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Gravitational Waves
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Colliding Gravity Waves
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In this section we consider the collision of two gravitational waves. Such collisions lead to many interesting effects such as the introduction of nonzero expansion and shear. To begin the study of this phenomenon, we consider the simplest case possible, the collision of two impulsive plane gravitational waves. An impulsive wave is a shock wave where the propagating disturbance is described by a Dirac delta function. That is, we imagine a highly localized disturbance propagating along the z direction. Since u = t z, we can create an idealized model of such a wave by taking h(u) = (u). This type of gravitational wave can be described with the metric ds 2 = (u) Y 2 X 2 du 2 + 2 du dr dX 2 dY 2 (13.42)
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In this metric, r is a spacelike coordinate. We can use a coordinate transformation (see Problem 7) to write this line element in terms of the null coordinates u and v, which gives ds 2 = 2 du dv [1 u (u)]2 dx 2 [1 + u (u)]2 dy 2 where (u) is the Heaviside step function. This function is de ned by (u) = 0 for 1 for u<0 u 0 (13.43)
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Therefore, in the region u 0, we can write the line element as ds 2 = 2 du dv (1 u)2 dx 2 (1 + u)2 dy 2 As an aside, note that the derivative of the Heaviside step function is the Dirac delta; i.e., d = (u) du (13.44)
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By considering the other null coordinate v, we can describe an opposing wave using ds 2 = 2 du dv [1 v (v)]2 dx 2 [1 + v (v)]2 dy 2 (13.45)
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Gravitational Waves
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Fig. 13-10. Dividing spacetime into four regions to study colliding waves. [Courtesy of
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and so for v 0, (13.45) assumes the form ds 2 = 2 du dv (1 v)2 dx 2 (1 + v)2 dy 2 Note that in either (13.43) or (13.45) if we set the step function the at space line element ds 2 = 2du dv dx 2 dy 2 = 0, we obtain (13.46)
This line element holds in the region u, v < 0. With these observations in place, we see that we can divide spacetime into four regions, as shown in Fig. 13-10. Region I, where both u, v < 0, is the at background spacetime described by (13.46). In Region II, v < 0 and u 0, and so this region contains the approaching wave characterized by (u). It is described by the line element (13.43). An analogous result holds for Region III, which contains the approaching wave (v) and is described by the line element (13.46). Finally, Region IV where both u 0 and v 0, is where the collision , of the two waves takes place. As an exercise in working with gravitational wave spacetimes using the Newman-Penrose formalism, we begin by considering a simple metric. We consider the case of the metric in Region III. EXAMPLE 13-1 Show that the metric in Region III describes an impulsive gravitational wave characterized by (v). Find the nonzero Weyl scalars and determine the Petrov type.
Gravitational Waves
SOLUTION 13-1 First let s write down some preliminaries we will need in calculations. The line element is given by ds 2 = 2 du dv (1 v (v))2 dx 2 (1 + v (v))2 dy 2 The nonzero components of the metric tensor are g uv = g vu = 1 gx x = 1 (1 v (v))
g yy =
1 (1 + v (v))2 (13.47)
guv = gvu = 1 gx x = (1 v (v))2 , g yy = (1 + v (v))2