PLATONIC SOLIDS AND
EINSTEIN THEORY OF
GRAVITY:
UNEXPECTED CONNECTIONS
«
If you plan to make a voyage of discovery,
choose a ship of small draught
»
Captain James Cook
rejecting the large ships
offered by the Admiralty
GRAVITY: AN ACTIVE FIELD OF
RESEARCH
Of all fundamental forces, gravity is probably the most familiar.
Its understanding has led to scientific revolutions that have shaped physics
•
Newton and his «
Principia
»
•
Einstein and general relativity
It is currently an area of intense research, both theoretically and experimentally.
Yet, it is fair to say that gravity still holds many theoretical mysteries.
There are important conceptual issues that we fail to understand about it.
CONTENTS
–
A brief survey of Einstein theory: gravitation is
spacetime geometry
–
Problems
–
String (M

) theory: the key?
–
Platonic solids: the golden gate to symmetry
–
Coxeter groups (finite and infinite)
–
Infinite

dimensional symmetry groups
–
Gravitational billiards
–
Conclusions
General relativity was born because
of a theoretical clash between the
principles of (special) relativity and
those of the Newtonian theory of
gravity.
GRAVITATION = GEOMETRY
•
Einstein revolution: gravity is spacetime geometry
•
Time + space = « spacetime »
•
Gravity manifests itself through the deformation (« curvature » or « warping ») of the
spacetime geometry
•
Because of this deformation, « straight lines » in spacetime have a relative acceleration.
From J. A. Wheeler,
A Journey into Gravity and
Spacetime
, Scientific American Library 1999
SPACETIME TELLS MATTER HOW TO MOVE, MATTER TELLS
SPACETIME HOW TO CURVE (J. A. Wheeler)
This accounts for all known gravitational phenomena
Matter curves spacetime
http://math.ucr.edu/home/baez/gr/gr.html
http://www.astro.ucla.edu/~wright/cosmolog.htm
http://home.fnal.gov/~dodelson/welcome.html
Deflection of light
A spectacular example of gravitational lensing:
the Einstein cross
http://hubblesite.org/newscenter
/
http://www.astr.ua.edu/keel/agn/qso2237.gif
GRAVITATIONAL CURVATURE OF TIME
phyun5.ucr.edu/~wudka/Physics7/ Notes_www/node89.html
Gravity slows down time
Clocks on first floor tick more slowly than
clocks on top of the building (roughly 1 s
per 3 x 10
6
years).
ILLUSTRATION OF THE WARPING OF TIME :
the Global Positioning System
http://www.ctre.iastate.edu/educweb/ce352/lec24/gps.htm
Key features of GPS
Altitude of satellites: 20,000 kms
Distance from satellite = c
D
t
D
t must be known with great accuracy
Clocks on earth tick slowlier than clocks on satellites («
curvature of time
»)
Clocks quickly get out of synchronism: 50 x 10

6
s per day: this is a distance of 15 kms!
Must be corrected: satellite clock frequency adjusted to 10.22999999545 MHz prior to launch (sea level
clock frequency: 10.23 MHz). This offset of the satellite clock frequency is necessary.
Absolute precision: 30 m
Relative precision: 1
–
2 m
Applications: navigation (planes, boats, cars), tunnel under the Channel, surveying …

multi million
Euros industry!
Unpredictable payback of fundamental science
General relativity has proved to
be remarkably successful …
but there are …
PROBLEMS
General relativity + Quantum Mechanics =
Inconsistencies (e.g., infinite probabilities!)
Synthesis of both should shed light on the first
moments of universe (« big bang »), on black
holes, and on the problem of why the vacuum
energy is so small.
Towards a solution: string (M

)theory?
In string theory, the fundamental quanta are extended, one

dimensional objects
(in original formulation)
String theory predicts gravity. It incorporates it in a manner which is perturbatively
consistent with quantum mechanics. It also contains the other fundamental forces,
thereby unifying all the fundamental interactions.
Supersymmetry is an important ingredient.
Beyond general relativity
Atom ~ 10

