Tensors and Relativity: Chapter 6

The curvature tensor

The curvature tensor

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Imagine in our manifold a very small closed loop whose four sides are the coordinate lines tex2html_wrap_inline3703 , tex2html_wrap_inline3705 , tex2html_wrap_inline3707 , tex2html_wrap_inline3709 .

  figure1582
Figure 6.3: Parallel transport around a closed loop ABCD.

A vector tex2html_wrap_inline3398 defined at A is parallel transported to B. From the parallel transport law

equation3073

it follows that at B the vector has components

  eqnarray3079

where the notation `` tex2html_wrap_inline3707 '' under the integral denotes the path AB. Similar transport from B to C to D gives

equation3081

and

equation3083

The integral in the last equation has a different sign because of the direction of transport from C to D is in the negative tex2html_wrap_inline3735 direction.

Similarly, the completion of the loop gives

  equation3085

The net change in tex2html_wrap_inline3737 is a vector tex2html_wrap_inline3739 , found by adding (93)-(96).

eqnarray3087

To lowest order we get

eqnarray3089

This involves derivatives of tex2html_wrap_inline3481 's and of tex2html_wrap_inline3743 . The derivatives of tex2html_wrap_inline3743 can be eliminated using for example

equation3091

This gives

equation3093

To obtain this, one needs to relabel dummy indices in the terms quadratic in tex2html_wrap_inline3481 .

Notice that this just turns out to be a number times tex2html_wrap_inline3749 summed on tex2html_wrap_inline3456 . Now the indices 1 and 2 appear because the path was chosen to go along those coordinates. It is antisymmetric in 1 and 2 because the change tex2html_wrap_inline3739 would have the opposite sign if one went around the loop in the opposite direction.

If we use general coordinate lines tex2html_wrap_inline3763 and tex2html_wrap_inline3765 , we find

equation3095

Defining

equation3097

we can write

equation3099

tex2html_wrap_inline3767 are the components of a 1/3 tensor. This tensor is called the Riemann curvature tensor .