black holes in general relativity and astrophysics theoretical physics colloquium on cosmology...
TRANSCRIPT
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Black Holes inGeneral Relativityand Astrophysics
Theoretical Physics Colloquium on Cosmology 2008/2009 Michiel Bouwhuis
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Content
Part 1: Introduction to Black Holes
Part 2: Stellar Collapse and Black Hole Formation
2Black Holes in General Relativity and Astrophysics
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Black Holes in General Relativity and Astrophysics
Part 1:Introduction to Black Holes
3Black Holes in General Relativity and Astrophysics
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Introduction to Black Holes
Outline
- The Schwarzschild Solution for a stationary,
non-rotating black hole
- Properties of Schwarzschild black holes
- Adding rotation: The Kerr metric
- Properties of Kerr black holes
- Adding charge: The Kerr-Newman metric
4Black Holes in General Relativity and Astrophysics
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Vacuum Einstein Field Equations
The Schwarzschild Metric
5Black Holes in General Relativity and Astrophysics
18 0
2R Rg GT R
Spherically symmetric solution1
2 2 2 2 22 21 1
GM GMds dt dr r d
r r
Describes space outside any static, spherically symmetric
mass distribution
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The Schwarzschild Metric
6Black Holes in General Relativity and Astrophysics
12 2 2 2 22 2
1 1GM GM
ds dt dr r dr r
- The parameter M can be identified with mass, as can be seen by
taking the weak field limit:
- By Birkhoff’s Theorem, the Schwarzschild solution is the
unique solution
- Taking M = 0 or r → ∞ recovers Minkowski space
00 1 2
1 2rr
g
g
GM
r
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The Schwarzschild Metric
7Black Holes in General Relativity and Astrophysics
12 2 2 2 22 2
1 1GM GM
ds dt dr r dr r
The metric becomes singular at r = 0 and r = 2GM
• r = 0 : True singularity of infinite space-time curvature
• r = 2GM : Singular only because of choice of coordinate system
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Motion of test particles
8Black Holes in General Relativity and Astrophysics
Solving the geodesic equations and using
symmetry and conservation laws we get:
22
2 2
2 3
1 1( )
2 2
1( )
2 2
drV r E
d
GM L GMLV r
r r r
This gives circular orbits at radius rc if2 2 23 0c cGMr L r GML
For massless particles (ε = 0) this gives
For massive particles (ε = 1) we have
3cr GM
2 4 2 2 212
2c
L L G M Lr
GM
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Event Horizon
9Black Holes in General Relativity and Astrophysics
If r < 2GM then dt2 and dr2
change sign!
All timelike curves will
point in the direction of
decreasing r
12 2 2 2 22 2
1 1GM GM
ds dt dr r dr r
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Eddington-Finkelstein Coordinates
10Black Holes in General Relativity and Astrophysics
Coordinate transform:
2 2 2 221 2
Mds dv dvdr r d
r
This gives the Eddington-Finkelstein Coordinates:
2 log 12
rt v r M
M
Nonsingular at r = 2M
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Radial Light Rays
11Black Holes in General Relativity and Astrophysics
For radial light rays we have ds2 = 0 and dθ = dφ = 0
2 2 2 221 2
Mds dv dvdr r d
r
22 1 2 0
Mdv dvdr
r
1st solution: (incomming light rays)
2nd solution:
constv
21 2 0
2 2 log 1 const2
Mdv dr
r
rv r M
M
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Radial Light Rays
12Black Holes in General Relativity and Astrophysics
Incomming lightrays always move inwards.
But for r < 2M ‘outgoing’ lightraysalso move inwards!
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Most general stationary solution to the Vacuum Einstein Field Equations
The Kerr Black Hole
13Black Holes in General Relativity and Astrophysics
2 22 2 2 2 2
2 2
2 22 2 2 2
2
2 4 sin1
2 sin sin
Mr Mards dt d dt dr d
Mrar a d
This describes space outside a stationary, rotating,
spherically symmetric mass distribution
Where: 2 2 2 2 2 2, cos , 2J
a r a r Mr aM
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The Kerr Black Hole
14Black Holes in General Relativity and Astrophysics
2 22 2 2 2 2
2 2
2 22 2 2 2
2
2 4 sin1
2 sin sin
Mr Mards dt d dt dr d
Mrar a d
Singularity at ρ = 0. This implies both r = 0 and θ = π / 2
Event Horizon at Δ = 0
Located at
The t coordinate becomes spacelike when
2 22 2 4
2
M M ar
2
21
Mr
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Inner and outer Event Horizon
15Black Holes in General Relativity and Astrophysics
Two solutions for
2 22 2 4
2
M M ar
2 22 4 0M a
An inner and an outer event horizon!
