(a) Determine the signature of the metric tensor $g_{\mu \nu}$ given by

$$g_{\mu \nu}=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right)$$

Is it Riemannian, Lorentzian, or neither?

(b) Consider a stationary black hole with the Schwarzschild metric:

$$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$$

These coordinates break down at the horizon $r=2 M$. By making a change of coordinates, show that this metric can be converted to infalling Eddington-Finkelstein coordinates.

(c) A spherically symmetric, narrow pulse of radiation with total energy $E$ falls radially inwards at the speed of light from infinity, towards the origin of a spherically symmetric spacetime that is otherwise empty. Assume that the radial width $\lambda$ of the pulse is very small compared to the energy $(\lambda \ll E)$, and the pulse can therefore be treated as instantaneous.

(i) Write down a metric for the region outside the pulse, which is free from coordinate singularities. Briefly justify your answer. For what range of coordinates is this metric valid?

(ii) Write down a metric for the region inside the pulse. Briefly justify your answer. For what range of coordinates is this metric valid?

(iii) What is the final state of the system?