Relativity, the Cosmological Constant, and the Spacetime Expansion of the Universe Issam Mohanna
[email protected] (April 23, 2016)
Abstract: A mathematical and physical study of the dynamic mechanism that governs the expansion of the universe via Einstein’s field equation and the Landau-Raychaudhuri equation is shown with the effective contribution of the cosmological constant , which has observationally and theoretically been calculated and found to be positively constant and small since the Planck epoch. The energy conditions, namely the NEC, WEC, SEC, and DEC, which the stress-energy tensor should satisfy, are applied in case of a perfect fluid. A comparison between the QFT vacuum ,the cosmological-constant vacuum, and the thermodynamic vacuum is presented leading to the same equation of state. I.INTRODUCTION Since the Big-Bang beginning, space has homogeneously and isotropically been expanding with time proportionally with the scale factor of the universe and started to accelerate at redshift before the equality of matter and dark energy at redshift . Einstein’s matrical field equation is a kinematic and dynamical equation that describes the expansion of the universe by relating matter and energy to the geometry of spacetime. One of the two essential structural aspects of Einstein’s equation is that the stress-energy tensor of matter and energy is restricted by 4 energy conditions in order to be of physical sense. The other aspect, which is indeed of mathematical effect, is that the cosmological constant, , is essentially on the left hand side of Einstein’s field equation as part of the geometry of spacetime ;it was physically interpreted by Einstein as a mathematical term that allowed the universe to be static, but later astronomical observations showed that a positive value of was needed to explain the accelerating universe.
required by relativity—is the FriedmannLemaître-Robertson-Walker metric, whose squared line element is (1)
where (2)
is the Gaussian curvature for a static universe of radius and is the time-dependent scale factor. Setting
, Eq. (1) becomes (3)
where (4)
II.COSMOLOGICAL LINE ELEMENTS FOR A SPHERICALLY SYMMETRIC UNIVERSE The most general metric that describes a homogeneous , isotropic, spherically symmetric, and expanding universe , and which is locally invariant under Lorentz transformation—as
1
is the angular separation, and (5)
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is the radius of the spherically shaped universe at any given time . Rotating the smaller circle in Fig. (1) round axis by radians, one gets a full sphere of radius for every fixed value of , whose range is (6)
with and as the correspondent polar and azimuthal angles respectively with
which describes the geometry of Einstein static universe. The observable universe is a spherical ball centered on the observer and whose age is estimated to be round 13.8 billion years with a radius approximately equal to 13.8 billion light-years in case it is not expanding. As theoretically and experimentally evidenced by Friedmann, Lemaître, Hubble, and Einstein, the universe is spherically expanding and curved putting the edge of the observable universe at about 46.5 billion light-years, which is the distance from where photons haven’t yet reached us.
(7)
III.EINSTEINFIELDEQUATION, FRIEDMANN EQUATION,AND LANDAURAYCHAUDHURI EQUATION. Assuming a non-rotational, spherically symmetric, homogeneous, and isotropic universe whose shape is unchangeable, stipulates a diagonal perfect fluid whose stress-energy tensor is matrically given by
Figure (1)
(11)
where The proper distance, which changes over time, between two celestial objects, such as two galaxies, moving with the Hubble flow is given by (8)
while is the commoving distance, which remains constant with time. Replacing the scale factor , which is always nonnegative, by the exponential , and substituting for in Eq. (1), one gets (9)
(12)
are respectively the diagonal temporal and spatial components with and as the mass-energy density and the hydrostatic and isotropic stress respectively as measured in its the rest frame. or is the fluid four-velocity, which in the rest frame is . According to Lovelock’s theorem in the case of four dimensions, if is a tensor constructed pointwise from the metric components and their first and second derivatives such that , then is a linear combination of the form [2] (13)
known as Tolman squared line element [1]. Setting or equal to one, one gets the Einstein squared line element [1] (10)
where
is the Einstein tensor ,and
are two real numbers.
