MATHEMATICAL TRIPOSPart IIIThursday, 4 June, 2009 1:30 pm to 4:30 pmPAPER 56BLACK HOLESAttempt no more than THREE questions.There are FOUR questions in total.The questions carry equal weight.
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21A variant of general relativity admits an electrically charged black hole solution withmetric,ds2=H(r)1/2F (r)dt2+ H(r)1/2F (r)1dr2+ r2dΩ2−−2m sinh2α−2mH(r) = 1 +,F (r) = 1 −,
rrwhere m and α are real constants with m > 0.(a) Calculate the Komar mass of this solution.(b) Show that one can define a quantity r∗such that u = t −r∗and v = t + r∗are constant on outgoing and ingoing radial null geodesics respectively. (You may express r∗as anintegral.)(c) Obtain the above metric in ingoing Eddington-Finkelsteincoordinates (v, r, θ, φ).Hence show that it can be analytically extended through r = 2m.(d) Define the black hole regionof an asymptotically flat spacetime. Prove that the region r < 2m (in ingoing Eddington-Finkelstein coordinates) is within the black hole region, and that the region r > 2m does not intersect the black hole region.