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In this section, we start by discussing the properties of the static configuration in a general n-dimensional diffeomorphism-covariant purely gravitational theory sourced by a self-gravitating perfect fluid. As mentioned above, we consider a static configuration in n-dimensional spacetime. In this situation, there exists a static Killing vector field
$ \xi^a $ satisfying$ {\cal{L}}_\xi g_{ab} = \nabla_{(a} \xi_{b)} = 0 $ . The integral curves of$ \xi^a $ are the worldlines of static observers in spacetime. The velocity vector field is given by$ u^a = \chi^{-1} \xi^a $ , in which$ \chi = \sqrt{-\xi^a\xi_a} $ is the red-shift factor for the static observers and$ u^a $ is also the normal vector to$ \Sigma $ . The induced metric$ h^{ab} $ in$ \Sigma $ is given by$ h^{ab} = g^{ab}+\chi^{-2}\xi^a\xi^b\,. $
(1) The Lagrangian n-form in this theory is given by
$ {\boldsymbol{L}} = {\boldsymbol{L}}_{\rm{grav}}+{\boldsymbol{L}}_{\rm{fluid}}\,, $
(2) in which
$ {\boldsymbol{L}}_{\rm{grav}} $ and$ {\boldsymbol{L}}_{\rm{mt}} $ are the gravitational part and fluid part of the Lagrangian, respectively. The gravitational part of the Lagrangian is a function of the metric$ g_{ab} $ , Riemann tensor$ R_{bcde} $ and its higher-order derivative$ \nabla_{a_1}\cdots \nabla_{a_k} R_{bcde} $ . Using the relationship$ 2\nabla_{[a}\nabla_{b]}T_{c_1\cdots c_k} = \sum\limits_{i = 1}^{i = k}R_{abc_i}{}^d T_{c_1\cdots d \cdots c_k} $
(3) to exchange the indices of the derivative operators, it is not hard to verify that the Lagrangian L can be reexpressed as
$ {\boldsymbol{L}}_{\rm{grav}} = {\boldsymbol{\epsilon}}{\cal{L}}_{\rm{grav}}(g_{ab}, R_{bcde}, \cdots, \nabla_{(a_1}\cdots \nabla_{a_k)}R_{bcde},\cdots)\,, $
(4) This is exactly the expression of the Lagrangian considered in [8] for the general diffeomorphism-covariant gravitational theory. In our discussion, we use boldface symbols to denote the differential forms in spacetime. The gravitational equations are given by variations of the Lagrangian and we can express them by [8]
$ {\cal{E}}_{ab} = H_{ab}-T_{ab} $
(5) with
$ H_{ab} = A_{ab}+P_{acde}R_b{}^{cde}-2\nabla^c\nabla^d P_{acdb}-\frac{1}{2} g_{ab}{\cal{L}}_{\rm{grav}} $
(6) and the stress-energy tensor of the perfect fluid
$ T_{ab} = \rho u_a u_b+p (g_{ab}+u_a u_b)\,, $
(7) in which
$ \rho $ and p are the energy density and pressure of the fluid, respectively, and we have denoted$ A_{ab} = \frac{\partial {\cal{L}}_{\rm{grav}}}{\partial g^{ab}}\,,\quad P^{abcd} = \frac{\delta {\cal{L}}_{\rm{grav}}}{\delta R_{abcd}}\,. $
(8) The expression of the stress-energy tensor implies that static observers are also comoving observers of the fluid.
