# 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 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 TeVeSmore »

- Authors:

- Enrico Fermi Institute and Department of Physics, University of Chicago, 5640 S. Ellis Ave., Chicago, Illinois, 60637 (United States)

- Publication Date:

- OSTI Identifier:
- 21020398

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 8; Other Information: DOI: 10.1103/PhysRevD.75.084029; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; COSMOLOGY; DISTURBANCES; EINSTEIN FIELD EQUATIONS; EQUATIONS OF MOTION; GRAVITATION; GRAVITATIONAL FIELDS; IDEAL FLOW; LAGRANGIAN FIELD THEORY; LAGRANGIAN FUNCTION; MATHEMATICAL SOLUTIONS; STABILITY; TEV RANGE; VARIATIONAL METHODS

### Citation Formats

```
Seifert, Michael D., and Wald, Robert M.
```*General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.084029.

```
Seifert, Michael D., & Wald, Robert M.
```*General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories*. United States. doi:10.1103/PHYSREVD.75.084029.

```
Seifert, Michael D., and Wald, Robert M. Sun .
"General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories". United States.
doi:10.1103/PHYSREVD.75.084029.
```

```
@article{osti_21020398,
```

title = {General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories},

author = {Seifert, Michael D. and Wald, Robert M.},

abstractNote = {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.},

doi = {10.1103/PHYSREVD.75.084029},

journal = {Physical Review. D, Particles Fields},

number = 8,

volume = 75,

place = {United States},

year = {Sun Apr 15 00:00:00 EDT 2007},

month = {Sun Apr 15 00:00:00 EDT 2007}

}