Nonlinear Perturbations and Conservation Laws on Curved Backgrounds in GR and Other Metric Theories

TL;DR

This paper reviews a gauge-invariant, field-theoretical approach to perturbations and conservation laws in general relativity and other metric theories, providing exact equations and applications to higher-dimensional gravity.

Contribution

It introduces a unified, gauge-invariant formalism for perturbations and conserved quantities applicable to arbitrary metric theories and dimensions.

Findings

Exact perturbation equations derived from variational principle.

Conserved currents expressed via superpotentials, linking local properties to quasi-local quantities.

Application demonstrated in Einstein-Gauss-Bonnet gravity.

Abstract

The field-theoretical approach is reviewed. Perturbations in general relativity as well as in an arbitrary $D$-dimensional metric theory are studied on a background, which is a solution (arbitrary) of the theory. Lagrangian for perturbations is defined, and field equations for perturbations are derived from the variational principle. These equations are exact and equivalent to the equations in the standard formulation, but can be approximate also. The field-theoretical description is invariant under gauge (inner) transformations, which can be presented both in exact and approximate forms. Following the usual field-theoretical prescription, conserved quantities for perturbations are constructed. Conserved currents are expressed through divergences of superpotentials -- antisymmetric tensor densities. This form allows to relate a necessity to consider local properties of perturbations with a theoretical representation of the quasi-local nature of conserved quantities in metric theories. Applications of the formalism in general relativity are discussed. Generalized formulae for an arbitrary metric $D$-dimensional theory are tested in the Einstein-Gauss-Bonnet gravity.