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
geometry = matter-energy
Gij = k Tij
General Relativity
spacetime metric
ds2 = gij dxixj
gravity = spacetime curvature
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
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
espace sphérique
espace Euclidien
espace hyperbolique
finite (no edge)
finite or infinite
finite or infinite
Homogeneity
=>
constant space
curvature !
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.
Cosmic Microwave Background
Observed on a 2-sphere
€
Cl =1
2l+1alm
2
−l
l
∑
Multipole moments
€
δT =l
∑ almYlmm
∑
Spherical harmonicsus
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!
Poincaré Dodecahedral Spherical space (PDS)
Luminet et al., Nature 425, 593 (2003)
Planck Surveyor (2007)
• fit low quadrupole• fit low octopole
• < tot < 1.02
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
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