The Shape of Space:The Shape of Space:from Black Holes from Black Holes

to the Universeto the UniverseJ.-P.Luminet

Observatoire de Paris (LUTH)

Imaging in Space and Time 28/8-1/9 2006 Brijuni

4 levels of geometry4 levels of geometry

?

Black holesCosmology

Cosmic topology

Quantum gravity

geometry = matter-energy

Gij = k Tij

General Relativity

spacetime metric

ds2 = gij dxixj

gravity = spacetime curvature

GravitationGravitational lensingal lensing

Einstein ring

If MIf M** > 30 M > 30 MS S

BLACK HOLE !BLACK HOLE !

Imaging Black HolesImaging Black Holes

Newtonian spacetime

curved spacetime

Image of a spherical black hole with thin accretion disk

(J.-P. Luminet, 1979)

Flight into a black hole(J.A.Marck, 1993)

Black hole in front of Black hole in front of Milky WayMilky Way

(Riazuelo, 2006)(Riazuelo, 2006)

Black hole in front of Black hole in front of ConstellationsConstellations

Orion

Sirius

Aldebaran

Capella

Castor & Pollux

Orion 1

Capella 1

Orion 2

Capella 2

Aldebaran 1

Aldebaran 2

Einstein ring

Imaging spacetime : light conesImaging spacetime : light cones

Black hole in front of Magellanic Black hole in front of Magellanic CloudsClouds

Achernar

& Cen

Canopus

Southern Cross

Southern Cross 2

& Cen 2

Southern Cross 1Canopus 1

Canopus 2

Achernar 1

Achernar 2

Einstein ring

Black hole in front of Magellanic Black hole in front of Magellanic CloudsClouds

See movie 1

Flat (Minskowski) spacetimeFlat (Minskowski) spacetime

Curved spacetimeCurved spacetime

Imaging spacetime : light conesImaging spacetime : light cones

Gravitational collapse to a Gravitational collapse to a Schwarzschild black holeSchwarzschild black hole

€

ds2 = −(1−2GM

rc 2)dt 2 +

dr2

1−2GM

rc 2

+ r2(dθ 2 + sin2 θdφ2)

metric:

Schwarzschild radius:

€

r =2GM

rc 2

Event horizon

€

ds2 = −(1−2M(r)

r)dt 2 +

dr2

1−2M(r)

r

+ r2(dθ 2 + sin2 θdφ2)

EmbeddingEmbedding

Schwarzschild metric outside mass M (G=c=1) :

Embedding in 3D Euclidian space

€

ds2 = dz2 + dr2 + r2dφ2

Equatorial section

Time section

€

θ =π /2

t = const

Step 1:Step 1:

Step 2:Step 2:

Step 3:Step 3:

Curved 2-geometry:

€

ds2 =dr2

1−2M(r)

r

+ r2dφ2

Result for ordinary star Result for ordinary star (R(R** > 2M) > 2M)

€

z(r) = 8M(r − 2M) for r ≥ R*

€

z(r) = 8M(r)(r − 2M(r)) for r < R*

Outer solution (asymptotically flat)

Inner solution (regular)

Result for black holeResult for black hole

€

z(r) = 8M(r − 2M) for r ≥ 2M Outer solution only

(Flamm paraboloid)

Spherical black hole in Kruskal coordinatesSpherical black hole in Kruskal coordinates

€

(r, t) → (u,v)

€

u2 − v 2 = (r

2M−1)exp(r /2M) ;

v

u=

coth(t /4M) if r < 2M

1 if r = 2M

th(t /4M) if r > 2M

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

u

v

What is seen in C

What is seen in E

Flight into a static black hole

Radial photons

(A.Riazuelo, 2006)

See movie 1

What is seen in C What is seen in E What is seen further

Flight into a static black hole

2Non-radial photons

See movie 2

Flight into a Kerr (rotating) black hole

no movie yet!

CosmologyCosmology

espace sphérique

espace Euclidien

espace hyperbolique

finite (no edge)

finite or infinite

finite or infinite

Homogeneity

=>

constant space

curvature !

Space-time curvature

==> a dynamical universe !

Expansion

closed

open

Big bang modelsBig bang models

What is the size What is the size and shape of space and shape of space

??

T

G

Horizon

Infini

Assumption 1Universe is infinite

T

G

Horizon

Assumption 2Universe is finite (without boundary) but greater than the observable one

Assumption 3Universe is finite (without boundary) and smaller than the observable one

T

G

Horizon

GG G

G G

G G G

Not testable (only L >> Rh)

May be testable • if L

>~ Rh

Testable• topological

lensing

Think finite space without edge

Sphere = 2D Surfacefinite area, no

edge

Hypersphere = 3D space finite

volume, no edge

Lignes droites

A finite flat space without a boundary

• Torus

QuickTime™ et undécompresseur codec YUV420

sont requis pour visionner cette image.

Topological lens effect

horizon

Hypertorus

Observed Space

Physical Space

The universe as a cosmic

« drumhead »

Cosmic Microwave Background

Cosmic Microwave Background

Observed on a 2-sphere

€

Cl =1

2l+1alm

2

−l

l

∑

Multipole moments

€

δT =l

∑ almYlmm

∑

Spherical harmonicsus

The CMB multipolesQuadrupole

Power spectrum

l=180°/θ

Doppler peaks(Boomerang, Archeops, etc.)

Large scales (COBE, WMAP)

Tl2 =

l(l+1)Cl/2π

WMAP power spectrum (2003- 2006)

flat infinite

universe

• Universe seems to be positively

curved = 1.02 ± 0.02

• Lack of power at large scales (> 60°)

Space might be finite with a special shape!Space might be finite with a special shape!

120 copies tessellate

S3

Poincaré Dodecahedral Space

FP : 12 faces regular dodecahedron

S3/I*

Poincaré Dodecahedral Spherical space (PDS)

Luminet et al., Nature 425, 593 (2003)

Planck Surveyor (2007)

• fit low quadrupole• fit low octopole

• < tot < 1.02

The « football Universe »

36°

Octahedral space

(tot > 1.015)

Tetrahedral space

(tot > 1.025)

Also compatible …

J. Weeks, 2006

Imaging Quantum Gravity

Quantum foam

(J. Wheeler)

Solution 1 : string theory

Price to pay : extra-dimensions

Closed string

Open string

Veneziano, Green, Schwarz, Witten,

etc.

bulk

Solution 2 : loop quantum gravity

Atoms of space: 10-99 cm3

Spin networkAtoms of time : 10-43

secSpin foam

Ashtekhar, Smolin, Rovelli, Bojowald

Knot theory

If God had consulted me before embarking upon Creation, I

should have recommended something

simpler. Alfonso X, King of Castile