Transcript The Big Four:

KERR BLACK HOLES (Roy Kerr 1963)
• 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
 aM
– “Kerr parameter”  a / M
a / M  0 No spin: Schwarzschild hole
a / M  1 maximal (or extreme) Kerr hole
STRUCTURE OF A KERR BH
• Horizon smaller
rS / 2 in max. Kerr limit
rS at equator, touches horizon at poles
• Static limit
Gas can orbit closer to horizon
without plunging in
Higher energy efficiency:
2
0
.
06
Mc
Schwarzschild:
Extreme Kerr: 0.42 Mc 2
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
Mc 2
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,
“BLACK HOLES HAVE NO HAIR”
• No-hair theorem: All traces of the matter
that formed a BH disappear except for:
MASS
ANGULAR MOMENTUM
CHARGE
HOW DOES ALL THE
“HAIR” DISAPPEAR?
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
• 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:
HORIZON AREA ~ ENTROPY
GRAV. ACCEL. AT HORIZON ~ TEMPERATURE
(Bardeen, Carter, Hawking 1972)
• This turns out to be more than “just an analogy”
(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?
“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
Creation of “virtual pairs”
of particles
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
66
– For 1 solar mass BH: 1.5  10 yr
– 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