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GRAVITY AS AN EMERGENT FORCE
Erik Verlinde
University of Amsterdam
ICHEP conference, Paris , 22/07/10
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Emergence
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Current Paradigm
FUNDAMENTAL FORCES: carried by elementary particles
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Emergence of Particles and Forces
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Gravity as an Emergent Force
At a microscopic scale Nature is described by many degrees of freedom, most of which are invisible and at first sight irrelevant for the observed macroscopic physics.
Gravity arises due to the fact that the amount
of phase space volume (“information”) occupied by these microscopic degrees of freedom is influenced by the observable macroscopic variables, like the positions of material objects.
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mBlack HoleHorizon
Black hole thought experiments.Consider a particle graduallylowered in to a black hole. Classically, the energy associated with the particle gets redshifted, and vanishes when the particle is at the horizon.
PenroseChristodoulouBekensteinHawking
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Black Hole Entropy
=> Holographic Principle
€
SBH = kBAc 3
4Gh
Maximal information associated with a part of space can be encoded in a # of bits equal to the area in Planck units
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ADS/CFT CORRESPONDENCE
EQUIVALENCE BETWEEN FIELD THEORY ON THE “BOUNDARY” AND GRAVITY IN THE “BULK”
ONE SPACE DIMENSION EMERGES CORRESPONDING TO THE “SCALE” OF THE BOUNDARY THEORY. RADIAL EVOLUTION IS LIKE RENORMALIZATION GROUP FLOW.
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Black Hole
In AdS
space
Bulk description
Thermal Heat Bath
€
TDelocalized state gets thermalized by heath bath
Boundary description:
Particle gets lowered in to black hole
Hot CFT
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Entropic force (wikipedia)
An entropic force is a macroscopic force whose
properties are determined not by the character of an underlying
microscopic force, but by the whole system's statistical tendency to increase its entropy.
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Heat Bath
EntropicForce
Polymer€
T
€
F = T∇xS
€
S(E,x) = kB logΩ(E, x)
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mBlack Hole
Horizon
Thought experiment
€
dx =dr
1− 2GM /r
€
E = m 1− 2GM /r
€
F =dEdx
=GMm
r2 “stretched horizon”
black hole
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m
Black HoleHorizon
Consistency with blackhole thermodynamicsimplies
€
FΔx = TH ΔSBH
€
TH =g
2π
€
ΔSBH = 2πmΔx
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information is stored on holographic screens moving a particle over one Compton wavelength leads to one more bit of information
€
ΔS = 2π kB
€
Δx
€
m
€
Δx =h
mc
A HEURISTIC DERIVATION
OF GRAVITY
€
ΔS = 2π kBmch
Δx
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To get a force one needs a temperature. By taking that temperature to be the Unruh temperature one finds Newton’s law of inertia
€
Δx
€
m
€
FΔx = TΔS€
T
€
kBT =1
2πhac
€
F = ma
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In order to get an entropic force I need a temperature
€
T
€
F
€
E = Mc 2
€
12 kBT = Mc 2 / # bits
€
# bits =Ac 3
Gh
€
FΔx = TΔS
€
F =GMm
R2
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Holographic screens at equipotential
(= equal redshift) surfaces
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What about General Relativity?
Surface of constant redshift
€
kBT =1
2πhc∇Φ
Komar mass => Einstein equation
€
dn =c 3
GhdA
€
∇Φ∫ dA = 8πGM
€
Φ =logξ aξa
€
ξa = timelike Killing vector
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m
€
T =h
2πkB
ac
€
h2πkB
∇xS = mc
€
F = maRindlerHorizon
€
F = T∇xS = ma
€
c →vSuggestive link with QM:
What is this velocity v ?
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m
CosmologicalHorizon
€
T =h
2πkB
a0
c
De Sitter Space
€
a0 = c 2 Λ
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m
CosmologicalHorizon
€
T =h
2πkB
a2 + a02
c
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m
CosmologicalHorizon
€
T =h
2πkB
a2 + a02
c
€
h2πkB
dSdx
= mca
a2 + a02
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m
€
T =h
2πkB
dvdx
€
h2πkB
∇xS = mv
€
Φ =v 2
2Equipotential surface
v = escape velocity
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Born-Oppenheimer & Adiabatic theorem
€
i∂∂t
ψ (t) = H x(t)( )ψ (t)
€
H x( )ψ n (x) = En (x)ψ n (x)
Schroedinger eqn with H depending on infinitely slow variable
Instantaneous eigenstates
Adiabatic Reaction Force
€
F =dEn
dx(x)
€
J = pdq∫ = 2πnh
Semiclassically
€
F =dEdJ
dJdx
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MicroscopicFast
Variables
Born-Oppenheimer & Entropic Force
€
ζ
€
xMacroscopic
Slow Variables
€
x€
E
The system stays in an energy eigenstate of the fast variables( adiabatic theorem).
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Born-Oppenheimer & Entropic Force
MacroscopicSlow Variables
€
x€
E
€
Ω(E,x) = dζ∫ Θ E − H(ζ,x)( )
€
ddx
logΩ E(x),x( ) = 0
Assuming eigenvalues don’t cross, the energy follows from
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What lives on the screens?
According to string theory: open strings.
Integrating out the UV open strings produces closed strings in the emerged space.
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Open closed string duality
€
(-1)F dss3/2 exp - s(mi
20
∞∫i
∑ + x 2)
€
(-1)Fmid -2 ds
s(5-d)/2 exp - s0
∞∫i
∑ x 2
€
(-1)Fmid -2 d˜ s dk∫ exp
0
∞∫i
∑ ikx − ˜ s k 2 ( )x
Open string one loop diagram
Massless pole in dual channel
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UV/IR correspondence
€
(-1)F dss3/2 exp - s(mi
20
1Λ∫
i∑ + x 2)
€
(-1)Fmid -2 d˜ s dk∫ exp
Λ
∞∫i
∑ ikx − ˜ s k 2 ( )
€
(-1)F dss3/2 exp - s(mi
21Λ
∞∫i
∑ + x 2)
Open string with UV cut off
Closed string / gravity with UV cut off
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Matrix description of gravity.
€
tr ˙ X I2( ) +̀tr [X I , X J ]2( )
=>
˙ z 2 + (x − y)2 z 2
€
X =
x11 .. x1N z1
: :: : :xN1 .. xNN zN
z1* .. zN
* yI
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
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Matrix description of gravity.
€
X =
x11 .. x1N z1
: :: : :xN1 .. xNN zN
z1* .. zN
* yI
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
T
€
F
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Gravity as an Emergent Force
At a microscopic scale Nature is described by many degrees of freedom, most of which are invisible and at first sight irrelevant for the observed macroscopic physics.
Gravity arises due to the fact that the amount
of phase space volume (“information”) occupied by these microscopic degrees of freedom is influenced by the observable macroscopic variables, like the positions of material objects.
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Berry Phase and Crossing Eigenvalues
€
x
€
E
€
H =z x + iy
x − iy −z ⎛ ⎝ ⎜
⎞ ⎠ ⎟= v x ⋅ r
σ
€
r B =
ˆ x 4π r x 2
Dirac monopool
At the locus of coinciding eigenvalues one can construct
Non-abelian Berry
€
Aij = ψ i dψ j
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