Appendix 2: The Riemann Tensor as a 6 by 6 Matrix
123
the Riemann tensor is antisymmetric in this pair only 6 values of the pair occur (Adler
1975)
tensor indices αβ = 23 31 12 01 02 03
matrix indices A = 1 2 3 4 5 6
(8.53)
Similarly for the pair γ, δ we may associate a 6 valued matrix indexB. This allows
us to think of the Riemann tensor as a 6 by 6 matrix R AB . But the symmetry (8.18c)
means that the matrix is symmetric in A and B so it has at most 21 independent
components. The final symmetry in (8.19) is one more relation on the components
and reduces the number of independent components to 20, much less than the total
of 256.
Exercises
8.1 How many independent components does the Riemann tensor have in two
dimensions, three dimensions, and four dimensions?
8.2 Consider a metric in two dimensions with coordinates x, y that has the special
form ds
2
= dx
2
+G
2
(x)dy
2
. Show that one of the components of the Riemann
tensor is
R
1
212 = G
d
2 G
dx 2
.
Obtain all the nonzero components from this one. The metric which we will
use for cosmology will be analogous to this. See also Exercise 8.7.
8.3 What is the Riemann tensor for the 2-dimensional surface of a sphere? What
is it for the surface of a cylinder? (Is Exercise 8.2 any help?)
8.4 What is the Riemann scalar for the surface of a sphere? Is this a surprise?
8.5 Prove that the Ricci tensor is symmetric.
8.6 Prove Theorem 5, that the Einstein tensor is zero if and only if the Ricci tensor
is zero.
8.7 Consider a 4-dimensional spacetime with a particularly simple metric form,
ds
2
=
dx
0
2 − g ik dx
i dx
k
, i = 1, 2, 3,
where the g ik are independent of the time marker x
0 . That is the 4-space
contains a 3-space in a simple way. How are the 4-space connections related
to the 3-connections?
8.8 For the metric of Exercise 8.7, what is the relation between the Riemann tensor
in 4-space and that in 3-space? What is the relation between the Ricci tensor in
4-space and that in 3-space? What is the relation between the Riemann scalar
in 4-space and that in 3-space?
8.9 Show that the gravity-free pseudo-Euclidean space of special relativity is a
solution of the Einstein equations in vacuum. (This is as easy as it sounds.)
123
the Riemann tensor is antisymmetric in this pair only 6 values of the pair occur (Adler
1975)
tensor indices αβ = 23 31 12 01 02 03
matrix indices A = 1 2 3 4 5 6
(8.53)
Similarly for the pair γ, δ we may associate a 6 valued matrix indexB. This allows
us to think of the Riemann tensor as a 6 by 6 matrix R AB . But the symmetry (8.18c)
means that the matrix is symmetric in A and B so it has at most 21 independent
components. The final symmetry in (8.19) is one more relation on the components
and reduces the number of independent components to 20, much less than the total
of 256.
Exercises
8.1 How many independent components does the Riemann tensor have in two
dimensions, three dimensions, and four dimensions?
8.2 Consider a metric in two dimensions with coordinates x, y that has the special
form ds
2
= dx
2
+G
2
(x)dy
2
. Show that one of the components of the Riemann
tensor is
R
1
212 = G
d
2 G
dx 2
.
Obtain all the nonzero components from this one. The metric which we will
use for cosmology will be analogous to this. See also Exercise 8.7.
8.3 What is the Riemann tensor for the 2-dimensional surface of a sphere? What
is it for the surface of a cylinder? (Is Exercise 8.2 any help?)
8.4 What is the Riemann scalar for the surface of a sphere? Is this a surprise?
8.5 Prove that the Ricci tensor is symmetric.
8.6 Prove Theorem 5, that the Einstein tensor is zero if and only if the Ricci tensor
is zero.
8.7 Consider a 4-dimensional spacetime with a particularly simple metric form,
ds
2
=
dx
0
2 − g ik dx
i dx
k
, i = 1, 2, 3,
where the g ik are independent of the time marker x
0 . That is the 4-space
contains a 3-space in a simple way. How are the 4-space connections related
to the 3-connections?
8.8 For the metric of Exercise 8.7, what is the relation between the Riemann tensor
in 4-space and that in 3-space? What is the relation between the Ricci tensor in
4-space and that in 3-space? What is the relation between the Riemann scalar
in 4-space and that in 3-space?
8.9 Show that the gravity-free pseudo-Euclidean space of special relativity is a
solution of the Einstein equations in vacuum. (This is as easy as it sounds.)