You've reached the end of your free preview.
Want to read all 145 pages?
Unformatted text preview: arXiv:gr-qc/9707012v1 4 Jul 1997 Black Holes .......
.......
.......
.......
.......
.......
.......
.............. .... .................. .................. .................. .................. .................. .................. .....................
.
.
.
.
.
.
.... ... ..........
.. .....
... ......
...
....
...
....
....
.
...
.
.
.
...
...
....
...
.
.
.
.
.
....
.
.
....
..
...
.
...
.
.
.
.
.
....
....
..
...
.
.
.
.
.
.
....
....
..
..
.
.
.
.
.
...
.
.
...
..
..
....
.
.
.
.
.
.
.
.
...
...
..
..
.
.
.
.
.
.
.
.
....
...
..
...
.
.
....
.
.
.
.
.
....
..
...
...
.
.
.
.
.
.
....
....
..
...
.
.
.
.
...
.
.
....
..
..
....
.
.
.
.
.
.
.
...
....
..
..
.
.
.
.
.
.
.
.
...
...
..
..
.
.
.
....
.
.
.
.
.
...
..
...
...
.
.
.
.
.
.
.
....
....
..
...
.
.
.
.
....
.
.
.... ...
.
..
.
.
.......
.
.......
.
...
.
....
.
....
.. .....
...
.
.
.
.
.
.
...
...
..
..
.
.
....
.
.
.
.
.
.
...
...
...
...
....
....
....
....
...
....
...
...
....
...
....
....
...
....
...
...
.
.
.
.
.
.
.
.
.
...
...
..
...
....
....
....
....
....
....
....
....
...
...
...
...
....
....
....
....
.
...
.
.
.
.
.
.
....
....
...
...
...
....
...
...
....
...
....
....
....
....
...
...
...
...
....
....
.
.
.
.
.
....
.
.
.
...
.
.
....
....
...
...
... ......
.
....
.... ...
.
.
.
.
.
.
. ... ...
............... ................. .................. .................. .................. .................. .................. ................................
........
.......
.......
.......
.......
.......
....... Lecture notes
by
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
Cambridge University Mathematical Tripos. There are occasional references
to questions on four ’example sheets’, which can be found in the Appendix.
The writing of these course notes has greatly benefitted 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was based on unpublished notes of Gary Gibbons and owes much to the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´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. 2 Contents
1 Gravitational Collapse
1.1 The Chandrasekhar Limit . . . . . . . . . . . . . . . . . . . .
1.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Schwarzschild Black Hole
2.1 Test particles: geodesics and affine parameterization
2.2 Symmetries and Killing Vectors . . . . . . . . . . . .
2.3 Spherically-Symmetric Pressure Free Collapse . . . .
2.3.1 Black Holes and White Holes . . . . . . . . .
2.3.2 Kruskal-Szekeres Coordinates . . . . . . . . .
2.3.3 Eternal Black Holes . . . . . . . . . . . . . .
2.3.4 Time translation in the Kruskal Manifold . .
2.3.5 Null Hypersurfaces . . . . . . . . . . . . . . .
2.3.6 Killing Horizons . . . . . . . . . . . . . . . .
2.3.7 Rindler spacetime . . . . . . . . . . . . . . .
2.3.8 Surface Gravity and Hawking Temperature .
2.3.9 Tolman Law - Unruh Temperature . . . . . .
2.4 Carter-Penrose Diagrams . . . . . . . . . . . . . . .
2.4.1 Conformal Compactification . . . . . . . . . .
2.5 Asymptopia . . . . . . . . . . . . . . . . . . . . . . .
2.6 The Event Horizon . . . . . . . . . . . . . . . . . . .
2.7 Black Holes vs. Naked Singularities . . . . . . . . . .
3 Charged Black Holes
3.1 Reissner-Nordstr¨om . . . . . . . . . . .
3.2 Pressure-Free Collapse to RN . . . . . .
3.3 Cauchy Horizons . . . . . . . . . . . . .
3.4 Isotropic Coordinates for RN . . . . . .
3.4.1 Nature of Internal ∞ in Extreme
3 . . .
. . .
. . .
. . .
RN . .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 6
6
8 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 11
11
13
15
18
20
24
26
27
29
33
37
39
40
40
47
49
53 .
.
.
.
