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(prime does not represent differentiation, this is just a new label). Furthermore, a we nd that the derivative of ab changes as b,a = a b,a . We are free to choose as long as the Riemann tensor keeps the same form. Therefore we a demand that W = b a,b , which leads to b,a = 0. When we substitute this into the Einstein tensor, we can write the full eld equations as 1 W ab = Tab 2 In vacuum, we obtain W ab = 0 (13.13) (13.12)
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Recalling the de nition of the d Alembertian operator, this is nothing more than the wave equation for waves traveling at c. This choice of gauge using the coordinate transformation goes under several names. Two frequently used names are the de Donder or Einstein gauge. We can write (13.13) in terms of (13.8), which gives 1 W ab = W h ab ab h 2 1 = W h ab ab W h = 0 2
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However, recalling (13.9) we can multiply (13.13) by ab to obtain 0 = W ab = ab W ab = W ab ab = W b b = W = W h So we can drop W h in the expansion of W ab , and the study of gravitational waves reduces to a study of the equation W h ab = 0 (13.14)
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A plane wave is characterized by uniform elds that are perpendicular to the direction of propagation. More speci cally, if the wave is traveling in the z direction, then the condition of uniformity means that the elds have no x and y dependence. Another way to think of plane waves that are familiar from electrodynamics is as follows: Plane waves are typically visualized as a wave whose surfaces of constant phase are in nite planes that are perpendicular to the direction of
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Fig. 13-1. A schematic representation of a plane wave. The wave vector k gives the direction of propagation of the wave. It is perpendicular to the wavefront, which is a plane and is a surface of constant phase.
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propagation. If we de ne a vector k that gives the direction of propagation of the wave, then the wavefront is de ned by the equation k r = const (see Fig. 13-1). We call k the wave vector. We will now consider plane gravitational waves traveling in the z direction. In this case h ab = h ab (t z) and the condition that the eld does not depend on x and y means that h ab h ab = =0 x y We will nd that the Riemann tensor is greatly simpli ed in this case. We won t go into the details (see Adler et al., 1975) but under these conditions the Riemann tensor can be shown to be a function of h xx , h xy , h yx , and h yy alone. The metric (1) (2) perturbation can then be split into two parts h ab = h ab + h ab , where 0 0 = 0 0 0 h xx h yx 0 0 h xy h yy 0 0 h tt h 0 xt (2) and h ab = 0 h yt h zt 0 h tx 0 0 h zx h ty 0 0 h zy h tz h xz h yz h zz
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h ab
(13.15)
It can be shown that we can nd a coordinate system such that the components (2) (1) in h ab vanish and h ab represent the entire perturbation. First, noting that the Einstein gauge condition requires that h a b,a 1 h ,b = 0, we can obtain further 2 simpli cation of the perturbation components. Applying the gauge condition,
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