Electrovacuum spacetime near an extreme horizon

Understand

We determine all infinitesimal transverse deformations of extreme horizons in Einstein-Maxwell theory that preserve axisymmetry.

• In particular, we show that the general static transverse deformation of the AdS(2) X S(2) near-horizon geometry is a two-parameter family, which contains the known extreme charged, accelerating, static black hole solution held in equilibrium by an external electric or magnetic field (Ernst solution) and a special case of the extreme Kerr-Newman-Melvin solution.
• More generally, we find a three-parameter family of deformations of the extreme Kerr-Newman horizon, which contains the extreme Kerr-Newman-Melvin solution and a rotating generalisation of the Ernst solution.
• We also consider vacuum gravity with a cosmological constant and prove uniqueness of axisymmetric transverse deformations of the extreme Kerr-AdS horizon.