The Theorem of the entropy principle shows that we need to consider an off/on-shell static configuration which satisfies the gravitational constraint equation (time-time component of the equation of motion)
$ \rho = H_{uu} = H_{ab}u^a u^b\,. $
(9) Therefore, “off-shell” refers specifically to the off-shell static configuration which does not obey the spatial gravitational equations
$ p h_{ab}\neq\hat{H}_{ab} = h_{ac}h_{bd}H^{cd}\,. $
(10) In the following, we consider the thermodynamics of the self-gravitating perfect fluid which satisfies Tolman's law
$ T\chi = T_0 $ in a static configuration, where$ T_0 $ is a constant and can be regarded as the red-shift temperature. Without loss of generality, we shall set$ T_0 = 1 $ , such that$ T = \chi^{-1}\,. $
(11) The entropy density s is a function of the energy density
$ \rho $ and the particle number density n, i.e.,$ s = s(\rho ,n) $ . From the familiar first law for region C, one can derive the local first law and the Gibbs-Duhem relation of the fluid [16],$ \begin{aligned}[b] {\rm d}\rho =& T{\rm d}s+\mu {\rm d}n\,,\\ \rho =& T s-p+\mu n\,, \end{aligned}$
(12) where
$ \mu $ is the chemical potential corresponding to the particle number density n. From the local first law, we can see that the entropy of a perfect fluid can be treated as a function of the energy density$ \rho $ and particle number density n, i.e.,$ s = s(\rho, n) $ . The conservation law$ \nabla_a T^{ab} = 0 $ for a perfect fluid results in$ {\rm d}p+\chi^{-1}(\rho+p){\rm d}\chi = 0\,. $
(13) Together with the local first law and the Gibbs-Duhem relation in Eq. (12), we can further obtain the result that
$ \mu \chi = {\rm{constant}}\,. $
(14) -
In this section, we would like to prove the entropy principle based on the Neother charge method proposed by Iyer and Wald [8]. We consider a one-family
$ \phi(\lambda) $ of the off/on-shell static field configurations, as described in the previous section, in which we denote$ \phi(\lambda) $ to metric$ g_{ab}(\lambda) $ and the shelf-gravitating perfect fluid with$ \rho(\lambda), p(\lambda) $ . That is to say,$ \phi(\lambda) $ satisfies the thermodynamical properties of the fluid, as described in the previous section, as well as the gravitational constraint equation$ H_{uu}(\lambda) = \rho(\lambda)\,. $
(15) For the off-shell configuration
$ \phi(\lambda) $ , the spatial gravitational equations are not satisfied, i.e.,$ \hat{H}_{ab}(\lambda)\neq p(\lambda) h_{ab}(\lambda)\,. $
(16) In the following, we will define the notations
$ \chi = \chi(0)\,,\quad\quad\delta\chi = \left.\frac{\partial \chi}{\partial\lambda}\right|_{\lambda = 0} $
(17) to denote the background quantity and its variation in the family
$ \phi(\lambda) $ . Considering the diffeomorphism covariance of the theory, we can choose a gauge to fix the static Killing vector$ \xi^a(\lambda) $ under the variation in the static configuration$ \phi(\lambda) $ , i.e.,$ \delta \xi^a = 0 $ . For each static configuration$ \phi(\lambda) $ , we have$ g^{ab}(\lambda) = -\chi(\lambda)^{-2}\xi^a\xi^b+h^{ab}(\lambda)\,. $
(18) Then, we have
$ \delta g^{ab} = 2\chi^{-3}\delta\chi\xi^a\xi^b+\delta h^{ab}\,. $
(19) In this family, the variation of the gravitational part of the Lagrangian gives
$ \delta {\boldsymbol{L}}_{\rm{grav}} = {\boldsymbol{E}}_{ab}^{\rm{grav}}\delta g^{ab}+d{\bf{\Theta}}^{\rm{grav}}(g, \delta g)\,, $
(20) in which
$ {\boldsymbol{E}}_{ab}^{\rm{grav}} = \frac{1}{2}{\boldsymbol{\epsilon}} H_{ab}, $
(21) denotes the gravitational part of the equation of motion, and
