Thermodynamics of Shearing Massless Scalar Field Spacetimes is Inconsistent With the Weyl Curvature Hypothesis
aa r X i v : . [ g r- q c ] A p r Thermodynamics of Shearing Massless Scalar Field Spacetimes is InconsistentWith the Weyl Curvature Hypothesis
Daniele Gregoris,
1, 2, ‡ Yen Chin Ong,
1, 2, l and Bin Wang
1, 2, † Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University,180 Siwangting Road, Yangzhou City, Jiangsu Province, P.R. China 225002 School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Our Universe has an arrow of time. In accordance with the second law of thermodynamics,entropy has been increasing ever since the Big Bang. The fact that matter is in thermal equilibriumin the very early Universe, as indicated by the cosmic microwave background, has led to the ideathat gravitational entropy must be very low in the beginning. Penrose proposed that gravitationalentropy can be quantified by the Weyl curvature, which increases as structures formed. A concreterealization of such a measure is the Clifton-Ellis-Tavakol gravitational entropy, which has been shownto be increasing in quite a number of cosmological models. In this work, we show a counter-exampleinvolving a class of inhomogeneous universes that are supported by a chameleon massless scalar fieldand exhibit anisotropic spacetime shearing effects. The topology and the values of the three freeparameters of the model are constrained by imposing a positive energy density for the cosmic fluid,and the thermodynamical requirements which follow from the cosmological holographic principleand the second law. It is shown that a negative deceleration parameter and a time decreasing Weylcurvature automatically follow from those conditions. As a result, the Clifton-Ellis-Tavakol entropydecreases despite the shear continues to grow in time.
I. INTRODUCTION: THE ARROW OF TIME AND GRAVITATIONAL ENTROPY
The notion of time has always been an intriguing subject in both science and philosophy. Time, unlike space, hasa direction , it inevitably “flows” from the past to the future. In terms of the second law of thermodynamics, thearrow of time is reflected in the fact that statistically speaking, entropy tends to increase rather than decrease. Thisis because, as Boltzmann and Gibbs tell us, entropy counts the number of microstates via the formula S = k B log W ,where W is the number of microstates in which the energy of the molecules in a system can be arranged. In the phasespace then, a system naturally evolves from a region of smaller phase space volume to one with larger volume. Thereis hardly any physics at this stage – what we are dealing with is combinatorics. For example, there are just a lot moreways to have the wires of an earphone all tangled up than not. Therefore it is not at all surprising that one expectsentropy to increase merely because there are more ways for the configurations of a closed system to be in high entropystates than in lower entropy ones. The surprising thing is that this argument is time symmetric and so by appealingto combinatorics alone we should also expect entropy to be increasing towards the past. The fact that it does not –for otherwise there would cease to be an arrow of time – means that the beginning of the Universe must have a verylow entropy in some sense. In other words, the second law tells us that, since entropy is increasing, it must have beenlower in the past, all the way back to the Big Bang (see, however, [1]). It is because of the very low entropy of thevery early Universe that we exist at all, if everything has been in equilibrium at the very beginning, nothing wouldhave happened.The physics thus comes in by demanding that the initial condition of the Universe must be such that it is at a verylow entropy state, and then the combinatorics nature of the second law takes over and naturally evolves it towards ahigher entropy future, governed by various laws of physics. The question is not why entropy increases, as that wassettled by Boltzmann already. The question is: why is the very early Universe at such a low entropy state? In otherwords, the problem of the arrow of time is the problem of the initial condition of the Universe. To quote RichardFeynman in his Lectures on Physics [2], “so far as we know, all the fundamental laws of physics, such as Newton’sequations, are reversible. Then where does irreversibility come from? It comes from order going to disorder, but we donot understand this until we know the origin of the order.” We know from observational data of the cosmic microwave ‡ Electronic address: [email protected] l Electronic address: [email protected] † Electronic address: wang [email protected] For more detailed discussions regarding the issue of the arrow of time and its cosmological origin, see [3–5]. For an introductory article,see [6]. background (CMB) that the matter at the end of the epoch of recombination was already at thermal equilibrium, asshown by the almost perfect Planck distribution of the CMB. Normally we associate thermal equilibrium as a highentropy state. Thus, in order to have a low total entropy back then, the gravitational entropy must be properly takeninto account. Indeed, a smoothly distributed matter field like the conditions in the early Universe (with the densityperturbation being as mere δρ/ρ ∼ − ) is a low entropy state as far as gravity is concerned – gravity tends to clumpand contract matter, so structure formation is in accordance with the second law.How then does one define or quantify gravitational entropy? Penrose proposed that Weyl curvature can be usedfor this exact purpose [7]. Indeed, Weyl curvature describes how the shape of a body is distorted by the “tidal force”of a gravitational field [8]. It tends to increase during structure formation and gravitational collapse. Penrose thusproposed the Weyl curvature hypothesis , which claims that near the past singularity (the Big Bang) the Weyl curvaturemust vanish, and then it started to rise monotonically thereafter as matter starts clumping, stars and galaxies forming,and so on. If there is a crunch, future singularity can be arbitrarily distorted and so has large Weyl curvature, unlikethe initial singularity. A concrete realization of the notion of gravitational entropy is the Clifton-Ellis-Tavakol (CET)entropy [9], which essentially also measures the Weyl curvature.The question of the arrow of time is a deep one: there have been many proposals in the literature that attemptto explain why the initial gravitational entropy is so low, including – but not limited to – weakening the strength ofgravity during the very early universe (so that the smooth initial state is not an unusually low entropy state) [10],constructing a time symmetric universe (this ranges from the early model of Gold [11], in which the entropy gets lowerin the future as the universe shrinks in size, to a more sophisticated model of time symmetric multiverse by Carrolland Chen [12]; see also [13–16]), Penrose’s conformal cyclic universe [17], and “creation on a torus” in stringy modelthat identifies gravitational entropy with some notion of “geometric entropy” [18, 19]. There is as yet no consensusto the solution of the arrow of time problem, and it is not our aim in this paper to provide a better explanation.Instead, we are interested in a more modest question: Is the Weyl curvature hypothesis correct? More specifically,does Weyl curvature always increase in any physically realistic universe?
