Black Hole (Part I)
Kinwah Wu
Mullard Space Science Laboratory
University College London
kinwah.wu@ucl.ac.uk
Black hole as known by radio observers
Image credit: (left) NRAO; (right) HST/STScI
Black hole according to some scientists
Image credit: NASA/JPL-Caltech
Black hole – the Hollywood version
Image credit: Interstellar
Black hole that I see (on wednesdays)
Image credit: London Underground
Black holes that I found on Youtube
Image credit: (left) MUSE;
(right) Sony Music
What does wikipedia say about “black
hole”?
“A black hole is a geometrically defined region of spacetime exhibiting
such strong gravitational effects that nothing including particles and
electromagnetic radiation such as light can escape from inside it.”
Content
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Astrophysical black holes
What is a black hole – the “standard” view?
Space-time around a black hole
Imaging black holes
Black-hole thermodynamics (very basic)
Spinning objects around a black hole
Black holes, white holes and worm holes
Kruskal construct, Penrose diagram … etc
More on black hole space-time and its stability
Physical system analogs of black holes
Hairy black holes and related issues
Quantum states of black holes
Holography and information paradox
1. Astrophysical black holes
Astrophysical black holes
Stellar mass black holes as end point of the evolution of very massive stars
Image credit: Wikipedia
Astrophysical black holes
Stellar mass black holes as end point of the evolution of very massive stars
Image credit: unknown
Astrophysical black holes
Stellar BH in X-ray binaries, e.g. GRS1915+105
Image credit: (left) NRAO, F. Mirabel &
L. Rodrigueez; (right) J. Blondin
Astrophysical black holes
Sgr A*:
A four-million solar-mass
black hole residing at the
Galactic Centre
(Ghez etal. 2008; Meyer et al. 2012)
Astrophysical black holes
Supermassive black holes that power quasars and AGN, e.g. in M87
Image credit: NASA & the Hubble Heritage Team (STScI/AURA)
Astrophysical black holes
Power produced by accretion of matter into a black hole
GMbh
E =
m
R
dE
GMbh
2
=
ṁc
=
dt
c2 R
dE
L ⇡
dt
ṁ = 10
8
M yr
ṁ = 1 M yr
1
⇡
1.3 ⇥ 10
⇠
L
1
⇠
L
Eddington luminosity
LEdd
⌘ ṁc2
38
✓
M
M
◆
⌘
6 ⇥ 1037 erg s
6 ⇥ 1045 erg s
erg s
1
⇠
1
1
0.1
Astrophysical black holes
Ultra-luminous X-ray sources
(ULX) as intermediate-mass
black hole-candidates, e.g.
ULX-1 in the M101 galaxy
Image credit: CXO/Spitzer/HST/GALEX
Astrophysical black holes
Black-hole systems as generators cosmic rays
Image credit: (left) IceCube Collaboration;
(right) I. Jacobsen
Astrophysical black holes
M-σ relation: indication of co-evolution of black holes and their hosts
(Gueltekin et al. 2009)
Astrophysical black holes
Black holes and cosmological reionisation
quasars
stars
(Thomas et al. 2009)
Astrophysical black holes
Neutron star – neutron star mergers and black hole – black hole mergers as
gravitational wave sources
Image credit: C. Reisswig & L. Rozella
2. What is a black hole?
What is a black hole?
Escape velocity
K
=
K
=
v2
GM
2
r
(M/R)
0 ) vesc = 617.5
km s
(M/R)
Earth
11.2 km s
Moon
2.4 km s
⇡ (500
Milky Way
1
1
1
600) km s
1
Event horizon: location where escape velocity is equal to the speed of light
!
vesc
rs
=
c = 3.0 ⇥ 105 km s
2GM
c2
1
Schwarzschild radius
What is a black hole?
Black hole – the basic idea (according to Laplace)
