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

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 otherwiseasymptotically-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.

On the stability of gravity with Dirichlet walls

“…The structure is similar to that of a two-sided wormhole where the ADM Energy is separately conserved in each asymptotic region. On (perhaps AdS or dS) Schwarschild, extending the computations of [39] to λ = 0 and using the sign appropriate to being outside the wall gives In a recent article [14], we used the small velocity approximation to study the motion of a spherical extreme electrically-charged black hole in an asymptotically flat spacetime that contains a flat Dirichlet wall. Our results suggested that system to be unstable due to the appearance of negative kinetic energies for black holes near the wall.…”

“…In the particular case where the intrinsic metric is fixed, we call this surface a "Dirichlet wall." With such motivations in mind, we recently considered the problem of a black hole moving through a spacetime with a Dirichlet wall [14]. In order to make the problem analytically tractable, we considered the extreme multi-black-hole solutions of [15,16].…”

$T\bar T$ and the mirage of a bulk cutoff

“…Dirichlet boundary conditions at finite bulk radius have been studied at classical level in[50][51][52][53].…”

Exact gravitational wave signatures from colliding extreme black holes

“…Since there is no dynamics at leading order, the relevant equations are the constraints 3. See[25] for a critical discussion of this approximation.…”