By physically realistic we do not mean thatit must satisfy all the observational data of the actual Universe , but only the weaker requirement that it shouldsatisfy well-established laws of physics in general, and notably the laws of thermodynamics in particular. If the Weylcurvature hypothesis is indeed correct, then it should hold in any logically consistent universe with a thermodynamicalarrow of time, as structures are formed. Structure formation comes in the form of inhomogeneities and anisotropies.In this work we investigate the joint effect of spatial inhomogeneities and of a cosmological shear, and we constrainthe model at the theoretical level by imposing a few physical conditions: the cosmic fluid in the model must have apositive energy density, the second law of thermodynamics must be obeyed in the matter sector, and the total matterentropy must be bounded by the area of the dynamical apparent horizon (the “cosmological holographic principle”,see more below) [20]. We then show that in this specific example – despite all these physically realistic requirements – the Weyl curvature hypothesis does not hold – the Weyl curvature is monotonically decreasing (and hence so is theCET gravitational entropy), while spacetime shear continues to increase as the universe expands.Let us now move on to explain the shearing spacetime cosmological model, before returning to the issue of grav-itational entropy. Our paper is organized as follows. In Sec. (II) we will provide further motivation to the modelfrom the viewpoint of cosmology, irrespective of the arrow of time issue. In other words, the model we examined isnot an exotic one cooked up just to serve as an ad hoc counter-example to the Weyl curvature hypothesis, but hasphysical motivation on its own. In Sec. (III) we shall review the most important physical properties of the modelunder analysis by computing its cosmological parameters. Then in Sec. (IV) we shall exhibit the constraints on theparameters of the model, which are derived from the cosmological holographic principle and from the second law ofthermodynamics. We also clarify the role of the position of the observer in such a universe. In Sec. (V) we shallreturn to compute the gravitational entropy, as defined by the CET proposal. In the concluding Sec. (VI), we willdiscuss the notion of gravitational entropy in general and what our finding might imply in the larger context of thearrow of time problem. In addition, we shall put our work in the context of the current research developments intheoretical cosmology, which are gradually appreciating the importance of constructing model-independent (i.e. notrelying on the Copernican principle in any step of the analysis) techniques for constraining cosmological models. This is so that it manifestly passes the “double standard test” made explicit by Price [3]: if a physical mechanism is supposed to explainpast low entropy (which gives rise to what we experience as the passing of time) without itself sneaking in time asymmetry, then thatmechanism should also be applicable to the “end state of time”, i.e. future conditions. In other words the scenario must make sense ifwe reverse the arrow of time, unless there is some a priori “natural” reason that breaks the symmetry and makes the past objectivelydistinct from the future (if so, one should explain this). We have reserved capital letter “Universe” for the actual one we live in, while lower case “universe” refers to any generic universe. Ofcourse, some statements regarding the latter might turn out to hold also for the former.
II. SHEARING SPACETIME AND THE EARLY UNIVERSE
It is widely believed that our Universe went through an exponentially accelerated expansion at the very early time.This process, known as “inflation” [21], explains the flatness problem (why is the spatial curvature so close to zero),the horizon problem (why is the CMB temperature isotropic in all directions that we look, despite those regionshaving no causal contact in a standard Big Bang cosmology without inflation), and the monopole problem (why thereis no magnetic monopole). Inflation explains these by essentially “washing away” all irregularities. Nevertheless, ithas been argued that inflation by itself does not explain the arrow of time [22–26], as usually inflation itself requiresspecial initial conditions to occur. For example, the simplest models with single inflaton field requires a “slow-roll”condition (see the discussions in [10, 19]). See, however, [27, 28].In addition to scalar fields, the inflationary epoch of the Universe may also involve spacetime shearing effects .Although the presence of initial inhomogeneities and anisotropies, if any, will likely be washed away by inflation (ifinflation can start), since not much is known about inflation, we cannot yet rule out models in which cosmic shearsremain after inflation (see the discussion involving “vector inflation” in [30]; also shear viscous effects can arise inwarm inflation [31]), although the measurement of an almost-isotropic distribution of the temperature of the cosmicmicrowave background radiation suggests that they are negligible in the present epoch [32]. In fact, from a purelymathematical perspective it can be proved that at least within the homogeneous but anisotropic Bianchi I models withregular matter content, a shear term dominates the primordial evolution, which is well approximated by the Kasnersolution, subsequently becoming negligibly small at late times [33, 34]. (In the presence of a cosmological constant,Bianchi models lack “primordial anisotropic hair” [35] . Such a “cosmic no-hair theorem” can be circumvented,however, in the presence of vector fields [36–40], for example). However, we stress that in general it is not enough toobserve the isotropy of one physical quantity, like the temperature of the cosmic microwave background radiation, forclaiming that all the other cosmological parameters should also be isotropic as well [41, 42]. For example, statisticalanisotropy can appear in the bispectrum of curvature perturbation even if it does not appear in the power spectrum[39]. In addition, large-scale asymmetries and alignments of astrophysical filaments along a preferred spatial directionhave been observed [43–47], e.g., the so-called “axis of evil” [48]. Nevertheless, there is as yet no consensus