Multipole hair of Schwarzschild-Tangherlini black holes

Abstract

We study the field of an electric point charge that is slowly lowered into an n + 1 dimensional Schwarzschild-Tangherlini black hole. We find that if n > 3, then countably infinite nonzero multipole moments manifest to observers outside the event horizon as the charge falls in. This suggests the final state of the black hole is not characterized by a Reissner-Nordström-Tangherlini geometry. Instead, for odd n, the final state either possesses a degenerate horizon, undergoes a discontinuous topological transformation during the infall of the charge, or both. For even n, the final state is not guaranteed to be asymptotically-flat.

Electric Multipole Fields of Higher-Dimensional Massive Bodies

“…Surprisingly, as shown in Ref. [25], one (or both) of these is compromised during the infall of the charge, 3 which means that the final state is not RNT in nature.…”

“…Outside the shell, Ψ satisfies the source-free Maxwell equations in an n + 1 dimensional ST spacetime [25],…”

“…5 However, the precise details (see Ref. [25]) are again unimportant because the requirement of regularity of Ψ as r → ∞ implies d k = 0, so the R k solution leaves the analysis entirely.…”

“…(4) is sensitive to something like this. Whereas the mathematical origin of this dependency is clear[25], the author is unaware of a satisfying physical explanation.…”

“…This assumes the spatial dimension n is odd. If n is even, then the energy density of the electric field diverges as the charge approaches the horizon, which imposes unbounded stresses on the horizon and leads to an apparent violation of asymptotic flatness[25]. In either case, a RNT black hole is not produced.…”