He also jumps back to section 3.1 and uses the equation there for the Riemann tensor. It is$$
R_{\rho\sigma\mu\nu}=\kappa\left(g_{\rho\mu}g_{\sigma\nu}-g_{\rho\nu}g_{\sigma\mu}\right)
$$where ##\kappa## is a constant. That equation easily becomes$$
R_{\ \ \ \sigma\mu\nu}^\rho=\kappa\left(\delta_\mu^\rho g_{\sigma\nu}-\delta_\nu^\rho g_{\sigma\mu}\right)
$$so it looks like there's a really way to calculate the fiendish Riemann tensor for these special cases. Unfortunately $$
\kappa=\frac{R}{n\left(n-1\right)}
$$where ##n## is the number of dimensions and ##R## is the curvature scalar (aka Ricci scalar) which is constant in a maximally symmetric universe. Nevertheless to calculate it you have to calculate the Riemann tensor first! B*gger.
R_{\rho\sigma\mu\nu}=\kappa\left(g_{\rho\mu}g_{\sigma\nu}-g_{\rho\nu}g_{\sigma\mu}\right)
$$where ##\kappa## is a constant. That equation easily becomes$$
R_{\ \ \ \sigma\mu\nu}^\rho=\kappa\left(\delta_\mu^\rho g_{\sigma\nu}-\delta_\nu^\rho g_{\sigma\mu}\right)
$$so it looks like there's a really way to calculate the fiendish Riemann tensor for these special cases. Unfortunately $$
\kappa=\frac{R}{n\left(n-1\right)}
$$where ##n## is the number of dimensions and ##R## is the curvature scalar (aka Ricci scalar) which is constant in a maximally symmetric universe. Nevertheless to calculate it you have to calculate the Riemann tensor first! B*gger.
I still wanted to check that it was all true. We already did it for a the surface of a unit sphere, S², and have found it was constant at 2 and its Riemann tensor does satisfy the formula above. So S² is maximally symmetric. All points on the surface of a sphere are equal. The metric for conformal anti de Sitter is$$
{ds}^2=\frac{\alpha^2}{\cos^2{\chi}}\left(-{dt^\prime}^2+{d\chi}^2+\sin^2{\chi}{d\Omega_2}^2\right)
$$where ##\alpha## is some constant. So its Riemann tensor is trickier!
{ds}^2=\frac{\alpha^2}{\cos^2{\chi}}\left(-{dt^\prime}^2+{d\chi}^2+\sin^2{\chi}{d\Omega_2}^2\right)
$$where ##\alpha## is some constant. So its Riemann tensor is trickier!
Calculating that Riemann tensor, which is a relatively easy one, took 9 hours 45 minutes over three days and six sittings. I got faster as I went along, notwithstanding long lunches preceded by a refreshing dry martini.
Read all about it at
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