended to analytical estimation by Tsupko in Ref. [24]. These observables namely, shadow radius and distortion parameter, were extensively used in the characterization of dozens of black holes shadow. However, it is found that the distortion parameter is degenerate with respect to the spin and possible deviations from the Kerr solution, a method for discriminating the Kerr black hole shadow from other rotating black holes shadows also has been reported [25]. Later on, some other possible ways of shadow characterization were also reported. An analytic description of distortion parameters of shadow is discussed in the coordinate independent manner in Ref. [26]. This is very legitimate to study the shadow observables of any general shape. In this paper, we discuss the shadow observables and their applicability in determining the black hole parameters. We emphasis on the characterization of various black hole shadows of arbitrary shape and size. The proposed observables do not presume any non-trivial symmetries in shadow and can be used without knowing the other features of shadows. Indeed, these observables completely depends upon the geometry of shadow and for the characterization of its distortion, it is no longer required to compare them with reference circle. We examine several known black hole shadows and show that these observables provides an accurate and robust determination of black hole parameters. The paper is organized as follow. In Sec. II, we discuss the propagation of light in rotating black hole spacetime. Further, in Sec.III, we present the shadow observables for its charac3
terization and examine them for some known black hole shadow. In Sec. IV, we summarize our main results.
II.
PHOTONS ORBITS AROUND BLACK HOLE AND SHADOW
The Carter’s prescription of the separability of the Hamilton-Jacobi equation [27] is adopted to study the complete geodesic motion, which with Jacobean action S = S(τ, xµ ) and a metric tensor gαβ can be read as ∂S 1 ∂S ∂S = − g αβ α β , ∂τ 2 ∂x ∂x
(1)
where τ is affine parameter. A test particle (of rest mass m0 ) moving in a general stationary, axially symmetric and asymptotically flat spacetime exhibits two conserved quantities energy E and angular momentum L associated with ∂t and ∂φ Killing vector fields. Apart from these, the four velocity norm is also conserved. This assist to write the Jacobi action in the separable form as follow [28] 1 S = m0 2 τ − Et + Lφ + Sr (r) + Sθ (θ). 2
(2)
The existence of spacetime symmetries in terms of conserved quantities allows the geodesics equations to be simplified in terms of first order differential form. Accordingly, with the choice of action in Eq. (2) and for a general Kerr-like metric in Boyer-Lindqiust coordinate, upon solving the Hamilton–Jacobi equation for null geodesics (m0 = 0) one can deduce the complete set of equations of motion [28] dt dτ dr Σ dτ dθ Σ dτ dφ Σ dτ Σ
r 2 + a2 2 2 E(r + a ) − aL − a(aE sin2 θ − L) , r 2 − 2m(r)r + a2 p = ± R(r) , p = ± Θ(θ) , a L 2 2 , = 2 E(r + a ) − aL − aE − r − 2m(r)r + a2 sin2 θ =
(3) (4) (5) (6)
the expressions for R(r) and Θ(θ) in Eq. (4) and (5) have the form
2 R(r) = (r 2 + a2 )E − aL − (r 2 − 2m(r)r + a2 )(K + (aE − L)2 ), 2 L 2 2 − a E cos2 θ. Θ(θ) = K − 2 sin θ 4
(7) (8)
Here, m(r) is the mass function such that limr→∞ m(r) = M and a is the spin parameter defined as a = J/M, with J and M are, respectively, the angular momentum and ADM mass of general rotating black hole. The presence of constants of motion E and L are not sufficient for the complete integrable solution of geodesic equations, until the finding of new conserved quantity Q associated with the hidden symmetry of conformal Killing tensor, which is related to the Carter integral of motion K through Q = K + (aE − L)2 [27]. Conclusively, this leads to a set of four constant of motion for particle geodesics. We can minimize the number of parameters by defining two dimensionless impact parameters η and ξ as ξ = L/E and η = K/E 2 , both remains constant along each and individual geodesic. Due to spacetime symmetries, geodesics along t and φ do not reveals non-trivial feature of orbits, therefore we will be mainly concerned for Eqs. (4) and (5). Equation (5) along with (8) give some measure of insight about the θ-motion. Writing the integral equation of motion in term of µ Σ
Z
dµ p = Θµ
Z
dτ ;
Θµ = η − (ξ 2 + η − a2 )µ2 − a2 µ4 .
(9)
Here, we identify µ = cos θ and obviously η ≥ 0 is must for possible θ motion i.e., Θµ ≥ 0 (cf. Fig. 1). In the Schwarzschild black hole, due to the Spherical symmetry all null geodesic orbits are planner i.e., orbits with θ˙ = 0. However, in Kerr black hole, the frame dragging may leads to non-planer orbits as well. Indeed, planer (or circular) orbits in Kerr black hole are possible only in the equatorial plane and characterized with zero Carter constant i.e., orbits at θ = π/2 ⇒ K = 0. Furthermore, generic bound orbits at plane other than θ = π/2 are non-planer (θ˙ 6= 0) and cross the equatorial plane while oscillating symmetrically about it. These orbits are identified by K > 0 (or η > 0) and commonly known as spherical orbits. Therefore, the θ−motion is freeze only for photons in the equatorial plane. Equation (9) reveals that latitude of orbits depends upon the angular momentum of photons, i.e., smaller the angular momentum of photons larger the latitude of orbits. The spherical orbits can have maximum latitude θmax = cos−1 (µmax ) for a non-zero angular momentum, where µmax correspond to the solution of Θµ = 0 for µ. In particular, only photon with zero angular momentum (ξ = 0) can reach the polar plane of black hole (θ = 0, µ = 1) and cover the entire span of θ coordinate. To get an insight about radial motion of photons Eqs. (4) and (7) can be rewritten in terms
5
1.0
Η
QΜ
0.5 0.0 Μmax
-0.5 -1.0 -Ξ 2
-1.5 0.0
0.2
0.4
0.6
0.8
1.0
Μ FIG. 1: Variation of Θµ with µ for η = 1 and ξ = 1.2. Horizontal dashed lines corresponds to the maximum and minimum values of Θµ .