$ {\bf{\Theta}}^{\rm{grav}}(g, \delta g) $ is the symplectic potential. After completing all indices, Eq. (21) is expressed as$ (E_{ab}^{\rm{grav}})_{a_1\cdots a_n} = \epsilon_{a_1\cdots a_n}H_{ab} $ .For any vector field
$ \zeta^a $ , we can define the Noether current$ (n-1) $ -form as$ {\boldsymbol{J}}_\zeta^{\rm{grav}} = {\bf{\Theta}}^{\rm{grav}}(g,{\cal{L}}_\zeta g)-\zeta\cdot {\boldsymbol{L}}_{\rm{grav}}\,. $
(22) It has been shown in [8] that it can be expressed as
$ {\boldsymbol{J}}_\zeta^{\rm{grav}} = {\boldsymbol{C}}_\zeta^{\rm{grav}}+d{\boldsymbol{Q}}_\zeta^{\rm{grav}}\,, $
(23) in which
$ C_\zeta^{\rm{grav}} = \zeta\cdot {\boldsymbol{C}}^{\rm{grav}} $ with$ {\boldsymbol{C}}^{\rm{grav}}_{aa_1\cdots a_{n-1}} = {\boldsymbol{\epsilon}}_{ba_1\cdots a_{n-1}}H_a{}^b $
(24) is the constraint of the gravitational theory and
$ {\boldsymbol{Q}}_\zeta^{\rm{grav}} $ is a Neother charge$ (n-2) $ -form of the vector field$ \zeta^a $ .Using the two expressions (22) and (23) of the Noether current, we have
$ \begin{aligned}[b] \delta{\boldsymbol{C}}_\zeta^{\rm{grav}}+\zeta\cdot {\boldsymbol{E}}_{ab}^{\rm{grav}}\delta g^{ab} =& d\Big[\zeta\cdot {\bf{\Theta}}^{\rm{grav}}(g,\delta g)-\delta {\boldsymbol{Q}}_\zeta^{\rm{grav}}\Big]\\&+{\boldsymbol{\omega}}^{\rm{grav}}\big(g, \delta g, {\cal{L}}_\zeta g\big) \end{aligned} $
(25) where
$ \omega^{\rm{grav}}(g,\delta_1 g,\delta_2 g) = \delta_1 {\bf{\Theta}}^{\rm{grav}}(g, \delta_2 g)-\delta_2 {\bf{\Theta}}^{\rm{grav}}(g, \delta_1 g)\quad\quad $
(26) is the symplectic current
$ (n-1) $ -form.After replacing
$ \zeta^a $ by$ \xi^a $ and noting that the configuration is static, such that$ {\cal{L}}_\xi g_{ab} = 0 $ , we have$ \omega(g,\delta g,{\cal{L}}_\xi g) = 0 $ . Then, integration of Eq. (25) with respect to C yields$ \begin{aligned}[b]& \int_C\delta{\boldsymbol{C}}_\xi^{\rm{grav}}+\int_C\xi\cdot {\boldsymbol{E}}_{ab}^{\rm{grav}} \delta g^{ab}\\ =& \int_{\partial C}\Big[\xi\cdot {\bf{\Theta}}^{\rm{grav}}\big(g,\delta g\big)-\delta {\boldsymbol{Q}}_\xi^{\rm{grav}}\Big]\,. \end{aligned} $
(27) For the first term in Eq. (27), using Eqs. (24) and (15), we have
$ \int_C\delta{\boldsymbol{C}}_\xi^{\rm{grav}} = -\int_{C}\delta (\chi{\tilde{\boldsymbol{\epsilon}}}H_{uu}) = -\int_{C}\delta (\chi{\tilde{\boldsymbol{\epsilon}}}\rho), $
(28) where we have denoted
$ {\tilde{\boldsymbol{\epsilon}}}(\lambda) $ as the volume element of$ \Sigma $ in the static configuration$ \phi(\lambda) $ . Substituting Eq. (19) into the second term in Eq. (27), we have$ \begin{aligned}[b] \int_C\xi\cdot{\boldsymbol{E}}_{ab}^{\rm{grav}} \delta g^{ab} = &\int_C {\tilde{\boldsymbol{\epsilon}}} H_{uu}\delta\chi+\frac{1}{2}\int_C\chi {\tilde{\boldsymbol{\epsilon}}} H_{ab} \delta h^{ab}\\ =& \int_C {\tilde{\boldsymbol{\epsilon}}} \rho\delta\chi+\frac{1}{2}\int_C\chi {\tilde{\boldsymbol{\epsilon}}} H_{ab} \delta h^{ab}\,. \end{aligned} $
(29) Combining the above results, we can further obtain
$ \begin{aligned}[b]& \frac{1}{2}\int_C\chi {\tilde{\boldsymbol{\epsilon}}} H_{ab} \delta h^{ab}-\int_C\chi \delta({\tilde{\boldsymbol{\epsilon}}} \rho)\\ =& \int_{\partial C}\Big[\xi\cdot{\bf{\Theta}}^{\rm{grav}}(g,\delta g)-\delta {\boldsymbol{Q}}_\xi^{\rm{grav}}\Big]\,. \end{aligned} $
(30) For the off-shell configuration
$ \phi = \phi(0) $ , the first term of the left side in Eq. (30) is not equal to$ p h_{ab} $ .Next, we would like to evaluate the variation in the total entropy of a perfect fluid inside C when the total particle number is fixed. The total entropy of a perfect fluid is given by