as to howmuch of these effects are due to systematical or contaminative errors in observation or in data analysis [49].An analytical and exact solution of the Einstein’s field equations of general relativity entirely written in termsof elementary functions describing both nontrivial shearing and expansion effects supported by a massless scalarfield in both open and closed topologies, has been investigated in [50–54] (see also page 261 of [55] for a summary).Therefore, it is important to discuss the physical viability of this class of cosmological spacetimes as a realistic modelof our Universe, at least at the early times. One such check is by investigating their thermodynamical properties.In the cosmological context, one of the theoretical thermodynamical constraint is formulated as the “cosmologicalholographic principle”, according to which the amount of matter entropy inside the region bounded by the dynamicalapparent horizon must not be larger than the area of the horizon itself [20]. Another requirement is that a physicalmodel should satisfy the second law of thermodynamics, which requires a non-decreasing entropy in time. Our goal isto clarify which, of any, of the two topologies is favoured by these two requirements and possibly set an upper boundfor the shear at early times with respect to the other cosmological parameters, namely the Hubble function and thematter parameter. In fact, we want to extend our way of thinking, which has already been proven as a valid toolfor constraining the strength of spatial inhomogeneities for the spherically symmetric Stephani universe (in whichpressure is a function of both space and time), to the inflationary epoch. In that case, the second law could be recastas an independent and complementary estimate of the present day “acceleration” of the Universe without relying onastrophysical measurements [56]. Morever, it should be emphasized that, the cosmological holographic principle is apowerful tool in testing dark energy models in late-time cosmology [57–59], inhomogeneous cosmological models suchas the Lemaˆıtre-Tolman-Bondi model (in which density is a function of space and time) [60], the number of spatialdimensions of the Universe [61], and the cosmic microwave background signatures [62, 63], just to mention a fewapplications.Furthermore, while primordial quasars and galaxies containing a supermassive black hole have been observed even atredshift z ∼
10 [64, 65], perturbation theory applied to a homogeneous and isotropic Friedmann universe and standardaccretion mechanisms cannot account for their existence [66, 67]. Thus, it has been argued that inhomogeneous Not to be confused with “cosmic shear”, which is the distortion of images of distant galaxies due to weak gravitational lensing by thelarge scale structure in the Universe [29]. In the commonly used Planck units, in which ~ = G = c = k B = 1, with ~ the reduced Planck constant, G the Newton’s constant, c thespeed of light, and k B the Boltzmann’s constant, the precise statement would be S A/ S denoting the entropy and A the area.For our purpose, it is enough to consider S . A . In this work, however, we do not employ the Planck units. shearing spacetimes supported by a massless scalar field may provide a valid framework for their description withoutthe need to invoke any quantum modification to general relativity [68, 69].The cosmological solutions we will study in this paper are algebraically Petrov type D, unlike the Friedmann metricwhich is of Petrov type O (conformally flat with only Ricci curvature). In other words, some Weyl curvature affectsthe evolution of the matter content and of the whole Universe. Therefore, we have in hands an exact frameworkfor testing the Weyl curvature conjecture, which states that the gravitational entropy in a non-stationary spacetimeshould be proportional to the square of the Weyl tensor, which consequently must grow during the time evolution[7, 70–72], for complementing the previous literature studies in homogeneous cosmologies [73], and black hole physics[74, 75], which include black rings [76].In this manuscript we will provide a more transparent physical interpretation of a class of mathematical solutionsof the Einstein’s equations of General Relativity found by Leibovitz-Lake-Van den Bergh-Wils-Collins-Lang-Maharaj[50–54] by proposing a novel set of conditions on the free parameters of their model. First and foremost, we require thatthe energy density of the cosmic fluid must be positive. Adopting a modern language, the cosmic fluid is interpretedas a so-called “chameleon field” [77, 78] because its equation of state parameter is energy-dependent, and as a masslessscalar field following the canonical formalism. After that, the evolution of these spacetimes is further constrained inlight of the cosmological holographic principle and the second law of thermodynamics, complementing our previousstudy of the shear-free and conformally flat Stephani model [56]. We will also comment on the consequences on thesign of the deceleration parameter, which will be shown to be negative after imposing those requirements, and beforerelying on any astrophysical datasets. III. SOME EXACT COSMOLOGICAL SHEARING SOLUTIONS WITH A MASSLESS SCALAR FIELD
In this section we will introduce the cosmological models that we want to investigate in light of the cosmologicalholographic principle, and of the second law of thermodynamics. Firstly, we will derive some constraints for their freeparameters by requiring that the energy density of the cosmic fluid must be non-negative, and then we will computethe kinematical variables characterizing the evolution of this spacetime.In a spherical coordinate system x µ =( t , r , θ , φ ), and adopting the Lorentzian segnature ( − , + , + , +), the spacetimemetric tensor d s = g µν d x µ d x ν = − (cid:16) cr l (cid:17) d t + d r ǫ + Cr + r h ǫ h ( t ) i (d θ + sin θ d φ ) , (1)with: h ( t ) = A sin( ct/l ) + B cos( ct/l ) if ǫ = − , (2) h ( t ) = − (cid:18) ct l (cid:19) + 2 Actl + B if ǫ = 0 , (3) h ( t ) = Ae ct/l + Be − ct/l if ǫ = 1 , (4)is an exact solution of the Einstein’s field equations of general relativity, G µν = (8 πG/c ) T µν , for a perfect fluid whoseequation of state relating pressure and energy density is [50–54] p = c ρ + 3 c C πG . (5)The former two account for an open topology of