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Gravitational Waves
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we nd that (remember, derivatives with respect to y and z vanish) 1 h tt,t h t z,z h ,t = 0 2 1 h zt,t h zz,z h ,z = 0 2 h xt,t h x z,z = 0 h yt,t h yz,z = 0 Now we de ne the new variable u = t z. Then h ab h ab u h ab = = = h ab t u t u h ab h ab u h ab = = = h ab z u z u and so writing the derivatives in terms of the new variable, we have 1 h tt + h tz h = 0 2 1 h zt + h zz + h = 0 2 h xt + h xz = 0 h yt + h yz = 0 Let s take the last equation. We have h yt + h yz = h yt + h yz = 0 This can be true only if h yt + h yz is a constant. We have an additional physical requirement: h ab must vanish at in nity. Therefore we must choose the constant to be zero. We then nd that h yt = h yz
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In addition, we nd that h xt = h zx . We are left with 1 h tt + h tz h = 0 2 1 h zt + h zz + h = 0 2 Adding these equations give h tz = 1 (h tt + h zz ). Now subtract the rst equa2 tion from the second one to get h zt + h zz + 1 h h tt + h tz 1 h = h h tt + 2 2 h zz = 0. Now, writing out the trace explicitly, we have h = h tt h xx h yy h zz . The end result is h h tt + h zz = h tt h xx h yy h zz h tt + h zz = h xx h yy Since this term vanishes, we conclude that h yy = h xx . The complete metric perturbation has now been simpli ed to h ab = h tt h tx h ty 1 (h tt + h zz ) 2 h tx h xx h xy h tx h ty h xy h xx h ty 1 (h tt + h zz ) 2 h tx h ty h zz
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(13.16)
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We can go further with our choice of gauge so that most of the remaining terms vanish (see Adler et al., 1975, or D Inverno, 1992, for details). We simply state the end result that we will then study. A coordinate transformation can always be found to put the perturbation into the canonical form, which means (1) that we need to consider only h ab in (13.15) with the metric written as in (13.16). That is, we take 0 0 h ab = 0 0 0 h xx h xy 0 0 h xy h xx 0 0 0 0 0
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(13.17)
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Two polarizations result for gravity waves in the canonical form. In particular, we can take h xx = 0 and h xy = 0, which lead to +-polarization, or we can set h xx = 0 and h xy = 0, which gives -polarization. We examine both cases in detail in the next section.
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y hxx > 0
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Fig. 13-2. When h xx > 0, the relative distance of two particles separated along the
y-axis decreases.
The Behavior of Particles as a Gravitational Wave Passes
To study the behavior of massive particles as a gravitational wave passes, we consider the two cases of polarization which are given by h xx = 0, h xy = 0 and h xx = 0, h xy = 0. Taking the former case rst with h xy = 0, we use (13.17) together with gab = ab + h ab to write down the line element, which becomes ds 2 = dt 2 (1 h xx ) dx 2 (1 + h xx ) dy 2 dz 2 (13.18)
As a gravitational wave passes, this metric tells us that the relative distances between the particles will change. The wave will have oscillatory behavior and so we need to consider the form of (13.18) as h xx changes from h xx > 0 to zero and then to h xx < 0. For simplicity we imagine particles lying in the x y plane. Furthermore, suppose that the separation between the particles lies on a line that is parallel with the y-axis. Then dx vanishes and at a xed time, we can write ds 2 = (1 + h xx ) dy 2 This tells us that when h xx > 0 the distance along y-axis between the particles decreases because ds 2 becomes more negative. This is illustrated in Fig. 13-2. On the other hand, when h xx < 0, we can see that the relative distance between the particles will increase. This is shown in Fig. 13-3. When the separation of the particles is along the x-axis, we can see from the line element that the behavior will be the opposite. In particular, the proper