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