Entropy of Self-Gravitating Anisotropic Matter

We examine the entropy of self-gravitating anisotropic matter confined to a box in the context of general relativity. The configuration of self-gravitating matter is spherically symmetric, but has anisotropic pressure of which angular part is different from the radial one. We deduce the entropy from the relation between the thermodynamical laws and the continuity equation. It is verified that variational equation for this entropy reproduces the gravitational field equation for the anisotropic matter. This result re-assures us the correspondence between gravity and thermodynamics. We apply this method to calculate the entropies of a few objects such as black holes and wormholes.


I. INTRODUCTION
Ever since Hawking deduced the thermodynamics of black holes [1], thermodynamic approach to gravitating systems [2][3][4][5][6] provides good insights on the systems because it considers a relatively few thermodynamic variables rather than solving complicated dynamic gravitational field equations. Getting thermodynamic quantities has thus been widely studied to understand gravitating systems.
Examining self-gravitating systems has been one of those efforts for decades, which helps to understand astrophysical systems. Especially, the entropy of a spherically symmetric self-gravitating radiation and its stability were calculated in a series of researches [7,8]. Those studies show that requiring maximum entropy of a spherical box of selfgravitating radiation reproduces the Tolman-Oppenheimer-Volkoff (TOV) equation for hydrostatic equilibrium [9]. This equivalence has been dubbed as 'Maximum Entropy Principle'(MEP). The total entropy is regarded as an action functional of mass, m(r), the energy density ρ(r) and the radius r. Varying the total entropy, one gets the hydrostatic equation. Hence, having exact entropy of a system enables one to get hydrostatic equation through MEP. In most of those studies, the gravitating matters are assumed to be perfect fluids of which pressure p(r) = wρ(r) is locally isotropic. The entropy density for perfect fluids can be deduced from the continuity equation with series of thermodynamic equation combined [10]. It is given by a function of the energy density and w, that is, s P ≡ s P (ρ, w).
When the energy distribution is smooth in a sufficiently small region of volume V , one can define energy density ρ from U (T, V ) = ρ(T )V for the total energy U inside the volume. Using the first law of thermodynamics, T dS = dU + pdV , we have where T, S denote the temperature and entropy of the system respectively. Since the entropy is a scalar quantity, the exactness condition of S gives ρ as a function of T : for a perfect fluid with p = wρ (w 1 = w 2 = w 3 ≡ w). The entropy density of the fluid can be deduced by considering an isochoric process, for which dU = T dS, 1 where the integral constant can be set to zero by the third law of thermodynamics. The constant α w depends on w and the physical nature of matter consisting the fluid. This entropy density was already given in Ref. [11,12]. For radiation with w = 1/3, the density ρ rad = σ rad T 4 and the constant σ rad is the Stefan-Boltzmann constant. 2 Although majority of researches focus on the dynamics of perfect fluid [13][14][15], anisotropic matter in cosmological configuration has recently drawn interests [16][17][18][19][20][21][22][23][24]. For example, the authors in [25] obtained stable black holes with anisotropic matter. It is well known that for a wormhole to be traversable, the throat of the wormhole needs to be made of anisotropic matter [26,27] in the context of classical general relativity. Thus studies on the dynamics of anisotropic matter including thermodynamics become more important than ever.
For an anisotropic matter satisfying the equation of state p k = w k ρ, k = 1, 2, 3, the entropy density s A will depend on ρ, and w k . That is, s A ≡ s A (ρ, w 1 , w 2 , w 3 ). When all w's are equal, s A goes to that of the perfect fluid. Since an entropy density is a scalar quantity, s A can be written as a product of s P and a scalar function φ that describes the effect of anisotropy: where φ(w k ) → 1 in the isotropic limit. The function φ may contain matter information which is not described by the equation of state.
In the above discussion, the pressure is divided into two parts. One is the isotropic part of pressure p iso and the other is the deviation from it, p − p iso . However, there is an arbitrariness in the choice of the isotropic part. One example is that one can divide the pressure into an isotropic part with an averaged pressure p iso ≡p = (p 1 + p 2 + p 3 )/3 and the difference from the isotropic pressure (p 1 −p, p 2 −p, p 3 −p). The other example could be that p iso ≡ p 1 , and so on. The entropy should be independent of this choice. This freedom of choice will be gauged by a compensation vector later in this work.