of η and ξ as 2 i 1 h 2 2 2 2 R(r) = 2 (r + a ) − aξ − (r − 2m(r)r) (a − ξ) + η . E
(10)
Depending upon the values of impact parameters η and ξ, photon’s orbits can be classified in three categories namely scattering, unstable spherical orbits, and plunging orbits; among all the unstable orbits play crucial role in determining the optical appearance of black holes. The unstable spherical photon orbit of constant radius (rp ) can be easily found by demanding the local maximum of effective potential, which is mathematically equivalent to ∂R = 0; R|(r=rp ) = ∂r (r=rp )
with
(11)
∂ 2 R ≤ 0. ∂r 2 (r=rp )
Solving Eq. (11) yield the critical locus (η, ξ) associated with these unstable orbits. These orbits are unstable in any radial perturbation and forcing photons to either plunge into black hole or scatter to the spatial infinity. A photon’s orbit around non-rotating black holes is at a fixed radial coordinate (rp = 3M for Schwarzschild black hole) and lies on a spherical photon sphere. Such that, a photon if radially crosses the photon sphere boundary will inexorably falls into the event horizon forming the dark region. For a rotating black hole photons moving on circular orbits on the 6
equatorial plane can have rotation either along black hole or against it. The radii rp− and rp+ , respectively, denotes co-rotating and counter-rotating circular orbits, which are explicit functions of spin a. They lie in the range M ≤ rp− ≤ 3M and 3M ≤ rp+ ≤ 4M for the Kerr
black hole. Clearly, rp− ≤ rp+ , which can be attributed to the Lens-Thirring effect. Further spherical photon orbits (orbits at θ 6= π/2), are no longer affix to a fixed plane, rather lies
on a three-dimensional surface and have radii in the interval [rp− , rp+ ], i.e. for η > 0 orbit’s radius is rp− < rp < rp+ . These orbits filled a region around rotating black hole, known as photon region. Rotating black holes generically have two distinct photon regions namely, inside the Cauchy horizon (r− ) and outside the event horizon (r+ ), however, for shadow study we will be only focusing on the later one with rp > r+ [29]. On the other hand, the
critical impact parameter ξ is a monotonically decreasing function of rp with ξ(rp− ) > 0 and ξ(rp+ ) < 0, such that at rp = rp0 (rp− < rp0 < rp+ ) ξ is vanishing. Even though, for orbit at rp0 the angular momentum of photons is zero, still they cross the equatorial plane with non-zero azimuthal velocity φ˙ 6= 0. Detailed discussion about geodesic around rotating black holes can be found in [28, 30, 31]. In principle, a black hole in the presence of a luminous background of stars or glowing accreting matter, is believed to appears as a dark spot, accounting for the photons which are unable to reach the observer, this dark region is popularly known as black hole shadow. Following ray tracing method, we propagate light rays emanating from observer backward in time and locate their origins. Accordingly, those photons which scattered to infinity after passing to a minimum distance to black hole are causing brightness on the observer sky. On the other hand, photons which plunge directly into the black hole create a deficit of photon and essentially form a dark region in observer sky. In the borderline case, photons follow the geodesics which form bound unstable orbits and locus of these photons form the shadow silhouette. A far distant observer perceive shadow as a projection of locus of points η and ξ on celestial sphere to a two-dimensional plane. Let α and β are celestial coordinates, which are related with the η and ξ as follow [23]. dφ 2 2 dθ α = lim −rs sin θ0 , β = lim rs , (12) rs →∞ rs →∞ dr dr where rs is the distance between the black hole and far distant observer. However, as long as the black hole spacetime is asymptotically flat, we can consider a static observer at abritrary large distance. Setting the inclination angle to θ0 = π/2 as to witness the 7
maximum distortion in shadow, √ β = ± η.
α = −ξ,
(13)
Though, the shape of black hole shadow is determined by the properties of null geodesics, but it is neither the Euclidean image of its horizon nor of its photon sphere, rather it is the gravitationally lensed image of region inside photon sphere. For instance, Sgr A* with M ≈ 4.3 × 106 M⊙ at a distance d ≈ 8.35 kpc span an angular size of 20µas whereas its shadow has an angular size of ≈ 53µas. III.
