skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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 » theory.« less

Authors:
;  [1]
  1. 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}
}
  • The theory of even-parity nonspherical perturbations of collisionless, isotropic, spherical star clusters in general relativity is developed for l> or =2. A variational principle for the associated normal modes of oscillation is derived. In this principle, an expression for the squared frequency of oscillation is stationary with respect to arbitrary acceptable variations of the independent dynamical variables if and only if the variations are carried out around the eigenfunctions of a normal mode. The stationary value of the frequency is the corresponding eigenfrequency. The form of the variational principle guarantees the applicability of a previous theorem, which implies that amore » cluster is unstable and only if it possesses a so-called unstable proper normal mode. Under a weak assumption, the form of the principle also guarantees that instabilities can possibly set in only through zero-frequency modes.« less
  • Perturbations of a general background space-time with a hypersurface of discontinuity, such as the history of a collapsing star, are considered. The junction conditions that these perturbations obey are expressed in terms of the perturbed first and second fundamental forms (intrinsic metric and extrinsic curvature) of the (perturbed) set of hypersurfaces one of which is the discontinuous one. These junction conditions are applied to the odd-parity metric and hydrodynamical asymmetries of a slightly aspherical but otherwise general and realistic spherically collapsing star. The junction conditions are stated in terms of those metric and matter perturbational objects that are the mostmore » natural, economic, and versatile: gauge-invariant geometrical objects. For odd-parity perturbations these are scalars and vectors on the totally geodesic submanifold spanned by the time and radial coordinates. The end result is simple: The junction conditions amount to the continuity of the gradient of a master gauge-invariant scalar wave function from which all other perturbational quantities can be derived.« less
  • A new highly efficient and versatile general relativistic perturbational formalism for general matter occupied spherically symmetric space--times is developed. The perturbations are geometrical objects on the two dimensional totally geodesic submanifold spanned by the radial and time coordinates. The geometrical objects are ''gauge invariant'' scalars, vectors, and tensors which are independent of infinitesimal coordinate transformations on the background space--time. This article gives the even parity gauge invariant perturbation objects for arbitrary background scalars, vectors, and symmetric tensors on a spherically symmetric space--time. In particular, metric, matter, first and second fundamental forms, as well as vacuum-matter interface gauge invariant perturbations formore » a collapsing star are given. In addition four even parity continuity conditions across discontinuous timelike hypersurfaces are given. Two are conditions on the metric gauge invariants, one is a condition on the perturbation away from the spherical contour of the interface, and the fourth couples that contour perturbation to the metric gauge invariants.« less
  • We study the most general case of spherically symmetric vacuum solutions in the framework of the covariant Horava-Lifshitz gravity, for an action that includes all possible higher order terms in curvature which are compatible with power-counting normalizability requirement. We find that solutions can be separated into two main classes: (i) solutions with nonzero radial shift function, and (ii) solutions with zero radial shift function. In the case (ii), spherically symmetric solutions are consistent with observations if we adopt the view of Horava and Melby-Tomson [P. Horava and C. M. Melby-Thompson, Phys. Rev. D 82, 064027 (2010).], according to which themore » auxiliary field A can be considered as a part of an effective general relativistic metric, which is valid only in the IR limit. On the other hand, in the case (i), consistency with observations implies that the field A should be independent of the spacetime geometry, as the Newtonian potential arises from the nonzero radial shift function. Also, our aim in this paper is to discuss and compare these two alternative but different assumptions for the auxiliary field A.« less
  • In this note we examine whether spherically symmetric solutions in covariant Horava-Lifshitz gravity can reproduce Newton's Law in the IR limit {lambda}{yields}1. We adopt the position that the auxiliary field A is independent of the space-time metric [J. Alexandre and P. Pasipoularides, Phys. Rev. D 83, 084030 (2011).][J. Greenwald, V. H. Satheeshkumar, and A. Wang, J. Cosmol. Astropart. Phys. 12 (2010) 007.], and we assume, as in [A. M. da Silva, Classical Quantum Gravity 28, 055011 (2011).], that {lambda} is a running coupling constant. We show that under these assumptions, spherically symmetric solutions fail to restore the standard Newtonian physicsmore » in the IR limit {lambda}{yields}1, unless {lambda} does not run, and has the fixed value {lambda}=1. Finally, we comment on the Horava and Melby-Thompson approach [P. Horava and C. M. Melby-Thompson, Phys. Rev. D 82, 064027 (2010).] in which A is assumed as a part of the space-time metric in the IR.« less