We obtain the explicit form of the entropy density of self-gravitating anisotropic matter confined to a spherically symmetric box of radius R from thermodynamic consideration up to a constant multiplication factor, which depends on the individual characteristics of matter. With this entropy density, we perform the variation of total entropy of the anisotropic matter and show that the variational equation reproduces a modified TOV equation. The result of this paper shows that MEP still holds. We also seek applications of this entropy to cosmological phenomena.
The order of the paper is the following: We review properties of anisotropic matter in Sec. II. In Sec. III, the entropy of an anisotropic matter is obtained by comparing the continuity equation with thermodynamic relations. The maximum entropy principle for a perfect fluid is briefly reviewed in Sec. IV. The MEP yields the exact dynamical equation (modified TOV) using the obtained entropy density of the anisotropic matter in Sec. III. Finally, in Sec. V, we obtain the total entropy of a ball of anisotropic matter. We apply this calculation to discuss properties of objects such as wormhole and black hole.

II. ANISOTROPIC MATTER
The stress-energy tensor T ab of a relativistic matter can be divided into a thermostatic part T ab 0 and a dissipative part T ab 1 as The dissipative part is written as where u a is a timelike unit normal vector field. The quantities, q is interpreted as a heat flux vector and π ab as a viscous shear tensor. The viscous shear tensor satisfies π ab u b = 0 or u a π ab u b = 0 for the Landau frame or the Eckart frame respectively [10]. The thermostatic part of a perfect fluid, pressure being isotropic, has the form where ρ is the rest frame energy density and p is the isotropic pressure. The set of four mutually orthogonal vector fields {u a , x a , y a , z a } conforms a frame of orthonormal vector fields. That is, x a x a = y a y a = z a z a = 1.
However, when matter is not perfect, more general tensor for anisotropic pressure replaces T 0 , for example, elastic solid has anisotropic stress tensor [28]. In the absence of sheer, off-diagonal components of π ab vanishes. The energymomentum tensor of the anisotropic matter being considered in this article has the form: 3 where A in T ab A represents 'anisotropic' property and ρ, the rest frame energy density, p k (k = 1, 2, 3) denotes the pressure measured in locally orthogonal rest frame.
A spherically symmetric, static metric can be written as In this coordinate system, by spherical symmetry, we have p θ = p φ . We denote p 1 ≡ p r and p 2 ≡ p θ = p ϕ from now on. The Einstein equations G t t = 8πT t t and G r r = 8πT r r read With these equations and G θ θ = 8πT θ θ gives the modified TOV equation: which resembles the TOV equation of the isotropic fluid [9] up to the last term:

III. ENTROPY DENSITY OF ANISOTROPIC MATTER
In the present article, we consider a spherically symmetric static configuration, for example, a solution in [25]. The temporal part of the continuity equation, ∇ a T ab = 0, for non-dissipative matter, can be interpreted as a thermal relation Here s A denotes the entropy density of anisotropic matter and n, µ N represents the number density of particles, chemical potential respectively. Because the system is static, we examine the case that the dissipative part T ab 1 of the stress-energy tensor vanishes. In the presence of the dissipative part, T ab 1 = 0, the entropy vector s a A can be defined [10], [29]: 3 The anisotropic matter can also be regarded as an imperfect fluid. As we discussed in Sec. I, there is no preferred decomposition of the isotropic part in (10). For example, the isotropic part of the pressure can be p or p + π: where the terms in the bracket represent the isotropic part and σ ab is the traceless part of π ab , i.e., π ab = π(x)h ab + σ ab . One can find that p, π can have any values provided that they satisfy p + π = (p 1 + p 2 + p 3 )/3.
where β is the inverse temperature (β ≡ T −1 ). The net entropy production can be written as [30], Hence, there are two conditions in which the entropy is conserved. One is that there is a Killing vector field ξ a ≡ βu a and the other is that, T 1 = 0 or T ab = T ab A , i.e., no dissipation. To analyse spherically symmetric configuration, we impose that the anisotropic pressure to be When the number of particles does not change, ∇ a (nu a ) = 0, the continuity equation reads: Let us write the entropy density of anisotropic matter as s A = φ · s P . Here, s P ∝ ρ 1 1+w u a is the entropy density for a perfect fluid with an isotropic pressure p iso = wρ, and the function φ ≡ φ(w 1 − w, w 2 − w, w 3 − w) describes how much the entropy deviates from that of the isotropic form in the presence of an anisotropy. Hence, φ = 1 when w 1 = w 2 = w 3 , i.e., when isotropic.
