Gravitational Waves in VS .NET

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Gravitational Waves
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Shearing and contracting congruence
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Parallel congruence =0 =0
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Coordinate singularity v= 2
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Gravitational wave 0 = a 2 (v) Wavefront v=0
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Fig. 13-12. An illustration of two colliding gravity waves. In the at region, the null congruence has parallel rays with no shear or contraction. In the region where the two gravity waves collide, there is shear and congruence. [Courtesy of J.B. Grif ths (1991).]
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Show that the congruence contracts and shears after passing the gravitational wavefront. Determine the Petrov type and interpret. Describe the focusing effect and determine the alignment of the shear axes. SOLUTION 13-2 The components of the metric tensor are given by g uv = g vu = 1 gx x = cos2 1 , (av) g yy = 1 cosh2 (av) (13.53)
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guv = gvu = 1 gx x = cos2 (av), g yy = cosh2 (av)
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Using (4.16), one can show that the nonzero Christoffel symbols are given by
u x xx xv
= a cos av sin av, = a tan av,
y yv
= a cosh av sinh av (13.54)
= a tanh av
Gravitational Waves
We de ne the null tetrad such that l a points along v, giving l a = (0, 1, 0, 0), m a = 0, 0, n a = (1, 0, 0, 0) 1 2 cos av , i 2 cosh av (13.55)
Lowering indices with the metric tensor (i.e., la = gab l b , etc.), we nd la = (1, 0, 0, 0), n a = (0, 1, 0, 0) (13.56)
cos av i cosh av m a = 0, 0, , 2 2
To show that the congruence contracts and shears, we must show that and are nonzero. Now la;b = b la c ab lc . The simplicity of la means that this expression will take a very simple form. In fact, since la has only a u component which is constant (and therefore b la = 0 for all a,b), we can write la;b =
u ab l u
We can calculate the spin coef cient representing contraction using = la;b m a m b , where m b is the complex conjugate of m a as given in (13.55). There are only two nonzero terms in the sum. We calculate each of these individually: l x;x = l y;y = and so we have = la;b m a m b = l x;x m x m x + l y;y m y m y = (a cos av sin av) 1 2 cos av 1 2 cos av i 2 cosh av
u u xx yy
= a cos av sin av = a cosh av sinh av
i + ( a cosh av sinh av) 2 cosh av
= = cos av sin av cos av cos av a 2
Gravitational Waves
cosh av sinh av cosh av cosh av
a sin av a sinh av 2 cos av 2 cosh av a = (tan av tanh av) 2 Next we compute the shear. This can be done by computing = la;b m a m b . Again, the only nonzero terms in the sum are those with l x;x , l y;y . And so we obtain = l x;x m x m x + l y;y m y m y 1 = (a cos av sin av) 2 cos av 1 2 cos av i 2 cosh av
i + ( a cosh av sinh av) 2 cosh av a a cos av sin av + 2 cos av cos av 2 a = (tan av + tanh av) 2 = cosh av sinh av cosh av cosh av
An exercise shows that the remaining spin coef cients vanish. Let s make a quick qualitative sketch of the shear.
Fig. 13-13. The tan function blows up at av = /2.
Gravitational Waves
As we can see from = a (tan av tanh av) and by looking at the sketch (see 2 Fig. 13-13), the shear blows up at av = /2. In other words, there is a singularity. Looking at the Newman-Penrose identities, we see that the only nonzero Weyl scalar is given by 0 .The only nonzero spin coef cients are , , and so we have
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