NEW OBSERVABLES
The black hole shadow which is a manifestation of strong gravity field around black hole and appears as an optical appearance, can be used to unveil some information of astrophysical relevance about them. In this section, we will discuss about the shadow observables and their potency to determine the black hole parameters. The shape and size of black hole shadow depends upon the black hole spacetime parameters and also upon the observer’s position and orientation. A non-rotating black hole cast a perfectly circular shadow, whereas, a general rotating black hole shadow silhouette appears as a distorted circle. In particular, for a rotating black hole, an observer placed at a position other than in the polar direction witness a off-center displacement in the shadow along the direction of black hole rotation, furthermore, for sufficiently large value of spin parameter a dent appears on the shadow edge. This distortion in shadow can be accounted as a manifestation of the Lense-Thirring effect. Hioki and Maeda [23] led the study to characterize this distortion by means of two observables namely Rs and δs . These observables characterize the size and shape of shadow and have been extensively used for large number of black holes. The shadow is approximated with a circle passing through three points located at top, bottom and right side of shadow, such that Rs is the radius of this circle and δs is the deviation of left points of shadow from circle boundary. In the subsequent studies it was found that their applicability is limited to a specific class of shadows demanding some symmetries in their shape, and may not work well for black hole in complex theories of gravity [26]. Later on, some other observables were also introduced and used to characterize the shadow [24–26, 32–35]. Considering a shadow of arbitrary shape and size, we proposed the observables, namely area A, circumference C, 8
and oblateness D of the shadow. As already stated in earlier section, shadow is the parametric curve between α and β as a function of rp ranging from rp− ≤ rp ≤ rp+ i.e., plot β(rp ) vs α(rp ). Therefore, the area enclosed by black hole shadow can be defined by Z Z rp+ dα(rp) A = 2 β(rp )dα(rp ) = 2 β(rp ) drp , drp rp− whereas, the circumference of shadow is defined by v ! Z q Z rp+ u u dβ(rp ) 2 dα(rp ) 2 2 2 t C=2 drp + (dβ(rp ) + dα(rp ) ) = 2 − drp drp rp
(14)
(15)
The prefactor-2 is due to the fact that rotating black hole shadow is generically symmetric along the axes perpendicular to the rotational axes viz, shadow is symmetric along α−axes. We can also characterize the shadow of rotating black hole through its oblatenes [24, 36, 37] by defining the parameter D as the ratio of horizontal diameter and vertical diameter as follow αr − αl . (16) βt − βb The subscript r, l, t, and b respectively denote right, left, top, and bottom. For spherically √ symmetric black hole shadow d = 1, whilst for a Kerr shadows 3/2 ≤ D ≤ 1. Therefore, D=
oblateness parameter can be identified as the measure of distortion in shadow viz, more the value of D is away from 1 more the shadow distorted. We must notice, the definition of these observables require neither any non-trivial symmetry in shadow shape nor any primary curve to approximate the shadow, and hence are valid for any general black hole shadow. Without a priory, this can be expected that an observer targeting the black hole shadow through astronomical observations can measure the area, length of shadow silhouette, and also horizontal and vertical diameters. Therefore, having the precise knowledge of these primary features of shadow is sufficient to uniquely determine the black hole parameters. With the intent of making acquaintance with these observables we will discuss some wellknown black hole shadows.
A.
Kerr black hole
Kerr black hole is the paradigmatic solution of Einstein GR and also one of the mostly studied vacuum solution due to its established astrophysical relevance by uniqueness theorem 9
[38]. The photon geodesic around Kerr black hole has been studied long back by Bardeen in 1973 [6] and since then by many others also. The value of radial effective potential has the following form R(r) = (r 2 + a2 − aξ)2 − (a2 + r 2 − 2Mr)(η + (ξ − a)2 ),
(17)
which gives the following solution for celestial coordinates α and β for shadow silhouette α = − β =
a2 M + a2 rp − 3Mrp 2 + rp 3 , a(M − rp )
rp 3 (−4a2 M + 9M 2 rp − 6Mrp 2 + rp 3 ) a2 (M − rp )2
1/2
.
(18)
The contour of the following equation in (α, β) plane traces the shadow silhouette of the Kerr black hole a2 M 2 + 2a2 Mrp + (a2 − 6M 2 )rp 2 + 2rp 4 α +β = , (M − rp )2 2
2
(19)
which retain the value for Schwarzschild black hole in the limit a = 0 and rp = 3M as follow α2 + β 2 = 27M 2 .
(20)
The spherically symmetric Schwarzschild black hole cast a perfect circular shadow which is also apparent from Eq. (20), whereas, the Kerr-like shadow appears as a distorted disk [6, 28]. This distortion appears as a dent on the left side of shadow, which is primarily a manifestation of black hole spin and related to the differences in the effective potential for co-rotating and counter-rotating photons [39]. It is found that this dent reduces as observer moves from the equatorial plane to the axis of black hole symmetry, and eventually disappear completely for θ = 0. In Fig. 2, we plotted the calculated area A, circumference C, and oblateness parameter D of shadow with varying spin parameter. This is clearly evident from figure that value of these observables decrease with increasing a, and a rapid fall can be noticed beyond a ≈ 0.5. The increasing distortion in shadow is causing these decrements. Since, the Kerr black holes have two parameters associated with them mass M and spin a, however, presuming the knowledge of mass through the stellar motion around black hole, leads to the ambiguity only in the spin. From Fig. 2, this is clear that with the knowledge of shadow observables A, C, and D, we can easily extract the precise value of spin.
10
84
A
82
80
78 0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.6
0.8
1.0
a 32.6 32.4
C
32.2 32.0 31.8 31.6 31.4 31.2 0.0
0.2
0.4
a 1.00 0.98
D
0.96 0.94 0.92 0.90 0.0
0.2
0.4
a FIG. 2: Kerr black hole shadow area A, circumference C and oblateness parameter D behaviour with varying spin parameter a. B.
Kerr-Newman black hole
The apparent shape of Kerr-Newman black hole shadows were studied and plotted for different values of spin parameter a and charge Q in [9]. The value of radial effective potential
11
has the following form R(r) = (r 2 + a2 − aξ)2 − (r 2 − 2Mr + a2 + Q2 )(η + (ξ − a)2 ).
