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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
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Gravitational Waves
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An exercise shows that all the remaining spin coef cients vanish. With only one nonzero spin coef cient to consider, the Newman-Penrose identities become rather simple. There are three identities we can use that contain D = ( + ) + ( + ) ( + ) + + + ( )
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(13.58) (13.59)
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= + + 2 + ( )
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2 NP 11
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= + ( ) + ( + ) + v
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(13.60) Here the Newton-Penrose 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
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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 in-depth 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 up-to-date 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 pp-Waves 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 13-3 was based.
Gravitational Waves
Quiz
1. Following the procedure used in Example 9-2, 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 Aichelburg-Sexl 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 9-1. You will nd (a) 22 = + + ( 3 ) (b) 22 = 2 + + ( 3 ) = 2 2 (c)
=
2
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