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

q = −amB Magnetised Kerr-Newman• With q = −amB, the problem of an ergoregion at infinity canbe avoided. Using the coordinate ˜φ = φ − Ω t, first look atthe off-diagonal metric component g t˜φat large z:g t˜φ = 2(8Ω + 12am2 B 4 + a 3 m 2 B 6 )ρ 2(4 + a 2 m 2 B 4 + B 2 ρ 2 ) 2 + O( 1 z ) .Choosing Ω = Ω s , wherewe find at large z thatΩ s = − 1 8 am2 B 4 (12 + a 2 B 2 ) ,g t˜φ = −8amB2 (4 + a 2 m 2 B 4 )ρ 2(4 + a 2 m 2 B 4 + B 2 ρ 2 ) 2 1z + O( 1 z 2) ,g tt = − 116 (4 + a2 m 2 B 4 + B 2 ρ 2 ) 2 + O( 1 z ) ,and so the metric near the axis is then genuinely asymptoticto the static Melvin universe.