(21)
The celestial coordinates α and β delineating the shadow are given by α = −
a2 M + a2 rp + 2Q2 rp − 3Mrp 2 + rp 3 , a(M − rp )
1/2 rp 2 (4a2 (Q2 − Mrp ) + (2Q2 + rp (−3M + rp ))2 ) . β = − a2 (M − rp )2
(22)
Contour of the following equation in (α, β) plane traces the shadow silhouette of the KerrNewman black hole α2 + β 2 =
a2 (M + rp )2 + 2rp (2MQ2 − 3M 2 rp + rp 3 ) , (M − rp )2
(23)
In Fig. 3, we plotted the calculated area A, circumference C, and oblateness parameter D of shadow with varying Q for different values of a. In Ref. [9] it was shown that the presence of electric charge decrease the size of apparent shadow, which is also clearly evident from Fig. 3 that the values of these observables decrease with increasing Q. For non-rotating charged black hole, shadow is spherically symmetric for arbitrary value of Q, i.e., 0 ≤ Q ≤ 1, which leads to D = 1. However, for rotating black holes, shadow possess a distortion which further increases with increasing Q or a, this increasing distortion in shadow is causing these decrements in values of A, C, and D. Since the charged black holes have not been completely ruled out by astrophysical observations, therefore, it is relevant to examine the possibility of extracting the black hole parameters through its shadow analysis. Kerr-Newman black hole have two associated parameters spin a and charge Q apart from mass M which can be fixed through other observations. This require atleast two independent shadow observables for spin and charge estimation. In Fig. 4, we make a contour plot of A and Q in the (a, Q) plane. The point of intersection of constant A and D curves impart the unique values of black hole spin and charge.
C.
Rotating Bardeen black hole
James Bardeen proposed the first ever regular black hole solution, widely known as Bardeen black hole [40], which are deprived of curvature singularity at the center. Later on, 12
85 80 75
A
70 65 60 55 50 0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.6
0.8
1.0
Q 32
C
30
28
26 0.0
0.2
0.4
Q 1.00 0.98 0.96
D
0.94 0.92 0.90 0.88 0.86 0.0
0.2
0.4
Q FIG. 3: Plot showing the Kerr-Newman black hole shadow area A, circumference C and oblateness parameter D behaviour with varying electric charge parameter Q, (solid black curve) for a = 0, (Solid green curve) for a = 0.3, (Dashed blue curve) for a = 0.5 and (Dotted red curve) for a = 0.8.
13
1.0 55
0.8 62 68
0.6 72
Q
76
0.4 79 82
0.2
0.99
0.999
0.0 0.0
0.2
0.92 0.98 0.97 0.96 0.94 0.9 0.95
0.4
0.6
0.8
1.0
a FIG. 4: Contour plots of the observables A and D in the plane (a, Q) for Kerr-Newman black hole. Each curve is labled with the corresponding values of A and D. (Solid red curve) corresponds to the area curve, and (Dotted blue curve) for oblateness parameter.
Ayon-Beato and Garcia explicitly showed that this metric can be interpreted as the solution of Einstein equation coupled to the nonlinear electrodynamics with a magnetic charge [41], meanwhile, in a recent work, Bardeen’s regular model is also re-interpreted as the solution of Einstein equation with an electric source [42]. This black hole solution is parameterized by two parameters spin a and magnetic charge g, and always been a topic of great attention. The detailed study of geodesics motion around rotating Bardeen black hole have been made in Ref. [43], whereas the shadow of these black hole and the impact of magnetic charge g on it is reported in [13]. It is found that the presence of magnetic charge g decreases the size of shadow whereas, increases the distortion. The radial effective potential has the following form
3/2 r2 r − a2 )(η + (ξ − a)2 ), (24) R(r) = (r + a − aξ) − (r − 2M r2 + g 2 which for the unstable photon orbits gives the following solution for celestial coordinates α 2
2
2
2
and β α = β =
(4a2 g 2 + a2 rp 2 − 3rp 4 ) Mrp 2 + (rp 2 + a2 )(g 2 + rp 2 )5/2 , a(g 2 + rp 2 )5/2 − Marp 2 (4g 2 + rp 2 )
−rp 4 (g 2 + rp 2 )5 − Mrp 4 (g 2 + rp 2 )5/2 (4a2 (2g 2 − rp 2 ) + 6rp 4 ) − 9M 2 rp 12 a(g 2 + rp 2 )5/2 − Marp 2 (4g 2 + rp 2 ) 14
1/2
. (25)
For non-rotating Bardeen black hole, shadow silhouette is described by the following equation α2 + β 2 =
2rp 2 (rp 10 + g 10 + 5g 2 rp 2 (rp 6 + g 6 ) + 10g 4rp 4 (rp 2 + g 2 ) − 3M 2 rp 6 (rp 2 + 4g 2 )) 2
((g 2 + rp 2 )5/2 − Mrp 2 (4g 2 + rp 2 ))
.
