The Schwarzschild Solution in Visual Studio .NET
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The Schwarzschild Solution
Taking the square root, expanding to rst order, and using conventional units (so we put ct in place of t), we obtain cdt = dr 2 1 r0 /r 2
2 2m mr0 3 r r
This result can be integrated. To consider the travel time of light between Earth and another planet in the solar system, we integrate between r0 to rp and r0 to re , where rp is the planet radius and re is the Earth radius. The result is 2 2 rp r0 + rp 2 r0 2 2 re r0 + re
ct =
2 2 rp r0 +
2 2 re r0 + 2m ln 2 2 re r0
m 2 2 rp r0
The ordinary at space distance between Earth and the planet is given by
2 2 2 2 the rst term, rp r0 + re r0 . The remaining terms indicate the increased distance caused by the curvature of spacetime (i.e., by the gravitational eld of the Sun). These terms cause a time delay that is measurable in the solar system. For example, radar re ections to Venus are delayed by about 200 s. Because of limited space our coverage of the Schwarzschild solution is incomplete. The reader is encouraged to consult the references listed at the end of the book for more extensive treatment. Quiz
1. Using the variational method described in Example 410, the nonzero Christoffel symbols for the Schwarzschild metric are (a) t r t = d dr r 2( ) d , r rr = d , r = r e 2 , r = r e 2 sin2 tt = e dr dr 1 r = r , = sin cos 1 r = r , = cot t rt r
The Schwarzschild Solution
= d dr
tt r
= e2( ) d , r rr = d , dr dr 1 r = r , = sin cos 1 r = r , = cot
= r e 2 , = r e 2 sin2
d , dr
= d , dr
r r
= r e 2 , = r e 2 cos2
1 = r, 1 = r, = sin cos = cot
Suppose that we were to drop the requirement of time independence and wrote the line element as ds 2 = e2 (r,t) dt 2 e2 (r,t) dr 2 r 2 d 2 + sin2 d 2 The Rr t component of the Ricci tensor is given by 1 (a) Rr t = r d dt (b) Rr t = (c) Rr t = 1 d r dt r12 d dt
For the following set of problems, we consider a Schwarzschild metric with a nonzero cosmological constant. We make the following de nition: f (r ) = 1 We write the line element as ds 2 = f (r ) dt 2 + 1 dr 2 + r 2 d 2 + r 2 sin2 d 2 f (r ) 2m 1 r 3 r2 When you calculate the Ricci rotation coef cients, you will nd (a) (b) (c) r r r tt tt tt
= = = r3 9r 18m 3 r 3
3 3m r 9r 18m 3 r 3
3m r 3 9r 18m
The Schwarzschild Solution
When you calculate the components of the Ricci tensor, you will nd (a) Rt t = Rr r = R = R = (b) Rt t = Rr r = R = R = r 3 (c) Rt t = Rr r = R = R = 0 The Petrov type of the Schwarzschild spacetime is best described as (a) type O (b) type I (c) type III (d) type D

