Noether Currents of Charged Spherical Black Holes

TL;DR

This paper computes Noether currents and charges for charged spherical black holes in Einstein-Maxwell theory, linking them to black hole mass, entropy, and energy concepts, with implications for extremal black holes.

Contribution

It introduces a method to derive Noether charges in spherical symmetry and relates these to black hole mass, entropy, and energy, including extremal cases.

Findings

The combined Einstein-Maxwell charge equals the black hole mass in static cases.

Entropy is proportional to a quarter of the horizon area.

Proposes identifying the Noether charge as an energy associated with the Kodama vector.

Abstract

We calculate the Noether currents and charges for Einstein-Maxwell theory using a version of the Wald approach. In spherical symmetry, the choice of time can be taken as the Kodama vector. For the static case, the resulting combined Einstein-Maxwell charge is just the mass of the black hole. Using either a classically defined entropy or the Iyer-Wald selection rules, the entropy is found to be just a quarter of the area of the trapping horizon. We propose identifying the combined Noether charge as an energy associated with the Kodama time. For the extremal black hole case, we discuss the problem of Wald's rescaling of the surface gravity to define the entropy.