The symbol $\nabla_{a}$ denotes the covariant derivative defined by the Christoffel connection $\Gamma_{b c}^{a}$ for a metric $g_{a b}$. Explain briefly why

$$\begin{gathered} \left(\nabla_{a} \nabla_{b}-\nabla_{b} \nabla_{a}\right) \phi=0 \\ \left(\nabla_{a} \nabla_{b}-\nabla_{b} \nabla_{a}\right) v_{c} \neq 0 \end{gathered}$$

in general, where $\phi$ is a scalar field and $v_{c}$ is a covariant vector field.

A Killing vector field $v_{a}$ satisfies the equation

$$S_{a b} \equiv \nabla_{a} v_{b}+\nabla_{b} v_{a}=0$$

By considering the quantity $\nabla_{a} S_{b c}+\nabla_{b} S_{a c}-\nabla_{c} S_{a b}$, show that

$$\nabla_{a} \nabla_{b} v_{c}=-R_{a b c}^{d} v_{d}$$

Find all Killing vector fields $v_{a}$ in the case of flat Minkowski space-time.

For a metric of the form

$$\mathrm{d} s^{2}=-f(\mathbf{x}) \mathrm{d} t^{2}+g_{i j}(\mathbf{x}) \mathrm{d} x^{i} \mathrm{~d} x^{j}, \quad i, j=1,2,3$$

where $\mathbf{x}$ denotes the coordinates $x^{i}$, show that $\Gamma_{00}^{0}=\Gamma_{i j}^{0}=0$ and that $\Gamma_{0 i}^{0}=\Gamma_{i 0}^{0}=$ $\frac{1}{2}\left(\partial_{i} f\right) / f$. Deduce that the vector field $v^{a}=(1,0,0,0)$ is a Killing vector field.

[You may assume the standard symmetries of the Riemann tensor.]