Energy Extremum Principle for Charged Black Holes

TL;DR

This paper proves an extremum principle for a set of charged black holes, showing that the extremal charge configuration can be derived by extremizing total energy with fixed charges and separations, extending variational principles in black hole physics.

Contribution

It introduces a new extremum principle for charged black holes, linking extremal charge configurations to energy extremization at fixed charges and separations.

Findings

Extremal charge configuration corresponds to a total energy extremum.

The principle is valid through second order in inverse separation expansion.

Results align with known BPS energy minimum.

Abstract

For a set of $N$ asymptotically flat black holes with arbitrary charges and masses, all initially at rest and well separated, we prove the following extremum principle: the extremal charge configuration ($|q_i|=m_i$ for each black hole) can be derived by extremizing the total energy, for variations of the black hole apparent horizon areas, at fixed charges and fixed Euclidean separations. We prove this result through second order in an expansion in the inverse separations. If all charges have the same sign, this result is a variational principle that reinterprets the static equilibrium of the Majumdar-Papapetrou-Hartle-Hawking solution as an extremum of total energy, rather than as a balance of forces; this result augments a list of related variational principles for other static black holes, and is consistent with the independently known Bogomol'nyi-Prasad-Sommerfield (BPS) energy minimum.