State without proof the properties of local inertial coordinates $x^{a}$ centred on an arbitrary spacetime event $p$. Explain their physical significance.

Obtain an expression for $\partial_{a} \Gamma_{b}{ }^{c}$ at $p$ in inertial coordinates. Use it to derive the formula

$$R_{a b c d}=\frac{1}{2}\left(\partial_{b} \partial_{c} g_{a d}+\partial_{a} \partial_{d} g_{b c}-\partial_{b} \partial_{d} g_{a c}-\partial_{a} \partial_{c} g_{b d}\right)$$

for the components of the Riemann tensor at $p$ in local inertial coordinates. Hence deduce that at any point in any chart $R_{a b c d}=R_{c d a b}$.

Consider the metric

$$d s^{2}=\frac{\eta_{a b} d x^{a} d x^{b}}{\left(1+L^{-2} \eta_{a b} x^{a} x^{b}\right)^{2}}$$

where $\eta_{a b}=\operatorname{diag}(1,1,1,-1)$ is the Minkowski metric tensor and $L$ is a constant. Compute the Ricci scalar $R=R_{a b}^{a b}$ at the origin $x^{a}=0$.