Post on 18-Dec-2015
transcript
« 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.htmlhttp://www.astro.ucla.edu/~wright/cosmolog.htmhttp://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 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 t
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
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)
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}
s2 = 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
I2(3), order 6 I2(4), order 8 I2(5), order 10
etc …
I2(6), order 12
1
2
3
(s1)2=1,
(s2)2=1,
(s1s2)p = 1(fundamental domain in red)
{3}
12
3
4
{4}
12
3
4
5
{5}
12
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 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: (s r)p = 1angles between reflection axes: /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 groupsp = 3, 4, 6
A2
B2 – C2
G2
A2 B2/C2 G2
|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
Tetrahedron24 6
Cube and octahedron 48 9
Icosahedron and dodecahedron 120 15
A3
B3/C3
5
H3
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)
H3 is not crystallographic
List of Finite Reflection Groups (= Finite Coxeter Groups)
|G| N
An (n+1)! n(n+1)/2
Bn/
Cn
2n n! n2
Dn 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
Coxeter graphs of finite Coxeter groups(source: J.E. Humphreys, Reflection Groups andCoxeter 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 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 GroupsIn previous cases, the hyperplanesof reflection contain the origin andthus leave the unit sphere invariant(« spherical case »)
One can relax this condition and consider reflections about arbitraryhyperplanes 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 andCoxeter 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.gifhttp://www.pps.jussieu.fr/~cousinea/Tilings/poisson.9.gif
www.dagonbytes.com/gallery/ escher/escher12.htm
ClassificationHyperbolic simplex reflection groups exist only in hyperbolic spacesof dimension < 10. In the maximum dimension 9, the groups are generatedby 10 reflections. There are three possibilities, all of which are relevant toM-theory . (See e.g. Humphreys, Reflection Groups and Coxeter Groups,for the complete list.)
E10
BE10 – CE10
DE10
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 inphysics so far are finite-dimensional (depend on a finite number of continuousparameters). A great mathematical achievement has been the completeclassification of all finite-dimensional, simple Lie groups (Lie algebras arethe vector spaces of « infinitesimal transformations »).
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.
The connection between crystallographic finite Coxeter groups and finite-dimensionalsimple 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 casethe corresponding Lie algebra is infinite-dimensional and of the Kac-Moodytype.
Infinite-dimensional Lie algebras (i.e., infinite-dimensional symmetries)are playing an increasingly important role in physics. In the gravitationalcase, the relevant Kac-Moody algebras are of hyperbolic or Lorentzian type (beyond the affine case).
These algebras are unfortunately still poorly understood.
Cosmological BilliardsInfinite Coxeter groups of hyperbolic (Lorentzian) type emerge when one investigates thedynamics 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 spacetimeDimensions.
The billiard is a triangle with angles /2, /3 and 0,corresponding to the Coxeter group (2,3,infinity).
The triangle is the fundamentalregion 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