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(when going through this calculation, be careful with the overall minus sign). A similar exercise shows that the spin coef cients = = 0. With all the vanishing spin coef cients, the Newman-Penrose identity we need to calculate, 4 , takes on a very simple form. The expression we need is given by (9.23): = ( + ) (3 ) + 3 + + In this case, we have
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Recalling (9.13), this equation becomes 4 = m a a . Note, however, that is a scalar we need to calculate only partial derivatives. The end result is that
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= m a a = m x x + m y y 1 = 2 i + 2 = 1 4 1 x 2 2 1 y 2 2 + H H +i x y H H +i x y i 4 y 2 H 2 H +i 2 x y y
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2 H 2 H +i x2 x y
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1 2 H i 2 H 1 2 H + 4 x2 2 x y 4 y2
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Null Tetrads and the Petrov
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and so we conclude that the Weyl scalar is given by
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(9.31)
In the exercises, if you are so inclined you can show that the only nonzero component of the Ricci tensor is given by
2 H 2 H + x2 y2
(9.32)
The fact that 4 is the only nonzero Weyl scalar tells us that the Petrov type is Type N, meaning that the principal null direction is repeated four times. Therefore, this metric describes transverse gravity waves.
Quiz
1. Using the Minkowski metric ab = diag(1, 1, 1, 1), A = (3, 0, 3, 0) is (a) timelike (b) not enough information has been given (c) null (d) spacelike Using the de nition of the complex vectors in the null tetrad, i.e., ma = j a + ika , 2 ma = j a ika 2
which one of the following statements is true (a) These are null vectors and that m m = 1. (b) These are timelike vectors and that m m = 1. (c) These are null vectors and that m m = 1. (d) These are null vectors and that m m = 0. 3. Consider Example 9-5. Which of the following relations is true (a) = b n a m a l b = 0 (b) = b n a m a l b = 1 b (c) = b n a m a l = 0 b (d) = b n a m a l = 1
Null Tetrads and the Petrov
Again, consider Example 9-5. Which of the following relations is true (a) = b la m a m b = 0 (b) = b n a m a m b = 0 (c) = b n a m a m b = 0 (d) = b n a m a m b = 0 A spacetime described as Petrov type I (a) has no null directions (b) has four distinct principal null directions (c) has a single null direction (d) has a vanishing Weyl tensor, but has two null directions
CHAPTER
The Schwarzschild Solution
When faced with a dif cult set of mathematical equations, the rst course of action one often takes is to look for special cases that are the easiest to solve. It turns out that such an approach often yields insights into the most interesting and physically relevant situations. This is as true for general relativity as it is for any other theory of mathematical physics. Therefore for our rst application of the theory, we consider a solution to the eld equations that is time independent and spherically symmetric. Such a scenario can describe the gravitational eld found outside of the Sun, for example. Since we might be interested only in the eld outside of the matter distribution, we can simplify things even further by restricting our attention to the matter-free regions of space in the vicinity of some mass. Within the context of relativity, this means that one can nd a solution to the problem using the vacuum equations and ignore the stress-energy tensor.
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