General relativity (Advanced pysics)

Consider the following metric: ds' = A(r)c' dt e A(r) (d +r do) +r sin 0do?. It is not spherically symmetric due to the factor exp- m sin' 0 , but the deviations from spherical symmetry become negligible for large values of the radial coordinate r. a) Determine the function A(r) so that the metric becomes a vacuum solution to Einstein's equations. Normalise the solution so that A(r) -> 1 when r-> co. The solution may be seen as the field from some non-spherical star. Determine the constant m in terms of the mass M of the star (and fundamental constants). 2.5p (Hint. Even if R. = 0 and G. =0 are equivalent for a vacuum metric, one of the two sets might give a simpler set of equations than the other.)