Copyright 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use. in .NET framework

Create QR Code in .NET framework Copyright 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use.

Copyright 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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The Schwarzschild Solution
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The solution we will obtain is known as the Schwarzschild solution. It was found in 1916 by the German physicist Karl Schwarzschild while he was serving on the Russian front during the First World War. He died from an illness soon after mailing his solution to Einstein, who was surprised that such a simple solution to his equations could be obtained.
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The Vacuum Equations
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The vacuum eld equations describe the metric structure of empty space surrounding a massive body. In the consideration of empty space where no matter or energy is present, we set Tab = 0. In this case, the eld equations become Rab = 0 (10.1)
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A Static, Spherically Symmetric Spacetime
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To obtain the form of the metric that represents the eld outside of a spherically symmetric body, we rst consider the limiting form that it must take. Far away from the body (as r becomes large), we expect that it will assume the form of the Minkowski metric. Since we are assuming spherical symmetry, we express the Minkowski metric in spherical coordinates ds 2 = dt 2 dr 2 r 2 d 2 r 2 sin2 d 2 (10.2)
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To obtain a general form of a time-independent, spherically symmetric metric that will reduce to (10.2) for large r , we rst consider the requirement of time independence. If a metric is time independent, we should be able to change dt dt without affecting the metric. This tells us that the metric will not contain any mixed terms such as dtdr, dtd , dtd . With no off-diagonal terms involving the time coordinate allowed in the metric, we can write the general form as ds 2 = gtt dt 2 + gij dx i dx j We also require that the components of the metric are time independent; i.e., gab =0 t A metric that meets these conditions is called static.
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The Schwarzschild Solution
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Our next task is to think about how spherical symmetry affects the form of a metric. A spherically symmetric metric is one that has no preferred angular direction in space, meaning that we should be able to change d d and d d without changing the form of the metric. In the same way that time independence eliminated mixed terms involving dt from consideration, we cannot have mixed terms such as dr d , dr d , d d that would be affected by the changes d d and d d . So we have arrived at a metric that must have an entirely diagonal form. We have already eliminated any explicit time dependence from the terms of the metric. Since we are imposing radially symmetry, each term in the metric can be multiplied by a coef cient function that depends only on r . Using (10.2) as a guide, we can write this metric as ds 2 = A (r )dt 2 B (r ) dr 2 C (r ) r 2 d 2 D (r ) r 2 sin2 d 2 (10.3)
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Spherical symmetry requires that the angular terms assume the normal form of d 2 that we are used to, and so we take C = D and write this as ds 2 = A (r ) dt 2 B (r ) dr 2 C (r ) r 2 d 2 + r 2 sin2 d 2 (10.4)
Now we can simplify matters even further by a change of radial coordinate to eliminate C. For the moment we will call the new radial coordinate and de ne it using = C (r ) r . Then we have 2 = Cr 2 and the angular part of the metric assumes the familiar form C (r ) r 2 d 2 + r 2 sin2 d 2 = Cr 2 d 2 + Cr 2 sin2 d 2 = 2 d 2 + 2 sin2 d 2 From the de nition = C (r ) r , we see that we can write 1 d = dC r + C dr 2 C = = 1 dC r + C dr 2 C dr C r dC r + 1 dr 2C dr 1 n
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