8.1 Curved Space and the Riemann Tensor
113
theorem is proved. Since the field as we have constructed it is independent of the
path used to parallel displace the vector to the desired point we say that the space is
integrable.
We may summarize the results of this section by saying that the following three
properties of a space are equivalent:
1. The space is Euclidean or pseudo-Euclidean, so there is a coordinate system in
which the metric is constant. It may, if desired, be put into the Cayley-Sylvester
canonical form with positive and negative ones on the diagonal.
2. The space is flat, or the Riemann tensor is zero.
3. The space is integrable, so we may set up a constant vector field by parallel
displacement, with a covariant derivative equal to zero.
The integrability property is noteworthy: in a curved space we cannot set up a
constant vector field. See Fig. 5.5 for an illustration on the surface of a sphere.
The vanishing of the Riemann tensor is a very useful characteristic indeed, and
in fact will lead us to the field equations of general relativity. In the next section we
will study the symmetries of this fourth rank tensor.
8.2 Symmetries of the Riemann Tensor
The Riemann tensor is the largest tensor we have encountered so far. It is extremely
important in Riemann geometry and in general relativity. In 4-dimensions it has
4
2
= 256 components. However there are a number of symmetries which reduce this
to only 20 independent components. These symmetries are easy to derive if we make
use of the special geodesic coordinate system in which the connections vanish. In
order to study the symmetries we must first lower an index on the Riemann tensor as
it is defined in (8.8), for tensors can have symmetry only among indices of the same
type. Thus we will study in the geodesic system the totally covariant R αβγ δ (Kenyon
1990).
Note that although the connections are zero at some selected point note their
derivatives are not zero. In the geodesic system the Riemann tensor as defined in
(8.8) has only the first two terms, instead of all 4 terms. That is
R
λ βγ δ =
λ
βγ,δ −
λ
βδ,γ , geodesic system.
(8.15)
There is yet another simplification in the geodesic system. Since the connections
vanish the ordinary derivatives of the metric tensor are equal to the covariant derivatives. But the covariant derivatives of the metric are zero by the Ricci theorem. Thus
all the first derivatives of the metric tensor vanish at the selected point in the geodesic
system. This is true for both the covariant and contravariant versions of the metric
tensor. (Note that the second derivatives do not in general vanish at the selected
point.) Because of this we can write out (8.15) as
113
theorem is proved. Since the field as we have constructed it is independent of the
path used to parallel displace the vector to the desired point we say that the space is
integrable.
We may summarize the results of this section by saying that the following three
properties of a space are equivalent:
1. The space is Euclidean or pseudo-Euclidean, so there is a coordinate system in
which the metric is constant. It may, if desired, be put into the Cayley-Sylvester
canonical form with positive and negative ones on the diagonal.
2. The space is flat, or the Riemann tensor is zero.
3. The space is integrable, so we may set up a constant vector field by parallel
displacement, with a covariant derivative equal to zero.
The integrability property is noteworthy: in a curved space we cannot set up a
constant vector field. See Fig. 5.5 for an illustration on the surface of a sphere.
The vanishing of the Riemann tensor is a very useful characteristic indeed, and
in fact will lead us to the field equations of general relativity. In the next section we
will study the symmetries of this fourth rank tensor.
8.2 Symmetries of the Riemann Tensor
The Riemann tensor is the largest tensor we have encountered so far. It is extremely
important in Riemann geometry and in general relativity. In 4-dimensions it has
4
2
= 256 components. However there are a number of symmetries which reduce this
to only 20 independent components. These symmetries are easy to derive if we make
use of the special geodesic coordinate system in which the connections vanish. In
order to study the symmetries we must first lower an index on the Riemann tensor as
it is defined in (8.8), for tensors can have symmetry only among indices of the same
type. Thus we will study in the geodesic system the totally covariant R αβγ δ (Kenyon
1990).
Note that although the connections are zero at some selected point note their
derivatives are not zero. In the geodesic system the Riemann tensor as defined in
(8.8) has only the first two terms, instead of all 4 terms. That is
R
λ βγ δ =
λ
βγ,δ −
λ
βδ,γ , geodesic system.
(8.15)
There is yet another simplification in the geodesic system. Since the connections
vanish the ordinary derivatives of the metric tensor are equal to the covariant derivatives. But the covariant derivatives of the metric are zero by the Ricci theorem. Thus
all the first derivatives of the metric tensor vanish at the selected point in the geodesic
system. This is true for both the covariant and contravariant versions of the metric
tensor. (Note that the second derivatives do not in general vanish at the selected
point.) Because of this we can write out (8.15) as