8
cm
Nucleus ~ 10

13
cm
String ~10

33
cm
Recent developments have merged known consistent string models into a single framework,
called « M

theory ».
String theory has revolutioned further our conceptions of space and time:
•
Extra spatial dimensions (total of 10, 11, 26 (?))
•
Number of spacetime dimensions depends on formulation
•
Topology can be changed
•
Impossibility to probe to arbitrarily small distance (minimum size)
… but we are still lacking a fundamental formulation of string theory that would enable us to
truly go beyond perturbation theory (non

perturbative techniques (eg dualities) still in infancy).
M

theory
SYMMETRIES: THE KEY?
Symmetry = invariance of the laws of physics under
certain changes in the point of view
Symmetries play a central role in the formulation of
fundamental theories (Lorentz invariance and
special relativity, internal symmetries and non

gravitational interactions, symmetry among
arbitrary reference frames and general relativity)
What are the underlying
symmetries of M

theory?
THE FIVE PLATONIC SOLIDS
Tetrahedron {3,3}
Octahedron {3,4}
Cube {4,3}
Icosahedron {3,5}
Dodecahedron {5,3}
http://home.teleport.com/~tpgettys/platonic.shtml
http://www.math.nmsu.edu/breakingaway/Lessons/barrels_casks_and_flasks/Local_images/shapes3.gif
(Convex) Regular polygons
{p}
s
2
= 1
Symmetry groups
All Euclidean isometries are products of reflections
Symmetry groups of regular polytopes are all finite reflection groups
(= groups generated by a finite number of reflections)
Number of generating reflections = dimension of space
Reflection in a line (hyperplane)
Dihedral groups
I
2
(3), order 6
I
2
(4), order 8
I
2
(5), order 10
etc …
I
2
(6), order 12
1
2
3
(s
1
)
2
=1,
(s
2
)
2
=1,
(s
1
s
2
)
p
= 1
(fundamental domain in red)
{3}
1
2
3
4
{4}
1
2
3
4
5
{5}
1
2
3
4
5
6
{6}
Coxeter Groups
The previous groups are examples of Coxeter groups: these are (by
definition) generated by a finite set of reflections s
i
obeying the
relations:
(s
i
)
2
= 1
;
(s
i
s
j
)
m
ij
= 1
with m
ij
= m
ji
positive integers (=1 for i = j and >1 for different i,j’s)
Notation: (s r)
p
= 1
angles between reflection axes:
p
/p
no line if p = 2
p not written when it is equal to 3
(2 lines if p = 4, 3 lines if p = 6)
p
s
r
Crystallographic dihedral groups
p = 3, 4, 6
A
2
B
2
–
C
2
G
2
A
2
B
2
/C
2
G
2
G
6
8
12
N
3
4
6
Hexagonal lattice
Square lattice
G = group order
N = number of reflections
Symmetries of Platonic Solids
G
N
Tetrahedron
24
6
Cube and
octahedron
48
9
Icosahedron and
dodecahedron
120
15
A
3
B
3
/C
3
5
H
3
G is in all cases a Coxeter group
{s
1
, s
2
, s
3
}; (s
i
)
2
= 1; (s
i
s
j
)
m
ij
= 1; m
ij
= 2,3,4,5 (i different from j)
H
3
is not crystallographic
List of Finite Reflection Groups
(= Finite Coxeter Groups)
G
N
A
n
(n+1)!
n(n+1)/2
B
n
/
C
n
2
n
n!
n
2
D
n
2
n

1
n!
n(n

1)
E
6
2
7
3
4
5
36
E
7
2
10
3
4
5 7
63
E
8
2
14
3
5
5
2
7
120
F
4
2
7
3
2
24
G
2
12
6
H
3
120
15
H
4
14400
60
Coxeter graphs of finite Coxeter groups
(source: J.E. Humphreys,
Reflection Groups and
Coxeter Groups
, Cambridge University Press 1990)
Comments
•
In dimensions > 4, there are only 3 regular polytopes: the regular n

simplex
(triangle, tetrahedron …), the cross polytope (square, octahedron …) and its
dual, the hypercube (square, cube …). The symmetry group of the regular n

simplex is A
n
, that of the cross polytope and of the hypercube is B
n
(~ C
n
).
•
In dimension 4, there are 6 (convex) regular polytopes. Besides the three just
mentioned, there are:

the 24

cell {3,4,3} with symmetry group F
4
(24 octahedral faces); and

the 120

cell {5,3,3} and its dual, the 600

cell {3,3,5}
with symmetry group H
4
(120 dodecahedra in
one case, 600 tetrahedra in the other).
•
H
3
and H
4
are not crystallographic.
•
D
n
, E
6
, E
7
and E
8
are finite reflection groups but are not symmetry groups of
regular polytopes (generalization).
•
Fundamental domain is always a (spherical) simplex
•
A very nice reference: H.S.M. Coxeter,
Regular polytopes
, Dover 1973
Affine Reflection Groups
In previous cases, the hyperplanes
of reflection contain the origin and
thus leave the unit sphere invariant
(«
spherical case
»)
One can relax this condition and
consider reflections about arbitrary
hyperplanes in Euclidean space
(«
affine case
»).
http://www.uwgb.edu/dutchs/symmetry/archtil.htm
Classification
Coxeter graphs of affine Coxeter groups
(source: J.E. Humphreys,
Reflection Groups and
Coxeter Groups
, Cambridge University Press 1990)
Remarks
•
Groups are infinite
•
Fundamental region is an
Euclidean simplex
Hyperbolic Reflection Groups
http://www.hadron.org/~hatch/HyperbolicTesselations/
One can also consider reflection groups in hyperbolic space.
These groups are also infinite.
Circle

