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 ...
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.
Asymptotically-Melvin Magnetised Kerr-Newman• In the asymptotically static Melvin frame with timelike Killingvector K = ∂/∂t + Ω s ∂/∂φ, the q = −amB magnetised Kerr-Newman-Melvin solution will still have an ergoregion in theform of an oblate spheroid outside the outer horizon (verylike in the Kerr solution):• In the Kerr solution, ∂/∂t is the unique Killing vector thatis timelike at infinity, and hence it is the unique choice asgenerator of time translations:• In the magnetised Kerr-Newman solution there is in fact arange of angular velocities Ω around Ω = Ω s for which theergoregion is of only finite extent, and confined to the vicinityof the black hole.
- 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 and 8: Magnetisation of Kerr-Newman Black
- Page 9 and 10: Magnetisation of Kerr-Newman Black
- Page 11: • This is evident in the cylindri
- 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
Asymptotically-Melv<strong>in</strong> Magnetised Kerr-Newman• In the asymptotically static Melv<strong>in</strong> frame with timelike Kill<strong>in</strong>gvector K = ∂/∂t + Ω s ∂/∂φ, the q = −amB magnetised Kerr-Newman-Melv<strong>in</strong> solution will still have an ergoregion <strong>in</strong> theform <strong>of</strong> an oblate spheroid outside the outer horizon (verylike <strong>in</strong> the Kerr solution):• In the Kerr solution, ∂/∂t is the unique Kill<strong>in</strong>g vector thatis timelike at <strong>in</strong>f<strong>in</strong>ity, <strong>and</strong> hence it is the unique choice asgenerator <strong>of</strong> time translations:• In the magnetised Kerr-Newman solution there is <strong>in</strong> fact arange <strong>of</strong> angular velocities Ω around Ω = Ω s for which theergoregion is <strong>of</strong> only f<strong>in</strong>ite extent, <strong>and</strong> conf<strong>in</strong>ed to the vic<strong>in</strong>ity<strong>of</strong> the black hole.
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