The equation (19) is where s a A ≡ s A u a and s a P ≡ α w ρ 1 1+w u a . In general, the energy-momentum tensor (10) can be divided as: The continuity equation reads The above equation can be written as a divergence relation of the entropy density vector of a perfect fulid s P ∝ ρ 1/(1+w) u a : The case when w = −1 or ρ + p = 0 will be treated separately. Comparing this equation with the Eq. (20) gives where Φ a (x) is a vector orthogonal to u a (Φ a u a = 0) and when matter is isotropic Φ a = 0. This is the compensation vector discussed in Sec. I. For the metric (11), a calculation in the orthonormal basis gives x · ∇x a = 0, y · ∇y a = − 1 r δ ar , z · ∇z a = − 1 r δ ar − cot θ δ aθ .
With w 2 = w 3 , one gets from (24), (26), Since the factor φ depends only on the radial coordinate, one can determine the components of Φ a as Φ t = 0, Φ θ = − cot θ, Φ ϕ = 0 and Φ r = Φ r (r) without loss of generality. Hence, a function of r is sufficient to gauge the difference between two isotropic pressures. Note that w 1 do not appear in Eq. (27). This indicates that the radial pressure is absorbed in the isotropic pressure part, namely in the w. Solving this equations gives where C is an integral constant, and Φ(r) ≡ e Φr is a function to be determined. We call this function Φ as a compensation factor. Therefore, in general, the anisotropic entropy density has a form: The entropy density is a function of energy density and other related parameters(w's here). Because entropy density is an intensive quantity, the r dependence above seems awkward at first sight. Actually, the entropy density can be written as s A ≡ s A (ρ), that is, a function of energy density. One can regard r as a function of ρ since ρ satisfies a first order differential equation for a given metric u, v in Eq. (11). For a specific u(r), v(r), the above equation solves r as a function of energy density, r = r(ρ).
If one knows a compensation factor Φ w (r) for a choice, p iso = wρ, the compensation factor Φ w ′ (r) in other choice p ′ iso = w ′ ρ can be calculated. By equating entropies one gets Φ w ′ in terms of Φ w . Obtaining a compensation factor Φ for a choice which solves to produce a modified TOV is not always an easy task. However, if there is a choice w 0 that Φ is a constant, general compensation factors will have the form: We will show that, with a choice p iso = p 1 or w 0 = w 1 , the MEP with a constant P hi produces the exact modified TOV equation. Consequently, one can write down the entropy density of anisotropic matter as: where α (w1,w2) is a constant that does not modify the equation of motion and is determined by the physical nature of anisotropic matter as we mentioned above in the footnote 2 (page 2). When ρ + p 1 = 0 or w 1 = −1, it is obvious that one cannot use the formula (33) directly. Thankfully, one can use the freedom of choice for the isotropic part when one encounters similar problems. As an example, we obtain an explicit form of the compensation factor for w 1 = −1 case in the next section.

IV. REQUIRING MAXIMUM ENTROPY
The local maximum of entropy of relativistic matter coincides with dynamically stable equilibrium configuration. (MEP) [7,8]. The premises in those papers about extrema of total entropy can also be applicable to our system. The system being considered here is spherically symmetric, time symmetric and static, and the only difference is that the pressure of matter is anisotropic. Thus, all the arguments used in [8] about extrinsic curvature K ab of a spacelike hyper-surface Σ holds here.
In this section, we briefly review the MEP for an isotropic matter(perfect fluid) and then show that it also holds for the anisotropic matter. Finally, for the exceptional case with w 1 = −1, we obtain the entropy density separately.

A. Isotropic matter case
The total entropy for the isotropic fluid in a spherical box of radius R is given by The total entropy can be regarded as an action integral I = For anisotropic matter with static, spherical symmetry, in accordance with MEP, variation of total entropy for the entropy density (33)(Φ = 1), gives the exact modified TOV equation (13). The proof is the following: The total entropy of a box of radius R is given by In this case, the action to be extremize is given by where L 0 is the Lagrangian for the perfect fluid (35). The Euler-Lagrange equation reproduces the modified TOV equation (13) where we use the relation m ′ = 4πr 2 ρ.
