Abstract
We prove that the only asymptotically flat spacetimes with a suitably regular event horizon, in a generalised Majumdar–Papapetrou class of solutions to higher-dimensional Einstein–Maxwell theory, are the standard multi-black holes. The proof involves a careful analysis of the near-horizon geometry and an extension of the positive mass theorem to Riemannian manifolds with conical singularities. This completes the classification of asymptotically flat, static, extreme black hole solutions in this theory.
Highlights
The Majumdar–Papapetrou solution to Einstein–Maxwell theory represents the static equilibrium of an arbitrary number of charged black holes whose mutual electric repulsion exactly balances their gravitational attraction [1]
We show that a similar result holds in all dimensions: any suitably regular asymptotically flat black hole solution in the generalised Majumdar– Papapetrou class (1) must have (i) a base (Σ, h) isometric to euclidean space and (ii) a harmonic function H of multi-centre type (2)
We have found that the near-horizon geometry must be a direct product of AdS2 and an Einstein space (S, σ) normalised as above, gNH = −f0λ2dv2 + 2dvdλ + (n − 3)2f0−1σabdyadyb, (18)
Summary
The Majumdar–Papapetrou solution to Einstein–Maxwell theory represents the static equilibrium of an arbitrary number of charged black holes whose mutual electric repulsion exactly balances their gravitational attraction [1]. We show that a similar result holds in all dimensions: any suitably regular asymptotically flat black hole solution in the generalised Majumdar– Papapetrou class (1) must have (i) a base (Σ, h) isometric to euclidean space (minus a point for each horizon) and (ii) a harmonic function H of multi-centre type (2).
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