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
Deflection of light
http://math.ucr.edu/home/baez/gr/gr.html
http://www.astro.ucla.edu/~wright/cosmolog.htm
http://home.fnal.gov/~dodelson/welcome.html
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
Gravity slows down time
phyun5.ucr.edu/~wudka/Physics7/ Notes_www/node89.html
Clocks on first floor tick more slowly than
clocks on top of the building (roughly 1 s
per 3 x 106 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 Dt
Dt 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?
Beyond general relativity
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.
Atom ~ 10-8 cm
Nucleus ~ 10-13 cm
String ~10-33 cm
M-theory
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).
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 nongravitational 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
(Convex) Regular polygons
{p}
http://www.math.nmsu.edu/breakingaway/Lessons/barrels_casks_and_flasks/Local_images/shapes3.gif
Symmetry groups
Reflection in a line (hyperplane)
s2 = 1
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
Dihedral groups
5
3
4
4
{3}
2
{5}
3
{4}
2
1
I2(3), order 6
3
2
1
I2(4), order 8
1
I2(5), order 10
(s1)2=1,
6
5
{6}
4
etc …
(s2)2=1,
3
2
1
I2(6), order 12
(s1s2)p = 1
(fundamental domain in red)
Coxeter Groups
The previous groups are examples of Coxeter groups: these are (by
definition) generated by a finite set of reflections si obeying the
relations:
(si)2 = 1;
(sisj)mij = 1
with mij = mji positive integers (=1 for i = j and >1 for different i,j’s)
Notation:
angles between reflection axes: p/p
p
(s r)p = 1
s
r
no line if p = 2
p not written when it is equal to 3
(2 lines if p = 4, 3 lines if p = 6)
Crystallographic dihedral groups
Hexagonal lattice
p = 3, 4, 6
A2
B2 – C2
G2
A2
B2/C2
G2
|G|
6
8
12
N
3
4
6
Square lattice
|G| = group order
N = number of reflections
Symmetries of Platonic Solids
G is in all cases a Coxeter group
{s1, s2, s3}; (si)2 = 1; (sisj)mij = 1; mij = 2,3,4,5 (i different from j)
|G|
N
24
6
48
9
120
15
Tetrahedron
A3
Cube and
octahedron
B3/C3
Icosahedron and
dodecahedron
5
H3
H3 is not crystallographic
List of Finite Reflection Groups
(= Finite Coxeter Groups)
Coxeter graphs of finite Coxeter groups
(source: J.E. Humphreys, Reflection Groups and
Coxeter Groups, Cambridge University Press 1990)
|G|
N
An
(n+1)!
n(n+1)/2
Bn/
Cn
Dn
2n n!
n2
2n-1 n!
n(n-1)
E6
27 34 5
36
E7
210 34 5 7
63
E8
214 35 52 7
120
F4
27 32
24
G2
12
6
H3
120
15
H4
14400
60
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 nsimplex is An, that of the cross polytope and of the hypercube is Bn (~ Cn).
• 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 F4
(24 octahedral faces); and
- the 120-cell {5,3,3} and its dual, the 600-cell {3,3,5}
with symmetry group H4 (120 dodecahedra in
one case, 600 tetrahedra in the other).
• H3 and H4 are not crystallographic.
• Dn, E6, E7 and E8 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
Remarks
• Groups are infinite
• Fundamental region is an
Euclidean simplex
Coxeter graphs of affine Coxeter groups
(source: J.E. Humphreys, Reflection Groups and
Coxeter Groups, Cambridge University Press 1990)
Hyperbolic Reflection Groups
One can also consider reflection groups in hyperbolic space.
These groups are also infinite.
http://www.hadron.org/~hatch/HyperbolicTesselations/
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.)
E10
BE10 –
CE10
DE1
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 »).
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.
Example: unitary symmetry and permutation group.
The Coxeter group An is isomorphic to the permutation group Sn+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 An 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.
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 E10 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 E10
Truncation to 11-dimensional supergravity
Billiard is fundamental Weyl chamber of E10
Is E10 the symmetry algebra (or a subalgebra of the symmetry
algebra) of M-theory? (perhaps E10(Z), E11, E11(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