rs
rs
2GM
=
c2
> Robj
event horizon:
a critical surface within
which light is trapped
Robj
rs
In a Laplacian black hole light can cross the event horizon from within but it is just
that the light from the region within is not allowed to escape to infinity
What is a black hole?
A “no-frills” black hole – the Schwarzschild black hole
o  spherical
o  a point singularity at the centre
o  an event horizon at the Schwarzschild radius
singularity
event horizon
In a relativistic black hole light cannot cross the event horizon from within and it is
trapped inside forever
What is a black hole?
No-hair theorem/conjecture:
we need only three classical parameters (mass, angular momentum and
electric charge) to specify a stationary black hole
Four kinds of black holes
non-rotating (J = 0 )
rotating (J > 0 )
uncharged (Q = 0)
Schwarzschild
Kerr
charged (Q > 0)
Reissner-Nordström
Kerr-Newman
•  Only Kerr black holes and Schwarzschild black holes are relevant in
astrophysics
•  If magnetic charges (monopoles) exit, there will more than 4 kinds of
black holes
What is a black hole?
Kerr black hole
o  not appear to be spherical
o  a ring singularity at the centre
o  two event horizons
ring singularity
static limit
inner event horizon
outer event horizon
ergosphere
What is a black hole?
Black hole, grey hole or pink hole?
rs
⇡
3 ⇥ 105
✓
M
M
◆
M = 1.989 ⇥ 1033 g
cm
M = 5.972 ⇥ 1027 g ⇡ 3 ⇥ 10
6
M
Collapsing an Earth-mass object makes a black hole with a Schwarzschild radius
rs
⇡
0.9 cm
Electromagnetic waves
= 12.2 cm
⇡ 6 cm
⇡ 5 ⇥ 10 5 cm
in a microwave oven
5 GHz radiation from a quasar
peak of the solar spectrum
Q: Can an Earth-mass black hole absorb all these light?
3. Space-time around a black hole
Space-time around a black hole
Space-time interval
ds2
=
gµ⌫ dxµ dx⌫
Schwarzschild metric
✓
◆
✓
◆
1
2M
2M
ds2 =
1
dt2 + 1
dr2 + r2 d⌦
r
r
d⌦ = d✓2 + sin2 ✓d 2
c = G = 1 (geometrical units)
singularity
r=0
event horizon
r = rs = 2M
Schwarzschild radius
Space-time around a black hole
Event horizon as a “frozen surface”
“observer” time
proper time
event horizon
R
r = (1 + cos ⌘)
2r
R
R
⌧=
(⌘ + sin ⌘)
2 2M
✓
◆
R
R
t=
+ 2M
⌘+
sin ⌘
2
2
+ tan(⌘/2)
+2M ln
tan(⌘/2)
r
R
=
1
2M
⌘ 2 [ 0, ⇡ ]
Space-time around a black hole
Scalar curvature
R
µ⌫↵
Rµ⌫↵
Rµ ⌫↵
µ
↵
48M 2
=
r6
= @↵ µ ⌫
@ µ ⌫↵ +
1 µ⌫
=
g (@↵ g⌫ + @ g⌫↵
2
µ
⇢↵
⇢
⌫
µ
⇢
⇢
⌫↵
@⌫ g↵ )
Event horizon is not a true singularity
Rµ⌫↵ Rµ⌫↵
=
0
! 0
Rµ⌫↵ Rµ⌫↵
!
=
Rµ⌫↵ Rµ⌫↵
=
1
3
4M 4
r
!
r
r
1
2M
Space-time at the event horizon of a very massive black hole is quite flat
Space-time around a black hole
Tortoise coordinate
r
r⇤ = r + 2M ln
2M
✓
◆ 1
dr⇤
2M
=
1
dr
r
1
r⇤ 2 ( 1, 1)
1
event horizon
r 2 [ 0, 1)
r⇤
r
0
2M
event horizon
Space-time around a black hole
Ingoing Eddington-Finkelstein coordinates
v
=
ds
2
=
t + r⇤
✓
1
2M
r
◆
dv 2 + 2dvdr + r2 d⌦
Outgoing Eddington-Finkelstein coordinates
u
ds
=
2
t
=
r⇤
✓
1
Finkelstein diagram
t̃
=
2M
r
◆
du2
(t̃, r)
r
t + 2M ln
2M
1
2dudr + r2 d⌦
Space-time around a black hole
Eddington-Finkelstein coordinate
a black hole
a stellar collapse forming a black hole
Image credit: L. Gualtieri & V. Fettari
Space-time around a black hole
Escape of photons from a
collapsing star
(Thorne 1967)
Space-time around a black hole
Kerr metric (in the Boyer-Lindquist coordinates)
ds
2
=
In the limit of
ds
2
=


