Curvature-based gauge-invariant perturbation theory for gravity: A new paradigm

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’être of causal hyperbolic formulations is to make the evolution of constraint violations well posed.