Gauge-invariant observables in gravity and electromagnetism: black hole backgrounds and null dressings

TL;DR

This paper explores gauge-invariant observables in gravity and electromagnetism within black hole backgrounds, proposing null geodesic dressings to simplify their construction and examining their properties and implications for black hole complementarity.

Contribution

It introduces null geodesic dressings for gauge-invariant observables in complex geometries, analyzing their properties and implications for black hole physics.

Findings

Null dressings simplify gauge-invariant observable construction.

Observables inside and outside black holes can be compatible.

Non-commutativity of observables is not significantly affected by horizons.

Abstract

We address questions regarding construction and implications of gauge-invariant "dressed" observables in nontrivial background geometries such as that of a black hole. Formally, such observables can be constructed, e.g. by locating points with geodesics launched from infinity. However, practical complications arise in non-trivial geometries, and in particular for observables behind black hole horizons. Greater simplicity can be achieved by considering null constructions where the dressing lies along a null geodesic, or null surface such as a cone. We first investigate basic properties of these null dressings in the simpler context of electromagnetism. Since null constructions provide simple dressings for gauge-invariant observables inside black holes, they also allow us to investigate the question of compatibility of observables inside and outside black holes, and in particular the idea of black hole complementarity. While such observables in general have non-vanishing state-dependent commutators, the failure to commute does not appear particularly enhanced by the presence of the horizon.