the universe, while the latter for a closed one. The stress-energytensor for the matter content is T µν = ( ρ + p/c ) u µ u ν + ( p/c ) g µν , in which we have introduced the observer four-velocity u µ = dx µ /dτ = cδ µt / √− g tt , u µ u µ = − c . Moreover, the constants A , B and C are the free parameters of themodel which are not constrained by the field equations, and ǫ accounts for the topology of the universe. Note that A , B , ǫ and ct/l are dimensionless quantities, while [C]=L − . In addition, l is a reference length scale which has beenintroduced for dimensional purposes, that from now on we will assume to be unity without loss of generality because The reader can find these information summarized on page 261 of [55]. it can be re-absorbed into the time coordinate as a rescaling factor. The equation of state of the cosmic fluid can bere-written in the form p = w ( ρ ) ρ , w ( ρ ) = c + 3 c C πGρ , (6)in which the constant equation of state parameter adopted in the standard cosmological modeling has been promotedto an energy-dependent chameleon field [77, 78], so named because the range of the force mediated by the scalarparticle becomes small in regions of high density, but shows its effect at large cosmic distances. In particular, thecomic fluid reduces to an ideal fluid with its energy density and pressure being directly proportional to each otherin the high energy regime, exhibiting the same asymptotic freedom which characterizes the bag model of quark-gluonplasma with the constant C playing the role of vacuum energy [79, 80]. This also leads to a similar notion of “bagenergy” as one finds in the contexts of quark physics, as will be discussed later. The chameleon properties of thecosmic fluid are suppressed in the limit C →
0. However, in this case the spacetime metric (1) would be ill-definedboth in the cases ǫ = − ǫ = 0 because of the unphysical Lorentz signature in its g rr component; while for thechoice ǫ = 1 such parameter would not play any role because it can be re-absorbed through a re-scaling of the radialcoordinate r .The adiabatic speed of sound within the fluid is c s = q ∂p∂ρ = c which is the same as a stiff fluid. Taking into accountthe canonical equations of the “fluid-scalar field correspondence” [81, 82], the Einstein’s equations for a scalar field Φminimally coupled to gravity can be derived by applying a variational principle to the total Lagrangian L = L EH + L m , (7)where L EH is the Einstein-Hilbert part, and the matter contribution can be written in terms of the kinetic andpotential energy of the scalar field L m = − g µν ∂ µ Φ ∂ ν Φ + V (Φ) . (8)In fact, the canonical equations, when the gradient of the scalar field is timelike, allow us to express the energy andpressure of a perfect fluid in terms of the kinetic and potential energy of the underlying scalar field as [83–85] c ρ = Φ ; µ Φ ; µ V (Φ) , p = Φ ; µ Φ ; µ − V (Φ) . (9)Thus, the equation of state (5) can be re-interpreted as describing a free inhomogeneous scalar field inside a constantpotential V (Φ) = − c C/ πG . As is well known, an additive constant entering the Lagrangian can be re-absorbed,shifting the zero energy level of the system, and does not affect the dynamical evolution of the scalar field, which is stillgoverned by the free Euler-Lagrange equation ✷ Φ := g µν ∇ µ ∂ ν Φ = 0. Therefore, the spacetime under investigation ispermeated by a fluid which behaves effectively as stiff matter, and consequently as a massless scalar field, from thehydrodynamic point of view. The relation (5) is named stiffened equation of state and constitutes a simplified versionof the Gr¨uneisen model, in which the constant C takes into account the deviations which occur at high pressure whichare likely to be realized in the early Universe [86–88]. Massless scalar fields (or equivalently stiff fluids) have alreadybeen adopted in the modeling of the early Universe. For example, they are a basic assumption in the formulation ofthe Belinskii-Khalatnikov-Lifschitz (BKL) locality conjecture for studying the Big Bang spacelike singularity [89–91].Furthermore, energy exchanges with a massless scalar field may cause the accretion of primordial black holes [92], andmore generally a stiff matter dominated era occurs both in the Zel’dovich model of a primordial universe constitutedby a cold gas of baryons [93], and when the cosmic fluid is represented by a relativistic self-gravitating Bose-Einsteincondensate [94].We can claim that the metrics (1) with the function h ( t ) defined by (2)-(3)-(4) are spatially inhomogeneous byconsidering the r -dependence affecting the Ricci scalar (which is a curvature invariant independent of the system ofcoordinates [95]): R = − ǫ + R + 6 Cr ( ǫ + 2 h ( t )) r ( ǫ + 2 h ( t )) , (10)in which R = 4( A + B ) for ǫ = − , (11) R = 4(4 A + B ) for ǫ = 0 , R = − AB for ǫ = 1 . We can compute the energy density of the cosmic fluid by inserting the equation of state (5) into the trace of theEinstein’s field equations: ρ = − c ( R + 18 C )16 πG , (12)where the Ricci scalar R has been given in (10). A well-defined (non-negative) energy density requires C ǫ + R r ( ǫ + 2 h ( t )) . (13)Furthermore, the energy density exhibits the two limiting behaviors ρ := lim r → ρ = ∞ · sgn( ǫ + R ) , ρ ∞ := lim r → + ∞ ρ = − Cc πG , (14)at the center of the configuration and in the far field limit, respectively, where sgn denotes the sign function. Thelatter expression shows that a non-negative energy density requires C
0, which opens up the possibility of having anegative pressure from (5) in certain spatial regions and/or at certain time intervals mimicking a cosmological constantterm. In fact, in this asymptotic regime the effective equation of state of the cosmic fluid reduces to p ∞ = − c ρ ∞ .Moreover, ρ > ǫ + R >
