A.4 Example Sheet 4

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1. Use the Komar integral,

J =

1

16_G

I

1

dS__D_m_ ;

for the total angular momentum of an asymptotically-at axisymmetric

spacetime (with Killing vector m) to verify that J = Ma for the Kerr-

Newman solution with parameter a.

2. Let l and n be two linearly independent vectors and ^B a second rank

tensor such that

^B

_

_l_ = ^B_

_n_ = 0 :

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

that

"____l_n_ ^B_

_(_(1)

_ _(2)

_ 􀀀 _(1)

_ _(2)

_ ) = _ "____ l_n__(1)

_ _(2)

_ :

where _ = ^B_

_.

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_]__

___ ;

141

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,

___[_@_]_ = _[_R_]

___ (A.1)

= 8_G_[_T_]

___ ; (A.2)

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

TH =

1

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.

142

A _nite reservoir of radiation of volume V at temperature T has an

energy, Eres and entropy, Sres given by

Eres = _V T4 Sres =

4

3

_V T3

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

Vc =

220_4E5

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

@SBH

@TH

____

Q

;

where TH is its Hawking temperature and SBH its entropy. Assuming that

the entropy of a black hole is given by SBH = 1

4A, where A is the area of

the event horizon, show that the speci_c heat of a Reissner-Nordstrom black

hole is

C =

2SBH

p

M2 􀀀 Q2

(M 􀀀 2

p

M2 􀀀 Q2)

:

Hence show that C􀀀1 changes sign when M passes through 2pjQj 3

.

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

the critical reservoir volume, Vc, is in_nite for jQj _ M _ 2pjQj 3

. Why is this

result to be expected from your previous result for C?

143

Index

acceleration horizon, 32

ADM energy, 85

a_ne parameter, 10, 92

asymptotically

empty, 43

simple, 42

weakly, 42

axisymmetric, 68

Beckenstein-Hawking entropy, 119

bifurcate Killing horizon, 28

bifurcation

2-sphere, 28

point, 28

Birkho_'s theorem, 12, 69

black body radiation, 118

black hole, 16

entropy, 119

Bogoliubov transformation, 113

Boyer-Kruskal axis, 23

Boyer-Linquist coordinates, 69

Carter-Penrose diagram, 38

Carter-Robinson theorem, 69

Cauchy

horizon, 60

surface

partial, 60

Cauchy surface, 60

Chandrasekhar limit, 7

co-rotating electric potential, 102

conformal compacti_cation, 36

congruence, 92

geodesic, 92

null, 94

cosmic censorship hypothesis, 51

degenerate pressure, 5

dominant energy condition, 89

Eddington-Finkelstein coordinates

ingoing, 15

outgoing, 16

einbein, 9

Einstein Static Universe, 38

Einstein-Rosen bridge, 22

ergoregion, 79

ergosphere, 79

Finkelstein diagram, 15, 16

_xed point, 23

_xed sets, 23

Frobenius' theorem, 26, 96

future event horizon, 44

geodesic, 9

congruence, 92

deviation, 93

global violation of causality, 74

graviton, 69, 85

Hawking

radiation, 114

temperature, 34, 118

imaginary time, 33

144

isotropic coordinates, 21

Israel's theorem, 69

Kaluza-Klein vacuum, 66

Kerr metric, 70

Kerr-Newman family, 69

Kerr-Schild coordinates, 71

Killing

horizon, 26

bifurcate, 28

degenerate, 65

vector, 11

Klein-Gordon equation, 109

Komar integrals, 87

Kruskal-Szekeres coordinates, 17

maximal analytic extension, 20

naked singularity, 47

null hypersurface, 24

parallel transport, 10

particle number operator, 112

Pauli-Fierz equation, 86

Penrose

process, 79

Planck distribution, 118

positive energy theorem, 91

proper time, 14

quantum gravity, 120

Raychaudhuri equation, 97

Regge-Wheeler radial coordinate,

15

Reissner-Nordstrom solution, 50

Rindler

metric, 31

spacetime, 30

Euclidean, 34

sandwich spacetime, 112

Schwarzschild metric, 12

singularity

conical, 35

Smarr's formula, 102

static, 68

stationary, 68

Stephan's law, 118

string theory, 120

strong energy condition, 90

super-radiance, 81

surface gravity, 26, 33

symplectic structure, 109

Tolman law, 35

totally-geodesic, 72

uniqueness theorems, 68

Unruh

e_ect, 35

temperature, 35

weak energy condition, 90

weak static dust, 86

white dwarf, 6

white hole, 17

145__