Page 171 - Intro to Tensor Calculus

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For these nonvanishing components only one independent component exists. By convention, the com-
ponent R 1212 is selected as the single independent component and all other nonzero components are
expressed in terms of this component.
Find the nonvanishing independent components R ijkl for i, j, k, l =1, 2, 3, 4and show that

R 1212 R 3434 R 2142 R 4124
R 1313 R 1231 R 2342 R 4314

R 2323 R 1421 R 3213 R 4234
R 1414 R 1341 R 3243 R 1324
R 2424 R 2132 R 3143 R 1432

can be selected as the twenty independent components.
I 21.
(a) For N =2 show R 1212 is the only nonzero independent component and
R 1212 = R 2121 = −R 1221 = −R 2112 .
2
2
(b) Show that on the surface of a sphere of radius r 0 we have R 1212 = r sin θ.
0
I 22. Show for N =2 that
2
2 ∂x
R 1212 = R 1212 J = R 1212
∂x

i
1 i
i
i
ik
I 23. Define R ij = R s .ijs as the Ricci tensor and G = R − δ R as the Einstein tensor, where R = g R kj
2 j
j
j
j
i
and R = R . Show that
i
ab
(a) R jk = g R jabk
2
∂ log √ g b ∂ log √ g ∂ a b a
(b) R ij = − − +
i
∂x ∂x j ij ∂x b ∂x a ij ia jb
(c) R i ijk =0
I 24. By employing the results from the previous problem show that in the case N =2 we have
R 11 R 22 R 12 R 1212
= = = −
g 11 g 22 g 12 g
where g is the determinant of g ij .
I 25. Consider the case N = 2 wherewehave g 12 = g 21 = 0 and show that
2R 1221
(a) R 12 = R 21 =0 (c) R =
g 11 g 22
(b) R 11 g 22 = R 22 g 11 = R 1221 1 ij
(d) R ij = Rg ij , where R = g R ij
2
i
1 i
i
The scalar invariant R is known as the Einstein curvature of the surface and the tensor G = R − δ R is
j j 2 j
known as the Einstein tensor.
I 26. For N =3 show that R 1212 ,R 1313 ,R 2323 ,R 1213 ,R 2123 ,R 3132 are independent components of the
Riemann Christoffel tensor.