0, which in turn is equivalent to the following constraints between the free parametersin the three topologies: 4( A + B ) − > ǫ = − , (15)4(4 A + B ) > ǫ = 0 , − AB > ǫ = 1 . Therefore, the two parameters A and B live in a phase region bounded by a circumference of a circle, a parabola,and a hyperbola, respectively. Defining the Big Bang time t BB as the time at which the energy density diverges, wecan conclude that it is given implicitly by the condition ǫ + 2 h ( t BB ) = 0. Therefore, in our model of the Universethe time at which the initial singularity occurs is affected both by the topology and by the parameters A and B ,but not C . A qualitative difference with the more popular Lemaˆıtre-Tolman-Bondi is that the Big Bang time is notspace-dependent [96, 97].The kinematical variables characterizing the spacetime described by metric (1) can be computed following [33]. TheHubble function is given by H := u µ ; µ h ( t )3( ǫ + 2 h ( t )) r , (16)which diverges and vanishes for small and large r respectively, and is monotonically decreasing in between. In thisformula an over dot denotes a derivative with respect to the coordinate time. Interestingly, in this model the Hubblefunction is inhomogeneous allowing us to complement our previous analysis based on the Stephani model, whichinstead exhibits a homogeneous rate of expansion [56, 98]. The shear tensor reads σ ij = diag h − h ( t )3( Cr + ǫ )( ǫ + 2 h ( t )) r , ˙ h ( t ) r , ˙ h ( t ) r sin θ i , i, j = r, θ, φ , (17)which implies σ := 12 σ ij σ ij = 4 ˙ h ( t ) ǫ + 2 h ( t )) r = 3 H . (18)Thus, the shear displays a more severe divergence towards the center of the configuration, and it asymptotes to zerofaster at spatial infinity than the Hubble function. Moreover, the spacetime shearing effects are bounded by the rateof expansion of the Universe σ/H < u µ := u ν ∇ ν u µ = Cr + ǫr c δ µr . (19)Then, the generalized Friedmann equation (which is the mixed-rank time-time component of the Einstein’s equations)allows us to compute the spatial curvature as [33]: R = 16 πGρc − H + 2 σ = − R − C − H " ǫ + R − h ( t ) ( ǫ + 2 h ( t )) r − C , (20)where in the last step we used (18) and (12). The first equality means that in this model of the Universe, unlike theFriedmann cosmology, the evolution of the Hubble function is affected not only by the energy density permeatingthe space, but also by a certain linear combination of the invariant shear and of the spatial curvature. The Stephaniuniverse exhibits a similar behavior because the evolution of the Hubble function is affected not only by the abundanceof regular matter within the spacetime, but also by the strength of spatial inhomogeneities which plays the role ofan effective mass-energy parameter as we discussed in our previous work [56]. However, an important difference isthat the former spacetime is shear-free. Adopting the standard terminology, we can introduce the matter densityparameter Ω m = 8 πG H ρ = 3 c
16 ˙ h ( t ) [ ǫ + R − C ( ǫ + 2 h ( t )) r ] , (21)using (12) and (16). We may note that the matter density parameter is regular even at r = 0 because the divergencein the Hubble function has canceled the divergence in the energy density.Unlike the Friedmann spacetime, which is conformally flat, the model under consideration (1) displays a non-trivialWeyl curvature tensor C µνρσ because it is of the algebraic Petrov type D. We quantify the strength of the Weylcurvature applying the Newman-Penrose formalism [55, 100, 101]. Let l a = √ cr t − d r p Cr + ǫ ) , (22) n a = √ cr t + d r p Cr + ǫ ) ,m a = r p ǫ + 2 h ( t )2 (d θ + i sin θ d φ ) , i = − , be a null tetrad such that l a l a = n a n a = m a m a = ¯ m a ¯ m a = 0 , − l a n a = 1 = m a ¯ m a , (23)where an over bar stands for complex conjugation, in terms of which the metric (1) can be written in the formd s = − l ( a n b ) + 2 m ( a ¯ m b ) , where round parentheses denote symmetrization. The coframe (22) provides the canonicalform of the Newman-Penrose scalars related to the Weyl curvature tensor because Ψ = Ψ = Ψ = Ψ = 0, andΨ = − R + ǫ r (2 h ( t ) + ǫ ) , (24)where R can be obtained from (11). Thus, the quantity Ψ contains all the information we need about the Weylcurvature. We note that Ψ accounts for a Coulomb-like gravitational potential [102] and it is related to the “electric”( E µν ) and “magnetic” ( B µν ) Weyl components through (since it is purely real in our case) [55]:Ψ = E µν E µν − B µν B µν . (25)Finally, the covariant deceleration parameter in this class of metrics is given by [33]: q = 1 H σ − ˜ ∇ µ ˙ u µ c + ˙ u µ ˙ u µ c ! + Ω m (cid:18) ωc (cid:19) , (26)in which we have introduced the notation ω = p/ρ for the equation of state parameter, and˜ ∇ µ ˙ u µ = h µν h τ µ ∇ τ ˙ u ν = h τ ν ∇ τ ˙ u ν , (27)for the fully orthogonally projected covariant derivative, where h µν = g µν + u µ u ν is the spatial metric. Therefore, thedeceleration parameter can be written explicitly as a function of ǫ and h ( t ), and the derivative of h ( t ): q = 3 (cid:20) − c ( ǫ + 2 h ( t ))( ǫ + 2 ǫh ( t ) − h ( t ) (cid:21) . (28)We note that the parameter C does not play any direct role. Furthermore, the deceleration parameter is spatiallyhomogeneous, contrary to the case of the Stephani universe [98]. IV. A THERMODYNAMICAL ESTIMATE OF THE COSMOLOGICAL PARAMETERS
We are now ready to investigate how the cosmological holographic principle and the second law of thermodynamicscan provide a set of constraints between various cosmological parameters (deceleration parameter, expansion, shear,matter-energy abundance, and curvature strength) complementary and independent to those which may come fromastrophysical observations.We start by recalling that the location of the dynamical apparent horizon follows from the condition ||∇ ˜ r || = 0,where ˜ r = r · q ǫ +2 h ( t )2 is the areal radius [103]. Explicitly we must solve the algebraic equation( Cr + ǫ ) c ( ǫ + 2 h ( t )) − h ( t ) = 0 , (29)which admits a non-complex solution only for the closed topology ǫ = 1 (taking into account that C < r > r − C . (30)In this latter case two mathematical solutions r , = ± c ( ǫ + 2 h ( t )) · s h ( t ) − c ǫ ( ǫ + 2 h ( t )) C (31)can be found, of which only the positive root is of physical interest. Therefore, the dynamical apparent horizon islocated at: r AH = vuut C " h ( t ) c ( ǫ + 2 h ( t )) − ǫ ( ǫ + 2 h ( t ))2 (32)= s − AB + 4 e tc + 4 e − tc + 12 C (2 Ae tc + 2 Be − tc + 1) = s R − h ( t ) − ǫ C ( ǫ + 2 h ( t )) , where in the last step we have specialized the result to the topology ǫ = 1. The existence of the square root canbe easily guaranteed by restricting both A and B to be positive. In this case, taking into account (15), we obtain afurther constraint A · B . (33)For the topology with ǫ = 1, the deceleration parameter can be re-written as: q = − − R / h ( t )) c h ( t ) = − AB + Ae tc + Be − tc )2( Ae tc − Be − tc ) , (34)which is automatically negative if both A and B are positive. Moreover, the choices of positive A and B – if taken atface value – make the model with ǫ = 1 come without a big bang singularity because ǫ + 2 h ( t ) = 0 ∀ t ∈ R , and furtherrequiring a negative C together with the condition (30) they preserve the Lorentzian signature and the causalitystructure of the spacetime (1). Nevertheless, as we shall see the model is only valid after some time t > A. Cosmological Constraints From the Cosmological Holographic Principle