2M r
4M ra sin2 ✓
2
1
dt
dt d
r2 + a2 cos2 ✓
r2 + a2 cos2 ✓
 2
⇥ 2
⇤ 2
r + a2 cos2 ✓
2
2
2
+ 2
dr + r + a cos ✓ d✓
2
r
2M r + a

2M ra2 sin2 ✓
2
2
2
2
+ r +a + 2
sin
✓
d
r + a2 cos2 ✓
a !0
✓
1
2M
r
◆
dr2
2
dt +
1
Schwarzschild metric
2M
r
+ r2 d✓2 + sin2 ✓ d
2
Space-time around a black hole
Kerr metric (in the Boyer-Lindquist coordinates)
ds
2
=


2M r
4M ra sin2 ✓
2
1
dt
dt d
r2 + a2 cos2 ✓
r2 + a2 cos2 ✓
 2
⇥ 2
⇤ 2
r + a2 cos2 ✓
2
2
2
+ 2
dr + r + a cos ✓ d✓
2
r
2M r + a

2M ra2 sin2 ✓
2
2
2
2
+ r +a + 2
sin
✓
d
r + a2 cos2 ✓
Asymptotic at the infinity
ds2
=

2M
1
+O r
r

2M
+ 1+
+O r
r
3
2

dt2
⇥
2
4M a sin2 ✓
+O r
r
2
2
2
dr + r (d✓ + sin ✓ d
2
3
)
⇤
dt d
4. Imaging black holes
Imaging black holes
Geodesics equation
d2 xµ
+
2
d⌧
µ
↵
d⌧ 2
dx⌫ dx
= 0
d⌧ d⌧
1 µ⌫
=
g (@↵ g⌫ + @ g⌫↵ @⌫ g↵ )
2
=
ds2 =
gµ⌫ dxµ dx⌫
µ
⌫
Lagrangian
L
=
1
gµ⌫ ẋµ ẋ⌫
2
ẋ
Euler-Lagrange equation
d @L
d⌧ @ ẋµ
@L
@xµ
=
0
µ
=
dxµ
d⌧
Imaging black holes
Lagrangian of particles
gµ⌫ ẋµ ẋ⌫
=
gµ⌫ ẋµ ẋ⌫
=
0
(massless particle)
1
(massive particle)
Equation of motion for particles around a Schwarzschild black hole
⇡
✓ =
2
˙ = L
r2
✓
ṫ = E 1
ṙ2
=
E2
◆
⌥
2M
⌥
r
✓
◆✓ 2
◆
2M
L
1
+⌥
r
r
1
=
=
0
1
(massless particle)
(massive particle)
Imaging black holes
Different photon orbits around a black hole
Schwarzschild black hole
Fast spinning Kerr black hole
Image credit: Z. Younsi
Imaging black holes
Photon orbits around a maximally spinning black hole (a 3D view)
Image credit: Z. Younsi
Imaging black holes
Black-hole shadow
movie credit: Z. Younsi
Imaging black holes
Torus around a black hole (frequency shift and intensity suppression/boost)
movie credit: Z. Younsi
Imaging black holes
Emission from optically thick and optically thin Tori around a black hole
movie credit: Z. Younsi
Imaging black holes
Lensing and frame dragging
back
front
( Younsi & Wu 2015)
Imaging black holes
Lensing and frame dragging
back
front
( Younsi & Wu 2015)
Imaging black holes
Lensing and frame dragging
movie credit: Z. Younsi
Imaging black holes
Lensing and frame dragging
movie credit: Z. Younsi
5. Black hole thermodynamics
Black hole thermodynamics
Laws of thermodynamics
•  Zeroth law of thermodynamics
The temperature T is the same everywhere in a system in a
thermal equilibrium
•  First law of thermodynamics
dE
=
T dS + ⌦ dJ + V dQ
•  Second law of thermodynamics
For adiabatic processes dS
>
0
•  Third law of thermodynamics
“ T = 0 “ cannot be attained via finite number of operations *
* Nernst postulate
Black hole thermodynamics
Laws of black-hole thermodynamics
•  Zeroth law of black-hole thermodynamics
The surface gravity K is the same everywhere on the event horizon
of a black hole in a steady state
•  First law of black-hole thermodynamics
dM
=
K
dA + ⌦ dJ + V dQ
8⇡
•  Second law of black-hole thermodynamics
dA >
0
generalised 2nd law
d(S + A)
>
0
•  Third law of black-hole thermodynamics
“ K = 0 “ cannot be attained with finite physical processes
Black hole thermodynamics
Area of black-hole event horizon
dA
=
d✓d (g✓✓ g
)
1/2
For a Kerr black hole
g✓✓
r2 + a2 cos2 ✓
=
2 2
r +a
sin2 ✓
rI2 + a2 cos2 ✓
2
g
=
A(r+ )
=
dA
r=r+
A
2
4⇡ r+
+ a2
=
r+
=
rg
=
rg2
rg +
a
rs
= M
2
=
h
8⇡rg rg + (rg2
2 1/2
a2 )1/2
i
radius of event horizon of a Kerr black hole
gravitational radius
Black hole thermodynamics
Area of black-hole event horizon
A
=
4⇡
2
r+
2
4⇡r+
A 6=
Irreducible mass
Mirr
A
+a
=
=
2
=
h
8⇡rg rg +
(rg2
a )
because space-time is curved
0
"
M
p @1 + 1
2
16⇡Mirr 2
a
61
rg
) Mirr 6 M
dA > 0
) dMirr > 0
2 1/2
i
(Kerr black hole)
11/2
#
1/2
✓ ◆2
a
A
rg
Schwarzschild black holes have “larger” surface
for their event horizon than Kerr black holes
Area increase theorem
Black hole thermodynamics
Merging of two Schawarzschild black holes
A1
=
16⇡M1 2
A2
=
16⇡M2 2
Atotal
=
16⇡Mtotal 2
Mtotal
=
M1 + M2
Atotal
>
A1 + A2
Total event horizon area increases
as a result of the merging process
Image credit: R. Hamerly
Black hole thermodynamics
Merging of two Schawarzschild black holes with an extreme mass ratio
(Hamerly & Chen 2011)
Black hole thermodynamics
Black hole’s first law of thermodynamics
A
=
2
4⇡ r+
+ a2
area of event horizon (general case)
1/2
1/2
rg + rg2 a2 Q2
= M + M 2 a2 Q2
J
1
a =
dr+ =
(r+ dM a da Q dQ)
M
r+ M
K
dM =
dA + ⌦ dJ + V dQ
8⇡
1
2
2
2 1/2
curvature at the event horizon
K =
M
a
Q
2
2
r+ + a
a
⌦ =
2 + a2
r+
r+ Q
V =
2 + a2
r+
r+
=
Black hole thermodynamics
Black hole entropy and temperature
K
dM =
dA + ⌦ dJ + V dQ
8⇡
✓ ◆
kB A
“Bekenstein-Hawking formula”
Sbh =
2
4 lp
✓
◆1/2
G~
Planck length
lp =
3
c
X
S = kB ln ⇠ =
kB
pi ln pi
i
1
2
2
2 1/2
K =
M
a
Q
2 + a2
r+
2
1/2
M 2 a2 Q2
~ 4
⇣
⌘
kB Tbh =
1/2
2⇡ 2M M + [M 2 a2 Q2 ]
Q2
3
5
Black hole thermodynamics
Hawking radiation 2
kB Tbh
=
1/2
M 2 a2 Q2
~ 4
⇣
⌘
2⇡ 2M M + [M 2 a2 Q2 ]1/2
Q2
3
5
•  This Hawking temperature is the temperature perceived by a distant
stationary observer
•  Hawking radiation must be thermal; otherwise, causality is violated
For Schwarzschild black holes
✓
◆
1
1
c3 ~
kB Tbh =
8⇡ M
G
✓
◆
1
=
[c ~] ⇡ 15.9 rs
max
kB Tbh
✓
◆ 6
1
1
c ~
4
P = A Tbh =
15360⇡ M 2
G2
Stefan-Boltzmann law
Black hole thermodynamics
Black hole evaporation
✓
dM
1
1
=
P =
dt
15360⇡ M 2
Z tev
Z 0
dt =
15360⇡
dM M 2
0

