# Riemann tensor notation and Christoffel symbol notation

In paper by Barnich and Brandt Covariant theory of asymptotic symmetries, conservation laws and central charges they defined the Riemann tensor like this:

$$R_{\rho\mu\nu}^{\quad \ \ \lambda}~=~\partial_\rho \Gamma_{\mu\nu}^{\ \ \ \ \ \lambda}+\Gamma_{\rho\sigma}^{\ \ \ \ \ \lambda}\Gamma_{\mu\nu}^{\ \ \ \ \ \sigma}-(\rho\leftrightarrow\mu).$$

Now I have taken the 'normal' definition of Riemann tensor and raised the first index, and lowered the last one, and if I do the same with original (lower the first one and raise the last one) I get the same expression, which means that this is ok.

But, why such definition? And does that mean that the Christoffel symbols have different definition compared to usual? I mean raised first and lowered last index.

• What is $(\rho\leftrightarrow\mu)$? Is this a notation indended to mean that you're antisymmetrizing on those two indices? – Ben Crowell Oct 11 '13 at 20:12
• It's just a notation meaning that you repeat the existing terms, but replace those two indices. But I think that, in the end, it's just the weird notation at play. Because if I calculate the Christoffel symbols of the second kind in the usual way, I get the same result as they do in the article. But I was confused why did they do that, that's all – dingo_d Oct 12 '13 at 8:53
• The standard notation for that would be the following. Rather than writing $T_{ab}-(a\leftrightarrow b)$, you would write $T_{[ab]}$. – Ben Crowell Oct 12 '13 at 16:15
• Oh, they used it in the article, not me :D And it means the following: $R_{\rho\mu\nu}^{\quad \ \ \lambda}=\partial_\rho \Gamma_{\mu\nu}^{\ \ \ \ \ \lambda}+\Gamma_{\rho\sigma}^{\ \ \ \ \ \lambda}\Gamma_{\mu\nu}^{\ \ \ \ \ \sigma}-\partial_\mu \Gamma_{\rho\nu}^{\ \ \ \ \ \lambda}+\Gamma_{\mu\sigma}^{\ \ \ \ \ \lambda}\Gamma_{\rho\nu}^{\ \ \ \ \ \sigma}$ – dingo_d Oct 12 '13 at 21:16
• Note that the notation $T_{[ab]}$ sometimes includes a half and sometimes don't, cf. this Phys.SE post. – Qmechanic Oct 13 '13 at 0:03

1. Beware that different authors have different conventions for the horizontal order of indices for the Christoffel symbols$^1$ $\Gamma^{\lambda}{}_{\mu\nu}$ and the Riemann curvature tensor $R^{\sigma}{}_{\mu\nu\lambda}$. Some may e.g. write $\Gamma_{\mu\nu}{}^{\lambda}$ and $R_{\mu\nu\lambda}{}^{\sigma}$ instead.
2. It may be useful to write objects such as $\Gamma^{\lambda}{}_{\mu\nu}$ and $R^{\sigma}{}_{\mu\nu\lambda}$ with covariant and contravariant indices not merely on top of each other a la $\Gamma^{\lambda}_{\mu\nu}$ and $R^{\sigma}_{\mu\nu\lambda}$ but horizontally displaced to keep track of the horizontal position of the indices. Then the indices can be raised or lowered by a metric $g_{\mu\nu}$ with no ambiguity in notation of which index was raised or lowered.
$^1$ It is covenient to call $\Gamma^{\lambda}{}_{\mu\nu}$ Christoffel symbols even if the tangent-space connection $\nabla$ is not torsionfree nor compatible with a metric.