The Singularity in .NET

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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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Black Holes
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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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Further Reading
The study of black holes is an interesting, but serious and complicated topic. A great deal of this chapter was based on a very nice introduction to the subject, Exploring Black Holes: An Introduction to General Relativity by Edwin F. Taylor and John Archibald Wheeler, Addison-Wesley, 2000. For a more technical and detailed introductory exposition on black holes consult D Inverno (1992). There one can nd a good description of black holes, charged black holes, and Kerr black holes. One interesting phenomenon associated with rotating black holes we are not able to cover owing to space limitations is the Penrose process. This is a method that could be used to extract energy from the black hole. See 15 of Hartle (2002) or pages F21 F30 of Taylor and Wheeler (2000) for more information. Our de nition of the orbital equations was taken from Lightman, Press, Price, and Teukolsky (1975), which contains several solved problems related to black holes.
Black Holes
To see how to choose an orthonormal tetrad to use with this metric and the results of calculations of the curvature tensor, consult http://panda.unm.edu/ courses/ nley/p570/handouts/kerr.pdf.
Quiz
1. Which of the following could not be used to characterize a black hole (a) Mass (b) Electron density (c) Electric charge (d) Angular momentum Using Eddington-Finkelstein coordinates, one nds that (a) the surface de ned by r = 2m is a genuine singularity (b) moving along the radial coordinate, in the direction of smaller r , light cones begin to tip over. At r = 2m, outward traveling photons remain stationary. (c) moving along the radial coordinate, in the direction of smaller r , light cones begin to become narrow. At r = 2m, outward traveling photons remain stationary. (d) moving along the radial coordinate, in the direction of smaller r , light cones remain stationary. At r = 2m, outward traveling photons remain stationary. In Kruskal coordinates, there is a genuine singularity at (a) r = 0 (b) r = 2m (c) r = m Frame dragging can be best described as (a) an intertial effect (b) a particle giving up angular momentum (c) a particle, initially having zero angular momentum, will acquire an angular velocity in the direction in which the source is rotating (d) a particle, initially having zero angular momentum, will acquire an angular velocity in the direction opposite to that in which the source is rotating The ergosphere can be described by saying (a) inside the ergosphere, all timelike geodesics rotate with the mass that is the source of the gravitational eld
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