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Tensors and Relativity: Chapter 6

# Properties of the Riemann curvature tensor

Next: Geodesic deviation Up: The curvature tensor and Previous: The curvature tensor

Recall that the Riemann tensor is

In a local inertial frame we have , so in this frame

Now

so

since i.e the first derivative of the metric vanishes in a local inertial frame. Hence

Using the fact that partial derivatives always commute so that , we get

in a local inertial frame. Lowering the index with the metric we get

So in a local inertial frame the result is

We can use this result to discover what the symmetries of are. It is easy to show from the above result that

and

Thus is antisymmetric on the final pair and second pair of indices, and symmetric on exchange of the two pairs.

Since these last two equations are valid tensor equations, although they were derived in a local inertial frame, they are valid in all coordinate systems.

We can use these two identities to reduce the number of independent components of from 256 to just 20 .

A flat manifold is one which has a global definition of parallelism: i.e. a vector can be moved around parallel to itself on an arbitrary curve and will return to its starting point unchanged. This clearly means that

i.e. the manifold is flat [ Assignment 6.5: try a cylinder! ].

An important use of the curvature tensor comes when we examine the consequences of taking two covariant derivatives of a vector field :

As usual we can simplify things by working in a local inertial frame. So in this frame we get

The third term of this is zero in a local inertial frame, so we obtain

Consider the same formula with the and interchanged:

If we subtract these we get the commutator of the covariant derivative operators and :

The terms involving the second derivatives of drop out because [ partial derivatives commute ].

Since in a local inertial frame the Riemann tensor takes the form

we get

This is closely related to our original derivation of the Riemann tensor from parallel transport around loops, because the parallel transport problem can be thought of as computing, first the change of in one direction, and then in another, followed by subtracting changes in the reverse order.

Next: Geodesic deviation Up: The curvature tensor and Previous: The curvature tensor