Paper56 (2) - MATHEMATICAL TRIPOS Part III Thursday 4 June 2009 1:30 pm to 4:30 pm PAPER 56 BLACK HOLES Attempt no more than THREE questions There are

Paper56 (2) - MATHEMATICAL TRIPOS Part III Thursday 4 June...

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MATHEMATICAL TRIPOS Part III Thursday, 4 June, 2009 1:30 pm to 4:30 pm PAPER 56 BLACK HOLES Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
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STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
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2 1 A variant of general relativity admits an electrically charged black hole solution with metric , ds 2 = H(r) 1 / 2 F (r)dt 2 + H(r) 1 / 2 F (r) 1 dr 2 + r 2 2 2m sinh 2 α 2m H(r) = 1 + , F (r) = 1 ,
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r r where m and α are real constants with m > 0. (a) Calculate the Komar mass of this solution. [4] (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 an integral.) [2] (c) Obtain the above metric in ingoing Eddington-Finkelstein coordinates (v, r, θ, φ). Hence show that it can be analytically extended through r = 2m. [2] (d) Define the black hole region of 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. [5]
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