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

TL;DR

This work presents a general, prescriptive method to analyze the stability of static, spherically symmetric solutions under spherical perturbations in diffeomorphism-covariant Lagrangian theories. By fixing a gauge and algebraically eliminating metric perturbations through the linearized constraints, it reduces the problem to matter perturbations and then uses the conserved symplectic current to construct a positive-definite inner product, yielding a self-adjoint evolution operator and a variational stability principle. The approach is exemplified by re-deriving Chandrasekhar's radial-oscillation variational principle for Einstein gravity with a perfect fluid, and the authors outline future applications to $f(R)$ gravity, Einstein-{}ther theory, and TeVeS. This framework offers a systematic, non-artistic pathway to viability checks for alternative gravity theories by connecting gauge-fixed perturbations, symplectic structure, and variational stability.

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.