$ S = \int_C{\tilde{\boldsymbol{\epsilon}}} s(\rho, n)\,. $
(31) The variation of the total entropy yields
$ \delta S = \int_{C} \left[s\delta {\tilde{\boldsymbol{\epsilon}}}+\left(\frac{\partial s}{\partial \rho}\delta \rho+\frac{\partial S}{\partial n}\delta n\right){\tilde{\boldsymbol{\epsilon}}}\right]\,. $
(32) From the local first law
$ {\rm d}s = \chi {\rm d}\rho-\chi \mu {\rm d}n $ , we have$ \frac{\partial s}{\partial \rho} = \chi\,,\quad \frac{\partial s}{\partial n} = -\chi \mu\,. $
(33) Then, Eq. (32) becomes
$ \delta S = \int_C\left[s \delta{\tilde{\boldsymbol{\epsilon}}}+\chi\left(\delta \rho-\mu \delta n\right){\tilde{\boldsymbol{\epsilon}}}\right]\,. $
(34) From the assumption that the total number of particles
$ N(\lambda) = \int_C{\tilde{\boldsymbol{\epsilon}}}(\lambda) n(\lambda) $
(35) are fixed inside C, we have
$ { \int_Cn\delta{\tilde{\boldsymbol{\epsilon}}} = -\int_C{\tilde{\boldsymbol{\epsilon}}}\delta n\,.} $
(36) Together with the fact that
$ \mu \chi = {\rm{constant}} $ , Eq. (34) reduces to$ \begin{aligned}[b] \delta S = &\int_C\Big[(s+\chi\mu n) \delta{\tilde{\boldsymbol{\epsilon}}}+{ {\tilde{\boldsymbol{\epsilon}}}}\chi\delta \rho\Big]\\ =& \int_C\left[\chi \delta({\tilde{\boldsymbol{\epsilon}}} \rho)-\frac{1}{2}{\tilde{\boldsymbol{\epsilon}}}\chi p h_{ab}\delta h^{ab}\right]\\ =& \int_C{\tilde{\boldsymbol{\epsilon}}}\frac{\chi}{2} (\hat{H}_{ab}-p h_{ab})\delta h^{ab}+\int_{\partial C}\Big[\delta {\boldsymbol{Q}}_\xi^{\rm{grav}}-\xi\cdot {\bf{\Theta}}^{\rm{grav}}(g,\delta g)\Big]\,. \end{aligned} $
(37) In the second step, we used the Gibbs-Duhem relation in Eq. (12) and
$ \delta {\tilde{\boldsymbol{\epsilon}}} = -(1/2){\tilde{\boldsymbol{\epsilon}}} h_{ab}\delta h^{ab} $ . In the last step, we used the variational identity in Eq. (30). The second part of the righthand side of Eq. (37) is only a boundary quantity for$ \partial C $ . Indeed, this boundary term corresponds to the variations in the quasi-local conserved charge corresponding to the static Killing vector$ \xi^a $ in the enclosed region C, which is defined as$ Q(\xi) = \int_{\partial C}\Big[ \Delta {\boldsymbol{Q}}_\xi^{\rm{grav}}-\xi\cdot\int^{\lambda}_{\lambda_0}{\rm d}\lambda{\bf{\Theta}}^{\rm{grav}}(g(\lambda),g'(\lambda))\Big], $
(38) with
$ \Delta {\boldsymbol{Q}}_\xi^{\rm{grav}} = {\boldsymbol{Q}}_\xi^{\rm{grav}}(g(\lambda))-{\boldsymbol{Q}}_\xi^{\rm{grav}}(g(\lambda_0))\,. $
(39) in which
$ \phi(\lambda_0) $ of the one-parameter family$ \phi(\lambda) $ is a vacuum solution in the diffeomorphism-covariant purely gravitational theory [36]. Then, neglecting this term amounts to the variations in the quasi-local conserved charge vanishing, i.e.,$ \delta Q(\xi) = \int_{\partial C}\Big[\delta {\boldsymbol{Q}}_\xi^{\rm{grav}}-\xi\cdot {\bf{\Theta}}^{\rm{grav}}(g,\delta g)\Big] = 0\,. $
(40) For the Einstein gravity in the static spherically symmetric spacetime [16] with the line element
$ {\rm d}s^2 = g_{tt}{\rm d}t^2+\left(1-\frac{2m(r)}{r}\right)^{-1}{\rm d}r^2+r^2{\rm d}\Omega^2\,. $
(41) Let C be in the compact region
