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Following the process outlined in the Schwarzschild case, we wish to move beyond the coordinate singularity and consider any singularity we can nd from invariant quantities. In this case we again consider the invariant quantity formed from the Riemann tensor R abcd Rabcd , which leads to a genuine singularity described by r 2 + a 2 cos2 = 0 In the equatorial plane, again we have = 0 and the singularity is described by the equation r 2 + a 2 = 0. This rather innocuous equation actually describes a ring of radius a that lies in the x y plane. So for a rotating black hole the intrinsic singularity is not given by r = 0 but is instead a ring of radius a that lies in the equatorial plane with z = 0.
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A Summary of the Orbital Equations for the Kerr Metric
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The equations that govern the orbital motion of particles in the Kerr metric are given by
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r = Vr = V a = a E L z / sin2 + P (11.24)
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t = a a E sin2 L z + r 2 + a2 P
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where the derivative is in respect to the proper time or an af ne parameter. The extra quantities de ned in these equations are P = E r 2 + a2 L z a Vr = P 2 2r 2 + (L z a E)2 + A V = A cos2 a 2 2 E 2 + L 2 / sin2 z
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where
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E = conserved energy L z = conserved z component of angular momentum A = conserved quantity associated with total angular momentum = particles rest mass
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