ds² = Consider the following spacetime: - (r² = 2M (v)) du² + 2 dv dr + r² (dx² + dy²), where M(v) is a function of the coordinate v. (a) Explain in your own words the notion of black hole based on the behaviour of the null geodesics. Does this spacetime have a horizon? Reason your answer. (b) Use the Euler-Lagrange equations for the geodesics to show that г =r+ M(v) 21 zz -= T=³-M(v) - - 2M(v)2 M'(v) ((4))--1 г — (r² + M(v)), (c) Using the fact that, in certain units, the Ricci scalar for this geometry evaluates to R = -12 and the Einstein equation with a cosmological constant is given by, Gab-3gab = 8πTab calculate the vv component of the stress tensor corresponding to the line element above. (Hint: recall that I = In √√9.)