◆
M0
2
G
tev = 5120⇡M0
⇡ 8.4 ⇥ 10
4
c ~
✓ ◆1/3
tev
8
M0 ⇡ 2.3 ⇥ 10
g
s

~G
tev = 5120⇡
Planck mass black hole
c5
3
Solar mass black hole
tev ⇡ 6.6 ⇥ 1074 s
26
✓
M
1g
◆3
s
1/2
⇡ 8.7 ⇥ 10
40
s
Black hole thermodynamics
Black holes not in dynamical and thermal equilibrium
tidal interaction with a companion
a vibrating black hole
The event horizon has non-uniform curvatures, i.e. thermally nonequilibrium
1
Normal mode virbrational frequency (Schwarzschild) f = (2⇡M )
tdyn ⇠ ([G]⇢)
1/2
image credit: K. Thorne
) f ⇠ ⇢1/2 ⇠ (M M
3 1/2
)
=M
1
Black hole thermodynamics
Stability of Schwarzschild space-time (“black hole perturbation”)
image credit: K. Thorne
r
r⇤ = r + 2M ln
2M
Regge and Wheeler equation (odd partity)
⇥
@t
V

2
2
(@r⇤
✓
=
1
d2
2
+
!
dr⇤2
V)
⇤
2M
r
◆
V (r⇤ )
=
0
l(l + 1) (1
+
2
r
=
0
spin of the field (e.g. 1 for
photon, 2 for graviton ….)
s2 )2M
r3
Schrödinger-like equation
1
Black hole (Parts II and III): TBC/TBD ……
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•
•
•
•
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•
•
•
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•
•
•
Astrophysical black holes
What is a black hole – the “standard” view?
Space-time around a black hole
Imaging black holes
Black-hole thermodynamics (very basic)
Spinning objects around a black hole
Black holes, white holes and worm holes
Kruskal construct, Penrose diagram … etc
More on black hole space-time and its stability
Physical system analogs of black holes
Hairy black holes and related issues
Quantum states of black holes
Holography and information paradox
Preview:
spinning objects around a black hole