Stability of general relativistic gaseous masses and variational principles

Abstract

The Einstein field equations for a self-gravitating fluid that obeys an equation of state of the formp=p(w),p the pressure andw the energy density may be derived from a variational principle. The perturbations of the metric tensor and the fluid dynamic variables satisfy equations which may be derived from a related variational principle, namely the principle associated with the “second variation problem.” It is shown that the variational principle given by Chandrasekhar from which a sufficient criterion may be obtained for deciding when a self gravitating spherical gaseous mass is unstable against spherically symmetric perturbations is that given by the “second variation problem”. It is further shown that this criterion is equivalent to requiring that the integral entering into the second variation be negative. The latter form of the criterion may be used in general situations.