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The following exercise will demonstrate that the cosmological constant has no effect over the scale of the solar system. Start with the general form of the
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Cosmology
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Schwarzschild metric ds 2 = e2 (r ) dt 2 e2 (r ) dr 2 r 2 d 2 + sin2 d 2
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With a nonzero cosmological constant, the Ricci scalar for this metric satis es (a) R = (b) R = 0 (c) R = 4 (d) R = It can be shown that (r ) = ln k (r ). If A and B are constants of integration, then (a) r e = A + Br kr 3 (b) r e = A + Br (c) r e = A + Br r 2
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By considering the eld equation R = (a) B = 0 (b) B = k (c) B = 4k
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With the previous results in mind, choosing k =1 we can write the spatial part of the line element as dr 2 (a) dl 2 = + r 2 d 2 + r 2 sin2 d 2 2m 1 1 r3 r 3 dr 2 2 (b) dl = + r 2 d 2 + r 2 sin2 d 2 1 1 r3 3 dr 2 2 (c) dl = + r 2 d 2 + r 2 sin2 d 2 1 1+ r3 3 If the cosmological constant is proportional to the size of the universe, say a =
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(a) the Ricci scalar vanishes (b) particles would experience additional accelerations as revealed in tidal effects (c) the presence of the cosmological constant could not be detected by observations within the solar system
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Consider a at universe with positive cosmological constant. Starting with a 2 = C + 3 a 2 use a change of variables u = 2 a 3 , it can be shown a 3C that 3C (a) a 3 = 2 cosh (3 )1/2 t 1 (b) a = (c) a =
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CHAPTER
Gravitational Waves
An important characteristic of gravity within the framework of general relativity is that the theory is nonlinear. Mathematically, this means that if gab and ab are two solutions of the eld equations, then agab + b ab (where a, b are scalars) may not be a solution. This fact manifests itself physically in two ways. First, since a linear combination may not be a solution, we cannot take the overall gravitational eld of two bodies to be the summation of the individual gravitational elds of each body. In addition, the fact that the gravitational eld has energy and special relativity tells us that energy and mass are equivalent leads to the remarkable fact that the gravitational eld is its own source. We don t notice these effects in the solar system because we live in a region of weak gravitational elds, and so the linear newtonian theory is a very good approximation. But fundamentally these effects are there, and they are one more way that Einstein s theory differs from Newton s. A common mathematical technique when faced with such a situation is to study a linearized version of the theory to gain some insight. In this chapter that is exactly what we will do. We consider a study of the linearized eld equations and will make the astonishing discovery that gravitational effects can
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Gravitational Waves
propagate as waves traveling at the speed of light. This will require a study of weak gravitational elds. From here we will develop the Brinkmann metric that is used to represent pp-wave spacetimes. We will study this metric and the collision of gravitational waves using the Newman-Penrose formalism. We conclude the chapter with a very brief look at gravitational wave spacetimes with nonzero cosmological constant. In this chapter we will use the shorthand notation ka,b to represent the covariant derivative of ka in place of the usual notation b ka . Therefore ka,b = b ka c ab kc .
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