. 56
56
65
67
70
74
3.4.2 Multi Black Hole Solutions . . . . . . . . . . . . . . . 4 Rotating Black Holes
4.1 Uniqueness Theorems . . . . . . . . . .
4.1.1 Spacetime Symmetries . . . . . .
4.2 The Kerr Solution . . . . . . . . . . . .
4.2.1 Angular Velocity of the Horizon
4.3 The Ergosphere . . . . . . . . . . . . . .
4.4 The Penrose Process . . . . . . . . . . .
4.4.1 Limits to Energy Extraction . . .
4.4.2 Super-radiance . . . . . . . . . . 75 .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. 76
76
76
78
84
88
88
89
90 5 Energy and Angular Momentum
5.1 Covariant Formulation of Charge Integral . . . . . . . .
5.2 ADM energy . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Alternative Formula for ADM Energy . . . . . .
5.3 Komar Integrals . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Angular Momentum in Axisymmetric Spacetimes
5.4 Energy Conditions . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
. .
.
.
.
.
. .
.
.
.
.
. 93
93
94
96
97
98
99 .
.
.
.
.
.
. 101
. 101
. 106
. 109
. 109
. 110
. 112
. 113 .
.
.
.
. 119
. 119
. 123
. 125
. 129
. 130 .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
. 6 Black Hole Mechanics
6.1 Geodesic Congruences . . . . . . . . . . . . . . . . .
6.1.1 Expansion and Shear . . . . . . . . . . . . . .
6.2 The Laws of Black Hole Mechanics . . . . . . . . . .
6.2.1 Zeroth law . . . . . . . . . . . . . . . . . . .
6.2.2 Smarr’s Formula . . . . . . . . . . . . . . . .
6.2.3 First Law . . . . . . . . . . . . . . . . . . . .
6.2.4 The Second Law (Hawking’s Area Theorem)
7 Hawking Radiation
7.1 Quantization of the Free Scalar Field . . . . . . . .
7.2 Particle Production in Non-Stationary Spacetimes
7.3 Hawking Radiation . . . . . . . . . . . . . . . . . .
7.4 Black Holes and Thermodynamics . . . . . . . . .
7.4.1 The Information Problem . . . . . . . . . . .
.
.
.
. .
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
. 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,
E = Egrav + Ekin (1.1) is a minimum. For a star of mass M and radius R
GM 2
R
3
∼ nR hEi Egrav ∼ −
Ekin (1.2)
(1.3) where hEi is average kinetic energy of atoms. Eventually, fusion at the
core must stop, after which the star cools and contracts. Consider the
possible final state of a star at T = 0. The pressure P does not go to zero
as T → 0 because of degeneracy pressure. Since me ≪ mp the electrons
become degenerate first, at a number density of one electron in a cube of
side ∼ Compton wavelength.
n−1/3
∼
e ~
,
hpe i hpi = average electron momentum (1.4) Can electron degeneracy pressure support a star from collapse
at T = 0?
Assume that electrons are non-relativistic. Then
hEi ∼ hpe i2
.
me (1.5)
6 So, since n = ne ,
2/3 Ekin ∼ ~2 R2 re
.
me (1.6) Since me ≪ mp , M ≈ ne R3 me , so ne ∼
Ekin Thus M
and
mp R 3
~2 M 5/3 1
∼
.
me mp
R2
|
{z
}
constant for
fixed M E∼− E α
β
− 2,
R R (1.7) α, βindependent of R. ...
......
........
....
...
...
...
...
2
−1/3
...
...
...
...
...
...
...
...
...
5/3
...
...
...
...
e p
...
...
...
...
...
....
..
..
..
..
...
..
..
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
..
...
......................................................................................................................................................................................................................................................................................................................................
..
....
..
..................
.....
...............
.....
.............
......
...........
.......
..........
.
.
.
.
.........
.
.
.
.
.
............................... (1.8) ~M
Rmin ∼
Gm m Rmin 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
M
ne ∼
3
mp Rmin 2/3
me G
M m2p
2
~ 3 . (1.9) But the validity of non-relativistic approximation requires that hpe i ≪ me c,
i.e.
