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The analysis is greatly simpli ed by the fact that n a only has a u component that is a constant. So this reduces to n a;b = u ab n u = u ab . Considering the mixed terms rst, we nd n x;y = n y;x = For the other two terms, we have n x;x = n y;y =
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= (1 v (v)) (v) = (1 + v (v)) (v)
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Now we write down the products m x m x and m y m y . mx mx = 1 2 (1 v (v)) 1 2 (1 + v (v)) 1 2 (1 v (v)) i 1 2 (1 + v (v)) = 1 2 (1 v (v))2 1 2 (1 + v (v))2
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and so the calculation for the spin coef cient becomes = n x;x m x m x + n y;y m y m y = (1 v (v)) (v) (1 + v (v)) (v) = = 1 2 (1 v (v))2 1 2 (1 + v (v))2
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The calculation for the remaining spin coef cient is similar. The only difference this time is that we use terms like m x m x instead of m x m x . We have = n x;x m x m x + n y;y m y m y = (1 v (v)) (v) (1 + v (v)) (v) = = 1 2 (1 v (v))2 1 2 (1 + v (v))2
(v) (v) 2 (1 v (v)) 2 (1 + v (v)) v (v) (1 v (v)) (1 + v (v))
Gravitational Waves
Looking at the expressions for the Weyl scalars, we can see that the only nonzero term we have in this case is going to be 4 . Since most of the spin coef cients vanish, the calculation of this term is relatively painless. First we need to compute the directional derivative of . Remember that is a scalar, and so the directional derivative is just = n a a = n a a because the covariant derivative reduces to an ordinary partial derivative when applied to a scalar. Looking at (13.51), the only nonzero component is the v component. Calculating the derivative, we nd = v v (v) 1 v 2 (v)
f g g f g2
We compute this derivative using ( f /g) = f = (v) f = (v)
. We take
g = 1 v 2 (v) g = 2v (v) v 2 (v) = 2v (v) Noting the overall minus sign, we obtain v = (v)(1 v 2 (v)) + (2v (v)) (v) (1 v 2 (v))
(v) + 2v (v) (1 v 2 (v))2
Using f (v) (v) = f (0) (v), we can simplify this term because (v) (1 v 2 (v))2 This allows us to write the derivative as v = (v) 2v (v) (1 v 2 (v))2 = (v)
All together, the Weyl scalar turns out to be = ( + ) (3 ) + 3 + +
= ( + ) = 2
= (v) +
Gravitational Waves
2v (v) (1 v 2 (v))2 2 (v) (1 v (v)) (1 + v (v))
v (v) (1 v (v)) (1 + v (v)) 2v (v) (1 v 2 (v))
= (v) + = (v)
2v (v) (1 v 2 (v))2
As expected, we obtain a Dirac delta function. Since 4 is the only nonzero Weyl scalar, we conclude that this spacetime is Petrov Type N.
The Effects of Collision
Referring once again to Fig. 13-10, Regions I, II, and III are at. In Region IV , which represents the interaction region of the two waves, spacetime is curved. Of speci c interest: is the interaction of the two waves causes a focusing effect that does two things. Since focusing means that the wave no longer has zero convergence, the waves are no longer plane waves in Region IV More interesting . is the fact that the focusing in this case results in a singularity described by u2 + v 2 = 1
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