A light-cone gauge for black-hole perturbation theory

TL;DR

This paper introduces a light-cone gauge for black-hole perturbation theory that preserves key geometrical properties of Schwarzschild coordinates, simplifying physical interpretation and calculations, exemplified by a magnetic field perturbation solution.

Contribution

The paper develops a new light-cone gauge for black-hole perturbations that maintains coordinate meaningfulness and provides explicit solutions, enhancing analysis of black-hole environments.

Findings

The light-cone gauge preserves properties (i)-(iii) of Schwarzschild coordinates in perturbations.

A new three-parameter family of solutions for a black hole in a magnetic field is constructed.

The gauge facilitates extracting physical insights from perturbed black-hole metrics.

Abstract

The geometrical meaning of the Eddington-Finkelstein coordinates of Schwarzschild spacetime is well understood: (i) the advanced-time coordinate v is constant on incoming light cones that converge toward r=0, (ii) the angles theta and phi are constant on the null generators of each light cone, (iii) the radial coordinate r is an affine-parameter distance along each generator, and (iv) r is an areal radius, in the sense that 4 pi r^2 is the area of each two-surface (v,r) = constant. The light-cone gauge of black-hole perturbation theory, which is formulated in this paper, places conditions on a perturbation of the Schwarzschild metric that ensure that properties (i)--(iii) of the coordinates are preserved in the perturbed spacetime. Property (iv) is lost in general, but it is retained in exceptional situations that are identified in this paper. Unlike other popular choices of gauge, the light-cone gauge produces a perturbed metric that is expressed in a meaningful coordinate system; this is a considerable asset that greatly facilitates the task of extracting physical consequences. We illustrate the use of the light-cone gauge by calculating the metric of a black hole immersed in a uniform magnetic field. We construct a three-parameter family of solutions to the perturbative Einstein-Maxwell equations and argue that it is applicable to a broader range of physical situations than the exact, two-parameter Schwarzschild-Melvin family.