m = g m = in .NET framework

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m = g m =
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1 m a = (0, 0, 1/r, i/r sin ) 2
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and so we have m m = ma ma = = 1
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Null Tetrads and the Petrov
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1 i (r ) + r r sin
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( ir sin ) =
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1 ( 1 1) 2
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And so we see, everything works out as expected.
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CONSTRUCTING A FRAME METRIC USING THE NULL TETRAD
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We can construct a frame metric a b using the null tetrad. The elements of a b = ea eb , and so we make the following designations: e0 = l, e1 = n, e2 = m, e3 = m
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Using the orthogonality relations together with (9.4) and (9.5), it is easy to see that 0 1 0 0 1 0 = 0 0 0 0 0 1 0 0 1 0
vab = v ab
(9.12)
Extending the Formalism
We are now going to introduce a new notation and method for calculation of the curvature tensor and related quantities, the Newman-Penrose formalism. This formalism takes advantage of null vectors and uses them to calculate components of the curvature tensor directly, allowing us to extract several useful pieces of information about a given spacetime. We will also be able to tie it together with a classi cation scheme called the Petrov classi cation that proves to be very useful in the study of black holes and gravitational radiation. First let s introduce some notation. As a set of basis vectors, the null tetrad can be considered as directional derivatives. Therefore, we de ne the following: D = l a a , = n a a , = m a a , = m a a (9.13)
Null Tetrads and the Petrov
Next we will de ne a set of symbols called the spin coef cients. These scalars (which can be complex) can be de ned in two ways. The rst is to write them down in terms of the Ricci rotation coef cients. = = = =
130 , 201 ,
= = + +
230 ) , 233 ) ,
131, 202 ,
= = 1 = ( 2 = 1 ( 2
133 , 200 , 101
= =
231 )
132 203
1 ( 2 1 ( 2
(9.14)
232 )
Better yet, these coef cients can also be de ned directly in terms of the null tetrad: = b n a m a l b , = b n a m a m b = b la m a n b , = b la m a m b , = = b l a m a l b , = b l a m a m b 1 1 b la n a l b b m a m a l b , = b l a n a n b b m a m a n b (9.15) 2 2 1 1 = b la n a m b b m a m a m b , = b la n a m b b m a m a m b 2 2 In the usual course of business after nding the Ricci rotation coef cients (or Christoffel symbols in a coordinate basis) we tackle the problem of nding the curvature tensor. We will take the same approach here, so once we have the spin coef cients in hand we can obtain components of the Ricci and curvature tensors. However, in the Newman-Penrose formalism, we will be focusing on the Weyl tensor. We recall the de nition from 4. In a coordinate basis, we de ned the Weyl tensor in terms of the curvature tensor as Cabcd = Rabcd + + 1 (gad Rcb + gbc Rda gac Rdb gbd Rca ) 2 = b n a m a n b , = b n a m a m b ,
1 (gac gdb gad gcb ) R 6
Null Tetrads and the Petrov
The Weyl tensor has 10 independent components that we will represent with scalars. These can be referred to as Weyl scalars and are given by
0 1 2 3 4
= Cabcd l a m b l c m d = Cabcd l a n b l c m d = Cabcd l a m b m c n d = Cabcd l a n b m c n d = Cabcd n a m b n c m d
(9.16) (9.17) (9.18) (9.19) (9.20)
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