C. Anisotropic matter when w1 = −1 The entropy density for anisotropic matter depends on energy density ρ(r) = 4πr 2 m ′ (r). Thus the Lagrangian for perfect fluid (35) can be generalised into the form: The Euler equation, can be rewritten as a form of modified TOV equation: Comparing this to the modified TOV equation (13) one obtains: For w 1 = −1, the Lagrangian exactly corresponds to the one for the entropy density (33) obtained in Sec.III, hence confirms our result. However, for w 1 = −1, the a does not exist for the relation (41) to have a form of TOV equation. Since the solution with w 1 = −1 was explicitly obtained in [25], it is sufficient that we use the results from the article: where M and K are constants. Putting this information into Eq. (40) instead of (41) gives a and b: 4 Therefore, the entropy density which satisfy the maximum entropy principle when w 1 = −1 is where s 0 is a constant. As we discussed in the previous section, one can determine 'the compensation factor' Φ(r) for the above entropy density. When w 1 = −1, to avoid the vanishing denominator, one can choose the isotropic part of the pressure as p iso ≡ p 1 − 2p 2 or w iso = w 1 − 2w 2 (= −1 − 2w 2 ). The entropy density using the anistropic factor (28) is obtained as by replacing −1 − 2w 2 in place of w 1 in (29). After requiring MEP, one obtains the factor Φ(r) = r −2w2 up to a constant. By using the relation ρ(r) ∝ r −2(1+w2) in (43), one reproduces the entropy in (45).

V. SUMMARY AND DISCUSSIONS
In this paper we considered a system of static, spherically symmetric matter of which pressure satisfies the equation of state p 1 = w 1 ρ, p 2 = p 3 = w 2 ρ. We obtained a form of entropy density of anisotropic matter for all w ′ s. We observed that the choice of an isotropic pressure from an anisotropic pressure has an arbitrariness. A compensation vector gauges this arbitrariness. This gauge freedom could be beneficial when we meet a problem. In the process, the MEP played a key role to obtain the exact form of the entropy. We showed that the requirement of local maximum of total entropy gives the exact Einstein equation of the system composed of anisotropic matter. This observation supports the correspondence between thermodynamics and gravity. By the guidance of this correspondence, the entropy density for p 1 = −ρ or w 1 = −1 was obtained separately.
The entropy density we obtained in this article has a multiplicative factor that describes the anisotropic effect. The quantity we obtained is a function of radius r and w 2 − w 1 . This comes from the use of the spherical coordinate system and the simple form of anisotropy between the radial and the angular directions. In a more general situation, the entropy density is expected to have various forms of functions of energy density and pressure.
Total entropy of radiation confined in a box of radius R was obtained [8] for a solution regular at the center r = 0. By using the same method, we obtained general form of total entropy of anisotropic matter confined in a box of radius R which is regular at r = 0: One can verify that derivative of the function on the right-hand side is the Lagrangian (37) by using the relation (40). We call the indefinite integral S(r) as 'entropy function'. One can see that the above total entropy reproduces that in [8] when w 1 = w 2 = w 3 = 1/3. In order for a solution to be regular at r = 0, the regularity at the centre requires that at r = 0, m(r) ∼ (4πr 3 /3)ρ(0) ∼ r 3 , hence, for ρ(r) ∼ r α , α > 0, otherwise we have singular solutions. To analyze singularity more explicitly, one needs to have an exact solution. The properties of solutions with w 1 = −1 at r = 0 is extensively analyzed in [25] with the exact solutions described in the previous section.
The entropy function above can be further simplified to by using m ′ = 4πr 2 ρ and the relations (18) and (33).
Recently, in Ref. [31] the author suggested conditions to remedy a naked singularity for a black hole system with an anisotropic fluid. In that article, one of the conditions is ρ + p 1 + 2p 2 < 0, which indicates a violation of the strong energy condition. Putting this into the relations (47) or (48), the entropy function takes a negative value. Since entropy is defined by a possible number of physical configurations of a system, negative value of entropy seems unreasonable. If we look into the derivation of total entropy carefully, the total entropy, as a definite integral is always positive since the entropy function S A (r) is monotonically increasing function of r even if S A (r) < 0. It is because S ′ A (r) = L A , the Lagrangian density, which is positive. Therefore, even though ρ + p 1 + 2p 2 < 0, the total entropy of the system is positive.