According to the cosmological holographic principle, the matter entropy, S m , inside the region bounded by thedynamical apparent horizon should be smaller than the area A AH of this spacetime region [20]. In the case of theuniverse (1), for the topology for which a dynamical apparent horizon indeed exists, its area is [104] A AH = 4 πr = 2 π ( R − h ( t ) − ǫ ) C ( ǫ + 2 h ( t )) = − π AB + Ae tc + Be − tc ) + 1 C (2 Ae tc + 2 Be − tc + 1) . (35)The entropy of the matter content inside the spacetime region bounded by the dynamical apparent horizon is [56]: S m = ˜ α r = ˜ α (cid:20) R − h ( t ) − ǫ C ( ǫ + 2 h ( t )) (cid:21) / = ˜ α (cid:20) − AB + Ae tc + Be − tc ) + 12 C (2 Ae tc + 2 Be − tc + 1) (cid:21) / . (36)The constant ˜ α = 4 k B (cid:18) πT c ~ (cid:19) s (1 + z e ) s , s = 6 , (37)summarizes all the information about the cosmic fluid. In more details, k B is the Boltzmann constant that enters theBoltzmann law of black body radiation, ~ is the reduced Planck constant, T is the temperature of the cosmic fluid,and z e is the redshift at the decoupling era. The power factor s = 3(1 + w ), which accounts for the stretching ofwavelengths in an expanding Universe (Hubble law), has been computed for the equation of state parameter w = 1that characterizes a stiff fluid. It is important not to confuse the factor s which depends on the type of the mattercontent inside the region bounded by the dynamical apparent horizon, and the geometrical factor 3 in the first equalityof (36), which instead is needed for computing the volume of this region.Mathematically, the condition which follows from the cosmological holographic principle is: S m A AH < L p = c G ~ ⇒ α π r AH < , (38)with α = 4 ˜ αL p , L p being the Planck length. Observing that the quantity on the left hand side of the latter inequalityis positive, taking into account that squaring both sides of that relation does not change the sense of the inequality,and that a multiplication by a negative factor (like C ) instead reverses it, we can re-write the condition provided bythe cosmological holographic principle as: R − (cid:18) π Cα (cid:19) h ( t ) − (cid:18) π Cα (cid:19) ǫ > . (39)Thus, regardless of the location of the observer within such a universe, the bag energy of the cosmic fluid is constrainedaccording to C < α ( R − ǫ − h ( t ))32 π ( ǫ + 2 h ( t )) . (40)The condition that C must be negative is automatically fulfilled, because it would require h ( t ) > − AB , (41)which is automatically guaranteed for positive A and B . B. Cosmological Constraints From the Second Law of Thermodynamics
In this subsection we will establish which relationships between the cosmological parameters characterizing thespacetime described by metric (1) are compatible with the second law of thermodynamics. In agreement with standardphysics, we will impose a time-increasing matter entropy for the cosmic fluid, and show that the further constraintsamong the free model parameters which would be derived do not contradict the ones already obtained. Therefore,in the class of models we are investigating, it is not necessary to weaken the second law of thermodynamics into the0so-called generalized second law which requires only the sum of the matter entropy and of the gravitational entropynot to decrease during the cosmological evolution. This latter modification was needed for preserving the physicalapplicability of a number of cosmological models based on a Friedmann metric supported by radiation [105–110], amixture of radiation and cosmological constant or a pressureless dark matter [111], even beyond general relativityimplementing torsion [112, 113] and braneworld [114] modifications. However, since we want to check the Weylcurvature hypothesis, it is most convenient to impose the second law on the matter sector, so that if the gravitationalentropy (measured in some way by the square of the Weyl curvature) does indeed decrease – which it does – we canstill have the possibility that the generalized second law may hold, from the matter contribution (otherwise we mayrule out this cosmology as thermodynamically unphysical).Although we will be returning to the issue of gravitatrional entropy later on, it is worth emphasing already atthis point that unlike in the case of stationary black holes [115], there is no agreement on a commonly accepteddefinition of gravitational entropy in cosmology. For example, the definition of cosmological entropy from the Weyltensor as S = C αβγδ C γδαβ fails when isotropic singularities occur [116]. Moreover, a normalized gravitational entropyof the form S = C αβγδ C γδαβ / ( R αβ R βα ), while addressing the previous issue, clearly diverges in vacuum [117], justto mention the limitations of a couple of approaches in the literature. Instead, in this section we will show thatour estimates on the size and age of the Universe are not affected by these uncertainties because just imposinga monotonically time-increasing matter entropy we can derive further realistic properties of the spacetimes underinvestigation.From (36) a time-increasing matter entropy would imply a time-increasing radius of the dynamical apparent horizon:d S m d t > ⇒ ˙ r AH > . (42)Then, using (32), the time evolution of the location of the dynamical apparent horizon can be computed explicitly as:˙ r AH = − ( R + ǫ ) ˙ h ( t )2 C r AH ( ǫ + 2 h ( t )) , (43)which gives the following inequality for accounting for the second law of thermodynamics :( R + ǫ ) ˙ h ( t ) > . (44)Implementing (15) and using (16), we can conclude that the second law of thermodynamics requires an expandinguniverse (i.e. with a positive Hubble function) in this model. A sharper condition would be: Ae ct − Be − ct > , (45)which imposes a lower limit on the size of the Universe h ( t ) > Be − ct , (46)or equivalently on its age t > c ln BA . (47)
