Black Holes in General Relativity and Astrophysics Theoretical Physics Colloquium on Cosmology...
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Transcript of Black Holes in General Relativity and Astrophysics Theoretical Physics Colloquium on Cosmology...
Black Holes inGeneral Relativityand Astrophysics
Theoretical Physics Colloquium on Cosmology 2008/2009 Michiel Bouwhuis
Content
Part 1: Introduction to Black Holes
Part 2: Stellar Collapse and Black Hole Formation
2Black Holes in General Relativity and Astrophysics
Black Holes in General Relativity and Astrophysics
Part 1:Introduction to Black Holes
3Black Holes in General Relativity and Astrophysics
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
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
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
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
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
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
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
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
Radial Light Rays
12Black Holes in General Relativity and Astrophysics
Incomming lightrays always move inwards.
But for r < 2M ‘outgoing’ lightraysalso move inwards!
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
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
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!
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
The Kerr Black Hole
17Black Holes in General Relativity and Astrophysics
Singularity
Inner event horizon
Outer event horizon
Killing horizon
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
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
Black Holes in General Relativity and Astrophysics
Part 2:Stellar
Collapse and Black Hole Formation
20Black Holes in General Relativity and Astrophysics
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
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
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
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
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.
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
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
Equations of Structure
28Black Holes in General Relativity and Astrophysics
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
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
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)
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)
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.
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
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
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
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
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
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
Do Black Holes Exist?
39Black Holes in General Relativity and Astrophysics