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