General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories

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.

Mass, charge, and motion in covariant gravity theories

“…Equation (20) shows that α a = ξ a L for infinitesimal diffeomorphisms. done in total generality, see reference [17].) In this case we have…”

“…While we will assume in some intermediate calculations that ω a takes the form of equation (17), the final results (120)-(123) are valid for any choice of ω a . (The invariance is explicitly ensured by Stokes' theorem.)…”

“…If the theory contained additional scalar or vector fields, this identity would be modified by the addition of appropriate terms of the form present above. If the theory contained higher rank fields, there would be additional terms of similar structure; the explicit form of these terms is given in [17].…”

“…(The invariance is explicitly ensured by Stokes' theorem.) Equation (17) holds only for theories with second-order field equations. For higher derivative theories, an analogous form would be obtained with higher derivatives of the perturbations appearing on the right-hand side.…”

“…In a series of papers spanning more than a decade [8][9][10][11][12][13][14][15][16][17], Wald and coauthors developed a systematic and mathematically rigorous approach to Lagrangian and Hamiltonian field theory of a general n-dimensional covariant theory, with applications to (e.g.) black hole thermodynamics, conserved quantities, and stability.…”