GRAVITY AS AN EMERGENT FORCE
Erik Verlinde
University of Amsterdam
ICHEP conference, Paris , 22/07/10
Emergence
Current Paradigm
FUNDAMENTAL FORCES: carried by elementary particles
Emergence of Particles and Forces
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.
mBlack Hole
Horizon
Black hole thought experiments.
Consider a particle gradually
lowered in to a black hole.
Classically, the energy
associated with the particle gets
redshifted, and vanishes when
the particle is at the horizon.
PenroseChristodoulouBekensteinHawking
Black Hole Entropy
=> Holographic Principle
€
SBH = kB
Ac 3
4Gh
Maximal information associated with a part of space can be encoded in a # of bits equal to the area in Planck units
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.
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
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.
Heat Bath
EntropicForce
Polymer€
T
€
F = T∇xS
€
S(E,x) = kB logΩ(E, x)
mBlack Hole
Horizon
Thought experiment
€
dx =dr
1− 2GM /r
€
E = m 1− 2GM /r
€
F =dE
dx=
GMm
r2 “stretched
horizon”
black hole
m
Black HoleHorizon
Consistency with black
hole thermodynamics
implies
€
FΔx = TH ΔSBH
€
TH =g
2π
€
ΔSBH = 2πmΔx
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 kB
mc
hΔx
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
2π
ha
c
€
F = ma
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
Holographic screens at equipotential
(= equal redshift) surfaces
What about General Relativity?
Surface of constant redshift
€
kBT =1
2π
h
c∇Φ
Komar mass => Einstein equation
€
dn =c 3
GhdA
€
∇Φ∫ dA = 8πGM
€
Φ =logξ aξa
€
ξa = timelike Killing vector
m
€
T =h
2πkB
a
c
€
h2πkB
∇xS = mc
€
F = maRindlerHorizon
€
F = T∇xS = ma
€
c →vSuggestive link with QM:
What is this velocity v ?
m
CosmologicalHorizon
€
T =h
2πkB
a0
c
De Sitter Space
€
a0 = c 2 Λ
m
CosmologicalHorizon
€
T =h
2πkB
a2 + a02
c
m
CosmologicalHorizon
€
T =h
2πkB
a2 + a02
c
€
h2πkB
dS
dx= mc
a
a2 + a02
m
€
T =h
2πkB
dv
dx
€
h2πkB
∇xS = mv
€
Φ =v 2
2Equipotential surface
v = escape velocity
Born-Oppenheimer & Adiabatic theorem
€
i∂
∂tψ (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 =dE
dJ
dJ
dx
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).
Born-Oppenheimer & Entropic Force
MacroscopicSlow Variables
€
x€
E
€
Ω(E, x) = dζ∫ Θ E − H(ζ, x)( )
€
d
dxlogΩ E(x),x( ) = 0
Assuming eigenvalues don’t cross, the energy follows from
What lives on the screens?
According to string theory: open strings.
Integrating out the UV open strings produces closed strings in the emerged space.
Open closed string duality
€
(-1)F ds
s3/2 exp - s(mi2
0
∞
∫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
UV/IR correspondence
€
(-1)F ds
s3/2 exp - s(mi2
0
1Λ∫
i
∑ + x 2)
€
(-1)Fmid -2 d˜ s dk∫ exp
Λ
∞
∫i
∑ ikx − ˜ s k 2 ( )
€
(-1)F ds
s3/2 exp - s(mi2
1Λ
∞
∫i
∑ + x 2)
Open string with UV cut off
Closed string / gravity with UV cut off
Matrix description of gravity.
€
tr ˙ X I2
( )`+tr [X I , XJ ]2( )
=>
˙ z 2
+ (x − y)2 z2
€
X =
x11 .. x1N z1
: :: : :
xN1 .. xNN zN
z1* .. zN
* yI
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
Matrix description of gravity.
€
X =
x11 .. x1N z1
: :: : :
xN1 .. xNN zN
z1* .. zN
* yI
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
T
€
F
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.
Berry Phase and Crossing Eigenvalues
€
x
€
E
€
H =z x + iy
x − iy −z
⎛
⎝ ⎜
⎞
⎠ ⎟=
v x ⋅
r σ
€
rB =
ˆ x
4πr x
2Dirac monopool
At the locus of coinciding eigenvalues one can construct
Non-abelian Berry
€
Aij = ψ i dψ j