$ r\leqslant R $ . With a straightforward calculation,$ \delta Q(\xi) = 0 $ implies that the total mass$ M = m(R) $ within R is fixed at the boundary$ r = R $ . The boundary condition for fixing the quasi-local conserved charge$ Q(\xi) $ is dependent on the explicit theories considered. For instance, in Einstein gravity or Lovelock gravity, the induced metric and its derivative at the boundary$ \partial C $ need to be fixed [16, 25]; in$ f(R) $ gravity, we need to fix the induced metric as well as the scalar curvature R and its derivative at the boundary$ \partial C $ [26].For a usual thermodynamic system, the entropy principle is satisfied when the system is isolated. However, for self-gravitating cases, the phrase “isolated system” becomes more ambiguous as the gravitational theory is diffeomorphism invariant. For a quasi-local system, i.e., C is a finite compact region, and the boundary condition of the isolated system should be quasi-locally imposed. By analogy to the usual cases, we should impose a boundary condition such that the variation inside the compact region C does not affect the dynamics outside C, i.e., the variation in spacetime will not affect the on-shell solution without the fluid outside region C. That is to say, for any element of the one-parameter, their geometries outside C only differ by a diffeomorphism. Under the above condition, using the on-shell variational identity (25) outside C, it is easy to determine
$ \begin{aligned}[b] \delta Q(\xi) =& \int_{\partial C}\Big[\delta {\boldsymbol{Q}}_\xi^{\rm{grav}}-\xi\cdot{\bf{\Theta}}^{\rm{grav}}(g,\delta g)\Big]\\ =& \int_{\infty}\Big[\delta {\boldsymbol{Q}}_\xi^{\rm{grav}}-\xi\cdot{\bf{\Theta}}^{\rm{grav}}(g,\delta g)\Big]\,, \end{aligned} $
(42) where “
$ \infty $ ” denotes a$ (n-2) $ -sphere at asymptotical infinity. If the spacetime is asymptotically flat,$ Q(\xi) $ can be regarded as the mass M of spacetime. As we assume that the variation inside the isolated region will not affect the on-shell geometry outside, it is natural to impose a condition such that the total mass of the spacetime is fixed under the variation, i.e., we have$ \delta Q(\xi) = 0 $ . Then, the second part of the righthand side of Eq. (37) vanishes and we have$ \delta S = \frac{1}{2}\int_C{\tilde{\boldsymbol{\epsilon}}}\chi(\hat{H}_{ab}-p h_{ab})\delta h^{ab}\,. $
(43) From the above result, we show that for the off-shell static configuration
$ \hat{H}_{ab}\neq p h_{ab} $ , the variation in the total entropy is nonvanishing. In other words, the extrema of the total entropy of a perfect fluid inside an isolated region C for a fixed total particle number demand that the static configuration is an on-shell solution. This completes the proof of the Theorem of the entropy principle.
Universality of entropy principle for a general diffeomorphism-covariant purely gravitational theory
- Received Date: 2021-05-28
- Available Online: 2021-10-15
Abstract: Thermodynamics plays an important role in gravitational theories. It is a principle that is independent of gravitational dynamics, and there is still no rigorous proof to show that it is consistent with the dynamical principle. We consider a self-gravitating perfect fluid system with the general diffeomorphism-covariant purely gravitational theory. Based on the Noether charge method proposed by Iyer and Wald, considering static off/on-shell variational configurations, which satisfy the gravitational constraint equation, we rigorously prove that the extrema of the total entropy of a perfect fluid inside a compact region for a fixed total particle number demands that the static configuration is an on-shell solution after we introduce some appropriate boundary conditions, i.e., it also satisfies the spatial gravitational equations. This means that the entropy principle of the fluid stores the same information as the gravitational equation in a static configuration. Our proof is universal and holds for any diffeomorphism-covariant purely gravitational theories, such as Einstein gravity,