PLATONIC SOLIDS AND

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Oct 29, 2013 (3 years and 5 months ago)

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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