Geometry and Thermodynamics of Black Holes in Magnetic Fields ...

Conical Singularities and Force Balancing• To avoid conical singularity at the North (South) pole, φshould be identified with different periods at each:N pole :S pole :∆φ = 2π ∏ i∆φ = 2π ∏ i(1 + 1 2 B ip i ) ,(1 − 1 2 B ip i ) .In general, there is no single choice of period ∆φ that eliminatesboth conical singularities. This reflects the fact thatthere is a net force on the black hole, and it must thereforebe supported by a “strut,” which is described by an energymomentumtensor with a delta-function singularity on theaxis. This happens, for example, for a magnetic Reissner-Nordström black hole in an external magnetic field. (Or theS-dual, an electric black hole in an external electric field.)• With four separate gauge fields in the STU model we cantune the fields and the charges to achieve a force balance:∏(1 + 1 2 B ip i ) = ∏ (1 − 1 2 B ip i ) .iiThis provides families of non-singular magnetic black holesin external magnetic fields. These provide interesting modelsfor investigating thermodynamics of systems in externalbackgrounds.