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Geodesics in the Schwarzschild Spacetime
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Now that we have the metric in hand, we can determine what the paths of particles and light rays are going to be in this spacetime. To nd out we need to derive and solve the geodesic equations for each coordinate. One way this can be done fairly easily is by deriving the Euler-Lagrange equations. We are not covering largrangian methods in this book, so we will
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The Schwarzschild Solution
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simply demonstrate the method (interested readers, please see the references). To nd the geodesic equations, we take the variation ds = 1 2m r t2 1 r 2 r 2 2 r 2 sin2 2 ds = 0 1 2m r
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Now we make the following de nition: F = 1 2m r t2 1 r 2 r 2 2 r 2 sin2 2 2m 1 r
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The Euler-Lagrange equations are d ds F xa F =0 xa (10.36)
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Starting with the time coordinate, we have 2m F =2 1 t r d ds F t = = t t +2 1 2m t r
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d d 2m 2m t = 2 1 2 1 ds r ds r t
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4m 2m rt + 2 1 2 r r
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Since there are no terms in F that contain t, we set this to zero and obtain the rst geodesic equation: t+ 2m rt = 0 r (r 2m) (10.37)
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Now we consider the radial coordinate. We have F = r d ds 2 r 2m 1 r = 2 4m r 2 ( )2 r 2m r 1 r
F r
F is r dependent and so we calculate 2m 2m F = 2 t2 + 1 r r r
The Schwarzschild Solution
2m 2 r 2r 2 2r sin2 2 r
Putting these results together using (10.36) gives the second geodesic equation m m (r 2m) t 2 r 2 (r 2m) 2 + sin2 2 = 0 3 r r (r 2m) (10.38) Next we nd r+ F d F = 2r 2 ds F = 2r 2 sin cos 2 = 4r r 2r 2
and so the geodesic equation for is given by 2 + r sin cos 2 = 0 r Finally, for the coordinate we nd F = 2r 2 sin2 d F = 4r r sin2 4r 2 sin cos 2r 2 sin2 ds F =0 Therefore 2 + 2 cot + r = 0 r (10.40) (10.39)
Particle Orbits in the Schwarzschild Spacetime
In the previous section we obtained a set of differential equations for the geodesics of the Schwarzschild spacetime. However, these equations are quite daunting and trying to solve them directly does not seem like a productive use of
The Schwarzschild Solution
time. A different approach to learn about the motion of particles and light rays in the Schwarzschild spacetime is available, and is based on the use of Killing vectors. Remember, each symmetry in the metric corresponds to a Killing vector. Conserved quantities related to the motion of a massive particle can be found by forming the dot product of Killing vectors with the four velocity of the particle. We will de ne vectors in terms of their components, using the coordinate ordering (t, r, , ). The four velocity of a particle in the Schwarzschild spacetime has components given by u= dt dr d d , , , d d d d (10.41)
Here is the proper time. The four velocity satis es u u = gab u a u b = 1 (10.42)
In a coordinate basis, the components of the metric tensor, which can be read off of (10.33), are gtt = 1 2m r , grr = 1 , 1 2m r g = r 2 , g = r 2 sin2 (10.43)
Using (10.41), we nd that (10.42) gives u u = gab u a u b 2m = 1 r r 2 sin2 dt d d d
2m 1 r =1
dr d
d d
(10.44)
This equation forms the basis that we will use to nd the orbits. It can be shown that in general relativity, as in classical mechanics, the orbit of a body in a central force eld lies in a plane. Therefore we can choose our axes such that = /2 d and set d = 0. Before analyzing this equation further, we will de ne two Killing vectors for the Schwarzschild spacetime and use them to construct conserved quantities. This is actually quite easy. When the form of the metric was derived, we indicated two important criteria: the metric is time independent and spherically symmetric.