This is the range of the validity of the model imposed by the second law , despite the model comes without a Big Bangsingularity .Therefore, the strength of the Weyl curvature (24) is decreasing with time because ˙ h ( t ) > C in the equation of state of the cosmic fluid (5) is not restricted by the the second law of thermodynamics either. Remember that C is negative, and that a multiplication by a negative factor switches the sense of the inequality. One could of course entertain the possibility that the second law can somehow be violated, so that the model can be extrapolated backin time to the infinite past, with the entropy being decreasing up to a certain point. Such a scenario has been contemplated, e.g., in thecontext of bouncing cosmology [118, 119]. The arrow of time problem would then require one to explain why the entropy shrinks downto such a small value during the bounce. See, however, [120]. Alternatively we can impose the second law and take the more pragmaticviewpoint that for time earlier than the inequality (47), the spacetime should be described by some other metric. V. SHEARING SPACETIME AND THE VIOLATION OF THE WEYL CURVATURE HYPOTHESIS
The
Clifton-Ellis-Tavakol entropy [9] (see also [121] for more explanations) is a concrete realization of the generalidea of Weyl curvature hypothesis. It appears to be a valuable proposal for a measure of the gravitational entropybecause it increases monotonically during the formation of cosmic structures, that is when a gravitational collapseoccurs [122, 123]. Moreover, the Clifton-Ellis-Tavakol proposal comes with many desirable features of a measure ofentropy, because it is always non-negative, it vanishes in – and only in – conformally flat spacetimes; it measures thestrength of the local anisotropies of the gravitational field; and it can reproduce the Bekenstein-Hawking entropy ofa black hole. The sturdiness of such proposal has been investigated explicitly in the inhomogeneous dust Lemaˆıtre-Tolman-Bondi universe and in the formation of local cosmic voids of about 50 −
100 Mpc size [124–126]. More generally,an appropriate notion for the gravitational entropy should be adopted for tracking the formation of cosmic structuresbecause we know from statistical mechanics that the entropy is nothing else than an estimate of how many differentmicrostates can realize the same macrostate, i.e. how many different inhomogeneous configurations on small scalesare compatible with the dynamics of the same homogeneous universe after appropriate coarse graining [127].In our spacetimes (1), the so-called “gravitational energy” [9] ρ grav = 16 πGc | Ψ | (48)is decreasing in time for an expanding universe with H >
0; in particular this is the case compatible with the secondlaw of thermodynamics, as previously discussed. We remark that the gravitational energy does not depend on thechameleon properties of the cosmic fluid, that is, on the parameter C .Writing the Einstein equations in the so-called “trace-reversed form” R µν = 8 πGc (cid:18) T µν − g µν T (cid:19) , (49)we can easily compute R µν R µν = (cid:18) πGc (cid:19) · T µν T µν = (cid:18) πGc (cid:19) · [( ρc ) + 3 p ] (50)= R + 18 CR + 108 C (51)= 108 C + 4[ ǫ + R + 6 Cr ( ǫ + 2 h ( t )) ] [ ǫ + R − Cr ( ǫ + 2 h ( t )) ] r ( ǫ + 2 h ( t )) . (52)Thus, following the line of thinking of [71, 128], we can estimate the relative strength of the Ricci curvature withrespect to the Weyl curvature (or equivalently of the “matter energy” vs. the gravitational energy): R µν R µν Ψ = 36( R + ǫ ) h ( R + ǫ ) + 3( R + ǫ ) Cr ( ǫ + 2 h ( t )) + 9 C r ( ǫ + 2 h ( t )) i , (53)which would be constant and time-independent (equal to 36) for a non-chameleon cosmic fluid. The condition R µν R µν Ψ > R + ǫ ) + 108( R + ǫ ) Cr ( ǫ + 2 h ( t )) + 324 C r ( ǫ + 2 h ( t )) ≡ [18 Cr ( ǫ + 2 h ( t )) + 3( R + ǫ )] + 26( R + ǫ ) > T grav = | u a ; b l a n b | π = c r π √ Cr + ǫ . (56)Interestingly, we note that for the choices ǫ = 0, and ǫ = −
1, the gravitational entropy is ill-defined because
C < C → − .2On the other hand, if we consider the ideal fluid limit together with ǫ = 1 we can eliminate this latter parameter byre-absorbing it into r . We stress that this is indeed in agreement with the discussion about the Lorentzian signatureof the spacetime metric below Eq.(6). The gravitational entropy according to the Clifton-Ellis-Tavakol proposal is S grav = Z Vol ρ grav T grav d V , (57)where d V = r sin θ ( ǫ + 2 h ( t ))2 √ Cr + ǫ d r d θ d φ (58)is the elementary volume form, in which we remind the reader that ǫ + 2 h ( t ) >
0. Thus, the gravitational entropyis decreasing as ∼ /h ( t ). We remark that this decreasing behavior just follows from the requirement of a positiveHubble function, which corresponds to an expanding Universe, and does not rely anyhow on our previous analysesabout the cosmological holographic principle and the second law of thermodynamics, making it a genuinely geometricproperty of the spacetime we are studying.From Eq.(18), and regardless of the particular choice of the value of ǫ , we obtain the time-evolution of the shear as:d σ d t = 4 c ˙ h ( t ) (2 h ( t ) − R )3 r ( ǫ + 2 h ( t )) . (59)Thus, σ is monotonically increasing for the case ǫ = 1, should the universe be expanding (recalling that R < A and B positive), although its corresponding gravitational entropy isdecreasing.A few different varieties of cosmological models were recently studied [129], which shows that the Clifton-Ellis-Tavakol gravitational entropy does start from zero at the Big Bang and monotonically grows afterwards in thosemodels. Mathematically, the crucial difference in our case is due to the fact that the gravitational temperature is aconstant in time, whereas in the examples studied in [129], both ρ grav and T grav diverge in the limit t →
0, allowingthe divergences to be canceled in a way such that the gravitational entropy vanishes. This does not happen in our casedue to the gravitational temperature being time-independent. In fact, for ǫ = 1, the gravitational entropy is neversingular in time. Thus we have demonstrated that in an accelerated expanding universe in which shear continues togrow, the CET gravitational entropy is nevertheless decreasing. More generally, this seems to be violating the Weylcurvature hypothesis. VI. DISCUSSION: WHAT IS GRAVITATIONAL ENTROPY?