A wormhole is a solution of general relativity that has a throat and two sides of entrance and exit [32][33][34]. In the Einstein gravity theory, the throat of a traversable wormhole needs to be made of exotic, anisotropic material which violates energy conditions (especially null energy condition [35] along with weak, strong and dominant energy conditions). If anisotropic matter (that creates a wormhole) is bound (r < R) and the wormhole's throat is located at r = B, the entropy of the wormhole is given by provided that both sides of the wormhole have the same shape. Morris and Thorne [26] suggested specific wormhole solutions which use exotic material minimally: the zero-tidal force solutions, a solution with a finite radial cutoff of the stress-energy and solutions with exotic matter limited to the throat vicinity. The first and second solutions satisfy the equation of state of type p k = w k ρ with 1 + w 1 + 2w 2 = 0. Although the equation of state of the third solutions do not have the linear form, the matter still satisfy ρ+p 1 +2p 2 = 0. Putting these conditions to the total entropy formula (47),(48), one immediately gets indefinite value for the entropy function. From the Einstein equation for the metric (11), one gets When u ′ = 0 throughout the region where exotic matter is used, ρ + p 1 + 2p 2 = 0. The three examples in [26] is the case. For the same metric, the entropy function can be written as where we use the relation (12). We may slightly perturb the above wormhole solutions to avoid indefinite value of the entropy function. If we compare two configurations of S(u ′ = 0) and S(|u ′ | = 0), one may tell which configuration is more favourable than the other. For a smooth function u ′ (r) (|u ′ | ≪ 1 with dimension of inverse length) near the throat vicinity, the quantities (u ′ ) 2 , u ′′ are much smaller than u ′ . The denominator of the above equation reduces to The sign of the first term may switch from positive to negative and vice versa. For example, in the third wormhole in [26] the mass is given by m(r) = b[1 − (r − b)/a] 2 , where u ′ = 0 in the region b ≤ r ≤ b + a. Provided that for a very small perturbation of u ′ = 0, the solution of m(r) has the same quadratic form, one can see the occasional sign change for various values of a and b. The term u ′ /2 may or may not prevent the denominator from the sign change.
The entropy function becomes extremely large when the denominator approaches zero. When there is a sign change, the indefinite value makes comparison difficult. As we discussed in Sec. III, the freedom of choice in isotropic pressure part may help us prevent this pathology. When we set the isotropic pressure part as P ≡ p 1 + 2p 2 , the process of getting the anisotropic factor φ suffers the same pathology when ρ + p 1 + 2p 2 = 0 or w 1 + 2w 2 = −1 as in the case when w 1 = −1. We expect that with proper 'compensation factor' Φ one can obtain finite form of entropy density by using a similar process in Sec. IV C.
After comparing the total entropy, differences of the entropy functions, ∆S = [S A (r, u ′ = 0) − S A (r, u ′ = 0)] b a , one can address thermodynamical stability of the configuration u ′ = 0. To this end, further analysis with wider range of exact solutions including u ′ = 0 is necessary.
Incidentally, we get another form of the entropy function S A (r) = −4π (m + 4πr 3 p 1 ) using R 0 0 = −4π(ρ + p 1 + 2p 2 ). Most of the issues about wormholes come from the fact that matter composing wormholes are not ordinary but exotic matter. There are various ways to overcome or to go around these issues. For a modified gravity theory, one may find wormholes without the use of exotic matter. There are many articles that a throat of traversable wormholes is composed of ordinary matter in various modified gravity theories, e.g., Einstein-Gauss-Bonet theory [36] or Lovelock theory [37], f (R) gravity [38], etc. [39,40]. Similar analysis on the entropy for the wormholes in those theories is worth trying. On the other hand, if we stick to the Einstein gravity, our result shows that building materials or conditions should be carefully chosen so that the total entropy (47) takes finite value.
The self-gravitating solutions for the anisotropic matter were classified in [41]. For the case of matter with non-linear equation of states, deeper studies on the entropy formula will be required.