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Tensors and Relativity: Chapter 6

The Bianchi identities; Ricci and Einstein tensors

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In the last section we found that in a local inertial frame the Riemann tensor could be written as

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Differentiating with respect to tex2html_wrap_inline3765 we get

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From this equation, the symmetry tex2html_wrap_inline3871 and the fact that partial derivatives commute, it is easy to show that

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This equation is valid in a local inertial frame, therefore in a general frame we get

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This is a tensor equation, therefore valid in any coordinate system. It is called the Bianchi identities , and will be very important for our work.