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B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle 1
Black Holes MysteriesClassical description
Schwartzchild radiusNo entropy, temperature, stable!
Quantum mechanicsThe smallest we can measure: Planck lengthHawking radiationEntropy of a black hole
Holographic principleMaximum amountAre our 3 dimensions an illusion?
cf J.D. Bekenstein Scientific American Reports 2007 p 67Leonard SusskindThe Black Hole War
Quantum gravity and the Big Bang
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Classical Black Holes
2
Horizonseparates inner part of black hole from which you cannot escape
from outer part
“No hair” theoremClassically a black hole has just mass, charge and spinNo structureNo temperature, entropy, does not decay!
Schwartzchild radiusFor non rotating black hole. If mass concentrated in less than rs=> black holeEscape velocity =velocity of light
http://en.wikipedia.org/wiki/Schwartzchild_radius
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Quantum Mechanics and Gravity
3
No unified theory yetBest candidate: string theory but plagued with large number of
ground states (“Landscape”)Meanwhile: Semiclassical arguments
General characterDiffraction <=wave character
Discrete quanta
Massive Particles
Photons
1st Quantification
Wave like
Discrete
states
Obvious
2nd Quantification
Discrete particles
Obvious Planck
# photons /mode
size ≈ h
p= λ (wave length)
quanta of energy e.g. photon ε=hv = hcλ
particle of mass m ε ≈ mc2 + p2
2m(non relativistic)
Minimum measurable length= Planck lengthminimum wave length that a photon can have without collision leading
to a black hole
Energy after collision ≈ ε = hcλm
Schartzchild radius rs ≈2Gmc2
c4=2Gc4
hcλm
≈ λm ⇒λm =2Ghc3
Planck length Lp =
Gh2πc3
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Hawking Radiation
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Fluctuations on the horizonsmall hair: extended horizonEntropy
Entropy S = kB log W( ) W=number of accessible quantum states
S = kB4πrs
2
4Lp2
Quantum mechanics and horizonVacuum is full of pairs of particles and anti particlesSometimes one of the members of the pair is gobbled up
=> one particle left: radiationAs you cannot extract energy from vacuum, black hole looses mass(if energy E is emitted, energy -E is absorbed)
Temperature T =hc3
16π 2GMkB
tevaporation =5120i2πG2M 3
hc4
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Entropy of a black hole
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http://en.wikipedia.org/wiki/Holographic_principle
Log(W) =surface area of horizon divided by 4 Lp2
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Bound on number of states
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Holographic Bound Entropy<Area/(4Lp2)
Squeeze object within surface area A to make it a black holeEntropy has to increase => initial entropy <entropy of black hole with
has surface area smaller => bound cf Hologram
Universal entropy bound Entropy of an object with mass m and diameter d Merge it with a black hole of the same diameter Entropy has to increase => bound on initial entropy
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Consequences
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Holographic bond much higher than practical devices
However seems to indicate mapping 3 dimensional world on 2 dimensional surface
Can be made formal in some string theory modelsAre 3 dimensions an illusion? http://en.wikipedia.org/wiki/
Holographic_principleCan we test experimentally? Hogan’s idea about holographic noise
B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Epilogue
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A:Bp
uns
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B.SadouletPhys24 11/16/11: Black Holes and Holographic Principle
Quantum Gravity and the Big Bang
Quantum Gravity is not yet a full theorybut conjectures likely to be true
Hawking’s guessSingularity smoothed by quantum effect
Absence of boundary: no “initial conditions”
Metaphysical consequencesIn his view solves the problem of the origin of time “the problem of God”. Much too simplistic!
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