No solutions for 2 22 4 0M a
No event horizon at all, but a naked singularity!
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The Ergosphere
16Black Holes in General Relativity and Astrophysics
We have re > r+. The ergosphere lies outside the event horizon
2 2 2
2
2 2 4 cos21
2e
M M aMrr
Within the ergosphere timelike curves must move in the direction
of you increasing θ
Known as Lense-Thirring effect, or Frame-Dragging
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The Kerr Black Hole
17Black Holes in General Relativity and Astrophysics
Singularity
Inner event horizon
Outer event horizon
Killing horizon
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Charged Black Holes
18Black Holes in General Relativity and Astrophysics
Reissner-Nordström metric
Kerr-Newman metric
2 2 1 2 2 2
2 2
2
21
ds dt dr r d
M p q
r r
Kerr Metric with 2Mr replaced by 2Mr – (p2 + q2).No new phenomena
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Types of Black Holes
19Black Holes in General Relativity and Astrophysics
Supermassive BH
Intermediate-mass BH
Stellar-mass BH
Micro BH
5 10~ 10 10M M
3~10M M
1.5 20M M
moonM M
- Found in centres of most Galaxies- Responsible for Active Galactic Nuclei- Might be formed directly and indirectly
- Possibly found in dense stellar clusters- Possible explanation of Ultra-luminous X-Rays- Must be formed indirectly
- Remants of very heavy stars- Responsible for Gamma Ray Bursts- Formed directly
- Quantum effects become relevant- Predicted by some inflationary models- Possibly created in Cosmic Rays- Will cause LHC to destroy the Earth
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Black Holes in General Relativity and Astrophysics
Part 2:Stellar
Collapse and Black Hole Formation
20Black Holes in General Relativity and Astrophysics
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Stellar Collapse and Black Hole Formation
Outline
- Collapse of Dust (Non-Interaction Matter)
- White Dwarfs
- Neutron Stars
- Do Black Holes exist?
21Black Holes in General Relativity and Astrophysics
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Collapse of Dust
22Black Holes in General Relativity and Astrophysics
All particles follow radial timelike geodesics
Dust: Pressureless relativistic matter
A little bit of math:
2 2
21
sin
M dte u
r d
dl u r
d
First normalize four-velocity 1u u g u u
From the Killing vectors we get:
This gives: 1
2 2 222 21 1 1t rM M
u u r ur r
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Collapse of Dust
23Black Holes in General Relativity and Astrophysics
A little bit of math:
1 2 22
2
2 2 1 1 1
M M dr le
r r d r
Radial timelike geodesics initially at rest: e =1, l = 0
21
02
dr M
d r
22 2
2
1 1 1 2 1 1 1
2 2 2
e dr M l
d r r
1/ 21/ 2 2r dr M d
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Collapse of Dust
24Black Holes in General Relativity and Astrophysics
Integration yields:
For the Schwarzschild time we find:
2/3 1/3 2 /3( ) 3 / 2 2r M
2
1/ 2 11
02 22
12
1
dr Mdt M Md rdr r rM dt
er d
Here integration gives:
1/ 23/ 2 1/ 2
1/ 2
/ 2 122 2 log
3 2 2 / 2 1
r Mr rt t M
M M r M
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Collapse of Dust
25Black Holes in General Relativity and Astrophysics
The surface of a collapsing star reaches the event horizon at r = 2M in a finite amount of proper time, but an infinite Schwarzschild time will have passed
Signals from the surface will become infinitely redshifted.
