Riemann tensor: symmetries

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

We can derive a few useful symmetries of the Riemann tensor by looking at its form in a locally inertial frame (LIF). At the origin of such a frame, all first derivatives of {g_{ij}} are zero, which means the Christoffel symbols are all zero there. However, the second derivatives of {g_{ij}} are not, in general, zero, so the derivatives of the Christoffel symbols will not, in general, be zero either.

Using the definition of the Riemann tensor:

\displaystyle R_{\; j\ell m}^{i}\equiv\partial_{\ell}\Gamma_{\; mj}^{i}-\partial_{m}\Gamma_{\;\ell j}^{i}+\Gamma_{\; mj}^{k}\Gamma_{\;\ell k}^{i}-\Gamma_{\;\ell j}^{k}\Gamma_{\; km}^{i} \ \ \ \ \ (1)


we can write it at the origin of a LIF:

\displaystyle R_{\; j\ell m}^{i}\equiv\partial_{\ell}\Gamma_{\; mj}^{i}-\partial_{m}\Gamma_{\;\ell j}^{i} \ \ \ \ \ (2)

The Christoffel symbols are

\displaystyle \Gamma_{\; ij}^{m}=\frac{1}{2}g^{ml}\left(\partial_{j}g_{il}+\partial_{i}g_{lj}-\partial_{l}g_{ji}\right) \ \ \ \ \ (3)

The symmetries of the Riemann tensor are easiest to write if we look at its form with all indices lowered, that is:

\displaystyle R_{nj\ell m} \displaystyle = \displaystyle g_{nk}R_{\; j\ell m}^{k}\ \ \ \ \ (4)
\displaystyle \displaystyle = \displaystyle g_{nk}\left(\partial_{\ell}\Gamma_{\; mj}^{k}-\partial_{m}\Gamma_{\;\ell j}^{k}\right) \ \ \ \ \ (5)

First, we calculate the derivative:

\displaystyle \partial_{\ell}\Gamma_{\; mj}^{k}=\frac{1}{2}\partial_{\ell}g^{ki}\left(\partial_{j}g_{mi}+\partial_{m}g_{ij}-\partial_{i}g_{jm}\right)+\frac{1}{2}g^{ki}\left(\partial_{\ell}\partial_{j}g_{mi}+\partial_{\ell}\partial_{m}g_{ij}-\partial_{\ell}\partial_{i}g_{jm}\right) \ \ \ \ \ (6)

At the origin of a LIF, the first term is zero since all first derivatives of {g_{ij}} are zero, so we’re left with

\displaystyle \partial_{\ell}\Gamma_{\; mj}^{k}=\frac{1}{2}g^{ki}\left(\partial_{\ell}\partial_{j}g_{mi}+\partial_{\ell}\partial_{m}g_{ij}-\partial_{\ell}\partial_{i}g_{jm}\right) \ \ \ \ \ (7)

Multiplying this by {g_{kn}} and using {g_{kn}g^{ik}=\delta_{n}^{i}}, we have

\displaystyle g_{kn}\partial_{\ell}\Gamma_{\; mj}^{k} \displaystyle = \displaystyle \frac{1}{2}\delta_{n}^{i}\left(\partial_{\ell}\partial_{j}g_{mi}+\partial_{\ell}\partial_{m}g_{ij}-\partial_{\ell}\partial_{i}g_{jm}\right)\ \ \ \ \ (8)
\displaystyle \displaystyle = \displaystyle \frac{1}{2}\left(\partial_{\ell}\partial_{j}g_{mn}+\partial_{\ell}\partial_{m}g_{nj}-\partial_{\ell}\partial_{n}g_{jm}\right) \ \ \ \ \ (9)

By substituting indices, we can get the second term in 5:

\displaystyle g_{nk}\partial_{m}\Gamma_{\;\ell j}^{k}=\frac{1}{2}\left(\partial_{m}\partial_{j}g_{\ell n}+\partial_{m}\partial_{\ell}g_{nj}-\partial_{m}\partial_{n}g_{j\ell}\right) \ \ \ \ \ (10)


Subtracting 10 from 9 we see that the middle terms cancel, so we’re left with

\displaystyle \boxed{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)} \ \ \ \ \ (11)


This equation is valid only at the origin on a LIF.

