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(13.32)
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(13.33)
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We will now proceed to write the plane wave metric given by (13.18) in terms of U and V to nd ds 2 = dt 2 (1 h xx ) dx 2 (1 + h xx ) dy 2 dz 2 = 1 dU 2 + 2dU dV + dV 2 (1 h xx ) dx 2 4 1 dU 2 2dU dV + dV 2 (1 + h xx ) dy 2 4
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= dU dV (1 h xx ) dx 2 (1 + h xx ) dy 2 Now, recalling that h xx = h xx (t z) = h xx (U ), we de ne a 2 (U ) = 1 h xx and b2 (U ) = 1 + h xx
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and we obtain the Rosen line element ds 2 = dU dV a 2 (U ) dx 2 b2 (U ) dy 2 (13.34)
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We choose the following basis of one forms for this metric: 0 = with
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1 (dU + dV ) , 2
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1 =
1 (dU dV ) , 2
2 = a (u) dx,
3 = b (u) dy (13.35)
a b
1 0 = 0 0
0 1 0 0
0 0 0 0 1 0 0 1
An exercise (see Quiz) shows that the nonzero components of the Ricci tensor in this basis are R00 = R01 = R10 = R11 = The vacuum equations Ra b = 0 give 1 d2 b 1 d2 a + =0 a dU 2 b dU 2
1 d 1 d Letting h(U ) = a dUa2 , we see that this equation implies that b dUb2 = h(U ) and so we can write the metric in terms of the single function h. Therefore (13.34) becomes
1 d2 b 1 d2 a + a dU 2 b dU 2
ds 2 = dU dV h 2 (U ) dx 2 h 2 (U ) dy 2 We now apply the following coordinate transformation. Let u = U, v = V + x 2 aa + y 2 bb , X = ax, Y = by
From the rst two equations, we obtain du = dU and V = v x 2 aa y 2 bb =v a 2 b 2 X Y a b
Inverting the last two equations for x and y gives
Gravitational Waves
1 1 X, y = Y a b a 1 dx = dX 2 X du a a x = Now we use these relations along with the de nition h(u) = a /a = b /b and obtain dV = dv 2 + (a )2 a b a b X dX X 2 du + 2 X 2 du 2 Y dY Y 2 du a a a b b
(b )2 2 Y du b2 (a )2 a b X dX h(u)X 2 du + 2 X 2 du 2 Y dY + h(u)Y 2 du a a b
= dv 2 +
(b )2 2 Y du b2
We also have a dx = a
2 2 2
a 1 2 X du + dX a a b 1 X du + dX 2 b b
(a )2 2 2 a = 2 X du 2 du dX + dX 2 a a = (b )2 2 2 b Y du 2 du dY + dY 2 2 b b
b2 dy 2 = b2
Substitution of these results into ds 2 = dU dV a 2 (U ) dx 2 b2 (U ) dy 2 gives the Brinkmann metric ds 2 = h(u) Y 2 X 2 du 2 + 2 du dv dX 2 dY 2 (13.36)
From here on we will drop the uppercase labels and let X x and Y y (just be aware of the relationship of these variables to the coordinates used in the original form of the plane wave metric given in (13.18)). More generally, if we de ne an arbitrary coef cient function H (u, x, y), we can write this metric as ds 2 = H (u, x, y) du 2 + 2 du dv dx 2 dy 2 (13.37)
Gravitational Waves
This metric represents a pp-wave spacetime; however, it does not necessarily represent plane waves, which are a special case. We will describe the form of H in the case of plane waves below. In Example 9-5, we found the Weyl and Ricci scalars for this metric. In that problem we choose la = (1, 0, 0, 0) as the covariantly constant null vector where the coordinates are given by (u, v, x, y) . The principal null direction is along la , which means that this vector de nes the direction along which the rays of the gravity wave are coincident. The coordinate v is another null coordinate while the coordinates x, y de ne the surface of the plane wave. It is obviously true that la,b = 0. To understand the implication of this requirement, we state some of the spin coef cients here: = la,b m a m b , = la,b m a m b , = la,b m a n b
We see that la,b = 0 implies that = = = 0. Therefore, as we explained in the beginning of this section, a pp-wave has no expansion, twist, or shear. In addition, the fact that = 0 tells us that the null rays de ned by la are parallel. This shows that this metric is a pp-wave spacetime. The form of the function H in the Brinkmann metric can be studied further. A generalized pp-wave is one for which H can be written in the form H = Ax 2 + By 2 + C x y + Dx + E y + F where A, B, C, D, E, and F are real valued functions of u.
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