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KERR BLACK HOLES

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KERR BLACK HOLES

- Generalized BH description includes spin
- Later researchers use it to predict new effects!!

- Two crucial surfaces
- inner surface = horizon: smaller than nonrot. hole of same mass
- outer surface = “static limit”

- Angular momentum of a black hole
- “Kerr parameter”

No spin: Schwarzschild hole

maximal (or extreme) Kerr hole

STRUCTURE OF A KERR BH

- Horizon smaller in max. Kerr limit
- Static limit at equator, touches horizon at poles

Gas can orbit closer to horizon

without plunging in

Higher energy efficiency:

Schwarzschild:

Extreme Kerr:

THE ERGOSPHERE

- Region between horizon and static limit
- Nothing can remain stationary in the ergosphere

Must rotate in direction of BH spin

because

BH spin “drags space” along with it

aka:

“dragging of inertial frames”

“Lense-Thirring effect”

Why “ERGOSPHERE”?

- “ERGO” = ENERGY
- All the spin energy of a black hole resides outside the horizon!! it can all be extracted (… in theory)
- For maximal Kerr hole with mass M:

SPIN ENERGY = 29% of

Two famous energy extraction schemes:

Penrose Process: particle splitting inside the ergosphere

Blandford-Znajek Process: BH spin twists magnetic field

UNIQUENESS OF KERR’S SOLUTION

- Kerr’s solution describes all black holes without electric charge
- More generally,
- No-hair theorem: All traces of the matter that formed a BH disappear except for:

“BLACK HOLES HAVE NO HAIR”

MASS

ANGULAR MOMENTUM

CHARGE

AREA THEOREM (Hawking 1970)

- The area of a black hole’s event horizon can stay the same or increase, but can never decrease
- Area increases when:
- mass increases
- spin decreases

- Area increases when:
- You can extract spin energy of a BH but cannot add spin energy without also adding mass
- Deep connection to thermodynamics
- horizon area ~ entropy

Theorem is true for classical GR, can be violated at quantum level

WHAT IS ENTROPY?

- Entropy = measure of disorder
- Example: entropy of a gas
- you see the overall properties (density, pressure, etc.), but don’t know the exact location , energy, etc., of each atom
- ENTROPY = logarithm of the number of diffferent ways you can relocate the atoms and redistribute their energies WITHOUT changing the overall properties of the gas

- 2nd law of thermodynamics:

The entropy of a self-contained system never decreases

BLACK-HOLE THERMODYNAMICS

- Area theorem “looks like” 2nd law of thermo.
- could BH horizon area really represent entropy? (Beckenstein 1972)

- Law for addition of mass, ang.mom., etc. to BH “looks like” 1st law of thermodynamics if:
- This turns out to be more than “just an analogy”

HORIZON AREA ~ ENTROPY

GRAV. ACCEL. AT HORIZON ~ TEMPERATURE

(Bardeen, Carter, Hawking 1972)

(Hawking 1974)

BLACK HOLES EVAPORATE

- If black holes really have a finite temperature, they must radiate
- Hawking calculates by working out quantum effects in a curved (non-quantum) space
- Finds that black holes must radiate according to “black body” law
- same form of radiation as any warm opaque body

- Where does BH entropy come from?
- No. of different ways of throwing stuff together to make the same BH

HOW DOES THE HAWKING EFFECT WORK?

Creation of “virtual pairs”

of particles

“VACUUM FLUCTUATIONS”

In the very short time

that virtual pairs can exist...

Tidal forces pull them apart

Makes some of them real

One falls in, one flies away

Black hole evaporates

HOW FAST DO BHs EVAPORATE?

- BH temperature inversely proportional to mass
- For 1 solar mass BH: 60 billionths of a degree Kelvin
- For BH temp. to exceed cosmic background radiation (3° K): < 20 billionths of a solar mass (<0.007 Earths)

- Evaporation time proportional to (mass)3
- For 1 solar mass BH:
- To evaporate in age of Universe (~ 15 billion yr):
need BH mass < 100 million tons (< cu. km of dirt)

(size ~ atomic nucleus, temp. ~ 200 billion degrees)

- At final stage (trillions of °K) BH explodes