Black Holes

145 Pages·1997·0.734 MB·English

by  Townsend P.K.

Black Holes
07012 v1 4 Jul 97 ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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i Lecture notes
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r by
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Dr. P.K. Townsend
DAMTP, University of Cambridge,
Silver St., Cambridge, U.K.
Acknowledgements
These notes were written to accompany a course taught in Part III of the
CambridgeUniversityMathematicalTripos. Thereareoccasionalreferences
to questions on four ’example sheets’, which can be found in the Appendix.
The writing of these course notes has greatly bene(cid:12)tted from discussions
with Gary Gibbons and Stephen Hawking. The organisation of the course
was based on unpublished notes of Gary Gibbons and owes much to the
1972 Les Houches and 1986 Carg(cid:19)ese lecture notes of Brandon Carter, and
to the 1972 lecture notes of Stephen Hawking. Finally, I am very grateful
to Tim Perkins for typing the notes in LATEX, producing the diagrams, and
putting it all together.
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Contents
1 Gravitational Collapse 6
1.1 The Chandrasekhar Limit . . . . . . . . . . . . . . . . . . . . 6
1.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Schwarzschild Black Hole 11
2.1 Test particles: geodesics and a(cid:14)ne parameterization . . . . . 11
2.2 Symmetries and Killing Vectors . . . . . . . . . . . . . . . . . 13
2.3 Spherically-Symmetric Pressure Free Collapse . . . . . . . . . 15
2.3.1 Black Holes and White Holes . . . . . . . . . . . . . . 18
2.3.2 Kruskal-Szekeres Coordinates . . . . . . . . . . . . . . 20
2.3.3 Eternal Black Holes . . . . . . . . . . . . . . . . . . . 24
2.3.4 Time translation in the Kruskal Manifold . . . . . . . 26
2.3.5 Null Hypersurfaces . . . . . . . . . . . . . . . . . . . . 27
2.3.6 Killing Horizons . . . . . . . . . . . . . . . . . . . . . 29
2.3.7 Rindler spacetime . . . . . . . . . . . . . . . . . . . . 33
2.3.8 Surface Gravity and Hawking Temperature . . . . . . 37
2.3.9 Tolman Law - Unruh Temperature . . . . . . . . . . . 39
2.4 Carter-Penrose Diagrams . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Conformal Compacti(cid:12)cation . . . . . . . . . . . . . . . 40
2.5 Asymptopia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 The Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Black Holes vs. Naked Singularities. . . . . . . . . . . . . . . 53
3 Charged Black Holes 56
3.1 Reissner-Nordstro(cid:127)m . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Pressure-Free Collapse to RN . . . . . . . . . . . . . . . . . . 65
3.3 Cauchy Horizons . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Isotropic Coordinates for RN . . . . . . . . . . . . . . . . . . 70
3.4.1 Nature of Internal in Extreme RN . . . . . . . . . . 74
1
3
3.4.2 Multi Black Hole Solutions . . . . . . . . . . . . . . . 75
4 Rotating Black Holes 76
4.1 Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . 76
4.1.1 Spacetime Symmetries . . . . . . . . . . . . . . . . . . 76
4.2 The Kerr Solution . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.1 Angular Velocity of the Horizon . . . . . . . . . . . . 84
4.3 The Ergosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 The Penrose Process . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Limits to Energy Extraction. . . . . . . . . . . . . . . 89
4.4.2 Super-radiance . . . . . . . . . . . . . . . . . . . . . . 90
5 Energy and Angular Momentum 93
5.1 Covariant Formulation of Charge Integral . . . . . . . . . . . 93
5.2 ADM energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Alternative Formula for ADM Energy . . . . . . . . . 96