1/3 hpe i
~ne
=
≪c
me
me
m c 2
e
.
or ne ≪
~ (1.10)
(1.11)
7 For a White Dwarf this implies
2/3
me G
me c
M m2p
≪
2
~
~
3/2
~c
1
.
or M ≪ 2
mp G (1.12)
(1.13) For sufficiently large M the electrons would have to be relativistic, in
which case we must use
⇒ hEi = hpe i c = ~cn1/3
e
3 (1.14) ~cR3 n4/3
e
4/3 Ekin ∼ ne R hEi ∼
M
3
∼ ~cR
mp R 3 ∼ ~c M
mp 4/3 (1.15)
1
R (1.16) So now,
E∼− γ
α
+ .
R R (1.17) Equilibrium is possible only for
3/2
~c
1
γ=α ⇒ M ∼ 2
.
mp G (1.18) 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. For larger M , R must continue to decrease, so electron degeneracy
pressure cannot support the star. There is therefore a critical mass MC
3/2
3 1/2
1
~c
1
~
MC ∼ 2
⇒ RC ∼
(1.19)
mp G
me mp Gc
above which a star cannot end as a White Dwarf. This is the Chandrasekhar
limit. Detailed calculation gives MC ≃ 1.4M⊙ . 1.2 Neutron Stars The electron energies available in a White Dwarf are of the order of the Fermi
energy. Necessarily EF <
∼ me c2 since the electrons are otherwise relativistic
and cannot support the star. A White Dwarf is therefore stable against
inverse β-decay
e− + p+ → n + νe (1.20)
8 since the reaction needs energy of at least (∆mn )c2 where ∆mn is the
neutron-proton mass difference. Clearly ∆m > me (β-decay would otherwise be impossible) and in fact ∆m ∼ 3me . So we need energies of order of
3me c2 for inverse β-decay. This is not available in White Dwarf stars but for
M > MC the star must continue to contract until EF ∼ (∆mn )c2 . At this
point inverse β-decay can occur. The reaction cannot come to equilibrium
with the reverse reaction
n + νe → e− + p+ (1.21) because the neutrinos escape from the star, and β-decay,
n → e− + p+ ν¯e (1.22) cannot occur because all electron energy levels below E < (∆mn )c2 are
filled when E > (∆mn )c2 . Since inverse β-decay removes the electron degeneracy 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 collapse?
The ideal gas approximation would give same result as before but with
me → mp . The critical mass MC is independent of me and so is unaffected,
but the critical radius is now
3 1/2
~
GMC
1
me
∼
(1.23)
RC ∼ 2
mp
mp Gc
c2
which is the Schwarzschild radius, so the neglect of GR effects was not
justified. Also, at nuclear matter densities the ideal gas approximation is
not justified. A perfect fluid approximation is reasonable (since viscosity
can’t help). Assume that P (ρ) (ρ = density of fluid) satisfies
i) P ≥ 0 (local stability).
′ 2 ii) P < c (1.24) (causality). (1.25) Then the known behaviour of P (ρ) at low nuclear densities gives
Mmax ∼ 3M⊙ . (1.26) 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 gravitational collapse
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-approximated by assuming the collapsing material to be
a pressure-free ball of fluid. We shall consider this case shortly. 10 Chapter 2 Schwarzschild Black Hole
2.1 Test particles: geodesics and affine parameterization Let C be a timelike curve with endpoints A and B. The action for a particle
of mass m moving on C is
Z B
2
I = −mc
dτ
(2.1)
A where τ is proper time on C. Since
p
p
p
dτ = −ds2 = −dxµ dxν gµν = −x˙ µ x˙ ν gµν dλ where λ is an arbitrary parameter on C and x˙ µ =
Z λB
p
dλ −x˙ µ x˙ ν gµν (c = 1)
I [x] = −m dxµ
dλ , (2.2)
we have
(2.3) λA The particle worldline, C, will be such that δI/δx(λ) = 0. By definition,
this is a geodesic. For the purpose of finding geodesics, an equivalent action
is
Z
1 λB
dλ e−1 (λ)x˙ µ x˙ ν gµν − m2 e(λ)
(2.4)
I [x, e] =
2 λA
where e(λ) (the ‘einbein’) is a new independent function. 11 Proof of equivalence
δI
=0
δe ⇒ e= and (exercise)
δI
=0
δxµ ⇒ (for m 6= 0) 1 dτ
1p µ ν
−x˙ x˙ gµν =
m
m dλ D(λ) x˙ µ = (e−1 e)
˙ x˙ µ (2.6) (2.7) where
D(λ) V µ (λ) ≡ (2.5) d µ
V + x˙ ν
dλ µ
ρν Vρ If (2.5) is substituted into (2.6) we get the EL equation δI/δxµ = 0 of the
original action I[x] (exercise), hence equivalence.
