Region IV Curvature singularity t Coordinate singularity in .NET

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Region IV Curvature singularity t Coordinate singularity
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x u =1 Region II Flat background Region I u=0 B A v=0 Region III u =1
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Fig. 13-11. The collision of two impulsive plane gravitational waves. Focusing effects
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induce a curvature singularity which is inevitable for Particle A, which crosses both wavefronts. [Courtesy of J.B. Grif ths (1991).]
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Gravitational Waves
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In Fig. 13-11, the world lines of two particles are portrayed. Noting that the wave approaching from Region II is characterized by the Dirac delta function (u), we note that a particle will cross the wave if it passes the line u = 0 and analogously for a particle crossing v = 0. Looking at the gure, we see that Particle B crosses the wave approaching from Region II, but it encounters the line u 2 + v 2 = 1 before encountering the second wave, which is characterized by (v). Therefore Particle B avoids the curvature singularity in Region IV For . Particle B, the singularity, which comes across in Region II, is just a coordinate singularity. Particle A, meanwhile, has a different fate. The world line of this particle indicates that it encounters both wavefronts before encountering the singularity. Unfortunately for Particle A it encounters a real curvature singularity in Region IV .
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More General Collisions
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We now leave impulsive waves behind and consider more general types of collision. We will again consider the collision of two waves. First we imagine a null congruence in vacuum and review its characteristics. The geodesics of the congruence are parallel and = = 0, meaning that the contraction, twist, and shear vanish for the congruence. For a more general type of collision than that considered in the last section, we can imagine that the gravity wave encounters either an electromagnetic wave (which has nonzero energy density and therefore is a source of gravitational eld) or a collision with a gravitational wave. The interaction can be described by the two Newman-Penrose equations D = 2 + + D = ( + ) +
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00 0
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The terms 00 and 0 can represent an opposing electromagnetic and gravitational wave, respectively. Initially, as the wave travels through a region with no other waves present, 00 = 0 and 0 = 0. Since the wave has zero expansion, shear, and twist, this situation is described by D = 0 D = 0
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Gravitational Waves
> 0,
If the wave encounters an electromagnetic wave, which means that then initially D = D = 0
This causes to increase, causing the congruence to converge since Re gives the expansion of the wave. Therefore as gets larger the expansion gets smaller. On the other hand, if the wave encounters another gravitational wave, then initially D =
D = and so we can see that shear caused by the collision induces a contraction via the rst equation. In other words, these equations represent the following effects:
If a congruence passes through a region with nonzero energy density (which means that 00 is nonzero), it will focus. If a gravitational wave collides with another gravitational wave, it will begin to shear. This induces a contraction in the congruence and it will therefore begin to focus. Taking these effects together, we see that the opposing gravity wave causes an astigmatic focusing effect.
In the next example, we imagine that a null congruence begins in vacuum. We take the region v < 0 to be a at region of spacetime. De ning a plane wave by v = const, we choose the null vector l a to point along v. The null hypersurface is given by v = 0. In the region past v = 0, an opposing wave is encountered (see Fig. 13-12). In the next example, we consider the line element in the region where the two waves interact and illustrate that the shear and convergence become nonzero. EXAMPLE 13-2 The region v > 0 is described by the line element ds 2 = 2 du dv cos2 av dx 2 cosh2 av dy 2 (13.52)
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