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

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Further SU(2, 1) Transformations• Amongst the other possible transformations within the SU(2, 1)symmetry group are electrifications (adding an external electricfield):U =⎛(B+i E)⎜⎝(B 2 +E 2 )41 0 0⎟⎠ .(B−i√E)1 2√21 0• Electric/magnetic duality transformations, which in four dimensionstake the formF −→ F ′ = F cos α + ∗F sin α ,This is implemented in 3 dimensions by the SU(2, 1) matrix⎛e − i ⎞3 α 0 0U =⎜ 0 e 2i3 α 0.⎝0 0 e − 3 i ⎟α ⎠⎞
Magnetisation of Kerr-Newman Black Hole• Applied to Schwarzschild, it indeed gives Schwarzschild-Melvin.• Applied to the Kerr-Newman metric (with electric and magneticcharges q and p, it gives a rather indigestible result:wherewithds 2 4 =[ ( drH − fdt 2 + R 2 2 )]∆ + dθ2A = Φ 0 dt + Φ 3 (dφ − ωdt) ,H = 1 + H (1)B + H (2) B 2 + H (3) B 3 + H (4) B 4R 2H (1) = 2aqr sin 2 θ − 2p(r 2 + a 2 ) cos θ ,H (2) = 1 2 [(r2 + a 2 ) 2 − a 2 ∆ sin 2 θ] sin 2 θ + 2˜q2 3 (a 2 + r 2 cos 2 θ) ,+ Σ sin2 θH R 2 (dφ − ωdt) 2 ,H (3) = −pa 2 ∆ sin 2 θ cos θ − qa∆r [r2 (3 − cos 2 θ) cos 2 θ + a 2 (1 + cos 2 θ)] + aq(r2 + a 2 ) 2 (1 + cos 2 θ)2r− 1 2 p(r4 − a 4 ) sin 2 θ cos θ + q˜q2 a[(2r 2 + a 2 ) cos 2 θ + a 2 ]− 12r2 p˜q2 (r 2 + a 2 ) cos 3 θ ,H (4) = 1 16 (r2 + a 2 ) 2 R 2 sin 4 θ + 1 4 ma2 r(r 2 + a 2 ) sin 6 θ + 1 4 ma2˜q 2 r(cos 2 θ − 5) sin 2 θ cos 2 θ+ 1 4 m2 a 2 [r 2 (cos 2 θ − 3) 2 cos 2 θ + a 2 (1 + cos 2 θ) 2 ]+ 1 8˜q2 (r 2 + a 2 )(r 2 + a 2 + a 2 cos 2 θ) sin 2 θ cos 2 θ + 116˜q4 [r 2 cos 2 θ + a 2 (1 + sin 2 θ) 2 ] cos 2 θand we have defined ˜q 2 ≡ q 2 + p 2 .The function ω is given by
Page 1 and 2: Geometry and Thermodynamics of Blac
Page 3 and 4: Exact Solution with Back Reaction?
Page 5 and 6: Rotating Generalisations?• Rotati
Page 7: Magnetisation of Kerr-Newman Black
Page 11 and 12: • This is evident in the cylindri
Page 13 and 14: Asymptotically-Melvin Magnetised Ke
Page 15 and 16: ξ = ∆φ2π e−2ϕ (dφ + 2Ā) a
Page 17 and 18: Magnetised Black Holes in STU Super
Page 19: Conclusions and Open Problems• A

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

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