Maximum Entropy Principle And Stability Criterion For Charged Perfect Fluid Star In Curved Space-time

In recent decades,the relationship between thermodynamics and gravity has attracted more and more attention.A large number of works have shown that the thermodynamic laws can be obtained from various gravity theories,and meanwhile the gravitational field equations can also be derived from the basic thermodynam-ic relations.These results reveal the deep relationship between gravity and ther-modynamics.The maximum principle of entropy shows that for self-gravitating perfect fluid,the corresponding gravitational field equation can be derived from the extrema principle of entropy under some certain conditions.And this principle also shows that the thermodynamic stability criterion is the same as dynamic sta-bility criterion can be derived from the maximum of the total entropy.However,whether the maximum entropy principle is still valid in various cases,especially in the case for perfect fluid with charge is a unsolved issue.In this paper,we briefly review the thermodynamics of perfect fluid in curved space-time is first introduced,and the equilibrium distribution(TOV)equation of a spherically symmetric charged perfect fluid.At the same time,we introduce the specific content of maximum principle of entropy,and how to build the relation-ship between the first order variation of total entropy and the gravitational field equation through extreme principle of entropy.It is well known that the dynam-ic stability equation of the perfect fluid in curved space-time is derived from the first order variation of the gravitational field equation,which implies that there may exist some deep relations between the dynamic stability equation and the thermodynamic stability equation.Secondly,the dynamic stability of charged perfect fluid is studied.We intro-duce the general theory of solving the criterion of dynamic stability by dynam-ic method.For an adiabatic and reversible process,the separation of variables method can always separate the time and space functions of the perturbation,so that using the variable separation method can always be separated into the time part and space part space,solving the space part problem can be transformed into Sturm-Liouville eigenvalue problem.According to the Rayleigh-Ritz principle,the eigenvalue can give the criterion of dynamic stability.Then we discuss the spheri-cal symmetric perturbation for charged perfect fluid is discussed in detail and the basic method of its criterion of dynamic stability is introduced and the concrete results are given.Next,the thermodynamic stability of charged perfect fluid is investigated.It is found that the temperature is no longer proportional to the chemical potential in the fundamental thermodynamic relationship due to the presence of electric charge.According to the extrema entropy principle,we can get the general formula second variation of isolated sphere total entropy~2under the static background,The specific expression of thermodynamic stability criterion is obtained by considering spherically symmetric perturbation.When the barotropic conditions are satisfied,it can be seen that the thermodynamic stability criterion is exactly the same as the dynamical stability criterion.This conclusion indicates that the gravity and thermodynamics are still deep relationship in the sense for perfect fluid with charge.