(26)
From Fig. 5 this is evident that area, circumference and oblateness all decrease with increasing g or a. The area of dark region covered by shadow as well as its silhouette length monotonically decreases in a similar fashion with increasing g, while keeping a fixed. For a particular value of a, the shadow of rotating Bardeen black hole is more distorted than corresponding Kerr black hole, i.e., D(g 6= 0) < D(g = 0). The cosmic censorship conjecture prevent the formation of naked singularity, which put the upper bound on the allowed range of a and g. Shadow observables decrease comparatively faster with increasing g for large value of g than for small value. Rotating Bardeen black hole has two parameters a and g, therefore their estimation require at least two shadow observables. Indeed, precise enough measurement of shadow observables could served to estimate both the spin a and charge g. In order to extract these parameter we plotted the contour curves of constant area A and oblateness parameter D in the plane (a, g) in Fig. 6. The points in the plane where these curves intersect gives the corresponding unique values of the a and g. The one-to-one correspondence between the values of shadow observables (A, D) and black hole parameters (a, g) is clearly evident from Fig. 6. Therefore, if these observables are obtained from observations, then we can precisely estimate the values of the black hole spacetime parameter. For instance, if we measure the values of dimensionless area and oblateness observables as A = 82M 2 and D = 0.98, we can conclude that a = 0.5396M and g = 0.2045M. The interesting comparison of Bardeen black hole shadow with Kerr was made in Ref. [25], and shown that for a particular choice of parameters Bardeen black hole (M = 1, a/M = 0.5286, g/M = 0.6) cast a very similar shadow as of Kerr black hole (M = 0.9311, a/M = 0.9189). We find that for these values of parameters, the observables for Bardeen black hole are A = 69.1445, C = 29.5269, D = 0.925402, whereas for Kerr black hole A = 68.68015, C = 29.4213, D = 0.925402. It is clear that area and circumference differ by less than 1% viz., 0.671% and 0.357%, respectively.
15
85
80
A
75
70
65 0.0
0.2
0.4
0.6
0.8
0.6
0.8
0.6
0.8
g 33 32
C
31 30 29 28 0.0
0.2
0.4
g 1.00 0.98 0.96
D
0.94 0.92 0.90 0.88 0.86 0.0
0.2
0.4
g FIG. 5: Bardeen black hole shadow area A, circumference C and oblateness parameter D behaviour with varying charge parameter g. (Solid black curve) correspond to non-rotating Bardeen black hole a = 0.0, (Solid green curve) for rotating Bardeen black hole with a = 0.3, (Dashed blue curve) for a = 0.5, (Dotted red curve) for a = 0.8.
16
67 70
0.6
73 76
g
79
0.4 82
0.2 84
0.999
0.0 0.0
0.99
0.2
0.96 0.94 0.9 0.95 0.92 0.98 0.97
0.4
0.6
0.8
1.0
a FIG. 6: Contour plots of the observables A and D in the plane (a, g) for Bardeen black hole. Each curve is labeled with the corresponding value of A and D. (Solid red curve) corresponds to the area curve, and (Dotted blue curve) for oblateness parameter. D.
Rotating Rastall black hole
Rastall theory of modified gravity is solely based on the non-conservation of energy momentum tensor in curved spacetime [44]. In recent past this non-conservative theory of gravity grabbed a great attention and several work devoted to this can be found in literature. The static black hole solution in this theory was obtained in [45], later on the rotating black hole solution were also reported [46, 47]. In Ref. [48] the shadow of rotating Rastall black hole surrounded by a perfect fluid, namely quintessence field is studied. Due to the presence of cosmological horizon, the static observer is considered within the range of outer communication i.e., between the event and cosmological horizon. It was shown that the size of rotating Rastall black hole shadow decreases monotonically with increasing field parameter Ns or Rastall coupling parameter ψ. However, for higher values of Ns , the shadow observed by an observer at position rO , becomes apparently more symmetric with increasing Rastall coupling parameter ψ. The radial potential is related to the function R(r) given as R(r) =
h
r 2 + a2 − aξ
2
− (r 2 − 2Mr + a2 − Ns r ζ ) (a − ξ)2 + η
17
i
.
(27)
Here, ζ = (1 − 3ωs )/(1 − 3ψ(1 + ωs )), with ωs as the equation of state parameter for the surrounding field. Solving Eq. (27) for photon unstable orbits leads to the locus of critical impact parameters ξ and η [48] 1
ξ =
h 3ωs +4 a3 Ns (3ωs − 1)rp σ
3ωs +1 2 3 rp σ a2 Ns (3ωs − 1)rp σ + 2σ(M − rp )rp σ +1 1 2σ+3ωs +4 3ωs 4 − aNs (2σ − 3ωs + 1)rp σ − 2aσrp σ +2 Ns rp σ − a2 + rp (rp − 2M) rp σ i 5 2 2 +1 σ + 2aM(a − rp )σrp ,
(28)
2
−rp 2− σ
h
2σ+3ωs +5
) 3 ( σ 2 − 4Ns a σ(2σ − 3ωs + 1)rp 3ωs +1 2 a3 Ns (3ωs − 1)rp σ + 2σ(M − r)rp σ +1 3ωs −1 3ωs −1 6 Ns (2σ − 3ωs + 1)rp σ −1 + 2Mσ + 4aσrp σ +3 Ns rp σ − a2 + rp (rp − 2M) 3ωs −1 3σ+3ωs +5 + aNs rp ( σ ) 4Mσ(6σ − 3ωs + 1) + Ns (2σ − 3ωs + 1)2 + 4σ 2 rp σ −1 i 6 4σ+3ωs +5) 6 ) σ + 4aσ 2 5M 2 − 4Mrp + rp 2 rp σ +4 − 8Ma3 σ 2 rp (3+ σ ) − 8aNs σ 2 rp ( , (29)
η =
where σ = 3ψ(1 + ωs ) − 1. Shadow is visualized as a parametric plot between X and Y , which reads as Φ(rO , rp ) sin(Ψ(rO , rp )), X(rO , rp ) = −2 tan 2 Φ(rO , rp ) Y (rO , rp ) = −2 tan cos(Ψ(rO , rp )). 2
(30)
The celestial coordinates Φ and Ψ are related to the coordinates of projected two-dimensional plane as sin Ψ(rO , rp ) = sin Φ(rO , rp ) =
! p , 2 (a − ξ) + η r=r ! O p ∆[(a − ξ)2 + η] (r 2 + a2 − aξ) ξ−a
.
