Gregorio Ricci-Curbastro |

Win shows that for a diagonal metric the diagonal components of the Ricci tensor are$$

4R_{\mu\mu}=\left(\partial_\mu\ln{\left|g_{\mu\mu}\right|}-2\partial_\mu\right)\partial_\mu\ln{\left|\frac{g}{g_{\mu\mu}}\right|}-\sum_{\sigma\neq\mu}\left[\left(\partial_\mu\ln{\left|g_{\sigma\sigma}\right|}\right)^2+\left(\partial_\sigma\ln{\frac{\left|g\right|}{{g_{\mu\mu}}^2}}+2\partial_\sigma\right)g^{\sigma\sigma}\partial_\sigma g_{\mu\mu}\right]

$$and the off diagonal components are $$

4R_{\mu\nu}=\left(\partial_\mu\ln{\left|g_{\nu\nu}\right|}-\partial_\mu\right)\partial_\nu\ln{\left|\frac{g}{g_{\mu\mu}g_{\nu\nu}}\right|}+\left(\mu\leftrightarrow\nu\right)-\sum_{\sigma\neq\mu,\nu}{\partial_\mu\ln{\left|g_{\sigma\sigma}\right|}\partial_\nu\ln{\left|g_{\sigma\sigma}\right|}}

$$where ##\mu\neq\nu##, there is no summation over ##\mu,\nu## and ##\left(\mu\leftrightarrow\nu\right)## stands for preceding terms with ##\mu,\nu## interchanged. The partial derivative operator ##\partial_\tau## usually only applies to what immediately follow. ##\partial_\tau\partial_\tau## is a second order derivative. The exception is the final ##2\partial_\sigma## in the first expression which applies to the whole of ##g^{\sigma\sigma}\partial_\sigma g_{\mu\mu}##.

4R_{\mu\mu}=\left(\partial_\mu\ln{\left|g_{\mu\mu}\right|}-2\partial_\mu\right)\partial_\mu\ln{\left|\frac{g}{g_{\mu\mu}}\right|}-\sum_{\sigma\neq\mu}\left[\left(\partial_\mu\ln{\left|g_{\sigma\sigma}\right|}\right)^2+\left(\partial_\sigma\ln{\frac{\left|g\right|}{{g_{\mu\mu}}^2}}+2\partial_\sigma\right)g^{\sigma\sigma}\partial_\sigma g_{\mu\mu}\right]

$$and the off diagonal components are $$

4R_{\mu\nu}=\left(\partial_\mu\ln{\left|g_{\nu\nu}\right|}-\partial_\mu\right)\partial_\nu\ln{\left|\frac{g}{g_{\mu\mu}g_{\nu\nu}}\right|}+\left(\mu\leftrightarrow\nu\right)-\sum_{\sigma\neq\mu,\nu}{\partial_\mu\ln{\left|g_{\sigma\sigma}\right|}\partial_\nu\ln{\left|g_{\sigma\sigma}\right|}}

$$where ##\mu\neq\nu##, there is no summation over ##\mu,\nu## and ##\left(\mu\leftrightarrow\nu\right)## stands for preceding terms with ##\mu,\nu## interchanged. The partial derivative operator ##\partial_\tau## usually only applies to what immediately follow. ##\partial_\tau\partial_\tau## is a second order derivative. The exception is the final ##2\partial_\sigma## in the first expression which applies to the whole of ##g^{\sigma\sigma}\partial_\sigma g_{\mu\mu}##.

It turns out that in two dimensions a diagonal metric does imply a diagonal Ricci tensor - but we knew that. In more dimensions it's not so simple. It is necessary to calculate the off diagonal components of the R-W Ricci tensor.

To calculate the Ricci tensor more directly one would use $$

R_{\mu\mu}=R_{\ \ \mu\sigma\mu}^\sigma=\partial_\sigma\Gamma_{\mu\mu}^\sigma-\partial_\mu\Gamma_{\sigma\mu}^\sigma+\Gamma_{\sigma\rho}^\sigma\Gamma_{\mu\mu}^\rho-\Gamma_{\mu\rho}^\sigma\Gamma_{\sigma\mu}^\rho

$$ and$$

R_{\mu\nu}=R_{\ \ \mu\sigma\nu}^\sigma=\partial_\sigma\Gamma_{\nu\mu}^\sigma-\partial_\nu\Gamma_{\sigma\mu}^\sigma+\Gamma_{\sigma\rho}^\sigma\Gamma_{\nu\mu}^\rho-\Gamma_{\nu\rho}^\sigma\Gamma_{\sigma\mu}^\rho

$$and once the Christoffel symbols have been calculated, which is quite easy in the R-W case, the former takes about the same number of lines as Win's formula but the latter is probably much less efficient than Win. However Win's method avoids calculating Christoffel symbols at all.

R_{\mu\mu}=R_{\ \ \mu\sigma\mu}^\sigma=\partial_\sigma\Gamma_{\mu\mu}^\sigma-\partial_\mu\Gamma_{\sigma\mu}^\sigma+\Gamma_{\sigma\rho}^\sigma\Gamma_{\mu\mu}^\rho-\Gamma_{\mu\rho}^\sigma\Gamma_{\sigma\mu}^\rho

$$ and$$

R_{\mu\nu}=R_{\ \ \mu\sigma\nu}^\sigma=\partial_\sigma\Gamma_{\nu\mu}^\sigma-\partial_\nu\Gamma_{\sigma\mu}^\sigma+\Gamma_{\sigma\rho}^\sigma\Gamma_{\nu\mu}^\rho-\Gamma_{\nu\rho}^\sigma\Gamma_{\sigma\mu}^\rho

$$and once the Christoffel symbols have been calculated, which is quite easy in the R-W case, the former takes about the same number of lines as Win's formula but the latter is probably much less efficient than Win. However Win's method avoids calculating Christoffel symbols at all.

Read the full details at Commentary 8.2 Diagonal metric and Ricci tensor.pdf (11 pages)

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