= r2−2M r+ a,Σ = r2+ a2cos2θ .What is the significance of the parameter aWhere is the event horizon? Where is the ergoregion? Explain briefly how the existence of an ergoregion allows energy to be extracted froma black hole (Penrose process).(iv) Raychaudhuri’s equation for a null geodesic congru-ence isdλdθ= −12θ2−σˆµνσˆµν+ ωˆµνωˆµν−Rtµtν,where λis an aﬃne parameter. Explain briefly thesignificance of (a) the vector t, (b) the scalar θ(c) the symmetric tensor σˆ and (d) the antisymmetric tensor ωˆ.(v) A Kerr black hole of mass Mand angular momentum J has “horizon area”√A = 8π M 2+M 4−J A collision of two Kerr black holes, of the same mass and angular momentum, produces a Schwarzschild black hole and gravitational radiation. What is the maximum fraction of the initial mass (2M) that can be radiated away without violating the second law of black hole 2? µν, 2.
mechanics?Part III, Paper 55[TURN OVER
4(vi) A quantum field Φ(x) takes the formΦ(x) =iaiui(x) +ai†ui∗(x)(∗)Xin the far past, where the spacetime is Minkowski and the functions ui(x) are the positive-frequency eigenstates of the standard Minkowski time-translation Killing vector field. The operator coeﬃcients (ai, a†i) satisfy the commutation relations[ai, aj] = 0 , ai, a†j=δijIn the far future, where the spacetime is again Minkowski, the quantum field again takes the form (∗) but with the operatorsaireplaced bya′i=XajAji+a†jBji∗, and hence a†iby (a′i)†. Find the restrictions on thematrices A andB that follow from the requirement that the primedoperators obey the same commutation relations as the unprimed operators. Show that if the initial state is a vacuum state then the final state will be one withparticles unless B= 0..j