Let $(\mathcal{M}, \boldsymbol{g})$ be a four-dimensional manifold with metric $g_{\alpha \beta}$ of Lorentzian signature.

The Riemann tensor $\boldsymbol{R}$ is defined through its action on three vector fields $\boldsymbol{X}, \boldsymbol{V}, \boldsymbol{W}$ by

$$\boldsymbol{R}(\boldsymbol{X}, \boldsymbol{V}) \boldsymbol{W}=\nabla_{\boldsymbol{X}} \nabla_{\boldsymbol{V}} \boldsymbol{W}-\nabla_{\boldsymbol{V}} \nabla_{\boldsymbol{X}} \boldsymbol{W}-\nabla_{[\boldsymbol{X}, \boldsymbol{V}]} \boldsymbol{W}$$

and the Ricci identity is given by

$$\nabla_{\alpha} \nabla_{\beta} V^{\gamma}-\nabla_{\beta} \nabla_{\alpha} V^{\gamma}=R_{\rho \alpha \beta}^{\gamma} V^{\rho} .$$

(i) Show that for two arbitrary vector fields $\boldsymbol{V}, \boldsymbol{W}$, the commutator obeys

$$[\boldsymbol{V}, \boldsymbol{W}]^{\alpha}=V^{\mu} \nabla_{\mu} W^{\alpha}-W^{\mu} \nabla_{\mu} V^{\alpha}$$

(ii) Let $\gamma: I \times I^{\prime} \rightarrow \mathcal{M}, \quad I, I^{\prime} \subset \mathbb{R},(s, t) \mapsto \gamma(s, t)$ be a one-parameter family of affinely parametrized geodesics. Let $\boldsymbol{T}$ be the tangent vector to the geodesic $\gamma(s=$ const, $t)$ and $\boldsymbol{S}$ be the tangent vector to the curves $\gamma(s, t=$ const $)$. Derive the equation for geodesic deviation,

$$\nabla_{T} \nabla_{T} \boldsymbol{S}=\boldsymbol{R}(\boldsymbol{T}, \boldsymbol{S}) \boldsymbol{T}$$

(iii) Let $X^{\alpha}$ be a unit timelike vector field $\left(X^{\mu} X_{\mu}=-1\right)$ that satisfies the geodesic equation $\nabla_{\boldsymbol{X}} \boldsymbol{X}=0$ at every point of $\mathcal{M}$. Define

$$\begin{array}{ll} B_{\alpha \beta}:=\nabla_{\beta} X_{\alpha}, & h_{\alpha \beta}:=g_{\alpha \beta}+X_{\alpha} X_{\beta}, \\ \Theta:=B^{\alpha \beta} h_{\alpha \beta}, \quad \sigma_{\alpha \beta}:=B_{(\alpha \beta)}-\frac{1}{3} \Theta h_{\alpha \beta}, & \omega_{\alpha \beta}:=B_{[\alpha \beta]} . \end{array}$$

Show that

$$\begin{gathered} B_{\alpha \beta} X^{\alpha}=B_{\alpha \beta} X^{\beta}=h_{\alpha \beta} X^{\alpha}=h_{\alpha \beta} X^{\beta}=0 \\ B_{\alpha \beta}=\frac{1}{3} \Theta h_{\alpha \beta}+\sigma_{\alpha \beta}+\omega_{\alpha \beta}, \quad g^{\alpha \beta} \sigma_{\alpha \beta}=0 \end{gathered}$$

(iv) Let $\boldsymbol{S}$ denote the geodesic deviation vector, as defined in (ii), of the family of geodesics defined by the vector field $X^{\alpha}$. Show that $\boldsymbol{S}$ satisfies

$$X^{\mu} \nabla_{\mu} S^{\alpha}=B_{\mu}^{\alpha} S^{\mu}$$

(v) Show that

$$X^{\mu} \nabla_{\mu} B_{\alpha \beta}=-B_{\beta}^{\mu} B_{\alpha \mu}+R_{\mu \beta \alpha}{ }^{\nu} X^{\mu} X_{\nu}$$