Tensors and Relativity: Chapter 6

Calculating from the metric

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Calculating tex2html_wrap_inline3384 from the metric

Since tex2html_wrap_inline3483 is a tensor we can lower the index tex2html_wrap_inline3432 using the metric tensor:


But by linearity, we have:


So consistency requires tex2html_wrap_inline3533 . Since tex2html_wrap_inline3398 is arbitrary this implies that


Thus the covariant derivative of the metric is zero in every frame.

We next prove that tex2html_wrap_inline3537 [ i.e. symmetric  in tex2html_wrap_inline3434 and tex2html_wrap_inline3436 ]. In a general frame we have for a scalar field tex2html_wrap_inline3392 :


in a local inertial frame, this is just tex2html_wrap_inline3545 , which is symmetric in tex2html_wrap_inline3434 and tex2html_wrap_inline3432 . Thus it must also be symmetric in a general frame. Hence tex2html_wrap_inline3551 is symmetric in tex2html_wrap_inline3434 and tex2html_wrap_inline3432 :


We now use this to express tex2html_wrap_inline3551 in terms of the metric. Since tex2html_wrap_inline3559 , we have:


By writing different permutations of the indices and using the symmetry of tex2html_wrap_inline3551 , we get


Multiplying by tex2html_wrap_inline3563 and using tex2html_wrap_inline3565 gives


Note that tex2html_wrap_inline3567 is not a tensor since it is defined in terms of partial derivatives.

In a local inertial frame tex2html_wrap_inline3569 since tex2html_wrap_inline3571 . We will see later the significance of this result.