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4
and
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Given that
(14)
known as Friedmann equation the cosmological constant. The spatial part of Eq. (16) is (23)
and setting
(15)
one gets the Einstein metrical field equation
Given that
(24)
(16)
Eq. (23) becomes where is the Ricci curvature tensor or simply the Ricci tensor, and is the Ricci scalar also known as the scalar curvature. The conservation of the energy-momentum tensor expressed by (17)
follows from Eq. (13) and (14) thanks to the two purely mathematical identities (18)
and the cosmological constant is a mathematical consequence of Lovelock’s theorem before being physically interpreted. The temporal part of Eq. (16) is (19)
Given that
(20)
and
(21)
(25)
Multiplying Eq. (25) by 2 and summing the result and Eq. (19) yield (26)
which is called the acceleration or the LandauRaychaudhuri equation for it describes the volume-expansion rate of space with respect to time. Indeed, it can be shown that (27)
where Equation (8) indicates that the change in size or the expansion of the universe is a second order effect in time. In fact, the distance of a celestial object at time with respect to an observer at any point in spacetime changes to at time not because of its radial normal velocity along the line of sight, which is actually not associated to the expansion of space in time, but because the scale factor changes to , and this changes was seen by Hubble as a recession velocity (28)
Eq. (19) becomes (22)
or (29)
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and measured as a redshift
(30)
where is the speed of light in vacuum and in flat space, and is the Hubble constant. IV.EINSTEIN STATIC UNIVERSE AND THE COSMOLOGICAL CONSTANT In 1917, years before the idea of an expanding universe was introduced by Friedmann in 1922 and Lemaître in 1927 and later empirically proved by Hubble in 1929, who could observationally discovered that the universe was curved at high redshifts , Einstein proposed a cosmological mathematical model that described a static universe whose temporal and spatial field equations, using Eq. (10),(12),and (16), are
gravitational attraction allowing for the steadystate universe sought by Einstein. For , a pressure-gradient force, , must be added in order to reach the hydrostatic balance or equilibrium. Actually, this is similar to and based on the physical idea of the balance between pressure and gravity found in planets ad stars where . Even apart from the case of the weak gravitational field, the cosmological constant of Einstein static universe can represent a form of spatial energy of antigravity nature whose density is regulated by (35)
where (36)
(31)
with and (32)
Equation (32) shows that Einstein had to add a cosmological constant to allow the existence of a nonnegative pressure so that both and could on physical backgrounds be acceptable. In the case of weak gravity where spacetime is slightly curved and with pressureless matter, also known as cold dust, or matter distribution having , the temporal component of Eq. (16) can be written as
V.ENERGY CONDITIONS AND THE COSMOLOGICAL CONSTANT The stress energy tensor of Eq. (11) should satisfy some energy conditions in order to physically describe how space-time gets curved or expands by matter ,energy, or vacuum. These energy conditions have the following forms: (37)
(33)
which is Poisson’s equation in Newtonian gravity modified by the cosmological constant and corresponds to a gravitational field (34)
where NEC, WEC, SEC, DEC, stand respectively for the null, weak, strong, and dominant energy conditions. From Eq. (19) and (23) with the NEC, one gets (38)
Equation (34) shows that represents an antigravity or repulsive force that balances the
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The NEC sets an upper bound on Using Eq. (19) and (23), the SEC yields
and the relevant negative stress or pressure is (46)
(39)
which limits the cosmic spatial acceleration and permits only if the cosmological constant acts repulsively The cosmological constant can be thought of as a form of spatial energy, called dark energy or ordinary quintessence, whose equation of state is (40)
and that doesn’t interact through any of the four fundamental forces of the standard model other than gravity, which makes its effect suitable to be added to the attractive gravitational effect in such a way to have an accelerated universe. The DEC with Eq. (19) and (23) gives (41)
