114
8 Curved Space and Gravity
R
λ βγ δ =
1
2
g
λτ
g γ τ,β + g βτ,γ − g βγ,τ
,δ
−
1
2
g
λτ
g δτ,β + g βτ,δ − g βδ,τ
,γ
=
1
2
g
λτ
g γ τ,β,δ − g βγ ,τ,δ − g δτ,β,γ + g βδ,τ,γ
, geodesic system. (8.16)
Thus we may lower an index to obtain the fully covariant Riemann tensor as
R αβγ δ =
1
2
g γ α,β,δ − g βγ ,α,δ − g δα,β,γ + g βδ,α,γ
, geodesic system.
(8.17)
This is now in a form where the symmetries are transparent.
The following symmetries follow by simply writing out the four terms of the
Riemann tensor from (8.17):
R αβγ δ = −R αβδγ antisymmetry in last pair of indices,
(8.18a)
R αβγ δ = −R βαγ δ antisymmetry in first pair of indices,
(8.18b)
R αβγ δ = R γ δαβ symmetry in interchange of index pairs.
(8.18c)
There is one more symmetry for the 4-dimensional case; this is easily verified also
by writing out all the terms using (8.17),
R 0123 + R 0231 + R 0312 = 0.
(8.19)
This completes the algebraic symmetries. We emphasize that the symmetries have
been obtained in the special geodesic coordinate system at the selected point, but a
symmetry property of a tensor holds in any coordinate system, so the symmetries
are generally true.
There is also a set of symmetries on the derivatives of the Riemann tensor that
is easy to derive using the geodesic coordinate system. From the definition of the
Riemann tensor in (8.8) we may differentiate it with respect to the coordinates.
The right side of the defining equation will have two connection second derivative
terms and four terms which involve the connections and their first derivatives. In the
geodesic system the last four terms are clearly zero and we have thus
R
α ηβγ ,μ =
α
βη,γ ,μ −
α
ηγ ,β,μ .
(8.20)
But since this is the geodesic system the ordinary derivatives are the same as the
covariant derivatives, so
R
α ηβγ ;μ =
α
βη,γ ,μ −
α
ηγ ,β,μ .
(8.21)
It follows from this that the following permuted combination is zero
8 Curved Space and Gravity
R
λ βγ δ =
1
2
g
λτ
g γ τ,β + g βτ,γ − g βγ,τ
,δ
−
1
2
g
λτ
g δτ,β + g βτ,δ − g βδ,τ
,γ
=
1
2
g
λτ
g γ τ,β,δ − g βγ ,τ,δ − g δτ,β,γ + g βδ,τ,γ
, geodesic system. (8.16)
Thus we may lower an index to obtain the fully covariant Riemann tensor as
R αβγ δ =
1
2
g γ α,β,δ − g βγ ,α,δ − g δα,β,γ + g βδ,α,γ
, geodesic system.
(8.17)
This is now in a form where the symmetries are transparent.
The following symmetries follow by simply writing out the four terms of the
Riemann tensor from (8.17):
R αβγ δ = −R αβδγ antisymmetry in last pair of indices,
(8.18a)
R αβγ δ = −R βαγ δ antisymmetry in first pair of indices,
(8.18b)
R αβγ δ = R γ δαβ symmetry in interchange of index pairs.
(8.18c)
There is one more symmetry for the 4-dimensional case; this is easily verified also
by writing out all the terms using (8.17),
R 0123 + R 0231 + R 0312 = 0.
(8.19)
This completes the algebraic symmetries. We emphasize that the symmetries have
been obtained in the special geodesic coordinate system at the selected point, but a
symmetry property of a tensor holds in any coordinate system, so the symmetries
are generally true.
There is also a set of symmetries on the derivatives of the Riemann tensor that
is easy to derive using the geodesic coordinate system. From the definition of the
Riemann tensor in (8.8) we may differentiate it with respect to the coordinates.
The right side of the defining equation will have two connection second derivative
terms and four terms which involve the connections and their first derivatives. In the
geodesic system the last four terms are clearly zero and we have thus
R
α ηβγ ,μ =
α
βη,γ ,μ −
α
ηγ ,β,μ .
(8.20)
But since this is the geodesic system the ordinary derivatives are the same as the
covariant derivatives, so
R
α ηβγ ;μ =
α
βη,γ ,μ −
α
ηγ ,β,μ .
(8.21)
It follows from this that the following permuted combination is zero