Cosmological models describing the evolution of the primordial Universe may rely on three parameters, with theone based on the Shan-Chen fluid picture being one example [130]. Cosmological models accounting for the late-timedynamics of the Universe may rely on even more arbitrary quantities; for example the models with interaction in thedark sector may require also five free parameters (one for the equation of state of dark matter, two for accounting fora dynamical equation of state for dark energy, and two more which parametrize the energy exchanges between thetwo dark fluids) [131–136].In this report we have assumed an inhomogeneous spherically symmetric spacetime admitting anisotropic shearingeffects and whose evolution is driven by a stiffened fluid as a possible model for the early Universe. Our proposal isbased on some mathematical solution of the Einstein’s field equations found by Leibovitz-Lake-Van den Bergh-Wils-Collins-Lang-Maharaj. The three parameters of the model are constrained together with its topology by imposing apositive energy density for the cosmic fluid and analyzing the evolution of the matter entropy. In fact, according tothe cosmological holographic principle the matter entropy inside a region bounded by the dynamical apparent horizonshould be smaller than the area of that region (in the Planck units), while the more well-known second law requiresa non-decreasing entropy for a universe with a physically realistic evolution. Therefore, we could evaluate the “bagenergy” of those universes, their age, and their size. A negative deceleration parameter, and a time-decreasing Weylcurvature are obtained without the need of imposing any further condition. Despite being inhomogeneous, the locationof the observer does not affect those estimates, unlike the case of the Stephani universe [56]. Moreover, the effectsof the Weyl curvature has been explored in light of the role of gravitational entropy in the formation of primordialcosmological structures; this analysis was not possible in the Stephani spacetime which is conformally flat.Curiously, we have found the Weyl curvature – and hence the Clifton-Ellis-Tavakol gravitational entropy – ismonotonically decreasing as the universe expands, although the spacetime shear, and therefore the anisotropies, is3increasing (corresponding to some sort of structure formation). This is despite the fact that we have constrainedthe model parameters with physical requirements that the matter (massless scalar) field must have positive energydensity, must satisfy the second law of thermodynamics, and must satisfy the cosmological holographic principle. Weemphasize that this behavior persists even if we switch off the chameleon property of the scalar field. This behavioris surprising, since previous investigations have checked that the CET gravitational entropy is indeed increasing intime in a variety of cosmological models [129]. What conclusions can be drawn from this?Unlike matter entropy, gravitational entropy is a tricky notion. This is partly due to the fact that we do not knowwhat is the underlying “atoms” of gravitational degrees of freedom [137]. A useful definition of gravitational entropymust, at the very least, recover the Bekenstein-Hawking entropy of a black hole. It has been a longstanding problemas to what the Bekenstein-Hawking entropy is actually an entropy of (and why would adding charge or rotation decrease the entropy, compared to a neutral non-rotating black hole of the same mass?). This is not the only problem,however. Black holes have a lot more entropy than a typical matter configuration of the same size and energy. Theformer has S ∼ A (in Planck units), whereas the latter has only S ∼ A / [138, 139]. Consequently, as a star of mass M collapses into a black hole, its entropy increases by a staggering factor of 10 ( M/M ⊙ ) / , where M ⊙ denotes asolar mass [139]. Perhaps the process of collapse involves a huge increase in gravitational entropy, for whatever reasonyet to be fully understood.If the notion of gravitational entropy is a good one , then it should start small or even zero at the Big Bang, and thenmonotonically grow as structures like stars and galaxies and eventually black holes formed. In other words, it should,in some way, measure the increase in anisotropy . Since Weyl curvature increases during structure formation, itmakes sense to define gravitational entropy such that said quantity does indeed increase. One of the most promosingcandidate is the Clifton-Ellis-Tavakol gravitational entropy. Nevertheless, we have seen that inhomogeneities thatarise together with a nontrivial spacetime shear, can correspond to a decrease in the Weyl curvature, and thus alsothe CET gravitational entropy. This could indicate that we need to have a better definition for gravitational entropy,and the role of Weyl curvature in gravitational entropy should also be further revised. That is to say, the relationbetween gravitational entropy and spacetime shear might not be so straightforward. Furthermore, in view of theBekenstein-Hawking entropy of a black hole can be interpreted as entanglement entropy [142–146], perhaps the roleof entanglement entropy should also be considered [147]. On the other hand, perhaps other notions of “geometricalentropy”, such as the “creation on a torus” scenario that makes use of deep results in global differential geometry[18, 19] is more useful to explain the initial low entropy state of the Universe.Finally, let us remark that, irrespective of the arrow of time issue, our paper fits inside the wider cosmologicalresearch which is trying to provide a critical assessment of the astrophysical datasets. In fact, many drawbacks ofthe Λ-Cold Dark Matter model, like the Hubble tension, the coincidence problem, and even the predicted existence ofdark energy, may be caused by interpreting the cosmological data which have been refined already implementing theCopernican principle [148–150]. Our results on the other hand are based on theoretical considerations, which shouldcomplement observationally obtained constraints. Acknowledgement
Y.C.O. thanks NNSFC (grant No.11922508 & No.11705162) and the Natural Science Foundation of Jiangsu Province(No.BK20170479) for funding support. D.G. acknowledges support from China Postdoctoral Science Foundation (grant Wallace has argued that gravitational entropy is irrelevant in most contexts except in black hole physics, and that it suffices to considerthe dynamics caused by gravitational interactions [140]. For our purpose, the mathematics is clear: Weyl curvature decreases in time.Whether this is really a measure of entropy caused by gravity acting on matter, or entropy of gravity, requires a deeper scrutiny.Essentially, this has to do with the decomposition of the Riemann curvature tensor into Ricci and Weyl part – the latter remains, inpart, free because its value is not provided by the field equations, but only some constraints must be accounted for through the Ricciidentities. Thus one may say that the Weyl tensor constitutes the “gravitational” or “geometrical” part of the theory. See [141] forfurther discussions and implications. At least during the matter domination epoch. Thereafter, the Universe becomes dark energy dominated, and eventually even blackholes would Hawking evaporate away, though the total entropy of the Universe, including that of all the Hawking quanta, should remainincreasing. [1] Sheldon Goldstein, Roderich Tumulka, Nino Zanghi, “Is the Hypothesis About a Low Entropy Initial State of the UniverseNecessary for Explaining the Arrow of Time?”, Phys. Rev. D (2016) 023520, [arXiv:astro-ph.CO/1602.05601].[2] Richard Feynman, Matthew Sands, Robert B. Leighton, The Feynmann Lectures on Physics , Addison Wesley PublishingCo., Reading, MA, 1963, pp. 46-146-9.[3] Huw Price, “Cosmology, Time’s Arrow, and That Old Double Standard”, in
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