What are "inertial coordinates" and what is their physical significance? [A proof of the existence of inertial coordinates is not required.] Let $O$ be the origin of inertial coordinates and let $\left.R_{a b c d}\right|_{O}$ be the curvature tensor at $O$ (with all indices lowered). Show that $\left.R_{a b c d}\right|_{O}$ can be expressed entirely in terms of second partial derivatives of the metric $g_{a b}$, evaluated at $O$. Use this expression to deduce that (a) $R_{a b c d}=-R_{b a c d}$ (b) $R_{a b c d}=R_{c d a b}$ (c) $R_{a[b c d]}=0$.

Starting from the expression for $R_{b c d}^{a}$ in terms of the Christoffel symbols, show (again by using inertial coordinates) that

$$R_{a b[c d ; e]}=0$$

Obtain the contracted Bianchi identities and explain why the Einstein equations take the form

$$R_{a b}-\frac{1}{2} R g_{a b}=8 \pi T_{a b}-\Lambda g_{a b},$$

where $T_{a b}$ is the energy-momentum tensor of the matter and $\Lambda$ is an arbitrary constant.