Geometry and Thermodynamics of Black Holes in Magnetic Fields ...
Geometry and Thermodynamics of Black Holes in Magnetic Fields ...
Geometry and Thermodynamics of Black Holes in Magnetic Fields ...
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Magnetisation <strong>of</strong> Kerr-Newman <strong>Black</strong> Hole• We start from the rotat<strong>in</strong>g, charged Kerr-Newman black hole:ds 2 4 = ( −fdt2 + R 2 dr2 )∆ + dθ2A = Φ 0 dt + Φ 3 (dφ − ωdt) ,where+ Σ s<strong>in</strong>2 θR 2 (dφ − ωdt) 2 ,R 2 = r 2 + a 2 cos 2 θ , ∆ = (r 2 + a 2 ) − 2mr + q 2 ,ω = a(2mr − q2 ), Σ = (r 2 + a 2 ) 2 − a 2 ∆ s<strong>in</strong> 2 θ ,Σf = R2 ∆Σ , Φ 0 = qr(r2 + a 2 ), Φ 3 = − aqr s<strong>in</strong>2 θΣR 2 .• We now reduce to three dimensions on ∂/∂φ, apply the magnetis<strong>in</strong>gtransformation, which turns out to beU =⎛⎜⎝1 0 0B√21 0B 24⎞⎟⎠ ,B√ 1 2<strong>and</strong> then retrace the steps back to four dimensions.
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