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An extremal horizon can be always embedded into spacetime as a Killing
horizon but usually not uniquely. For example, extremal
Reissner-Nordstrom and Majumdar-Papapetrou black holes are two different
asymptotically flat spacetimes with the same geometry induced on their
horizons. We consider an infinitesimal version of this
problemâ€”classification of connections admissible on the horizon in
Einstein-Maxwell theory. We can interpret them as specific zero modes in
the corresponding Near Horizon spacetime. Surprisingly enough, for
spherically symmetric backgrounds those connections are basically unique
unless we fine-tune the charge and cosmological constant. We also
discuss how this uniqueness extends into the whole spacetime under an
assumption of analyticity.
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