Stability of Fluid Motions and Variational Principles

Abstract

It is the purpose of this paper to show that the equations satisfied by the difference of two “nearly equal” solutions of the Einstein field equations are derivable from a variational principle and indicate how this principle may be used to study the time dependence of the difference. The source of the gravitational fields will be assumed to be a perfect fluid obeying an equation of state. That is, the pressure p of the fluid will be assumed to be a function of only the energy density w. It will be further assumed that the perturbations in the fluid motions will be adiabatic. That is if the entropy for one solution is S and that for the nearby solution is S + e S′ then S′ = 0. It will not be assumed that S = constant for either solution although this condition does provide one of the possible equations of state that the fluid is required to obey.

This work was supported in part by the United States Atomic Energy Commission under contract number AT(11-1)-34, Project Agreement No. 125. It was performed while the author was on sabbatical leave from the University of California, Berkeley and in residence at the Collége de France, Paris.