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Tensors and Relativity: Chapter 6

Properties of the Riemann curvature tensor

Properties of the Riemann curvature tensor

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Recall that the Riemann tensor is

equation3101

In a local inertial frame we have tex2html_wrap_inline3773 , so in this frame 

equation3103

Now

equation3105

so

equation3107

since tex2html_wrap_inline3775 i.e the first derivative of the metric vanishes in a local inertial frame. Hence

equation3109

Using the fact that partial derivatives always commute so that tex2html_wrap_inline3777 , we get

equation3111

in a local inertial frame. Lowering the index tex2html_wrap_inline3432 with the metric we get

eqnarray3113

So in a local inertial frame the result is

equation3115

We can use this result to discover what the symmetries of tex2html_wrap_inline3781 are. It is easy to show from the above result that 

equation3117

and

equation3119

Thus tex2html_wrap_inline3781 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 tex2html_wrap_inline3781 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

equation3121

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 tex2html_wrap_inline3398 : 

eqnarray3125

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

eqnarray3127

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

equation3129

Consider the same formula with the tex2html_wrap_inline3432 and tex2html_wrap_inline3434 interchanged:

equation3131

If we subtract these we get the commutator of the covariant derivative operators tex2html_wrap_inline3797 and tex2html_wrap_inline3799 :

eqnarray3133

The terms involving the second derivatives of tex2html_wrap_inline3749 drop out because tex2html_wrap_inline3803 [ partial derivatives commute ].

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

equation3135

we get

equation3137

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 tex2html_wrap_inline3398 in one direction, and then in another, followed by subtracting changes in the reverse order.


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