The Riemann Curvature Tensor and Geodesic Coordinates

Riemann Tensor

Covariant differentiation, unlike partial differentiation, is not in general commutative.  For any tensor , we define its commutator to be

 (5.1)

Let us work out the commutator in the case of a vector .  Using the definition for covariant differentiation of a contravariant rank one tensor we see that

 (5.2)

This is a tensor of mixed tensor of type (1,1). Taking the covariant derivative once again we get

(5.3)

with a similar expression for , namely,

(5.4)

Subtracting these last two equations and assuming that

(5.5)

we obtain the result

(5.6) where  is defined by

(5.7)

Moreover, since we are only interested in torsion-free connections, the last term in (5.6) vanishes. Using the notation for antisymmetric tensors we get can rewrite (5.6) as follows:

 (5.8)

Since the left-hand side of (5.8) is a tensor, it follows that  is a tensor of type (1,3).  It is called the Riemann tensor.  It can be shown that, for a symmetric connection, the commutator of any tensor can be expressed in terms of the tensor itself and the Riemann tensor.  Thus, the vanishing of the Riemann tensor is a necessary and sufficient condition for the vanishing of the commutator of any tensor.

Geodesic Coordinates

We now prove a very useful result.  At any point P in a manifold, we can introduce a special coordinate system, called a geodesic coordinate system, in which

 (5.9)

Here we are using a particular coordinate system so we use the notation where equal signs have an asterisk * above them to indicate that the result is not general but is wholly reliant upon the characteristics of the coordinate system we evaluate with respect to.

We can, without loss of generality, choose P to be at the origin of coordinates  and consider a transformation to a new coordinate system

 (5.10)

where  are constants to be determined.  Differentiating (5.10), we get

 (5.11)
 (5.12)

Then, since  vanishes at P, we have

 (5.13)

from which it follows immediately that the inverse matrix

 (5.14)

We can now use the above results in the affine connection transformation law (3.17)

(5.15)       ,

We find the following relation between the affine connection in the two coordinate systems:

 (5.16)

Since the connection is symmetric, we can choose the constants so that

 (5.17)

and hence we obtain the promised result

 (5.18)

Many tensorial equations can be established most easily in geodesic coordinates.  Note that, although the connection vanishes at P, the derivative of the affine connection may not.

 (5.19)

It can be shown that the result can be extended to obtain a coordinate system in which the connection vanishes along a curve, but not in general over the whole manifold.  If, however, there exists a special coordinate system in which the connection vanishes everywhere, then the manifold is called affine flat or simply flat.  This is intimately connected with the vanishing of the Riemann tensor. The following theorem holds in this respect for Riemann tensors.

Theorem: A necessary and sufficient condition for a manifold to be affine flat is that the Riemann tensor vanishes.

[For a proof of this theorem see section 6.7 of the book Introducing Einstein's Relativity by Ray d'Inverno.]

06/10/2004 12:17 PM