5.3 Komar Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 Angular Momentum in Axisymmetric Spacetimes . . . 98
5.4 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Black Hole Mechanics 101
6.1 Geodesic Congruences . . . . . . . . . . . . . . . . . . . . . . 101
6.1.1 Expansion and Shear . . . . . . . . . . . . . . . . . . . 106
6.2 The Laws of Black Hole Mechanics . . . . . . . . . . . . . . . 109
6.2.1 Zeroth law . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.2 Smarr’s Formula . . . . . . . . . . . . . . . . . . . . . 110
6.2.3 First Law . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.4 The Second Law (Hawking’s Area Theorem) . . . . . 113
7 Hawking Radiation 119
7.1 Quantization of the Free Scalar Field . . . . . . . . . . . . . . 119
7.2 Particle Production in Non-Stationary Spacetimes . . . . . . 123
7.3 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Black Holes and Thermodynamics . . . . . . . . . . . . . . . 129
7.4.1 The Information Problem . . . . . . . . . . . . . . . . 130
A Example Sheets 132
A.1 Example Sheet 1 . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.2 Example Sheet 2 . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.3 Example Sheet 3 . . . . . . . . . . . . . . . . . . . . . . . . . 138
4
A.4 Example Sheet 4 . . . . . . . . . . . . . . . . . . . . . . . . . 141
5
Chapter 1
Gravitational Collapse
1.1 The Chandrasekhar Limit
A Star is a self-gravitating ball of hydrogen atoms supported by thermal
pressure P nkT where n is the number density of atoms. In equilibrium,
(cid:24)
E = E +E (1.1)
grav kin
is a minimum. For a star of mass M and radius R
GM2
E (1.2)
grav
(cid:24) (cid:0) R
E nR3 E (1.3)
kin
(cid:24) h i
where E is average kinetic energy of atoms. Eventually, fusion at the
h i
core must stop, after which the star cools and contracts. Consider the
possible (cid:12)nal state of a star at T = 0. The pressure P does not go to zero
as T 0 because of degeneracy pressure. Since m m the electrons
e p
! (cid:28)
become degenerate (cid:12)rst, at a number density of one electron in a cube of
side Compton wavelength.
(cid:24)
~
n 1=3 ; p = average electron momentum (1.4)
e(cid:0) (cid:24) p h i
e
h i
Can electrondegeneracypressure support a star from collapse
at T = 0?
Assume that electrons are non-relativistic. Then
p 2
e
E h i : (1.5)
h i(cid:24) m
e
6
So, since n = n ,
e
~2R2re2=3
E : (1.6)
kin
(cid:24) m
e
M
Since m m , M n R3m , so n and
e (cid:28) p (cid:25) e e e (cid:24) m R3
p
~2 M 5=3 1
E : (1.7)
kin (cid:24) m m R2
e (cid:18) p(cid:19)
constant for
(cid:12)xed M
| {z }
Thus
(cid:11) (cid:12)
E ; (cid:11);(cid:12)independent of R: (1.8)
(cid:24) (cid:0)R (cid:0) R2
E .........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................R........................R..............................................m........................m............................i..................n......i.................n......................................................(cid:24)........................................................................G.........~.................................2....m.....................M.................................e......................m.........(cid:0)..................................1...........5p...........=..........=...........3...........3............................................................................................................................................................................ R
The collapse of the star is therefore prevented. It becomes a White Dwarf
or a cold, dead star supported by electron degeneracy pressure.
At equilibrium
3
M m G
n e Mm2 2=3 : (1.9)
e (cid:24) mpR3min (cid:18) ~2 p (cid:19)
(cid:0) (cid:1)
But the validityof non-relativisticapproximationrequires that p m c,
e e
h i(cid:28)
i.e.
