Velocity-of-Light Surfaces in Kerr and Extreme Kerr
The extreme limit of the Kerr solution has recently attracted much attention, see (Compère, arXiv:1203.3561 [hep-th]) and references therein. We have investigated hypersurfaces called velocity-of-light surfaces for extreme and near extreme Kerr.
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Abstract The extreme limit of the Kerr solution has recently attracted much attention, see (Compère, arXiv:1203.3561 [hep-th]) and references therein. We have investigated hypersurfaces called velocity-of-light surfaces for extreme and near extreme Kerr.
1 Killing Vector Fields in Kerr The Kerr solution possesses two linearly independent Killing vector fields, namely @t and @ in Boyer–Lindquist coordinates. Constant linear combinations of these are also Killing vectors. The linear combination D @t C 2mra C @ is known to be null at the outer horizon r D rC . The norm of is kk D @t C 2
2 2 a a r rC a @ D gtt C gt C g D 2 2 2 f .r; /; 2mrC mrC 2mrC 4m rC
p p m2 a2 , 2 D r 2 Ca2 cos2 , where rC D m C m2 a2 , r D m 2 2 2 2 2 f .r; / D 4m rC .r r / a sin 4m .r rC / C .r r / 2 C 2mr . However, an interesting fact is that there are also solutions to f .r; / D 0 which correspond to other hypersurfaces where the Killing vector field becomes null; this hypersurface is called the velocity-of-light surface [1].
J.E. Åman () Department of Physics, Stockholm University, 106 91 Stockholm, Sweden e-mail: [email protected] H.F. Rúnarsson Department of Physics, Stockholm University, 106 91 Stockholm, Sweden, Current address: Physics Department Aveiro University, Campus de Santiago, 3810-183 Aveiro, Portugal e-mail: [email protected] A. García-Parrado et al. (eds.), Progress in Mathematical Relativity, Gravitation and Cosmology, Springer Proceedings in Mathematics & Statistics 60, DOI 10.1007/978-3-642-40157-2__70, © Springer-Verlag Berlin Heidelberg 2014
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J.E. Åman and H.F. Rúnarsson
Fig. 1 Polar and orthogonal plots for extreme Kerr, a D m. In our plots we take m D 1. The blue p line(s) represent f .r; / D 0. The green line is the outer limit of the ergosphere re D m C m2 a2 cos2 . In the orthogonal plot also the solution where r < 0 is shown
The general solution of f .r; / D 0 gives three roots for r, one of them for r < 0 [2]. The solutions for r are somewhat long and complicated, involving third degree roots, but the solution for can easily be presented [3]: s D arcsin
h
p h2 g ; 2 2a
where h D .r 2 C a2 /2 4m2 .a2 C 2rC .r m//; g D 16m2 2 .2mrC a2 /; D r 2 C a2 2mr:
2 Extreme Kerr, a D m For extreme Kerr a D m, f .r; / factorizes into: f .r; / D m2 .r m/.m sin 2 C.r Cm/ sin 2m/.m sin 2 .r Cm/ sin 2m/: Thus the Killing vector is also null when
r=m D
2 sin sin2 ; sin
and r=m D
2 sin C sin2 : sin
Velocity-of-Light Surfaces in Kerr and Extreme Kerr
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Fig. 2 Polar plot for Kerr with a=m D 999999=1000000. The null hypersurfaces are close to the a D m line, except near the horizons
Fig. 3 Polar and orthogonal plots for Kerr with a=m D 999=1000. The red lines represent the outer and inner horizons, rC and r . The inner velocity-of-light hypersurface (blue) always touches the inner horizon at the north pole
Fig. 4 Polar and orthogonal plots for Kerr with a=m D 99=100
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