From this we can get some symmetry properties. First, if we interchange the first two indices {n} and {j} in the tensor we see that the first and third terms on the RHS in 11 swap, as do the second and fourth, so we end up with the negative of what we started with. That is

\displaystyle \boxed{R_{jn\ell m}=-R_{nj\ell m}} \ \ \ \ \ (12)

If we interchange the last two indices {\ell} and {m}, again the first term swaps with the fourth, and the second with the third, so we get the same result:

\displaystyle \boxed{R_{njm\ell}=-R_{nj\ell m}} \ \ \ \ \ (13)

If we swap the first and third indices, and also the second and fourth, we get

\displaystyle R_{\ell mnj} \displaystyle = \displaystyle \frac{1}{2}\left(\partial_{n}\partial_{m}g_{j\ell}+\partial_{j}\partial_{\ell}g_{mn}-\partial_{n}\partial_{\ell}g_{mj}-\partial_{j}\partial_{m}g_{n\ell}\right)\ \ \ \ \ (14)
\displaystyle \displaystyle = \displaystyle R_{nj\ell m} \ \ \ \ \ (15)

the Riemann tensor is symmetric under interchange of its first two indices with its last two:

\displaystyle \boxed{R_{\ell mnj}=R_{nj\ell m}} \ \ \ \ \ (16)

A final symmetry
property is a bit more subtle. If we cyclically permute the last 3 indices {j}, {\ell} and {m} and add up the 3 terms, we get

\displaystyle R_{nj\ell m}+R_{n\ell mj}+R_{nmj\ell} \displaystyle = \displaystyle \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)+\nonumber
\displaystyle \displaystyle \displaystyle \frac{1}{2}\left(\partial_{m}\partial_{\ell}g_{jn}+\partial_{j}\partial_{n}g_{\ell m}-\partial_{m}\partial_{n}g_{\ell j}-\partial_{j}\partial_{\ell}g_{mn}\right)+\ \ \ \ \ (17)
\displaystyle \displaystyle \displaystyle \frac{1}{2}\left(\partial_{j}\partial_{m}g_{\ell n}+\partial_{\ell}\partial_{n}g_{mj}-\partial_{j}\partial_{n}g_{m\ell}-\partial_{\ell}\partial_{m}g_{jn}\right)\nonumber

Using the symmetry of {g_{ij}=g_{ji}} and the fact that partial derivatives commute, we find that the first two terms in the first line cancel with the last two terms in the second line, the first two in the second line cancel with the last two in the third line, and the first two in the third line cancel with the last two in the first line, giving the result:

\displaystyle \boxed{R_{nj\ell m}+R_{n\ell mj}+R_{nmj\ell}=0} \ \ \ \ \ (18)

We’ve derived these results for the special case at the origin of a LIF. However, the origin of a LIF defines one particular event in spacetime and since all these symmetries are tensor equations, they must be true for that particular event, regardless of which coordinate system we’re using. Further, in our discussion of LIFs, we showed that we could define a LIF with its origin at any point in spacetime, provided that point is locally flat (that is, that there is no singularity at that point). So the argument shows that these symmetries are true for all non-singular points in spacetime.

Incidentally, it might be confusing that we can say that these symmetries are universally valid at all points in all coordinate systems just because they are tensor equations, while we say that 11 is valid only at the origin of a LIF. The difference is that 11 is written explicitly in terms of a particular metric {g_{ij}} and that metric is defined precisely so that all its first derivatives are zero at the origin of the LIF. If we wanted an equation for {R_{nj\ell m}} at some other point in spacetime, we could write it in the same form, but we’d need to find a different metric {g_{ij}} whose first derivatives are zero at this other point. If we wanted to use the original metric, then since this other point is not at the origin of the original LIF, the {\Gamma_{\; k\ell}^{j}} would not be zero at this point since the derivatives of {g_{ij}} wouldn’t be zero there, and the expression for {R_{nj\ell m}} would be more complicated in terms of the original metric.

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