Einstein-Maxwell Dirichlet walls, negative kinetic energies, and the adiabatic approximation for extreme black holes

TL;DR

This paper investigates the Einstein-Maxwell Dirichlet boundary problem for extreme black holes, revealing issues with negative kinetic energies and potential ill-definedness of the adiabatic approximation in this context.

Contribution

It introduces a moduli space of static solutions with Dirichlet walls and analyzes the emergence of negative kinetic energies, highlighting challenges in the classical black hole adiabatic approximation.

Findings

Negative kinetic energy eigenvalues near the wall

Regulator dependence affects the moduli space metric

Potential unboundedness of the Hamiltonian

Abstract

The gravitational Dirichlet problem -- in which the induced metric is fixed on boundaries at finite distance from the bulk -- is related to simple notions of UV cutoffs in gauge/gravity duality and appears in discussions relating the low-energy behavior of gravity to fluid dynamics. We study the Einstein-Maxwell version of this problem, in which the induced Maxwell potential on the wall is also fixed. For flat walls in otherwise-asymptotically-flat spacetimes, we identify a moduli space of Majumdar-Papapetrou-like static solutions parametrized by the location of an extreme black hole relative to the wall. Such solutions may be described as balancing gravitational repulsion from a negative-mass image-source against electrostatic attraction to an oppositely-signed image charge. Standard techniques for handling divergences yield a moduli space metric with an eigenvalue that becomes negative near the wall, indicating a region of negative kinetic energy and suggesting that the Hamiltonian may be unbounded below. One may also surround the black hole with an additional (roughly spherical) Dirichlet wall to impose a regulator whose physics is more clear. Negative kinetic energies remain, though new terms do appear in the moduli-space metric. The regulator-dependence indicates that the adiabatic approximation may be ill-defined for classical extreme black holes with Dirichlet walls.