Plane Waves in .NET

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Plane Waves
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A pp-wave is a plane wave if we can write H in the form H (u, x, y) = a (u) x 2 y 2 + 2b (u) x y + c (u) x 2 + y 2 (general plane wave) (13.38)
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(compare with our derivation of (13.36)). The functions a and b describe the polarization states of the gravitational wave, while c represents waves of other types of radiation. To represent a plane gravitational wave in the vacuum, we let c vanish; that is, H (u, x, y) becomes H (u, x, y) = a (u) x 2 y 2 + 2b (u) x y (plane wave in vacuum) (13.39)
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Gravitational Waves
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Since these are plane waves, we expect that we can write the wave as an expression of the form Aei , where A is the amplitude of the wave. In fact we can, and it turns out that the wave is described by the Weyl scalar 4 . That is, we write the gravity wave as
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= A ei
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In this case represents the polarization of the wave. A linearly polarized wave is one for which is a constant. In Example 9-5, we found that = 1 4 2 H 2 H 2 H + 2i x2 y2 x y
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Let s consider this with the form of H given for a plane wave in vacuum (13.39). We write the metric as ds 2 = h 11 x 2 + h 22 y 2 + 2h 12 x y du 2 + 2 du dv dx 2 dy 2 The form of the Weyl scalar for this metric will lead to a plane wave solution. We will also demonstrate that the vacuum equation using the Ricci scalar we found in Example 9-5 leads to the relations among h 11 , h 22 , and h 12 for plane waves that we found in the previous section. To begin let s write down the Weyl scalar when H = h 11 x 2 + h 22 y 2 + 2h 12 x y. We have = = 1 4 2 H 2 H 2 H + 2i x2 y2 x y
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1 (2h 11 2h 22 + 4i h 12 ) 4 1 = (h 11 h 22 + 2i h 12 ) 2 In 9 we learned that the only nonzero component of the Ricci tensor was
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2 H 2 H + x2 y2
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Gravitational Waves
In this case taking H = h 11 x 2 + h 22 y 2 + 2h 12 x y, we have
1 (h 11 + h 22 ) 2
Now consider the vacuum eld equations, which are de ned by Rab = 0 or 22 = 0. This leads us to conclude that h 22 = h 11 . The curvature tensor then becomes
1 (2h 11 + 2i h 12 ) = h 11 + i h 12 2
Now let s consider the case of a plane wave with linear polarization, which means that h 12 is proportional to h 11 . In that case we expect that we can write i i 4 = h(u) e . Using Euler s formula to expand 4 = h(u) e , we have
= h(u) ei = h(u) (cos + i sin ) = h(u) cos + i h(u) sin
Comparison with 4 = h 11 + i h 12 tells us that we can write h 11 = h(u) cos and h 12 = h(u) sin . The meaning of is taken to be that the polarization vector of the wave is at an angle with the x-axis. Summarizing, in the case of constant linear polarization in the vacuum we can write H as H = h 11 x 2 + h 22 y 2 + 2h 12 x y = h(u) cos x 2 h(u) cos y 2 + 2 sin x y = h(u) cos (x y ) + 2 sin x y
(13.40)
The Aichelburg-Sexl Solution
An interesting solution involving a black hole passing nearby was studied by Aichelburg and Sexl. The metric is given by ds 2 = 4 log(x 2 + y 2 )du 2 + 2dudr dx 2 dy 2 (13.41)
This metric is clearly in the form of the Brinkmann metric and so represents a pp-wave spacetime. However, in this case H = 4 log(x 2 + y 2 ), which is not in the form given by (13.38); therefore, it does not represent plane waves.
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