VI. VACUUM ENERGY AND THE COSMOLOGICAL COSNTANT. In empty space, a space with no matter and no energy, (42)
and Lovelock’s theorem of equation (3) becomes (43)
which is a tensile stress or tension corresponding to the pressure experienced by empty space in its volume expansion due to the cosmological constant. The cosmological constant is experimentally measured and theoretically calculated to be [5] (47)
which is called the bare value of the cosmological constant and corresponds to an empty-space density (48)
. In quantum field theory of second quantization, a physical field is an infinite and continuous set of quantum harmonic oscillators located at every point in space, and each is labeled by a wave number or a momentum; particles whether bosons or fermions are nothing but excited states or field quanta. For any excitation to be of fermionic or bosonic physical sense, (49)
where is the wavelength of the particle produced by excitation and is Planck length defined as (50)
The cosmological constant is associated to an energetic and isotropic stress tensor denoted by (44)
A field is in its vacuum state when there are no particles and all harmonic oscillators are in their energetic ground states, which corresponds to the universe when it was 1 Planck length in diameter with a volume and an energy equal to Planck energy defined as
where the spatial vacuum density is
(51)
(45)
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so that the natural field vacuum energy density is the Planck energy density (52)
but for all fundamental or elementary particles since the rest mass is held fixed while is increased to its largest or maximum value corresponding to so that . Indeed, Eq. (56) can be written as (61)
(62)
The QFT vacuum energy can also be calculated by assigning a harmonic oscillator with a mode to every point in space. For a discrete set of quantum harmonic oscillators, each in its ground state, the total vacuum energy density is given by
Let
(63)
so
(64)
(53) (65)
In case of a continuous and infinite number of quantum harmonic oscillators (54)
and
but (66)
since
becomes (55)
whose value is unchanging is so that the Planck energy is almost completely due to the momentum. So (67)
(56)
(68) (57) (69) (58) (70) (59)
where and are respectively the ultraviolet cutoff wavenumber and the ultraviolet cutoff angular frequency. (60) It is worth noting that
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which is the same result as Eq. (59). The great discrepancy in value between or and ,which is known as vacuum catastrophe, is due to the fact that or is the lowest-energy-state density, also known as the ground state energy or the zero-point energy density, of a second-quantization quantum field,
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such as the electron or the neutrino field, with no particles or excitations ,and the quantum field is located in space and not in empty space, knowing that the universe is made of space and all things in space, whilst is strictly related to space itself whether nonempty or empty with no matter and no energy including all ground state energy of all quantum fields, as shown in LandauRaychaudhuri equation of empty space, (71)
cosmological constant and not from an unphysical, negative mass-energy density. Since the Planck epoch, the universe has been attached to a positive cosmological constant, which is of second-order effect in time and related to the volume change in size of the universe as pointed out by Eq. (26) and (71). In thermodynamics, a system of different constituent species with i-th species having particles at thermal and dynamic equilibrium is described by the Euler equation (76)
In a static universe, the temporal and spatial parts of Einstein field equation can be written as (72)
and (73)
which relates seven different thermodynamic variables. The thermodynamic vacuum of the system in equilibrium corresponds to its minimum energy at Substituting zero for in Eq. (76), the Euler equation becomes
In static and empty space, Eq. (3-5) and (3-6) becomes (74)
(77)
so that (78)
According to Eq.(71) and (74) , the cosmological constant, , can be considered or defined as the curvature by which the shape of the universe is spherical and the cause that makes spacetime ,and not what is in space, expand by changing its scale factor. The expansion of spacetime and its curvature are simultaneous.
