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The Singularity in .NET
The Singularity QRCode Reader In .NET Framework Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. Quick Response Code Creator In Visual Studio .NET Using Barcode creation for VS .NET Control to generate, create QR Code image in VS .NET applications. 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. QR Code ISO/IEC18004 Reader In VS .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Encoding Barcode In VS .NET Using Barcode encoder for Visual Studio .NET Control to generate, create bar code image in .NET applications. A Summary of the Orbital Equations for the Kerr Metric
Barcode Recognizer In Visual Studio .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Printing QRCode In Visual C# Using Barcode generator for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications. The equations that govern the orbital motion of particles in the Kerr metric are given by
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Generating USS Code 39 In Java Using Barcode printer for Java Control to generate, create Code 3 of 9 image in Java applications. Drawing Code 3/9 In None Using Barcode generator for Online Control to generate, create USS Code 39 image in Online applications. E = conserved energy L z = conserved z component of angular momentum A = conserved quantity associated with total angular momentum = particles rest mass ECC200 Maker In Java Using Barcode creation for Android Control to generate, create Data Matrix image in Android applications. Printing Code 128 Code Set C In Visual Studio .NET Using Barcode encoder for ASP.NET Control to generate, create Code 128B image in ASP.NET applications. 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, AddisonWesley, 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 EddingtonFinkelstein 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

