5. Gravitational Collapse and Black Holes

Question 1

Describe the properties of the interior of a star

Answer 1

The stress energy momentum tensor is not 0 (like it is for a vacuum)
- Approximated as a perfect fluid
- Made of fermions (electrons, protons, neutrons)

Question 2

What is hydrostatic equilibrium?

Answer 2

When the pressure gradient balances with the gravitational force of attraction

Question 3

How do stars generate their massive pressure to support it against the gravitational force?

Answer 3

Burn nucler fuel
- Gives a high temperature and a corresponding high pressure

Question 4

What state is an old star in at the end of its life?

Answer 4

In its quantum ground state

Question 5

What provides the pressure against the gravitational forces for old stars in their quantum ground state?

Answer 5

The fermions form a degenerate Fermi gas
- Gives a degenerate pressure which opposes gravity

Question 6

What defines a white dwarf?

Answer 6

When the degeneracy pressure is balanced against the gravitational pressure (equilibrium)

Question 7

What is a consequence of the equation of state for a white dwarf in relation to the mass of the star?

Answer 7

That more massive stars are more dense
- Fermions are more tightly confined
- Momenta from the uncertainty principle puts them into a relativistic regime

Question 8

What is the Chandrasekhar limit?

Answer 8

1.44 Mass of the sun

Question 9

What is a consequence of the Chandrasekhar limit?

Answer 9

Stars more massive than the limit have no known mechanism for supporting themselves against the gravity
- Stars more massive than the limit cannot end up as white dwarfs

Question 10

What is the mass limit for a neutron star?

Answer 10

2 -> 3 stellar masses
- Heavier stars than this collapse into a black hole

Question 11

State the two extra parameters in the metric for the Kerr black hole

Answer 11

• mass m = GM/c^2
• spin a = J/Mc

Question 12

State what happens to the Kerr metric if you set the values of a or m to 0

Answer 12

• a=0 you recover the Schwarz. solution
• m=0 you recover the Minkowski metric in bispherical coordinates

Question 13

State the symmetry properties of the Kerr metric

Answer 13

• Axissymmetric meaning it is constant under translations of the phi angle
• Not spherically symmetric
• Also stationary (constant under time translations), but not static (not time reversal)

Question 14

What are the necessary conditions for event horizon to exist, and when is there an extremal black hole?

Answer 14

Only exists when the spin a <= m
- Extremal when a=m

Question 15

What is the consequence of the Penrose conjecture and the properties of a black hole?

Answer 15

Penrose conjected that all black holes must have event horizons
- This means the spin is always less than or equal to the mass of the black hole implying a limit on the angular momentum of a BH

Question 16

Describe the structure of the ergosurface and ergosphere

Answer 16

Central blackhole enclosed in an oval like shape
- Oval is called the ergosurface
- Inside between the surface and BH is the ergosphere

Question 17

Describe the motion when inside the ergosphere

Answer 17

All observers including light must corotate with the black hole
- Impossible to counter-rotate or stand still

Question 18

Describe the areas of the masses of a black hole when there is no spin, and when the spin is at a maximum value

Answer 18

No spin: A = 16 pi m^2
a=n: A=8 pi m^2
- Faster spinning black holes are more compact objects

Question 19

State Stephen Hawking’s Area Theorem:

Answer 19

The total area of the event horizon does not decrease. dA >= 0

Question 20

What is the fundamental implication of the Area Theorem?

Answer 20

It is impossible to increase the spin of the black hole without adding mass
- Also places a limit on the amount of energy that can be radiated away from a BH in the accretion of mass/merger event

Question 21

What is the fundamental upper limit on the mass radiated away in the black hole merger?

Answer 21

0.29

Question 22

Describe the general process of accretion

Answer 22

Mass falls into the black hole
- Area increases and it spins faster

Question 23

Describe the analogy of the process of accretion and black hole thermodynamics

Answer 23

• Small changes in mass are now changes in energy
• Increasing the spin of the BH is like the work done
• The area of the event horizon is the BH entropy

Question 24

What is the second law of thermodynamic equivalent?

Answer 24

Area theorem is the second law

Question 25

Describe the Hawking Temperature

Answer 25

Particle emission from the black hole radiate like a black body

Question 26

What is the main problem with the observational evidence for black holes?

Answer 26

Receive no direct signals from the event horizon or within it
- Evidence is indirect

Question 27

State the main sources of evidence for BHs

Answer 27

Compacy X ray binaries, gravitational waves, AGN

Question 28

Describe the evidence of BHs from XRBs

Answer 28

Cosmic X-rays come from normal stars
- Spectra show periodic variation associated to orbital motion of a binary system
- If the mass is large enough, it is a BH

Question 29

Describe the evidence of BHs from AGN

Answer 29

Highly luminous (outshine galaxies) and compact source
- Narrow jets extending 10s of 1000s of light years
- Only plausible explanation is accretion onto a SMBH