Abstract
We study all transverse deformations of the extremal Reissner-Nordström–(A)dS horizon in the Einstein-Maxwell theory. No symmetry assumptions are needed. It is shown, that for the generic values of a charge, the only allowed deformation is spherically symmetric. However, it is shown that for fine-tuned values of the charge, the space of deformations is larger, yet still finite-dimensional.
Highlights
We consider the following solution to the Einstein-Maxwell equations:2 g = 2dv dr + r+2 dx2 1 − x2 (1 x2)dφ2 F = Ψdv ∧ dr, (2.1) where Ψ Q r+2Q is an electric charge
We study all transverse deformations of the extremal Reissner-Nordström– (A)dS horizon in the Einstein-Maxwell theory
As checked in [16], their leading-order deformation is exactly spherically symmetric, so our result cannot tell anything about the existence of such black holes
Summary
On the neighborhood of the extremal horizon, one can introduce a null gaussian coordinate system in which the metric reads:. Since φ is no longer diffeomorphism in this limit, it may be an entirely new (yet much simpler) solution to the Einstein-Maxwell equations. Where all objects h(a0), F (0), Ψ(0), Wa(0), Ba(0b) are r independent, and they are given by the limits as r → 0 of associated objects in the original spacetime. Where D is the number of spacetime dimensions, Da is the covariant derivative associated with γ(0) and all indices are lowered and raised using γ(0). Those equations, easier than the full set of Einstein-Maxwell equations, are still being studied, especially in.
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