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 bymany 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”) occupiedby these microscopic degrees of freedom is influenced by the observable macroscopicvariables, like the positions of material objects.
Black HoleHorizon
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
Ac3
4 Gh
Maximal information associated witha part of space can be encoded in a # of bits equal to the area in Planckunits
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 AdSspace
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 TxS
S( E , x ) kB
log ( E , x )
Black HoleHorizon
Thought experiment
dxdr
1 2GM / r
E m 1 2GM / r
FdE
dx
GMm
r2
“stretched
horizon”
black hole
Black HoleHorizon
Consistency with black
hole thermodynamics
implies
F x TH
SBH
TH
g
2S
BH2 m x
information is stored on holographic screensmoving a particle over one Compton wavelengthleads to one more bit of information
S 2 kB
x
m
xh
mc
A HEURISTIC DERIVATION
OF GRAVITY
S 2 kB
mc
hx
To get a force one needs a temperature. By taking that temperature to be the Unruh temperatureone finds Newton’s law of inertia
x
m
F x T S
T
kBT
1
2
ha
c
F ma
In order to get an entropic force I need a temperature
T
F
E Mc2
1
2k
BT Mc
2/ # bits
# bitsAc
3
G h
F x T SFGMm
R2
Holographic screens at equipotential(= equal redshift) surfaces
What about General Relativity?
Surface of constant redshift
kBT
1
2
h
c
Komar mass
=> Einstein equation
dnc
3
GhdA
dA 8 GM
loga
a
a timelike Killing vector
Th
2 kB
a
c
h
2 kB
xS mc
F maRindlerHorizon F T
xS ma
c vSuggestive link with QM:
What is this velocity v ?
CosmologicalHorizon
Th
2 kB
a0
c
De SitterSpace
a0
c2
CosmologicalHorizon
Th
2 kB
a2
a0
2
c
CosmologicalHorizon
Th
2 kB
a2
a0
2
c
h
2 kB
dS
dxmc
a
a2
a0
2
Th
2 kB
dv
dx
h
2 kB
xS mv
v2
2Equipotentialsurface
v = escape velocity
Born-Oppenheimer & Adiabatic theorem
it
( t ) H x ( t ) ( t )
H xn( x ) E
n( x )
n( x )
Schroedinger eqn with H depending on infinitely slow variable
Instantaneous eigenstates
Adiabatic Reaction Force
FdE
n
dx( x )
J pdq 2 nh
Semiclassically
FdE
dJ
dJ
dx
MicroscopicFast Variables
Born-Oppenheimer & Entropic Force
x
MacroscopicSlow Variables
x
E
The system stays in an energy eigenstateof the fast variables( adiabatic theorem).
Born-Oppenheimer & Entropic Force
MacroscopicSlow Variables
x
E
( E , x ) d E H ( , x )
d
dxlog E ( x ), x 0
Assuming eigenvaluesdon’t cross, the energy follows from
What lives on the screens?
According to string theory: open strings.
Integrating out the UV open stringsproduces closedstrings in the emerged space.
Open closed string duality
(-1)F ds
s3/2
exp - s(mi
2
0i
x2)
(-1)F
mi
d -2 ds
s(5 -d)/2
exp - s0
i
x2
(-1)Fm
i
d -2d ˜ s dk exp
0i
ikx ˜ s k2 x
Open string one loop diagram
Massless pole in dual channel
UV/IR correspondence
(-1)F ds
s3/2
exp - s(mi
2
0
1
i
x2)
(-1)Fm
i
d -2d ˜ s dk exp
i
ikx ˜ s k2
(-1)F ds
s3/2
exp - s(mi
2
1
i
x2)
Open string with UV cut off
Closed string / gravity with UV cut off
Matrix description of gravity.
tr Ý X I
2` tr [ X
I, X
J]
2
Ý z 2
( x y )2
z2
X
x11
.. x1N
z1
: :: : :
xN 1
.. xNN
zN
z1
*.. z
N
*yI
Matrix description of gravity.
X
x11
.. x1N
z1
: :: : :
xN 1
.. xNN
zN
z1
*.. z
N
*yI
T
F
Gravity as an Emergent Force
At a microscopic scale Nature is described bymany 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”) occupiedby these microscopic degrees of freedom is influenced by the observable macroscopicvariables, like the positions of material objects.
Berry Phase and Crossing Eigenvalues
x
EH
z x iy
x iy z
v x
r
r B
ˆ x
4r x
2
Diracmonopool
At the locus of coinciding eigenvaluesone can construct
Non-abelian BerryA
ij id
j