(31)
r=rO
Accordingly, the area and circumference enclosed by Rastall black hole shadow can be defined by rp+
dX(rp ) A = 2 Y (rp )dX(rp ) = 2 Y (rp ) drp , drp rp− v ! Z q Z rp+ u u dY (rp ) 2 dX(rp ) 2 2 2 t drp . (32) + C = 2 (dY (rp ) + dX(rp ) ) = 2 − drp drp rp Z
Z
18
2.0
A
1.5
1.0
0.5 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.08
0.10
0.12
0.08
0.10
0.12
Ψ 5.0 4.5
C
4.0 3.5 3.0 2.5 0.00
0.02
0.04
0.06
Ψ 1.00 0.99
D
0.98 0.97 0.96 0.95 0.00
0.02
0.04
0.06
Ψ FIG. 7: The variation of area A, circumference C and oblateness parameter D for Rastall Black hole surrounded by quintessence (Ns = 0.05, ωs = −2/3) with different spin parameter is shown. (Solid black curve) for a = 0.0, (Dashed blue curve) for a = 0.5, (Solid green curve) for a = 0.7 and (Dotted red curve) for a = 0.9.
19
0.12 0.10
0.4 0.7 1
0.08 1.2
Ψ
1.4
0.06 1.6
0.04 1.8
0.02
0.95
2 0.00 0.9999 0.998 0.995
0.0
0.2
0.99
0.4
0.98
0.6
0.97 0.96 0.93
0.8
1.0
a FIG. 8: Contour plots of the observables A and D in the plane (a, ψ) for Rastall black hole. Each curve is labeled with the corresponding value of A and D. (Solid red curve) corresponds to the area curve, and (Dotted blue curve) for oblateness parameter.
The area, circumference and oblateness parameter for Rastall black hole shadow is plotted in Fig. 7 for an observer placed at rO = 6M at equatorial plane. At first, we can easily see that the size of shadow decreases with increasing Rastall coupling parameter ψ while keeping the other parameters fixed. The area and circumference of black hole decrease more rapidly with increasing ψ for small value of spin parameter. Rastall parameter plays significantly opposite role in distortion of shadow as it make the shadow more symmetric along rotational axis [48] viz. the value of oblateness parameter increases with ψ. In Fig. 8, we plotted the A and D in the a, ψ plane, the point where the contours of A and D intersects, give the value of black hole spin parameter and Rastall coupling parameter. Hence, if one measure the dimensionless area and oblateness of shadow through observation, the spin and Rastall coupling parameter could be easily determined, e.g., if A = 1.4 and D = 0.98 then a = 0.7426 and ψ = 0.06566.
E.
Rotating nonsingular black hole
With the advent of regular black hole model by Bardeen, other regular black holes solutions were also start getting studied widely. One of the rotating non-singular black hole 20
solution was reported in [49], later on geodesics motion and shadow were also studied [50]. The value of radial effective potential has the following form R(r) = (r 2 + a2 − aξ)2 − (r 2 − 2Mre−k/r + a2 )(η + (ξ − a)2 ),
(33)
which gives the following solution for celestial coordinates α and β rp 2 (a2 + rp 2 )ek/rp + M((k − 3rp )rp 2 + a2 (k + rp )) , −arp 2 ek/rp + aM(k + rp ) 1/2 rp 4 (4a2 M(k − rp )ek/rp + (kM + rp (−3M + rp ek/rp ))2 ) β = − , a2 (kM + rp (M − rp ek/rp ))2
α =
(34)
for a = 0, the shadow is determine through α2 + β 2 =
2e(2k)/rp rp 6 + 2M 2 (k − 3rp )rp 2 (k + rp ) . (kM + rp (M − ek/rp rp ))2
(35)
The behavior of observables A, C, D with varying k for different values of a are shown in Fig. 9, and clearly each decreases with increasing k. The behaviour of area and circumference is very similar. However, oblateness falls very rapidly with increasing k for near extremal black hole. For the estimation of black hole parameters a and k, we make a contour plot of A and D in (a, k) plane. The point where curve of constant A and D intersects leads to the non-degenerate values of a, k. Next, we obtained the following approximate, dimensionless expressions of area, circumference and oblateness through the best fit of numerical data of observables A(a, k) = 84.823 − 0.013004a − 4.34942a2 − 2.36382a3 + 4.47457a4 − 4.83904a5 − 56.3846k − 0.418641ak + 6.05534a2k − 32.2151a3k + 24.8359a4 k − 5.25777k 2 + 4.46953ak 2 − 142.575a2k 2 + 320.995a3k 2 − 301.588a4 k 2 + 5.63201k 3 − 9.23287ak 3 + 309.192a2k 3 − 276.264a3k 3 − 14.1295k 4 + 7.04183ak 4 − 303.691a2k 4 , C(a, k) = 32.6484 − 0.002221a − 0.84867a2 − 0.374395a3 + 0.67452a4 − 0.774287a5 − 10.8185k + 0.493049ak − 1.0606a2 k − 5.8057a3k + 3.84901a4k − 3.24875k 2 − 6.12361ak 2 − 5.1476a2k 2 + 59.5101a3k 2 − 51.3798a4k 2 + 1.99375k 3 + 23.4929ak 3 − 20.4438a2 k 3 − 61.0496a3k 3 − 5.67007k 4 − 26.7951ak 4 + 28.1407a2k 4 , D(a, k) = 1. − 0.000544168a − 0.0377214a2 − 0.172164a3 + 0.719728a4 − 1.57686a5 + 1.6425a6 − 0.683473a7 + 0.014452ak − 0.490382a2k + 3.7850a3 k − 10.804a4 k + 11.709a5 k − 5.37221a6k − 0.071081ak 2 + 1.20427a2 k 2 − 19.6311a3k 2 + 43.3697a4k 2 − 19.823a5k 2 + 0.0486ak 3 + 1.1621a2k 3 + 28.0021a3k 3 − 52.7191a4 k 3 + 0.1211ak 4 − 7.30097a2k 4 (36) 21