The freedom in the choice of parameter λ is equivalent to the freedom in
the choice of function e. Thus any curve xµ (λ) for which tµ = x˙ µ (λ) satisfies
D(λ) tµ V µ = f (x)tµ (arbitrary f ) is a geodesic. Note that for any vector field on C, V µ (x(λ)),
µ
tν Dν V µ ≡ tν ∂ν V µ + tν
Vρ
νρ
µ
d µ
ν
V + x˙
Vρ
=
dλ
νρ
= D(λ) V µ (2.8) (2.9)
(2.10)
(2.11) Since t is tangent to the curve C, a vector field V on C for which
D(λ) = f (λ)V µ (arbitrary f ) (2.12) is said to be parallely transported along the curve. A geodesic is therefore a
curve whose tangent is parallely transported along it (w.r.t. the affine connection).
A natural choice of parameterization is one for which
D(λ) tµ = 0 (tµ = x˙ µ ) (2.13) This is called affine parameterization. For a timelike geodesic it corresponds
to e(λ) = constant, or
λ ∝ τ + constant (2.14)
12 The einbein form of the particle action has the advantage that we can
take the m → 0 limit to get the action for a massless particle. In this case δI
= 0 ⇒ ds2 = 0 (m = 0)
(2.15)
δe
while (2.6) is unchanged. We still have the freedom to choose e(λ) and the
choice e = constant is again called affine parameterization. Summary
Let tµ dxµ (λ)
and σ =
=
dλ Then 1 m 6= 0
0 m=0
. t · Dtµ ≡ D(λ) tµ = 0
ds2 = −σdλ2 (2.16) are the equations of affinely-parameterized timelike or null geodesics. 2.2 Symmetries and Killing Vectors Consider the transformation
xµ → xµ − αkµ (x), (e → e) (2.17) Then (Exercise)
I [x, e] → I [x, e] − α
2 Z λB λA dλ e−1 x˙ µ x˙ ν (£k g)µν + O α2 where
(£k g)µν (2.18) = kλ gµν,λ + kλ,µ gλν + kλ,ν gλµ (2.19) = 2D(µ kν) (2.20) (Exercise) Thus the action is invariant to first order if
£k g = 0 (2.21) A vector field kµ (x) with this property is a Killing vector field. k is associated with a symmetry of the particle action and hence with a conserved
charge. This charge is (Exercise)
Q = k µ pµ (2.22)
13 where pµ is the particle’s 4-momentum.
∂L
= e−1 x˙ ν gµν
∂ x˙ µ
dxν
= m
gµν when m 6= 0
dτ pµ = Exercise (2.23)
(2.24) Check that the Euler-Lagrange equations imply dQ
=0
dλ
Quantize, pµ → −i∂/∂xµ ≡ −i∂µ . Then
Q → −ikµ ∂µ (2.25) Thus the components of k can be viewed as the components of a differential
operator in the basis {∂µ }.
k ≡ kµ ∂µ (2.26) It is convenient to identify this operator with the vector field. Similarly for
all other vector fields, e.g. the tangent vector to a curve xµ (λ) with affine
parameter λ.
t = tµ ∂µ = d
dxµ
∂µ =
dλ
dλ (2.27) For any vector field, k, local coordinates can be found such that
k = ∂/∂ξ (2.28) where ξ is one of the coordinates. In such a coordinate system
£k gµν = ∂
gµν
∂ξ (2.29) So k is Killing if gµν is independent of ξ.
e.g. for Schwarzschild ∂t gµν = 0, so ∂/∂t is a Killing vector field. The
conserved quantity is
mkµ dt
dxν
gµν = mg00
= −mε (ε = energy/unit mass)
dτ
dτ 14 (2.30) 2.3 Spherically-Symmetric Pressure Free Collapse While it is impossible to say with complete confidence that a real star of mass
M ≫ 3M⊙ will collapse to a BH, it is easy to invent idealized, but physically
possible, stars that definitely do collapse to black holes. One such ‘star’ is
a spherically-symmetric ball of ‘dust’ (i.e. zero pressure fluid). Birkhoff ’s
theorem implies that the metric outside the star is the Schwarzschild metric.