On the other hand, in QFT, (79)
is the expectation value of [12] (80)
The effect of adding
(75)
to the Landau-Raychaudhuri equation is to slow down the expansion or the volume change in size. Supernovae observations combined with the cosmic-microwave-background-radiation measurements indicate that the expansion of the universe is currently accelerating and can only be generated by a non-negligible, positive
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which is indeed the Hamiltonian operator of the vacuum state at , where and are respectively the second-quantized Hamiltonian and the second-quantized number operator in function of the creation and annihilation operators, and Eq. (78) can be written as (81)
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Eq. (78) and (81) are also valid in the thermodynamic limit where
Whether in general relativity, quantum field theory, or thermodynamics, vacuum is described by the same equation of state: (72)
The quantum-field-theory vacuum density doesn’t take any part in the spacetime volume expansion because it isn’t included in the Landau-Raychaudhuri equation; only the QFT excitations are involved in addition to the cosmological constant. VII.REFERENCES [1] “Relativity, Thermodynamics, and Cosmology”, Richard Tolman. [2] “General Relativity with Applications to Cosmology”, N.Straumann. [3] “Applied Quantum Mechanics”, Walter Harrison. [4] “Relativistic Fluid Dynamics”, Jason Olsthoorn, University of Waterloo. [5] “Value of the Cosmological Constant: Theory versus Experiment”, Moshe Carmeli and Tanya Kuzmenko,Department of Physics, Ben Gurion University, Beer Sheva 84105,Israel. [6] “Dark Energy and the Accelerating Universe”, Joshua Frieman, Michael Turner, and Dragan Huterer. [7] “General Relativity and The Newtonian Limit”, Alexander Tolish. [8] “General Relativity: An Introduction for Physicists”,Hobson,Efstathhiou,and Lasenby. [9] “Vacuum Energy and the Cosmological Constant”, Steven Bass. [10] “The Quantum Vacuum and the Cosmological Constant Problem”, S.E.Rugh and H.Zinkernagel. [11] “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant”, Adam Riess and Alejandro Clocchiatti.
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[12] “Methods of Quantum Field Theory in Statistical Physics”, Abrikosov, Gorkov, and Dzyaloshinski. [13] “Second Quantization”, Mike Hermele. [14] “Quantum Field Theory”, David Tong. [15] “A5682:Introduction to Cosmology”, David H.Weinberg,Department Chair:OSU Astronomy. [16] “A Dynamical Study of The Friedmann Equations”, Uzan and Lehoucq. [17] “Introduction to Second Quantization”, Christophe Mora. [18] “Fluid Equations”, Joseph B.Keller. [19] “Generalized Raychaudhuri Equations for Strings and Membranes”, Sayan Kar,Institute of Physics,Sachivalaya Marg,Bhubaneswar751005,India. [20] “Canonical Quantization C6,HT 2016”, Uli Haisch, Rudolf Peierls Centre for Theoretical Physics, University of Oxford, United Kingdom. [21] “Of Some Theoretical Significance: Implications of Casimir Effects”,G.Jordan Maclay. [22] “Astronomy 401:The Cosmological Constant”, Dawn Erb, University of Wisconsin Milwaukee, Department of Physics. [23] “Lectures on Gravitation”, Apostolos Pilaftsis, School of Physics and Astronomy, University of Manchester. [24] “Space-time Singularities and Raychaudhuri Equations”, Haradhan Kumar Mohajan, Journal of Natural Sciences,Vol.1,No.2,December 2013. [25] “Energy Conditions and Current Acceleration of the Universe”, Gong and Wang, ScienceDirect,Physics Letters B 652 (2007) 6368. [26] “Energy Conditions Bounds and Their Confrontation with Supernovae Data”, M.P. Lima, S.Vitenti, M.J.Rebouças. [27] “Energy Conditions and Scalar Field Cosmology”, Shawn Westmoreland. [28] “Cosmology with a Time-Variable Cosmological Constant’, Peebles and Ratra, Joseph Henry Laboratories, Princeton University. [29] “The Cosmological Constant”, Sean M.Carroll. [30] “Derivation of Friedman Equations”, Joan Arnau Romeu, Department of Physics, University of Barcelona.
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[31] “A No-Nonsense Introduction to general Relativity”, Sean M.Carroll. [32] “Is the Zero-Point Energy Real?”, Simon Saunders. [33] “Introduction to Cosmology”,Luca Amendola, University of Heidelberg, [34] “Brief Introduction to FRW Cosmology”, Gregory Levine, Hofstra University. [35] “Introduction to Tensor Calculus for General Relativity”, Edmund Bertschinger. [36] “Density of States and Fermi Energy Concepts”, Dr.Alan Doolittle. [37] “Quantum Harmonic Oscillator with Ladder Operators”, Kenneth Taliaferro. [38] “The Meaning of Einstein’s Equation”, John C.Baez and Emory F. Bunn. [39] “On the Evolution of Large-Scale Structure in a Cosmic Void”, Sean Philip February. [40] “ Superluminal Recession Velocities”, Tamara M.Davis and Charles H.Lineweaver.
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