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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”
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
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 /
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
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