(i) What is a "stationary" metric? What distinguishes a stationary metric from a "static" metric?

A Killing vector field $K^{a}$ of a metric $g_{a b}$ satisfies

$$K_{a ; b}+K_{b ; a}=0$$

Show that this is equivalent to

$$g_{a b, c} K^{c}+g_{a c} K_{, b}^{c}+g_{c b} K_{, a}^{c}=0$$

Hence show that a constant vector field $K^{a}$ with one non-zero component, $K^{4}$ say, is a Killing vector field if $g_{a b}$ is independent of $x^{4}$.

(ii) Given that $K^{a}$ is a Killing vector field, show that $K_{a} u^{a}$ is constant along the geodesic worldline of a massive particle with 4-velocity $u^{a}$. Hence find the energy $\varepsilon$ of a particle of unit mass moving in a static spacetime with metric

$$d s^{2}=h_{i j} d x^{i} d x^{j}-e^{2 U} d t^{2}$$

where $h_{i j}$ and $U$ are functions only of the space coordinates $x^{i}$. By considering a particle with speed small compared with that of light, and given that $U \ll 1$, show that $h_{i j}=\delta_{i j}$ to lowest order in the Newtonian approximation, and that $U$ is the Newtonian potential.

A metric admits an antisymmetric tensor $Y_{a b}$ satisfying

$$Y_{a b ; c}+Y_{a c ; b}=0$$

Given a geodesic $x^{a}(\lambda)$, let $s_{a}=Y_{a b} \dot{x}^{b}$. Show that $s_{a}$ is parallelly propagated along the geodesic, and that it is orthogonal to the tangent vector of the geodesic. Hence show that the scalar

$$\phi=s^{a} s_{a}$$

is constant along the geodesic.