Emergent Space-Time and and Induced Gravity

Erik VerlindeUniversity of Amsterdam

Madrid , November 17th, 2006

Some (Speculative) Ideas on

“Strings versus Cosmology”

Standard (inflationary) cosmology is successfull.It uses low energy effective action, and needs very little input from string theory.

My personal view:

Don’t be satisfied to with using low energy action,

but use the complete (microscopic) string theory

to challenge the basic assumptions on which

standard cosmology (including inflation) is based.

Strings versus Cosmology

String theory needs concrete problems

• Black Holes: led to important progress (AdS/CFT).

drastic departure from old views.

(complementarity, holography, unitarity.)

• Cosmology: still in its infancy, no breakthrough yet.

expect drastic departure from old views.

(initial conditions, inflation, multiverse.)(complementarity, holography, unitarity.)

(collapse, information loss, baby universes.)

Strings versus Cosmology

String theory indicates that

• Space-time is emergent• Gravity is induced

Strings versus Cosmology

• What does this mean for cosmology?

What was the Big Bang? (if it ever happened)

How does space-time emerge?

Is the emergent space-time observer dependent?

Is there a unitary quantum system underlying all this?

Will this be at all important for observations?

Outline

Part I: “Observer complementarity ”. Parikh , EV (’04)

A Model for de Sitter space

Part II: “Emergent Space-Time”

The Matrix Big Bang Craps, Sethi, EV (’05)

Part III: “Induced Gravity”

The Black Hole Farey Tail. Dijkgraaf, Moore,Maldacena, EV (’00) de Boer, Cheng, Dijkgraaf, Manschot, EV (’06)

Part IV: “A Heretic View on Cosmology”

I: A Model for de Sitter space

Every eternal observer has complete knowledge about the quantum state of the Universe.

Observer complementarity

No quantum states correspond to physics outside the maximal causal diamant.

Observers agree on probabilities for events, but not necessarily on their interpretation.

Classical space-time is only an approximate notion and may be different for different observers.

Model for de Sitter space

• Every observer has a finite dimensional Hilbert space

• The probability is given by

P S

S• The state is de Sitter invariant and is the analogue of the S-matrix.

H

*H H

• These form a de Sitter representation.

• Events are described by a tensor product state

Model for de Sitter space

• The Hilbert space is reps of SO(d-1)

forms a reps of SO(d,1).*H H

H

• A concrete model can be made using a spinor field on the (d-1)-dim spatial sphere 1dS

1,

i

i j

x

x x

5

3i

i j

x

x x

x x x

or

( )x

II: The Matrix Big Bang

Lightlike Linear Dilaton

22 2 idxdxdxds

Qx

for

Qxsg e

x

22211 23

234

iQxQx dxdxdxedyeds

in new lightcone coordinates

222211 2 iudxdudvdyuds

Qxeu 32

xv Q23

Lift to M-theory:

10d metric+dilaton

uR

uR yuyuiuiu

2

4

1

0u

0u

null singularity

/

Matrix dual of lightlike linear dilaton in DLCQ

Matrix String = (1+1)d super Yang-Mills

string coupling

light-cone momentum

],[],[)(tr 122222 ii

Tsjiss

TiMS XgXXgFgDDXS dd

hgeg YM

RQs det2/22

forward quadrant of Milne space

time dependent worldsheet metric

)( 22/22.. ddeds RQ

sw .constgYM

RQe /)(

21

flat world sheet coordinates

ddds sw 22..

RQe /2

R

Np

],[],[)(tr 222222 ii

TYMjiYMYM

TiMS XgXXgFgDDXS d /

III: The Black Hole Farey Tail

Et

EtEt

Et

Extremal Black Hole

r Q

2

2 2 2 2 2 2( ) sin( )E

drds H r dt r d d

H r

Et

2

( ) 1Q

H rr

22 2 2 2 2 2 2

2sinE

dds Q dt d d

Near horizon geometry:

22AdS S

Q

Black holes in string and M-theory.

Et

23AdS S

Et

M-theory on CY M2-branes wrapping 2-cycles M5-branes wrapping 4-cycles 5d black strings

4d black hole = 5d black string wrapping circle.

World volume theory = 2d CFT (Maldacena, Strominger,Witten)

Holographic dual to near horizon geometry:

( ) ( )wCFT CFT

a bZ Z c d

c d

Partition function

obeys

242 ( )

0

( )Ci N

CFTN

Z D N e

An Exact Asymptotic Formula

2

,

24

( )( )

a bi

c d

CFT wc dC

N

eZ D N

c d

Then we have

1,

0

ad bc

a c

Thermal AdS3 vs. BTZ

2 22 2 2 2 2 2

2 2( )AdS E

drds r dt r d

r

Et

E Et i t i n Periodic identification

/ 3=AdS

Et

cigar

2 22 2 2 2 2

2( ) ( )

( )BTZ E E

drds N r dt r d N dt

N r

2 2 2 22 2 1

2

( )( )( )

r rN r

r

1 2

2( )N r

r

SL(2,Z) orbit of AdS Black Holes

Different euclidean black holes distinguished by non-contractible cycle:

Euclidean action

Et

/ 3=AdS

8 N

a b

G c d

i

S=

Maldacena, Strominger

B cA+dB

3

2 N

cG

AdS3/CFT2

Farey tail: Z()= sum over SL(2,Z) orbit of black holes

Et

24

24

2 ( )

( ) ( )C

C

a bi N

wc d

N

D N e c d

EtEt

Et

contribution of each black hole geometry

subleading corrections: black hole ‘dressed’ withlight particle states that do not form black holes

Black holesMass

IV: A Heretic View on Cosmology

Standard Big Bang

Model

Geometry of Universe is derived from OUR OBSERVATIONS

From our perspective we are in the middle of our Universe

Can one interprete the cosmological data in an STATIC isotropic but non-homogenous (!) cosmological model?

brane worlds: can live in a static background.

one adds “scale” as fifth dimension.

Idea: We live in a static five dimensional space. The apparent expansion of the Universe is caused by the fact that for more distant objects the observed signals are coming from bigger scales.

RULE: at every time step re-throw one dice.

QUESTION: What is the most likely state at the following time step?previous

The End