1=3
pe ~ne
h i = c (1.10)
m m (cid:28)
e e
m c 2
e
or n : (1.11)
e
(cid:28) ~
(cid:16) (cid:17)
7
For a White Dwarf this implies
m G m c
e Mm2 2=3 e (1.12)
~2 p (cid:28) ~
(cid:0) 1(cid:1) ~c 3=2
or M : (1.13)
(cid:28) m2 G
p (cid:18) (cid:19)
For su(cid:14)ciently large M the electrons would have to be relativistic, in
which case we must use
hEi= hpeic = ~cn1e=3 (1.14)
) Ekin (cid:24) neR3hEi(cid:24) ~cR3n4e=3 (1.15)
4=3 4=3
M M 1
~cR3 ~c (1.16)
(cid:24) m R3 (cid:24) m R
(cid:18) p (cid:19) (cid:18) p(cid:19)
So now,
(cid:11) (cid:13)
E + : (1.17)
(cid:24) (cid:0)R R
Equilibrium is possible only for
3=2
1 ~c
(cid:13) = (cid:11) M : (1.18)
) (cid:24) m2 G
p (cid:18) (cid:19)
For smaller M, R must increase until electrons become non-relativistic,
in which case the star is supported by electron degeneracy pressure, as we
just saw. ForlargerM, Rmustcontinue todecrease, so electrondegeneracy
pressure cannot support the star. There is therefore a critical mass M
C
1 ~c 3=2 1 ~3 1=2
M R (1.19)
C (cid:24) m2 G ) C (cid:24) m m Gc
p (cid:18) (cid:19) e p (cid:18) (cid:19)
abovewhichastarcannotend asaWhiteDwarf. Thisisthe Chandrasekhar
limit. Detailed calculation gives M 1:4M .
C
’ (cid:12)
1.2 Neutron Stars
TheelectronenergiesavailableinaWhiteDwarfareoftheorderoftheFermi
energy. Necessarily E < m c2 since the electrons are otherwise relativistic
F e
(cid:24)
and cannot support the star. A White Dwarf is therefore stable against
inverse (cid:12)-decay
e +p+ n+(cid:23) (1.20)
(cid:0) e
!
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since the reaction needs energy of at least ((cid:1)m )c2 where (cid:1)m is the
n n
neutron-proton mass di(cid:11)erence. Clearly (cid:1)m > m ((cid:12)-decay would other-
e
wise be impossible) and in fact (cid:1)m 3m . So we need energies of order of
e
(cid:24)
3m c2 forinverse(cid:12)-decay. Thisisnotavailablein WhiteDwarfstarsbutfor
e
M > M the star must continue to contract until E ((cid:1)m )c2. At this
C F n
(cid:24)
point inverse (cid:12)-decay can occur. The reaction cannot come to equilibrium
with the reverse reaction
n+(cid:23) e +p+ (1.21)
e (cid:0)
!
because the neutrinos escape from the star, and (cid:12)-decay,
n e +p+(cid:23)(cid:22) (1.22)
(cid:0) e
!
cannot occur because all electron energy levels below E < ((cid:1)m )c2 are
n
(cid:12)lled when E > ((cid:1)m )c2. Since inverse (cid:12)-decay removes the electron de-
n
generacy pressure the star will undergo a catastrophic collapse to nuclear
matter density, at which point we must take neutron-degeneracy pressure
into account.
Can neutron-degeneracy pressure support the star against col-
lapse?
The ideal gas approximation would give same result as before but with
m m . The critical mass M is independent of m and so is una(cid:11)ected,
e p C e
!
but the critical radius is now
me 1 ~3 1=2 GMC
R (1.23)
m C (cid:24) m2 Gc (cid:24) c2
(cid:18) p(cid:19) p (cid:18) (cid:19)
which is the Schwarzschild radius, so the neglect of GR e(cid:11)ects was not
justi(cid:12)ed. Also, at nuclear matter densities the ideal gas approximation is
not justi(cid:12)ed. A perfect (cid:13)uid approximation is reasonable (since viscosity
can’t help). Assume that P((cid:26)) ((cid:26)= density of (cid:13)uid) satis(cid:12)es
i) P 0 (local stability). (1.24)
(cid:21)
ii) P <c2 (causality). (1.25)
0
Then the known behaviour of P((cid:26)) at low nuclear densities gives
M 3M : (1.26)
max
(cid:24) (cid:12)
More massive stars must continue to collapse either to an unknown new
ultra-high density state of matter or to a black hole. The latter is more
9
likely. In any case, there must be some mass at which gravitationalcollapse
to a black hole is unavoidable because the density at the Schwarzschild
radius decreases as the total mass increases. In the limit of very large mass
the collapse is well-approximatedby assuming the collapsing materialto be
a pressure-free ball of (cid:13)uid. We shall consider this case shortly.
10