Consider a family of geodesics with $s$ an affine parameter and $V^{a}$ the tangent vector on each curve. The equation of geodesic deviation for a vector field $W^{a}$ is

$$\frac{D^{2} W^{a}}{D s^{2}}=R_{b c d}^{a} V^{b} V^{c} W^{d}$$

where $\frac{D}{D s}$ denotes the directional covariant derivative $V^{b} \nabla_{b}$.

(i) Show that if

$$V^{b} \frac{\partial W^{a}}{\partial x^{b}}=W^{b} \frac{\partial V^{a}}{\partial x^{b}}$$

then $W^{a}$ satisfies $(*)$.

(ii) Show that $V^{a}$ and $s V^{a}$ satisfy $(*)$.

(iii) Show that if $W^{a}$ is a Killing vector field, meaning that $\nabla_{b} W_{a}+\nabla_{a} W_{b}=0$, then $W^{a}$ satisfies $(*)$.

(iv) Show that if $W^{a}=w U^{a}$ satisfies $(*)$, where $w$ is a scalar field and $U^{a}$ is a time-like unit vector field, then

$$\begin{gathered} \frac{d^{2} w}{d s^{2}}=\left(\Omega^{2}-K\right) w \\ -\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \quad \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \end{gathered}$$

$$\begin{aligned} & \text { where } \quad \Omega^{2}=-\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \text {. } \end{aligned}$$

[You may use: $\nabla_{b} \nabla_{c} X^{a}-\nabla_{c} \nabla_{b} X^{a}=R_{d b c}^{a} X^{d}$ for any vector field $\left.X^{a} .\right]$