#2 – Hilbert – Einstein action – Part 2

Now we compute \frac{\delta R}{\delta g_{\mu\nu}}. First, the Riemann tensor has the following definition:

{R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}{\Gamma^{\rho}}_{\nu\sigma}-\partial_{\nu}{\Gamma^{\rho}}_{\mu\sigma}+ {\Gamma^{\rho}}_{\mu\lambda}{\Gamma^{\lambda}}_{\nu\sigma}-{\Gamma^{\rho}}_{\nu\lambda}{\Gamma^{\lambda}}_{\mu\sigma},

so

\delta{R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\delta{\Gamma^{\rho}}_{\nu\sigma}-\partial_{\nu}\delta{\Gamma^{\rho}}_{\mu\sigma}+\delta{\Gamma^{\rho}}_{\mu\lambda}{\Gamma^{\lambda}}+{\Gamma^{\rho}}_{\mu\lambda}\delta{\Gamma^{\lambda}}_{\nu\sigma}-\delta{\Gamma^{\rho}}_{\nu\lambda}{\Gamma^{\lambda}}_{\mu\sigma}-{\Gamma^{\rho}}_{\nu\lambda}\delta{\Gamma^{\lambda}}_{\mu\sigma}.

Continue reading “#2 – Hilbert – Einstein action – Part 2”

Advertisements