limits (M.C. Escher)
http://www.dartmouth.edu/~matc/math5.pattern/circlelimitI.gif
http://www.pps.jussieu.fr/~cousinea/Tilings/poisson.9.gif
www.dagonbytes.com/gallery/ escher/escher12.htm
Classification
Hyperbolic simplex reflection groups exist only in hyperbolic spaces
of dimension < 10. In the maximum dimension 9, the groups are generated
by 10 reflections. There are three possibilities, all of which are relevant to
M

theory . (See e.g. Humphreys,
Reflection Groups and Coxeter Groups
,
for the complete list.)
E
10
BE
10
–
CE
10
DE
1
0
Crystallographic Coxeter Groups
and Kac

Moody Algebras
There is an intimate connection between crystallographic Coxeter groups
and Lie groups/Lie algebras.
Lie groups are continuous groups (e.g. SO(3)). The ones usually met in
physics so far are finite

dimensional (depend on a finite number of continuous
parameters). A great mathematical achievement has been the complete
classification of all finite

dimensional, simple Lie groups (Lie algebras are
the vector spaces of «
infinitesimal transformations
»).
Example
: unitary symmetry and permutation group.
The Coxeter group A
n
is isomorphic to the permutation group S
n+1
of n+1 objects.
Consider the group SU(n+1) of (n+1)

dimensional unitary matrices (of unit determinant).
SU(n+1) acts on itself:
U
U’= M* U M
(unitary change of basis, adjoint action)
By a change of basis, one can diagonalize U («
U is conjugate to an element in the Cartan
subalgebra
»). The Weyl = Coxeter group A
n
is what is left of the original unitary symmetry once U
has been diagonalized since the diagonal form of U is determined up to a permutation of the n+1
eigenvalues.
The connection between crystallographic finite Coxeter groups and finite

dimensional
simple Lie algebras is that the Coxeter groups are the «
Weyl groups
» of the Lie algebras.
Coxeter groups may thus signal a much bigger symmetry.
Infinite Coxeter groups
The same connection holds for infinite Coxeter groups; but in that case
the corresponding Lie algebra is infinite

dimensional and of the Kac

Moody
type.
Infinite

dimensional Lie algebras (i.e., infinite

dimensional symmetries)
are playing an increasingly important role in physics. In the gravitational
case, the relevant Kac

Moody algebras are of hyperbolic or Lorentzian
type (beyond the affine case).
These algebras are unfortunately still poorly understood.
Cosmological Billiards
Infinite Coxeter groups of hyperbolic (Lorentzian) type emerge when one investigates the
dynamics of gravity in extreme situations. For M

theory, it is E
10
that is relevant.
Dynamics of scale factors is chaotic in
the vicinity of a cosmological
singularity.
It is the same dynamics as that of a
billiard motion in the fundamental
Weyl chamber of a Kac

Moody
algebra.
Reflections against the billiard walls =
Weyl reflections
Source: H.C. Ohanian and R. Ruffini,
Gravitation and Spacetime
, Norton 1976
Examples
Pure gravity in 4 spacetime
Dimensions.
The billiard is a triangle
with angles
p
/2,
p
/3 and 0,
corresponding to the
Coxeter group (2,3,
infinity).
The triangle is the fundamental
region of the group
PGL(2,Z).
Arithmetical chaos
http://www.hadron.org/~hatch/HyperbolicTesselations/
M

theory and E
10
Truncation to 11

dimensional supergravity
Billiard is fundamental Weyl chamber of E
10
Is E
10
the symmetry algebra (or a subalgebra of the symmetry
algebra) of M

theory?
(perhaps E
10
(Z), E
11
, E
11
(Z))
Conclusions
•
Gravity is a fascinating and very lively area of research
•
It has many connections with other disciplines (geometry, group
theory, particle physics and the theory of the other fundamental
interactions, cosmology, astrophysics, nonlinear dynamics (chaos) …)
•
There are, however, major theoretical puzzles
•
As in the past, symmetry ideas will probably be a crucial ingredient
in the resolution of these puzzles
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