P.K.Townsend - Black Holes

A.4 Example Sheet 4

for the total angular momentum of an asymptoticallyat axisymmetric spacetime (with Killing vector m) to verify that J = Ma for the KerrNewman solution with parameter a.

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2. Let l and n be two linearly independent vectors and B a second rank tensor such that

Given that (i) (i = 1; 2) are two further linearly independent vectors, show that

3. Let N be a Killing horizon of a Killing vector eld , with surface

gravity . Explain why, for any third-rank totally-antisymmetric tensor A, the scalar = A ( D ) vanishes on N. Use this to show that

( [ D ] )(D ) = [ D ] (on N) ; ( )

where the square brackets indicate antisymmetrization on the enclosed indices.

>From the fact that vanishes on N it follows that its derivative on N is normal to N, and hence that [ @ ] = 0 on N. Use this fact and the Killing vector lemma of Q.II.1 to deduce that, on N,

( R [ ] + R [ ] + R [ ]) :

Contract on and and use the fact that 2 = 0 on N to show that

[ R ] = [ R ] (on N) ; (y)

where R is the Ricci tensor.

For any vector v the scalar = ( D ) v vanishes on N. It follows that [ @ ] jN = 0. Show that this fact, the result (*) derived above and the Killing vector lemma imply that, on N,

[ @ ] = R [ ] = [ R ] ;

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where the second line is a consequence of the cyclic identity satis ed by the Riemann tensor. Now use (y) to show that, on N,

where the second line follows on using the Einstein equations. Hence deduce the zeroth law of black hole mechanics: that, provided the matter stress tensor satis es the dominant energy condition, the surface gravity of any Killing vector eld is constant on each connected component of its Killing horizon (in particular, on the event horizon of a stationary spacetime).

4. A scalar eld in the Kruskal spacetime satis es the Klein-Gordon

equation

D2 m2 = 0 :

Given that, in static Schwarzshild coordinates, takes the form

= R`(r)e i!tY`( ; )

where Y` m is a spherical harmonic, nd the radial equation satis ed by R`(r). Show that near the horizon at r = 2M, e i!r , where r is the

Regge-Wheeler radial coordinate. Verify that ingoing waves are analytic, in Kruskal coordinates, on the future horizon, H+ , but not, in general, on the past horizon, H , and conversely for outgoing waves.

Given that both m and ! vanish, show that

R` = A`P`(z) + B`Q`(z)

for constants A`; B`, where z = (r M)=M, P`(z) is a Legendre Polynomial and Q`(z) a linearly-independent solution. Hence show that there are no non-constant solutions that are both regular on the horizon, H = H+ [H , and bounded at in nity.

5. Use the fact that a Schwarzschild black hole radiates at the Hawking temperature

1

TH = 8 M

(in units for which ~, G, c, and Bolzmann's constant all equal 1) to show that the thermal equilibrium of a black hole with an in nite reservoir of radiation at temperature TH is unstable.

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where is a constant. A Schwarzschild black hole of mass M is placed in the reservoir. Assuming that the black hole has entropy

SBH = 4 M2 ;

show that the total entropy S = SBH + Sres is extremized for xed total energy E = M +Eres, when T = TH , Show that the extremum is a maximum if and only if V < Vc, where the critical value of V is

220 4E5

Vc = 55

What happens as V passes from V < Vc to V > Vc, or vice-versa?

6. The speci c heat of a charged black hole of mass M, at xed charge Q, is

C TH

where TH is its Hawking temperature and SBH its entropy. Assuming that the entropy of a black hole is given by SBH = 14 A, where A is the area of the event horizon, show that the speci c heat of a Reissner-Nordstrom black

Repeat Q.5 for a Reissner-Nordstrom black hole. Speci cally, show that

2jQj

the critical reservoir volume, Vc, is in nite for jQj M p3 . Why is this result to be expected from your previous result for C?

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