We show that for any perfect fluid in a static spacetime, if the Einstein constraint equation is satisfied and the temperature of the fluid obeys Tolman's law, then the other components of Einstein's equation are implied by the assumption that the total entropy of the fluid achieves an extremum for fixed total particle number and for all variations of metric with certain boundary conditions. Conversely, one can show that the extrema of the total entropy of the fluid are implied by Einstein's equation. Compared to previous works on this issue, we do not require spherical symmetry for the spacetime. Our results suggest a general and solid connection between thermodynamics and general relativity. PACS number(s): 04

We consider a static self-gravitating charged perfect fluid system in the Einstein-Maxwell theory.

According to the maximum entropy principle, it has been proved that the gravitational field equations could be derived by the extrema of the total entropy for a perfect fluid, which implies that thermodynamic relations contain information as regards gravity. In this manuscript, we obtain a criterion for the thermodynamical stability of an adiabatic, self-gravitating perfect fluid system by the second variation of the total entropy. We show, for Einstein's gravity with spherical symmetry spacetime, that the criterion is consistent with that for the dynamical stability derived by Chandrasekhar and Wald. We also find that the criterion could be applied to cases without spherical symmetry, or under general perturbations. The result further establishes the connection between thermodynamics and gravity.

The discussions on the connection between gravity and thermodynamics attract much attention recently. We consider a static self-gravitating perfect fluid system in f (R) gravity, which is an important theory could explain the accelerated expansion of the universe. We first show that the Tolman-Oppenheimer-Volkoff equation of f (R) theories can be obtained by thermodynamical method in spherical symmetric spacetime. Then we prove that the maximum entropy principle is also valid for f (R) gravity in general static spacetimes beyond spherical symmetry. The result shows that if the constraint equation is satisfied and the temperature of fluid obeys Tolmans law, the extrema of total entropy implies other components of gravitational equations. Conversely, if f (R) gravitational equation hold, the total entropy of the fluid should be extremum. Our work suggests a general and solid connection between f (R) gravity and thermodynamics.

We have studied the quasinormal modes (QNMs) of a slowly rotating black hole with Lorentz-violating parameter in Einstein–Bumblebee gravity. We analyse the slow rotation approximation of the rotating black hole in the Einstein–Bumblebee gravity, and obtain the master equations for scalar perturbation, vector perturbation and axial gravitational perturbation, respectively. Using the matrix method and the continuous fraction method, we numerically calculate the QNM frequencies. In particular, for scalar field, it shows that the QNMs up to the second order of rotation parameter have higher accuracy. The numerical results show that, for both scalar and vector fields, the Lorentz-violating parameter has a significant effect on the imaginary part of the QNM frequencies, while having a relatively smaller impact on the real part of the QNM frequencies. But for axial gravitational perturbation, the effect of increasing the Lorentz-violating parameter $$\ell $$ ℓ is similar to that of increasing the rotation parameter $${\tilde{a}}$$ a ~ .

We investigate the stability criterions for perfect fluid in f (R) theories which is an important generalization of general relativity. Firstly, using Wald's general variation principle, we recast Seifert's work and obtain the dynamical stability criterion. Then using our generalized thermodynamical criterion, we obtain the concrete expressions of the criterion. We show that the dynamical stability criterion is exactly the same as the thermodynamical stability criterion to spherically symmetric perturbations of static spherically symmetric background solutions. This result suggests that there is an inherent connection between the thermodynamics and gravity in f (R) theories. It should be pointed out that using the thermodynamical method to determine the stability for perfect fluid is simpler and more directly than the dynamical method.

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