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Realistic Matter
26Black Holes in General Relativity and Astrophysics
Assumptions:
- Non-rotating, spherically symmetric star
- Interior is a perfect fluid
- Known equation of state
- Static
2 ( ) 2 ( ) 2 2 2v r rds e dt e dr r d
/ 2 ,0vu e
( )p p
( )T p u u g p
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Realistic Matter
27Black Holes in General Relativity and Astrophysics
We need to solve the Einstein equations
18
2G R g R T
Four unknown functions - v(r)- λ(r)- p(r)- ρ(r)
It is costumary to replace:( ) 2 ( )
1r m re
r
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Equations of Structure
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2
3
2
3
2
( )4 ( )
( ) ( ) 4 ( )( ) ( )
1 2 ( ) /
1 ( ) 1 ( ) ( ) 4 ( )
2 ( ) ( ) 1 2 ( ) /
dm rr r
dr
dp r m r r p rr p r
dr r m r r
dv r dp r m r r p r
dr r p r dr r m r r
Equations describing relativistic hydrostatic equilibrium
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Gravitational Collapse
29Black Holes in General Relativity and Astrophysics
- Unchecked gravity causes stars to collapse
- Ordinary stars are balanced against this by the pressure due to thermonuclear reactions in the core
- Once a star runs out of fuel, this process can no longer support it, and it starts to collapse
- White dwarfs are balanced by the pressure of the Pauli Exclusion Principle for electrons
- Neutron stars are balanced by the pressure of the Pauli Exclusion Principle for neutrons
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White Dwarfs (or Dwarves)
30Black Holes in General Relativity and Astrophysics
Single fermion in a box2
2k
k
pE
m
For N fermions we have3 3
22 30
12 4
8 3
Fp FpLN p dp n
The energy density is given by
32
0
1/ 22 4 2 2
12 4 ( )
8
( )
FpLp E p dp
E p m c p c
22/32 2 5/3
1/32 4/3
33
10
33
4
mc n nm
c n
(nonrelativistic)
(relativistic)
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White Dwarfs
31Black Holes in General Relativity and Astrophysics
To find the pressure, use
dE pdV
22/32 5/3
1/32 4 /3
13
5
13
4
p nm
p c n
(nonrelativistic)
(relativistic)
where andE V /V N n
This gives dp n
dn
Giving us for the pressure
We now have both density and pressure in terms of n.Eliminate n to find equation of state p = p(ρ)
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White Dwarfs
32Black Holes in General Relativity and Astrophysics
Now all that’s left to do is solving some integrals!
Easiest to do numerically: Pick a core density ρc and integrate outward.
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White Dwarfs
33Black Holes in General Relativity and Astrophysics
Plotting R as a function of M we find
White Dwarfs have a maximum mass! - Chandrasekhar mass 1.4M M
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Neutron Stars
34Black Holes in General Relativity and Astrophysics
- As a White Dwarf compresses further the electrons gain more and more energy
- At electrons and protons combine to form neutrons
- As collapse continues the neutrons become unbound and form a neutron fluid
- Density becomes comparable or even greater than nuclear density. Strong interaction dominant source of pressure
- Upperbound on mass of about 2M○ based on theoretical models of the equation of state
2 2 1.3MeVe n pE m c m c
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Neutron Stars
35Black Holes in General Relativity and Astrophysics
Goal: Upperbound on mass based on GR alone
Assumptions:
- Equation of State satisfies
- Equation of State known up to density
0
0
/ 0
p
dp d
14 30 2.9 10 g / cm
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Neutron Stars
36Black Holes in General Relativity and Astrophysics
Goal: Upperbound on mass based on GR alone
Recall 3
2
( ) ( ) 4 ( )( ) ( )
1 2 ( ) /
dp r m r r p rr p r
dr r m r r
This implies ( ) ( )0 0
dp r d r
dr dr
We have a core with r < r0 and ρ > ρ0 and unknown equation of stateand a mantle with r > r0 and ρ > ρ0 where the equation of state is known
For the mass of the core we have
0 02 20 0 00 0
( ) 4 ( ) 4r r
M m r dr r r dr r
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Neutron Stars
37Black Holes in General Relativity and Astrophysics
Goal: Upperbound on mass based on GR alone
So we have for the core mass0 2 3
0 0 0 00
44
3
rM dr r r
But core can’t be in its own Schwarzschild radius 0 02M r
So
1/ 2
00
1 38.0
2 8M M
Any heavier compact object MUST be a Black Hole
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Do Black Holes Exist?
38Black Holes in General Relativity and Astrophysics
Name BHC Mass(solar masses)
Companion Mass (solar masses)
Orbital period (days)
Distance from Earth (103 ly)
A0620-00 9−13 2.6−2.8 0.33 ~3.5
GRO J1655-40 6−6.5 2.6−2.8 2.8 5−10
XTE J1118+480 6.4−7.2 6−6.5 0.17 6.2
Cyg X-1 7−13 ≥18 5.6 6−8
GRO J0422+32 3−5 1.1 0.21 ~ 8.5
GS 2000+25 7−8 4.9−5.1 0.35 ~ 8.8
V404 Cyg 10−14 6.0 6.5 ~ 10
GX 339-4 5−6 1.75 ~ 15
GRS 1124-683 6.5−8.2 0.43 ~ 17
XTE J1550-564 10−11 6.0−7.5 1.5 ~ 17
XTE J1819-254 10−18 ~3 2.8 < 25
4U 1543-475 8−10 0.25 1.1 ~ 24
GRS 1915+105 >14 ~1 33.5 ~ 40
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Do Black Holes Exist?
39Black Holes in General Relativity and Astrophysics