Curvature-based Gauge-Invariant Perturbation Theory for Gravity: A New Paradigm

TL;DR

This paper introduces a novel gauge-invariant perturbation theory for gravity based on the Einstein-Ricci system, emphasizing a hyperbolic formulation that isolates physical curvature perturbations and addresses constraint violations.

Contribution

It develops a new gauge-invariant perturbation framework from the fourth-order Einstein-Ricci system, applicable to arbitrary backgrounds, and clarifies the evolution of gauge and constraint violations.

Findings

Physical perturbations satisfy the Regge-Wheeler equation.

Unphysical gauge degrees evolve as a closed subsystem.

Hyperbolic formulation ensures well-posed evolution of constraints.

Abstract

A new approach to gravitational gauge-invariant perturbation theory begins from the fourth-order Einstein-Ricci system, a hyperbolic formulation of gravity for arbitrary lapse and shift whose centerpiece is a wave equation for curvature. In the Minkowski and Schwarzschild backgrounds, an intertwining operator procedure is used to separate physical gauge-invariant curvature perturbations from unphysical ones. In the Schwarzschild case, physical variables are found which satisfy the Regge-Wheeler equation in both odd and even parity. In both cases, the unphysical "gauge'' degrees of freedom are identified with violations of the linearized Hamiltonian and momentum constraints, and they are found to evolve among themselves as a closed subsystem. If the constraints are violated, say by numerical finite-differencing, this system describes the hyperbolic evolution of the constraint violation. It is argued that an underlying raison d'\^etre of causal hyperbolic formulations is to make the evolution of constraint violations well-posed.