ICHEP conference, Paris , 22/07/10

GRAVITY AS AN EMERGENT FORCE

Erik Verlinde

University of Amsterdam 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. Black Hole Horizon

. Black hole thought experiments. Consider a particle gradually lowered in to a black hole. Classically, the energy Penrose Christodoulou associated with the particle gets Bekenstein redshifted, and vanishes when Hawking the particle is at the horizon. Black Hole Entropy

Ac 3 S k BH B 4 Gh

=> Holographic Principle

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. Bulk description Black Particle gets lowered Hole in to black hole In AdS space

Hot CFT

Boundary description: Delocalized state gets Thermal T thermalized by heath bath Heat Bath 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 F T x S T Entropic Force

Polymer

S( E, x) k B log ( E, x) . Thought experiment Black Hole black hole Horizon

dr dx 1 2GM / r

E m 1 2GM / r

“stretched dE GMm F horizon” dx r 2 . Consistency with black Black Hole hole thermodynamics Horizon implies

F x TH S BH

g T S 2 m x H 2 BH x A HEURISTIC DERIVATION

OF GRAVITY m

information is stored on holographic screens moving a particle over one Compton wavelength leads to one more bit of information

mc h SS 22 kk x x BB h mc 1 ha x k B T 2 c m T

To get a force one needs a temperature. By taking that temperature to be the Unruh temperature one finds Newton’s law of inertia

F x T S F ma GMm FF x T S F R 2 T

In order to get an entropic force I need a temperature

1 2 2 k B T Mc / # bits

2 E Mc Ac 3 # bits G h Holographic screens at equipotential (= equal redshift) surfaces What about General Relativity?

a 1 h log a k T B 2 c a timelike Killing vector c 3 Surface of constant redshift dn dA Gh

dA 8 GM

Komar mass => Einstein equation Rindler F ma Horizon F T x S ma

h a T 2 k B c

h x S mc 2 k B

Suggestive link with QM: c v What is this velocity v ? Cosmological De Sitter Horizon Space

h a T 0 2 k B c

2 a 0 c Cosmological Horizon h a 2 a 2 T 0 2 k B c Cosmological Horizon h a 2 a 2 T 0 2 k B c

h dS a mc 2 k dx 2 2 B a a 0 v 2 2 Equipotential h dv surface T 2 k B dx

h x S mv 2 k B

v = escape velocity Born-Oppenheimer & Adiabatic theorem

Schroedinger eqn with H depending on infinitely slow variable

i (t) H x (t) (t) t

Instantaneous eigenstates

H x n ( x) E n ( x ) n ( x)

Adiabatic Reaction Force Semiclassically

dE n dE dJ F ( x ) F dx dJ dx

J pdq 2 nh Born-Oppenheimer & Entropic Force

The system stays in an energy eigenstate Microscopic of the fast variables( adiabatic theorem). Fast Variables Macroscopic x Slow Variables

E Born-Oppenheimer & Entropic Force

Assuming eigenvalues don’t cross, the ( E , x ) d E H ( , x) energy follows from d log MacroscopicE ( x ), x 0 dx Slow Variables

E

x 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

ds (-1) F exp - s(m 2 x 2 ) 0 3/2 i i s Open string one loop diagram

ds (-1) F m d -2 exp - sx 2 i 0 (5 -d)/2 i s

Massless pole in dual channel

x (-1) F m d -2 ds˜ dk exp ikx s˜ k 2 i 0 i UV/IR correspondence

ds (-1) F exp - s(m 2 x 2 ) 1 3/2 i i s

Open string with UV cut off

1 ds (-1) F exp - s(m 2 x 2 ) 0 3/2 i i s

F d -2 2 (-1) m i ds˜ dk exp ikx s˜ k i

Closed string / gravity with UV cut off Matrix description of gravity.

x11 .. x1N z1 : :: : : X x N 1 .. x NN z N * * z1 .. z N yI

Ý 2 2 tr X I ` tr [ X I , X J ]

2 2 zÝ ( x y ) 2 z Matrix description of gravity.

x11 .. x1N z1 : :: : : X x N 1 .. x NN z N * * z1 .. z N 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

z x iy v r H x E x iy z

r xˆ Dirac x B r 2 monopool 4 x

At the locus of coinciding eigenvalues one can construct A d Non-abelian Berry ij i j