The Bianchi identity for the Riemann tensor

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 19; d.

Another relation of the Riemann tensor involves the covariant derivative of the tensor, and is known as the Bianchi identity (actually the second Bianchi identity; the first
identity is the symmetry
{R_{nj\ell m}+R_{n\ell mj}+R_{nmj\ell}=0} that we saw earlier). The identity is easiest to derive at the origin of a locally inertial frame (LIF), where the first derivatives of the metric tensor, and thus the Christoffel symbols, are all zero. At this point, we have

\displaystyle R_{nj\ell m}=\frac{1}{2}\left(\partial_{\ell}\partial_{j}g_{mn}+\partial_{m}\partial_{n}g_{j\ell}-\partial_{\ell}\partial_{n}g_{jm}-\partial_{m}\partial_{j}g_{\ell n}\right) \ \ \ \ \ (1)

If the Christoffel symbols are all zero, then the covariant derivative becomes the ordinary derivative

\displaystyle \nabla_{j}A^{k}\equiv\partial_{j}A^{k}+A^{i}\Gamma_{\; ij}^{k}=\partial_{j}A^{k} \ \ \ \ \ (2)

Therefore, we get, at the origin of a LIF:

\displaystyle \nabla_{k}R_{nj\ell m} \displaystyle = \displaystyle \partial_{k}R_{nj\ell m}\ \ \ \ \ (3)
\displaystyle \displaystyle = \displaystyle \frac{1}{2}\left(\partial_{k}\partial_{\ell}\partial_{j}g_{mn}+\partial_{k}\partial_{m}\partial_{n}g_{j\ell}-\partial_{k}\partial_{\ell}\partial_{n}g_{jm}-\partial_{k}\partial_{m}\partial_{j}g_{\ell n}\right) \ \ \ \ \ (4)

By cyclically permuting the index of the derivative with the last two indices of the tensor, we get

\displaystyle \nabla_{\ell}R_{njmk} \displaystyle = \displaystyle \partial_{\ell}R_{njmk}\ \ \ \ \ (5)
\displaystyle \displaystyle = \displaystyle \frac{1}{2}\left(\partial_{\ell}\partial_{m}\partial_{j}g_{kn}+\partial_{\ell}\partial_{k}\partial_{n}g_{jm}-\partial_{\ell}\partial_{m}\partial_{n}g_{jk}-\partial_{\ell}\partial_{k}\partial_{j}g_{mn}\right)\ \ \ \ \ (6)
\displaystyle \nabla_{m}R_{njk\ell} \displaystyle = \displaystyle \partial_{m}R_{njk\ell}\ \ \ \ \ (7)
\displaystyle \displaystyle = \displaystyle \frac{1}{2}\left(\partial_{m}\partial_{k}\partial_{j}g_{\ell n}+\partial_{m}\partial_{\ell}\partial_{n}g_{jk}-\partial_{m}\partial_{k}\partial_{n}g_{j\ell}-\partial_{m}\partial_{\ell}\partial_{j}g_{kn}\right) \ \ \ \ \ (8)

By adding up 4, 6 and 8 and using the commutativity of partial derivatives, we see that the terms cancel in pairs, so we get

\displaystyle \boxed{\nabla_{k}R_{nj\ell m}+\nabla_{\ell}R_{njmk}+\nabla_{m}R_{njk\ell}=0} \ \ \ \ \ (9)

As usual we can use the argument that since we can set up a LIF with its origin at any non-singular point in spacetime, this equation is true everywhere and since the covariant derivative is a tensor, this is a tensor equation and is thus valid in all coordinate systems. This is the Bianchi identity.

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