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 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?

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 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} Icosahedron {3,5} Cube {4,3} 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) s 2 = 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

3 2 1

I 2 (3), order 6 {3}

4 3 2 1

I 2 (4), order 8 {4}

5 4 3 2 1

I 2 (5), order 10 {5}

6 5 4 3 2 1

I 2 (6), order 12 {6} etc …

(s

1

)

2

=1, (s

2

)

2

=1, (s

1

s

2

)

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 s relations: i obeying the

(s i ) 2 = 1 (s i s j ) mij

;

= 1

with m ij = m ji positive integers (=1 for i = j and >1 for different i,j’s) Notation: (s r) p = 1

s p r

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)

Crystallographic dihedral groups

Hexagonal lattice p = 3, 4, 6 A 2 B 2 – C 2 G 2 Square lattice |G| N A 2 6 3 B 2 /C 2 8 4 G 2 12 6 |G| = group order N = number of reflections

Symmetries of Platonic Solids 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) |G| N Tetrahedron 24 6

A 3

Cube and octahedron 48 9 Icosahedron and dodecahedron

B 3 /C 3 5 H 3

120 15 H 3 is not crystallographic

List of Finite Reflection Groups (= Finite Coxeter Groups) A n B n / C n D n E 6 E 7 E 8 F 4 G 2 H 3 H 4 |G| (n+1)!

2 n n!

2 n-1 n!

2 7 3 4 5 2 10 3 4 5 7 2 14 3 5 5 2 7 2 7 3 2 12 120 14400 N n(n+1)/2 n 2 n(n-1) 36 63 120 24 6 15 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

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.)

E 10 BE 10 CE 10 – 0 DE 1

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 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.

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 http://www.hadron.org/~hatch/HyperbolicTesselations/

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

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