Exact horizon response and charge ensembles of Majumdar--Papapetrou black holes

Abstract

We determine the electrostatic Dirichlet-to-Neumann map of an arbitrary four-dimensional Majumdar--Papapetrou spacetime. For a minimally coupled spectator Maxwell field, the potentials of the disconnected extremal horizons may be prescribed independently, and the resulting flux charges are related by an exact capacitance matrix. If the centers have masses $M_A$ and coordinate separations $R_{AB}$, its entries are $C_{AA}=M_A+\sum_{B\ne A}M_AM_B/R_{AB}$ and $C_{AB}=-M_AM_B/R_{AB}$. The matrix is a positive diagonal term plus the weighted Laplacian of the complete graph of horizons, giving an exact common-mode and differential-mode decomposition of the field energy. The grounded Dirichlet Green function shows that a point charge partitions its flux among infinity and the horizons according to elementary harmonic measures. A direct translation of the multicenter Green function of Frolov and Zelnikov gives the diagonal completion used in their published Eq.~(68), whose component-resolved Gauss flux preserves the total horizon charge but generally transfers charge between horizons. The symmetric Green function that preserves every horizon charge instead contains the inverse of the full capacitance matrix. For a binary, the difference gives a finite, negative-semidefinite shift of the electrostatic self-energy and a corresponding exact self-force difference without an additional ultraviolet subtraction. We finally extend the response matrix, flux partition, and fixed-charge Green function to $D=n+3$ dimensions, where the graph weights acquire the universal factor $π^{n/2}/Γ(n/2)$.