A general variational principle for spherically symmetric perturbations in diffeomorphism covariant theories

TL;DR

This paper introduces a unified variational approach to analyze the stability of static, spherically symmetric solutions in various diffeomorphism covariant theories, simplifying the process of stability testing through a systematic reduction and bilinear form analysis.

Contribution

It develops a general method for stability analysis in covariant theories by fixing gauge, eliminating metric perturbations, and deriving a variational principle applicable to multiple gravity theories.

Findings

Method applies to f(R) gravity, Einstein-aether, and TeVeS theories.

Provides a systematic way to derive stability criteria.

Re-derives Chandrasekhar's variational principle for stellar oscillations.

Abstract

We present a general method for the analysis of the stability of static, spherically symmetric solutions to spherically symmetric perturbations in an arbitrary diffeomorphism covariant Lagrangian field theory. Our method involves fixing the gauge and solving the linearized gravitational field equations to eliminate the metric perturbation variable in terms of the matter variables. In a wide class of cases--which include f(R) gravity, the Einstein-aether theory of Jacobson and Mattingly, and Bekenstein's TeVeS theory--the remaining perturbation equations for the matter fields are second order in time. We show how the symplectic current arising from the original Lagrangian gives rise to a symmetric bilinear form on the variables of the reduced theory. If this bilinear form is positive definite, it provides an inner product that puts the equations of motion of the reduced theory into a self-adjoint form. A variational principle can then be written down immediately, from which stability can be tested readily. We illustrate our method in the case of Einstein's equation with perfect fluid matter, thereby re-deriving, in a systematic manner, Chandrasekhar's variational principle for radial oscillations of spherically symmetric stars. In a subsequent paper, we will apply our analysis to f(R) gravity, the Einstein-aether theory, and Bekenstein's TeVeS theory.