Second and higher-order perturbations of a spherical spacetime

TL;DR

This paper extends the Gerlach and Sengupta formalism to second-order perturbations of spherical spacetimes, providing a covariant, gauge-fixed framework optimized for computer algebra implementation.

Contribution

It generalizes GS harmonics to higher orders and derives evolution and conservation equations for second-order perturbations in a covariant form.

Findings

Provides explicit formulas for second-order perturbation harmonics.

Derives covariant evolution equations in Regge-Wheeler gauge.

Optimizes formalism for computer algebra systems.

Abstract

The Gerlach and Sengupta (GS) formalism of coordinate-invariant, first-order, spherical and nonspherical perturbations around an arbitrary spherical spacetime is generalized to higher orders, focusing on second-order perturbation theory. The GS harmonics are generalized to an arbitrary number of indices on the unit sphere and a formula is given for their products. The formalism is optimized for its implementation in a computer algebra system, something that becomes essential in practice given the size and complexity of the equations. All evolution equations for the second-order perturbations, as well as the conservation equations for the energy-momentum tensor at this perturbation order, are given in covariant form, in Regge-Wheeler gauge.