Holography and the Emergence

of Gravity

Dennis Dieks, Jeroen van Dongen,

Sebastian de Haro

Reduction and Emergence, MCMP,

Munich, 2013

“Starting from first principles and general assumptions, we present a heuristic argument that shows that Newton’s law of gravitation naturally arises in a theory in which space emerges through a holographic scenario. Gravity is identified with an entropic force caused by changes in the information associated with the position of material bodies.”

Erik Verlinde, 2010

Philosophical concerns regarding quantum

gravity holography:

• Can one point to a fundamental ontology

with holographically related theories?

• Is one facing emergence of space, time

and/or gravity?

Overview

• Introduction to holography (Jeroen)

• AdS/CFT and emergence (Sebastian)

• Emergence and Verlinde’s holographic

scenario (Dennis)

Black hole thermodynamics and

quantum gravity degrees of freedom

• Bekenstein entropy: 𝑆 = 𝐴/4𝐺 (1972)

• Hawking radiation: 𝑇 = 1/8𝜋𝐺𝑀 (1974)

• Information paradox (1985)

• Holographic hypothesis (1993)

• “Maldacena” conjecture: AdS/CFT (1997)

Holographic hypothesis of ’t Hooft

• The total number of degrees of freedom, 𝑛, in a region of

spacetime containing a black hole is:

𝑛 =𝑆

log 2=

𝐴

4𝐺log 2

• Hence, “we can represent all that happens inside [a volume] by

degrees of freedom on the surface”

• “This suggests that quantum gravity should be described

entirely by a topological quantum field theory, in which all

degrees of freedom can be projected on to the boundary”

Holographic hypothesis of ’t Hooft

• “We suspect that there simply are no more

degrees of freedom to talk about than the ones

one can draw on a surface [in bit/Planck

length2]. The situation can be compared with a

hologram of a three dimensional image on a

two dimensional surface”

• Fundamental ontology, emergence?

AdS/CFT

• 𝐷-dim. anti-de Sitter space

• In local coordinates:

d𝑠2 =ℓ2

𝑟2d𝑟2 − d𝑡2 + d𝐱2

• Fields 𝜙 𝑟, 𝑥

• CFT on ℝ𝐷−1

• Operators 𝒪 𝑥

Duality Statement

• String theory in AdS space = CFT on boundary

• Fields 𝜙 𝑟, 𝑥 ↔ Operators 𝒪 𝑥

• Partition function 𝑑 = 𝐷 − 1 :

𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥 𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

• One-to-one map of observables.

• Physical equivalence, mathematical structure

different

• Large distance ↔ high energy divergences

Renormalization Group

• Radial integration: • Wilsonian

renormalization:

Λ 𝑏Λ 0

𝑘

integrate out

New cutoff 𝑏Λ

rescale 𝑏Λ → Λ until 𝑏 → 0

AdS𝑟

𝜕AdS𝑟 𝜕AdS𝜖

new boundary condition

integrate out

IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT

Philosophical Questions

• Is one side of the duality more

fundamental?

– If QFT more fundamental, space-time could be

‘emergent’

– If duality not exact: room for emergence (e.g.

thermodynamics vs. atomic theory)

• Exact duality: one-to-one relation between

the values of physical quantities

Remarks

• External view: meaning of observables

externally fixed, map relates different

physical quantities

– No empirical equivalence, numbers correspond

to different physical quantities

• No external point of view:

– How to decide which description to choose?

– Equivalence of descriptions

• Holography and emergence of gravity?

• Erik Verlinde’s proposal

Motivating Thoughts

• Hints from string theory, the holographic

conjecture/principle: there are solid indications

from quantum gravity research that gravitational

theories within a volume correspond to a theory

without gravity on the boundary of the volume

• AdS/CFT duality gives a concrete and detailed

example of the idea. Renormalization steps in the

CFT on the surface correspond to different sizes of

the bulk

Could it be that gravity is “just

the bulk-description” of a world

without gravity?

Further Motivating Thoughts

• Gravity is special: it is universal. It applies to all

matter and energy, regardless of specific

interactions; it seems to relate to space itself.

• This universality reminds one of the universal

character of thermodynamical behavior, which is

independent of microscopic details

• Gravity distinguishes itself from other forces

because it is difficult to quantize; is it

fundamentally different?

Program of Research

• Start with a theory without gravity on a two-

dimensional screen, e.g. the surface of a sphere

• Holography: this theory codifies information about

matter in an additional spatial dimension (“in the

bulk”)

• The microscopic details of this gravitation-less

theory remain unspecified: it is a mere information

processing device, a theory of holographic “bits”

• Make gravity appear as a macroscopic

thermodynamic phenomenon

Guiding idea about force as a

thermodynamic phenomenon

• Entropic processes: as a result of random motion of

its microscopic constituents a physical system will

end up in a state of greater entropy, i.e. higher

probability: the system seems to be directed

• Although there are no forces on the microscopic

level, on the thermodynamic level the system

appears driven, and this can be described by a

“macroscopic force”

• Like an ink droplet in water, or stretched polymer

Working this out

• Imagine a sphere, whose area is divided into small

cells with each one “bit”. This information suffices

to describe the inside (holography)

• On the sphere an entropic process takes place: the

distribution of 0-s and 1-s tends to an equilibrium

• This process will correspond to gravitational

motion inside the sphere

Appearance of Space

• In the surface theory, there are no spatial dimensions

other than those within the surface itself

• Consider several spheres, namely different surface

theories that relate to each other via

“renormalization” (or “coarse-graining”)

• “Coarse-grained” theories encode less information,

i.e. describe less space

• Thus, a spatial dimension x appears as a

bookkeeping device that records the level of coarse

graining on the sphere

The appearance of gravity

• Number of bits on the sphere:

N ~ A = 4πR2

• Equipartition: E = Mc2 ~ N. T

• F = T ∆S/∆x

• ∆S ~ m.∆x

• From which we get: F ~ M.m/R2

Microscopic theory

on sphere

Microscopic theory

in bulk

Macroscopic theory

in bulk: Gravity

Macroscopic theory

on sphere: no gravity

holography

Thermodynamic

limit

• The holographic duality relation may well be a

bijective mapping

• There is no reason in this case to think that one side

is more fundamental than the other (left-right)

• But the thermodynamic limit introduces the

emergence of gravity in an uncontroversial sense

(top-bottom)

Conclusions

• In the holographic scenario, the microscopic surface

theory is not necessarily more fundamental than the

microscopic bulk theory

• However, the appearance of gravity in the

thermodynamic limit makes it a clear case of

emergence, connected with robustness and novelty

of behavior. This robustness explains the

universality of gravitation

• That gravity is emergent could give rise to new

predictions: the law of gravity is not exact but

subject to fluctuations