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c xu m c in .NET
c xu m c QRCode Decoder In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Draw QR Code ISO/IEC18004 In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create QR image in VS .NET applications. All the Christoffel symbols of the form c xu are zero for this spacetime, and i since m x = 2 but = 1 + 2 (x 2 + y 2 ), the derivative with respect to u vanishes, and so m x;u vanishes as well. The same argument holds for each term. Therefore we conclude that = 1 1 v lu;u n u n u = 2 2 Recognize QR Code ISO/IEC18004 In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. Bar Code Drawer In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create bar code image in Visual Studio .NET applications. Gravitational Waves
Bar Code Reader In Visual Studio .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. QRCode Drawer In C# Using Barcode drawer for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. An exercise shows that all the remaining spin coef cients vanish. With only one nonzero spin coef cient to consider, the NewmanPenrose identities become rather simple. There are three identities we can use that contain D = ( + ) + ( + ) ( + ) + + + ( ) Quick Response Code Creation In .NET Framework Using Barcode drawer for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QR Code 2d Barcode Encoder In Visual Basic .NET Using Barcode printer for VS .NET Control to generate, create QRCode image in .NET applications. (13.58) (13.59) Barcode Drawer In .NET Using Barcode generation for VS .NET Control to generate, create bar code image in VS .NET applications. Generate EAN 128 In Visual Studio .NET Using Barcode generator for Visual Studio .NET Control to generate, create EAN / UCC  13 image in Visual Studio .NET applications. = + + 2 + ( ) 2D Barcode Generation In VS .NET Using Barcode creation for .NET Control to generate, create Matrix Barcode image in Visual Studio .NET applications. Bookland EAN Generation In .NET Framework Using Barcode maker for .NET Control to generate, create ISBN  10 image in .NET framework applications. 2 NP 11
Data Matrix ECC200 Creation In Java Using Barcode encoder for Java Control to generate, create Data Matrix image in Java applications. Bar Code Recognizer In Java Using Barcode Control SDK for Eclipse BIRT Control to generate, create, read, scan barcode image in BIRT applications. = + ( ) + ( + ) + v
UPCA Supplement 2 Reader In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Bar Code Recognizer In Visual C# Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET applications. (13.60) Here the NewtonPenrose scalar is related to the Ricci scalar via Looking at the left side of (13.58), the only nonzero term is D = l a a = v = v 1 v 2 = 1 2 Encode EAN13 In None Using Barcode drawer for Online Control to generate, create EAN13 Supplement 5 image in Online applications. Creating Bar Code In ObjectiveC Using Barcode drawer for iPhone Control to generate, create barcode image in iPhone applications. 1 R. 24 ECC200 Drawer In None Using Barcode creator for Microsoft Excel Control to generate, create Data Matrix 2d barcode image in Excel applications. USS128 Recognizer In Visual C#.NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Putting this together with the nonzero terms on the right side gives
(13.61) Moving to the next equation, everything vanishes except the last three unknown terms. Therefore (13.59) becomes (13.62) Finally, all terms vanish in (13.60) with the exception of the last two, giving us
= 2 (13.63) Using (13.62) and (13.63) in (13.61), we nd 1 2 =
NP 2
= 2 = 6 1 = 12 (13.64) Gravitational Waves
Back substitution of this result into (13.62) and (13.63) gives
(13.65) It can be shown that the other Weyl scalars vanish. Therefore, with 2 = 0, we conclude that this spacetime is Petrov Type D. This means there are two principal null directions, each doubly repeated. The fact that the spacetime contains 2 and not 4 or 0 indicates that this spacetime describes electromagnetic elds and not gravitational radiation. This spacetime represents a vacuum universe that contains electromagnetic elds with no matter. Further Reading
The study of gravitational radiation is an active and exciting area. With LIGO coming into operation, exciting experimental results will soon complement the theory. Unfortunately we can scratch only the surface in this brief treatment. Limited space precluded us from covering experimental detection of gravitational waves, and energy and power carried by the waves. The interested reader is encouraged to consult Misner et al. (1973) for an indepth treatment, or Schutz (1985) for a more elementary but thorough presentation. The Bondi metric is important for the analysis of radiating sources (see D Inverno, 1992). Hartle (2002) has uptodate information on the detection of gravity waves, and active area of research in current physics. The section on the collision of gravitational waves relied on Colliding Plane Waves in General Relativity by J.B. Grif ths (1991), which, while out of print, is available for free download at http://wwwstaff.lboro.ac.uk/ majbg/jbg/book.html. The reader is encouraged to examine that text for a thorough discussion of gravitational wave collisions. In addition, this chapter also relied on Generalized ppWaves by J.D. Steele, which the mathematically advanced reader may nd interesting. It is available at http://web.maths.unsw.edu.au/ jds/Papers/gppwaves.pdf. The reader interested in gravitational waves and the cosmological constant should consult Generalized Kundt waves and their physical interpretation, J.B. Grif ths, P. Docherty, and J. Podolsk , Class. Quantum Grav., 21, 207 222, 2004 (gry qc/0310083), or on which Example 133 was based. Gravitational Waves
Quiz
1. Following the procedure used in Example 92, consider the collision of a gravitational wave with an electromagnetic wave. The line element in the region v 0 is given by ds 2 = 2 du dv cos2 av dx 2 + dy 2 By calculating the nonzero spin coef cients, one nds that (a) there is pure focusing (b) the Weyl tensor vanishes (c) there is twist and shear 2. Consider the AichelburgSexl metric given in (13.41). The only nonzero spin coef cient is given by x (a) = 2 2 x 2 +y 2 (x+iy) (b) = 2 2 x 2 +y 2 (x+iy) (c) = 2 x 2 +y 2 Compute the Ricci scalar 22 for the metric used in Example 91. You will nd (a) 22 = + + ( 3 ) (b) 22 = 2 + + ( 3 ) = 2 2 (c) = 2

