2006/12/31 by Michael Seifert, Michael D. Seifert, Robert M. Wald
Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Classical mechanics #Cosmology and Gravitation Theories #Covariant transformation #Equations of motion #Gravitation #Mathematical physics #Physics #Variational principle #gr-qc
paper · pdf · doi:10.1103/physrevd.75.084029
published as Phys.Rev.D75:084029,2007 · 13 pages; submitted to Phys. Rev. D. v2: changed formatting, added conclusion, corrected sign conventions
arxiv created 2007/03/09 · openalex publication_date 2007/04/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 variables 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 rederiving, 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.