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Null Tetrads and the Petrov
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In the context of gravitational radiation, we have the following interpretations:
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transverse wave component in the n a direction longitudinal wave component in the n a direction transverse wave component in the l a direction longitudinal wave component in the l a direction
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Finally, we mention the physical meaning of some of the spin coef cients. First we consider a congruence of null rays de ned by l a : =0 Re ( ) Im ( ) | | l a is tangent to a null congruence expansion of a null congruence (rays are diverging) twist of a null congruence shear of a null congruence
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For a congruence de ned by n a , these de nitions hold for , , and . EXAMPLE 9-5 Consider the Brinkmann metric that describes plane gravitational waves: ds 2 = H (u, x, y) du 2 + 2 du dv dx 2 dy 2 Find the nonzero Weyl scalars and components of the Ricci tensor, and interpret them. SOLUTION 9-5 One can calculate these quantities by de ning a basis of orthonormal one forms and proceeding with the usual methods, and then using (9.22) and related equations to calculate the desired quantities directly. However, we will take a new approach and use the Newman-Penrose identities to calculate the Ricci and Weyl scalars directly from the null tetrad. The coordinates are (u, v, x, y) where u and v are null coordinates. Specifically, u = t x and v = t + x. The vector v de nes the direction of propagation of the wave. Therefore, it is convenient to take l a = (0, 1, 0, 0) = v We can raise and lower the indices using the components of the metric tensor, so let s begin by identifying them in the coordinate basis. We recall that we can
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write the metric as
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Null Tetrads and the Petrov
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ds 2 = gab dx a dx b Therefore we can just read off the desired terms. Note that the cross term is given by 2 du dv = guv du dv + gvu dv du and so we take guv = gvu = 1. The other terms can be read off immediately and so ordering the coordinates as [u, v, x, y], we have H 1 (gab ) = 0 0 1 0 0 0 0 0 0 1 0 0 0 1
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The inverse of this matrix gives the components of the metric tensor in the coordinate basis with raised indices: 0 1 = 0 0 1 H 0 0 0 0 1 0 0 0 0 1
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g ab
With this information in hand, we can calculate the various components of the null vectors. We start with l a = (0, 1, 0, 0). Lowering the index, we nd la = gabl b = gav l v The only nonzero term in the metric of this form is guv = 1, and so we conclude that lu = guv l v = 1 la = (1, 0, 0, 0) (9.25)
To nd the components of the rest of the tetrad, we can apply (9.8) and (9.9). To avoid ipping back through the pages, let s restate the identity for gab here: gab = la n b + lb n a m a m b m b m a
Null Tetrads and the Petrov
Once we have all the components with lowered indices, we can raise indices with g ab to obtain all the other terms we need. Since there are not many terms in the metric, this procedure will be relatively painless. Since we also have lu = 1 as the only nonzero component of the rst null vector, we can actually guess as to what all the terms are without considering every single case. Starting from the top, we have guu = H = 2lu n u 2m u m u guv = 1 = lu n v + lv n u m u m v m u m v = lu n v m u m v m u m v gvu = 1 = lv n u + lu n v m v m u m v m u gxx = 1 = 2l x n x 2m x m x = 2m x m x g yy = 1 = 2l y n y 2m y m y = 2m y m y We are free to make some assumptions, since all we have to do is come up with a null tetrad that gives us back the metric. We take m u = m v = m u = m v = 0 and assume that the x component of m is real. Then from the rst and second equations, since lu = 1, we obtain nu = 1 H 2 and nv = 1
Now n is a null vector so we set n x = n y = 0. The fourth equation yields 1 mx = mx = 2 These terms are equal to each other since they are complex conjugates and this is the real part. The nal equation gives us the complex part of m. If we choose 1 1 m y = i 2 then the fth equation will be satis ed if m y = i 2 , which is what we expect anyway since it s the complex conjugate. Anyway, all together we have la = (1, 0, 0, 0) , na = 1 H, 1, 0, 0 2 1 m a = (0, 0, 1, i) 2
1 m a = (0, 0, 1, i) , 2
(9.26)
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