80
A
70 60 50 40 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
k 32
C
30 28 26 24 22 0.0
0.1
0.2
0.3
0.4
k 1.00 0.98 0.96
D
0.94 0.92 0.90 0.88 0.86 0.0
0.1
0.2
0.3
0.4
k FIG. 9:
Plot showing the variation of area A, circumference C and oblateness parameter D for
non-singular black hole with different spin parameter. (Solid black curve) for a = 0.0, (Solid green curve) for a = 0.2, (Dashed blue curve) for a = 0.5 and (Dotted red curve) for a = 0.8.
which gives the values for Kerr black hole in the limit k = 0. For Schwarzschild black hole in the limit a = 0, observables take the values A = 84.823, C = 32.6484 and D = 1. For a = 0, the non singular black hole cast a perfect circular shadow [50], which is also fully 22
0.7 41 45
0.6 50 0.5 55
k
60
0.4 65
0.3
70
0.2 75 0.1 80 84
0.0
0.0
0.86 0.999
0.99
0.2
0.98 0.97
0.4
0.95 0.92
0.6
0.8
1.0
a FIG. 10: Contour plots of the observables A and D in the plane (a, k) for non-singular black hole. Each curve is labelled with the corresponding value of A and D. (Solid red curve) corresponds to the area curve and (Dotted blue curve) for oblateness parameter.
consistent from Eq. (36), i.e., D(0, k) = 1.
The exact nature of supermassive black hole Sgr A* at the galactic center is still elusive, albeit astronomical observations could place constraint on its mass and distance from earth as M = 4.3 × 106 M⊙ and d = 8.35 kpc [53]. Presuming the exact Kerr nature of Sgr A* and knowing the area covered by its shadow, we can calculate the solid angle made by it on the telescopes and also the angular size. Since the Kerr black hole shadow has a deformation from perfect circle and has oblateness D ≤ 1, therefore this necessitate to define its vertical and horizontal angular sizes i.e., major and minor angular size. We define the solid angle Ω and major ϑM , minor ϑm angular size by Ω= 0.2
βt − βb A , ϑM = , 2 d d
0.3
0.4
0.5
ϑm = 0.6
αr − αl d 0.7
(37)
a/M
0.1
0.8
0.9
1.0
Ω
2.14796 2.14433 2.13816 2.12922 2.11718 2.10146 2.0811 2.05426 2.01649 1.93185
ϑm 52.3092 52.2205 52.0686 51.8467 51.5432 51.1389 50.5998 49.8579 48.7346 45.3263 For a = 0, major and minor angular diameters are same viz., ϑM = ϑm = 52.3383, also the major angular size is independent of black hole spin. Solid angular size Ω is in units of 23
10−3 µas2 whereas major and minor angular size have units µas.
IV.
CONCLUSION
In our study we have shown that the black hole spacetime parameters can be easily obtained from the geometrical properties of its shadow. Namely, knowing the area covered by the dark region of shadow, its silhouette length and ratio of its horizontal and vertical diameters are sufficient to exactly determine the spacetime parameters. Furthermore, for the sake of examples we discuss the cases of Kerr, Kerr-Newman, rotating Bardeen, rotating Rastall and rotating non-singular black holes shadows. Though, each independent observable can at best determine only one parameter, a set of observables can uniquely determine the all parameters of spacetime. The advantage of these observables can be noticed from the fact that in these calculation we do not require any non-trivial symmetry in the shadow shape, therefore, they will be valid for shadow of any general shape and size, and also we no longer require a reference circle to characterize them. The area and circumference observable gives an insight about size of shadow, whereas oblateness is related to its shape. In particular, as the shadow is more distorted from perfect circle, its oblateness parameter is more deviated from 1.
V.
ACKNOWLEDGEMENTS
S.G.G. would like to thank SERB-DST Research Project Grant No.
SB/S2/HEP-
008/2014 and DST INDO-SA bilateral project DST/INT/South Africa/P-06/2016 and also to IUCAA, Pune for the hospitality while this work was being done. R.K. would like to thank UGC for providing Senior Research Fellowship, and also Md. Sabir Ali and Balendra Pratap Singh for fruitful discussion.
[1] http://eventhorizontelescope.org/. [2] V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009). [3] S. Hod, Phys. Rev. D 80, 064004 (2009).