Choose units for which
G = 1,
Then c = 1. (2.31)
2M
2M −1 2
dr + r 2 dΩ2
ds2 = − 1 −
dt2 + 1 −
r
r (2.32) dΩ2 = dθ 2 + sin2 θdϕ2 (2.33) where (metric on a unit 2-sphere) This is valid outside the star but also, by continuity of the metric, at the
surface. If r = R(t) on the surface we have
#
"
d
2M −1 ˙ 2
2M
2
2
2
2
˙
R = R (2.34)
R dt +R dΩ ,
− 1−
ds = − 1 −
R
R
dt
On the surface zero pressure and spherical symmetry implies that a point on
the surface follows a radial timelike geodesic, so dΩ2 = 0 and ds2 = −dτ 2 ,
so
"
#
2M
dt 2
2M −1 ˙ 2
1=
1−
R
(2.35)
− 1−
R
R
dτ
But also, since ∂/∂t is a Killing vector we have conservation of energy:
dt
2M dt
ε = −g00
= 1−
(energy/unit mass)
(2.36)
dτ
R
dτ
ε is constant on the geodesics. Using this in (2.35) gives
"
#
2M
2M −2 2
2M −1 ˙ 2
1=
1−
ε
1−
R
− 1−
R
R
R (2.37) or 1
R˙ 2 = 2
ε 2M
1−
R 2 2M
− 1 + ε2
R (ε < 1 for gravitationally bound particles).
15 (2.38) R˙ 2 ................. ..
...
..
...
...
... ..
...
...
...
...
...
...
...
...............................
...
...
......
......
...
...
.....
......
...
.....
.....
...
...
....
.....
.
...
.
...
.....
..
.
.
...
....
...
..
.
.....
..
.
.
...
.
....
.
...
.
...
....
..
....
...
...
.
....
..
...
...
.
....
.
...
.
.....
...
.
...
..
.....
.
...
.
...
....
..
.
....
.
...
...
.
...
....
.....
.
.
...
.
.
.
.
..
............................................................................................................................................................................................................................................................................................................................
....
...
....
..
....
....
....
.....
max
....
....
....
....
....
....
....
2
. •
2M R • 2M
=
1−ε R R˙ = 0 at R = Rmax so we consider collapse to begin with zero velocity at
this radius. R then decreases and approaches R = 2M asymptotically as
t → ∞. So an observer ‘sees’ the star contract at most to R = 2M but no
further.
However from the point of view of an observer on the surface of the star,
the relevant time variable is proper time along a radial geodesic, so use
−1
d
dt
d
d
1
2M
=
=
(2.39)
1−
dt
dτ
dτ
ε
R
dτ
to rewrite (2.38) as
dR
dτ dR
dτ 2 = 2M
− 1 + ε2
R 2 = (1 − ε )
Rmax
−1
R (2.40) 2 .
........
........
...
...
....
...
...
...
....
...
..
...
...
...
...
....
...
..
...
...
...
....
...
..
...
...
...
...
....
...
..
...
...
...
....
...
....
...
....
..
.....
.....
...
......
..
......
......
...
.......
....
...
..
.. ..............
........
.
...
........
....
.........
...
........
..
.........
...
.........
....
.........
....
..
.........
.
..
..........
..........
...
....
.........
..........
....
.....
.
.
.
.........................................................................................................................................................................................................................................................................................................................................................................................................................
...........
...........
.....
...........
...........
...........
max
...........
...........
...... 0 R 2M 16 R Surface of the star falls from R = Rmax through R = 2M in finite proper
time. In fact, it falls to R = 0 in proper time
τ= πM
(1 − ε)3/2 (Exercise) (2.41) Nothing special happens at R = 2M which suggests that we investigate the
spacetime near R = 2M in coordinates adapted to infalling observers. It is
convenient to choose massless particles.
On radial null geodesics in Schwarzschild spacetime
dt2 = 1
1− where dr
2M 2
r 2 ≡ (dr ∗ )2 (2.42) r − 2M r = r + 2M ln 2M ∗ (2.43...
View
Full Document
- Fall '17
