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