MATHEMATICAL TRIPOSPart IIIFriday 30 May 2008 1.30 to 4.30PAPER 66BLACK HOLESAttempt no more thanTHREEquestions.There areFOURquestions in total.The questions carry equal weight.STATIONERY REQUIREMENTSSPECIAL REQUIREMENTSCover sheetNoneTreasury TagScript paperYou may not start to read the questionsprinted on the subsequent pages untilinstructed to do so by the Invigilator.
21(a)State (without explanation) which of the space-times listed below is (i) extendible;(ii) geodesically complete; (iii) globally hyperbolic; (iv) asymptotically simple. (Presentyour answers in the form of a table.)The regionr >2Mof the Schwarzschild solution (withM >0 ),Kruskal,the maximal analytic extension of extremal Reissner-Nordstrom,Rindler,the Einstein static universe.(b)The Majumdar-Papapetrou (MP) solution has metricds2=-H-2dt2+H2(dx2+dy2+dz2),H(x,y,z) = 1 +NXi=1Mi|x-xi|whereMi>0 ,x= (), andxiare constant vectors inR3.