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momentum. That is, we set
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The Einstein Field Equations
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1 G ab = Rab gab R 2 and then arrive at the eld equations G ab = Tab where is a constant that turns out to be 8 G. The vacuum equations are used to study the gravitational eld in a region of spacetime outside of the source i.e., where no matter and energy are present. For example, you can study the vacuum region of spacetime outside of a star. We can set Tab = 0 and then the vacuum Einstein equations become Rab = 0 (6.5)
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The cosmological constant was originally added to the equations by Einstein as a fudge factor. At the time, he and others believed that the universe was static. As we shall see Einstein s equations predict a dynamic universe, and so Einstein tinkered with the equations a bit to get them to t his predispositions at the time. When the observations of Hubble proved beyond reasonable doubt that the universe was expanding, Einstein threw out the cosmological constant and described it as the biggest mistake of his life. Recently, however, observation seems to indicate that some type of vacuum energy is at work in the universe, and so the cosmological constant is coming back in style. It is possible to include a small cosmological constant and still have a dynamic universe. If we de ne the vacuum energy of the universe to be v = 8 G
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then including this term, Einstein s equations can be written as 1 Rab gab R + gab 2 or G ab + gab = 8 GTab = 8 GTab (6.6)
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Therefore with this addition the Einstein tensor remains unchanged, and most of our work will involve calculating this beast. We demonstrate the solution of Einstein s equations with a cosmological constant term in the next example.
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We now consider an example that considers Einstein s equations in 2+1 dimensions. This means that we restrict ourselves to two spatial dimensions and time. Models based on 2+1 dimensions can be used to simplify the analysis while retaining important conceptual results. This is a technique that can be used in the study of quantum gravity for example. For a detailed discussion, see Carlip (1998). In this example, we consider the gravitational collapse of an inhomogeneous, spherically symmetric dust cloud Tab = u a u b with nonzero cosmological constant < 0. This is a long calculation, so we divide it into three examples. This problem is based on a recently published paper (see References), so it will give you an idea of how relativity calculations are done in actual current research. The rst example will help you review the techniques covered in 5. EXAMPLE 6-2 Consider the metric ds 2 = dt 2 + e2b(t,r ) dr 2 + R(t, r ) d 2 and use Cartan s structure equations to nd the components of the curvature tensor. SOLUTION 6-2 With nonzero cosmological constant, Einstein s equation takes the form G ab + gab = 8 Tab
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For the given metric, we de ne the following orthonormal basis one forms: t = dt, r = eb(t,r ) dr, = R(t, r ) d
(6.7)
These give us the following inverse relationships which will be useful in
calculations: dt = t ,
The Einstein Field Equations
dr = e b(t,r ) r ,
d =
1 R(t, r )
(6.8)
With this basis de ned, we have a b 1 = 0 0 0 1 0 0 0 1
which we can use to raise and lower indices. We will nd the components of the Einstein tensor using Cartan s methods. To begin, we calculate the Ricci rotation coef cients. Recall that Cartan s rst structure equation is
d a = a b
b
(6.9)
Also recall that
a b
bc
(6.10)
The rst equation gives us no information, since we have d t = d (dt) = 0 Moving to r , we nd b b(t,r ) b b(t,r ) d r = d eb(t,r ) dr = e e dt dr + dr dr t r b b(t,r ) e dt dr = t Using (6.8), we rewrite this in terms of the basis one forms to get d r = b b(t,r ) b t b dt dr = e r = r t t t t (6.12)
(6.11)
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