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= sin cos =
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= cot
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Tensor Calculus
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Using the fact that the nonzero components of the Riemann tensor in two dizmensions are given by R1212 = R2121 = R1221 = R2112 , using (4.41) we calculate R = = = Since
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= 0, this simpli es to R =
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= ( sin cos ) (cot ) ( sin cos ) = sin2 cos2 + cos sin (sin cos )
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= sin2 cos2 + cos2 = sin2 The other nonzero components can be found using the symmetry R1212 = R2121 = R1221 = R2112 .
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The Ricci Tensor and Ricci Scalar
The Riemann tensor can be used to derive two more quantities that are used to de ne the Einstein tensor. The rst of these is the Ricci tensor, which is calculated from the Riemann tensor by contraction on the rst and third indices: Rab = R c acb (4.46)
The Ricci tensor is symmetric, so Rab = Rba . Using contraction on the Ricci tensor, we obtain the Ricci scalar R = g ab Rab = R a a (4.47)
Tensor Calculus
Finally, the Einstein tensor is given by G ab = Rab 1 Rgab 2
(4.48)
EXAMPLE 4-12 Show that the Ricci scalar R = 2 for the unit 2-sphere. SOLUTION 4-12 In the previous example, we found that R = sin2 . The symmetry conditions of the Riemann tensor tell us that R = R = R = R The components of the metric tensor are g = g = 1, g = sin2 , g = 1 sin2
Applying the symmetry conditions, we nd R = sin2 R = sin2 R = sin2 Now we need to raise indices with the metric. This gives R = g R = 1 sin2 1 sin2 sin2 = 1
R = g R = sin2 R = g R = sin2 = 1
The components of the Ricci tensor are given by R = R c c = R + R = 1
Tensor Calculus
R = R c c = R + R = sin2 R = R = R c c = R + R = 0 Now we contract indices to get the Ricci scalar R = g ab Rab = g R + g R = 1 + 1 sin2 sin2 = 1 + 1 = 2
The Weyl Tensor and Conformal Metrics
We brie y mention one more quantity that will turn out to be useful in later studies. This is the Weyl tensor that can be calculated using the formula (in four dimensions) Cabcd = Rabcd + + 1 (gad Rcb + gbc Rda gac Rdb gbd Rca ) 2 (4.49)
1 (gac gdb gad gcb ) R 6
This tensor is sometimes known as the conformal tensor. Two metrics are conformally related if gab = 2 (x) gab (4.50)
for some differentiable function (x). A metric is conformally at if we can nd a function (x) such that the metric is conformally related to the Minkowski metric gab = 2 (x) ab (4.51)
A nice property of the Weyl tensor is that C a bcd is the same for a given metric and any metric that is conformally related to it. This is the origin of the term conformal tensor.
Tensor Calculus
Quiz
For Questions 1 6, consider the following line element: ds 2 = dr 2 + r 2 d 2 + r 2 sin2 d 2 1. The components of the metric tensor are (a) grr = r, g = r sin , g = r 2 sin2 (b) grr = r, g = r 2 , g = r 2 sin2 (c) grr = 1, g = r 2 , g = r 2 sin2 (d) grr = r, g = r 2 , g = r 2 sin2 Compute the Christoffel symbols of the rst kind. (a) r 2 sin cos (b) r sin cos (c) r 2 sin2 (d) sin cos
Now calculate the Christoffel symbols of the second kind. 1 (a) r 1 (b) r cos sin (c) cot (d) r12 Calculate the Riemann tensor. Rr is (a) sin (b) r 3 sin (c) rcos sin (d) 0 The determinant of the metric, g, is given by (a) r 2 sin4 (b) r 4 sin2 (c) r 4 sin4 (d) 0
Now let w a = (r, sin , sin cos ) and v a = r, r 2 cos , sin . 6. The Lie derivative u = L v w has u given by (a) r 2 cos2 cos sin (b) r cos2 cos sin
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