Imaging in Space and Time
28/8-1/9 2006 Brijuni
The Shape of Space:
from Black Holes
to the Universe
J.-P.Luminet
Observatoire de
Paris (LUTH)
4 levels of geometry
Cosmic topology
Cosmology
Black holes
?
Quantum gravity
General Relativity
Gij = k Tij
geometry = matter-energy
ds2 = gij dxixj
spacetime metric
gravity =
spacetime curvature
Einstein ring
Gravitational
lensing
If M* > 30 MS
BLACK HOLE !
Imaging 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 Milky Way
(Riazuelo, 2006)
Capella
Castor & Pollux
Aldebaran
Orion
Sirius
Black hole in front of Constellations
Capella 1
Imaging spacetime : light cones
Aldebaran 2
Orion 2
Capella 2
Orion 1
Aldebaran 1
Einstein ring
Southern Cross
Canopus
a & b Cen
Achernar
Black hole in front of Magellanic Clouds
Southern Cross 1
Canopus 1
Achernar 2
a & b Cen 2
Southern Cross 2
Canopus 2
Einstein ring
Achernar 1
See movie 1
Black hole in front of Magellanic Clouds
Imaging spacetime : light cones
Curved spacetime
Flat (Minskowski) spacetime
Gravitational collapse to a
Schwarzschild black hole
metric:
2GM 2
dr2
2
2
2
2
ds  (1
)dt


r
(d


sin

d

)
2GM
rc 2
1
rc 2
2
Schwarzschild radius:
r
2GM
rc 2
Event horizon


Embedding
Step 1:
Schwarzschild metric outside mass M (G=c=1) :
2M(r) 2
dr2
ds  (1
)dt 
 r 2 (d 2  sin2 d 2 )
2M(r)
r
1
r
2
Step 2:
Step 3:
Equatorial section    /2
Time section
t  const
 in 3D
Embedding
Euclidian space
Curved 2-geometry:
dr2
ds 
 r 2 d 2
2M(r)
1
r
2
ds2 dz2  dr2  r 2 d 2
Result for ordinary star
(R* > 2M)
z(r)  8M(r  2M) for r  R* Outer solution (asymptotically flat)
z(r)  8M(r)(r  2M(r)) for r  R* Inner solution (regular)
Result for black hole
z(r)  8M(r  2M) for r  2M
Outer solution only
(Flamm paraboloid)
Spherical black hole in Kruskal coordinates
(r,t) (u,v)
coth(t /4 M) if r  2M 
r
v 

u2  v 2  (
1)exp(r /2M) ;
 1
ifr  2M 
2M
u
th(t /4 M) if r  2M

v
u
Flight into a
static black hole
Radial photons
(A.Riazuelo, 2006)
What is seen in C
See movie 1
What is seen in E
Flight into a
static black hole
2
Non-radial photons
What is seen in C
What is seen in E
See movie 2
What is seen further
Flight into a Kerr
(rotating) black hole
no movie yet!
Cosmology
finite (no edge)
Homogeneity
=>
constant
space
curvature !
espace sphérique
espace Euclidien
espace hyperbolique
finite or
infinite
finite or
infinite
Space-time
curvature ==>
a dynamical
universe !
Expansion
Big bang models
open
closed
What is the size and
shape of space ?
G
Horizon
G
T
Horizon
G
Assumption
Not
testable 1
(only L >>
Universe
is Rinfinite
h)
G
if L >~
Universe
is Rfinite
h
(without boundary)
but greater than
the observable one
G
G
T
G
Assumption
2 •
May
be testable
G
Horizon
T
Infini
G
G
G
Assumption 3
Testable
Universe is finite
(without boundary)
and smaller than
the observable one
• topological lensing
Think finite space without edge
Sphere = 2D Surface
finite area, no edge
Lignes
droites
Hypersphere = 3D space
finite volume, no edge
A finite flat space without a boundary
• Torus
QuickTime™ et un
décompresseur codec YUV420
sont requis pour visionner cette image.
Topological lens effect
horizon
Hypertorus
Physical Space
Observed Space
Cosmic Microwave Background
The universe as a cosmic « drumhead »
Cosmic Microwave Background
Observed on a 2-sphere
T  
l
us
a
Ylm
lm
m
Spherical harmonics
1
Cl 
2l  1
l
a
2
lm
l
Multipole moments
The CMB multipoles
Quadrupole
Power spectrum
Tl2 = l(l+1)Cl/2
Doppler peaks
(Boomerang,
Archeops, etc.)
l=180°/
Large scales
(COBE, WMAP)
WMAP power spectrum
(2003- 2006)
• Universe seems to be
positively curved
W
= 1.02 ± 0.02
flat infinite
universe
• Lack of power at large
scales (> 60°)
Space might be finite with a special shape!
Poincaré Dodecahedral Space
FP : 12 faces regular dodecahedron
S3/I*
120 copies
tessellate S3
Poincaré Dodecahedral Spherical
space (PDS)
• fit low quadrupole
• fit low octopole
•
< Wtot < 1.02
Luminet et al., Nature 425, 593
(2003)
Planck Surveyor
(2007)
The « football Universe »
36°
Also compatible …
Octahedral
space
Tetrahedral
space
(Wtot > 1.015)
(Wtot > 1.025)
J. Weeks, 2006
Imaging Quantum Gravity
Quantum foam
(J. Wheeler)
Solution 1 : string theory
Veneziano, Green,
Schwarz, Witten, etc.
Price to pay :
extra-dimensions
Closed string
Open string
bulk
Solution 2 : loop quantum gravity
Ashtekhar, Smolin, Rovelli, Bojowald
Atoms of space: 10-99 cm3
Spin network
Knot theory
Atoms of time : 10-43 sec
Spin foam
If God had consulted me
before embarking upon
Creation, I should have
recommended something
simpler.
Alfonso X, King of Castile