What are local inertial co-ordinates? What is their physical significance and how are they related to the equivalence principle?

If $V_{a}$ are the components of a covariant vector field, show that

$$\partial_{a} V_{b}-\partial_{b} V_{a}$$

are the components of an anti-symmetric second rank covariant tensor field.

If $K^{a}$ are the components of a contravariant vector field and $g_{a b}$ the components of a metric tensor, let

$$Q_{a b}=K^{c} \partial_{c} g_{a b}+g_{a c} \partial_{b} K^{c}+g_{c b} \partial_{a} K^{c}$$

Show that

$$Q_{a b}=2 \nabla_{(a} K_{b)},$$

where $K_{a}=g_{a b} K^{b}$, and $\nabla_{a}$ is the Levi-Civita covariant derivative operator of the metric $g_{a b}$.

In a particular co-ordinate system $\left(x^{1}, x^{2}, x^{3}, x^{4}\right)$, it is given that $K^{a}=(0,0,0,1)$, $Q_{a b}=0$. Deduce that, in this co-ordinate system, the metric tensor $g_{a b}$ is independent of the co-ordinate $x^{4}$. Hence show that

$$\nabla_{a} K_{b}=\frac{1}{2}\left(\partial_{a} K_{b}-\partial_{b} K_{a}\right)$$

and that

$$E=-K_{a} \frac{d x^{a}}{d \lambda}$$

is constant along every geodesic $x^{a}(\lambda)$ in every co-ordinate system.

What further conditions must one impose on $K^{a}$ and $d x^{a} / d \lambda$ to ensure that the metric is stationary and that $E$ is proportional to the energy of a particle moving along the geodesic?