24
[4] I. Z. Stefanov, S. S. Yazadjiev and G. G. Gyulchev, Phys. Rev. Lett. 104, 251103 (2010). [5] J.L. Synge, Mon. Not. R. Astron. Soc. 131, 463 (1966). [6] J. M. Bardeen, Black Holes, Edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973, p. 215). [7] T. Johannsen, Phys. Rev. D 87, no. 12, 124017 (2013). [8] T. Johannsen, Class. Quant. Grav. 33, no. 12, 124001 (2016). [9] A. de Vries, Classical Quantum Gravity 17, 123 (2000). [10] L. Amarilla, E. F. Eiroa and G. Giribet, Phys. Rev. D 81, 124045 (2010). [11] L. Amarilla and E. F. Eiroa, Phys. Rev. D 87, 044057 (2013). [12] A. Yumoto, D. Nitta, T. Chiba and N. Sugiyama, Phys. Rev. D 86, 103001 (2012). [13] A. Abdujabbarov, M. Amir, B. Ahmedov and S. G. Ghosh, Phys. Rev. D 93, 104004 (2016). [14] M. Amir and S. G. Ghosh, Phys. Rev. D 94, 024054 (2016). [15] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Phys. Rev. D 90, 024073 (2014). [16] A. Abdujabbarov, F. Atamurotov, N. Dadhich, B. Ahmedov and Z. Stuchlik, Eur. Phys. J. C 75, 399 (2015). [17] M. Amir, B. P. Singh and S. G. Ghosh, arXiv:1707.09521 [gr-qc]. [18] B. P. Singh and S. G. Ghosh, arXiv:1707.07125 [gr-qc]. [19] T. Ohgami and N. Sakai, Phys. Rev. D 91, no. 12, 124020 (2015). [20] R. Shaikh, Phys. Rev. D 98, no. 2, 024044 (2018). [21] M. Amir, K. Jusufi, A. Banerjee and S. Hansraj, arXiv:1806.07782 [gr-qc]. [22] M. Amir, A. Banerjee and S. D. Maharaj, arXiv:1805.12435 [gr-qc]. [23] K. Hioki and K. i. Maeda, Phys. Rev. D 80, 024042 (2009). [24] O. Y. Tsupko, Phys. Rev. D 95, no. 10, 104058 (2017). [25] N. Tsukamoto, Z. Li and C. Bambi, J. Cosmol. Astropart. Phys. 1406, 043 (2014). [26] A.A. Abdujabbarov, L. Rezzolla, and B. J. Ahmedov, Mon. Not. R. Astron. Soc. 454, 2423 (2015). [27] B. Carter, Phys. Rev. D 174, 1559 (1968). [28] S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1992). [29] A. Grenzebach, V. Perlick, and C. L¨ ammerzahl, Phys. Rev. D 89, 124004 (2014). [30] C. Liu, C. Ding and J. Jing, Sci. China Phys. Mech. Astron. 62, no. 1, 10411 (2019).
25
[31] Daniel C. Wilkins, Phys. Rev. D 5, 4 (1972). [32] P.V. P. Cunha, C.A. R. Herdeiro, E. Radu, and H.F. Runarsson, Phys. Rev. Lett. 115, 211102 (2015). [33] J. Schee and Z. Stuchlik, Int. J. Mod. Phys. D 18, 983 (2009). [34] T. Johannsen, Astrophys. J. 777, 170 (2013). [35] M. Wang, S. Chen and J. Jing, arXiv:1801.02118 [gr-qc]. [36] R. Takahashi, J. Korean Phys. Soc. 45, S1808 (2004) [Astrophys. J. 611, 996 (2004)]. [37] A. Grenzebach, V. Perlick and C. Lammerzahl, Int. J. Mod. Phys. D 24, no. 09, 1542024 (2015). [38] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963). [39] C. Bambi and K. Freese, Phys. Rev. D 79, 043002 (2009). [40] J. M. Bardeen, in Conference Proceedings of GR5, Tbilisi, URSS, p. 174 (1968). [41] E. Ayon-Beato and A. Garcia, Phys. Lett. B 464, 25 (1999). [42] M. E. Rodrigues and M. V. d. S. Silva, JCAP 1806, no. 06, 025 (2018). ˜ and J. Schee, Int. J. Mod. Phys. D 24, no. 02, 1550020 (2014). [43] Z. StuchlAk [44] P. Rastall, Phys. Rev. D 6, 3357 (1972). [45] Y. Heydarzade and F. Darabi, Phys. Lett. B 771, 365 (2017). [46] R. Kumar and S. G. Ghosh, arXiv:1711.08256 [gr-qc]. [47] Z. Xu and J. Wang, arXiv:1711.04542 [gr-qc]. [48] R. Kumar, B. P. Singh, M. S. Ali and S. G. Ghosh, arXiv:1712.09793 [gr-qc]. [49] S. G. Ghosh, Eur. Phys. J. C 75, no. 11, 532 (2015) [50] M. Amir and S. G. Ghosh, Phys. Rev. D 94, no. 2, 024054 (2016) [51] H. Falcke and S. B. Markoff, Class. Quant. Grav. 30, 244003 (2013) [52] http://blackholecam.org/. [53] H. Falcke and S. B. Markoff, Class. Quant. Grav. 30, 244003 (2013). [54] C. Bambi and N. Yoshida, Class. Quant. Grav. 27, 205006 (2010). [55] F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88, 064004 (2013).
26
