{centering}
Poincaré, Relativity, Billiards and Symmetry
Thibault Damour
Institut des Hautes Etudes Scientifiques,
35, Route de Chartres, F91440 BuressurYvette, France
This review is made of two parts which are related to Poincaré in different ways. The first part reviews the work of Poincaré on the Theory of (Special) Relativity. One emphasizes both the remarkable achievements of Poincaré, and the fact that he never came close to what is the essential conceptual achievement of Einstein: changing the concept of time. The second part reviews a topic which probably would have appealed to Poincaré because it involves several mathematical structures he worked on: chaotic dynamics, discrete reflection groups, and Lobachevskii space. This topic is the hidden role of KacMoody algebras in the billiard description of the asymptotic behaviour of certain Einsteinmatter systems near a cosmological singularity. Of particular interest are the Einsteinmatter systems arising in the lowenergy limit of superstring theory. These systems seem to exhibit the highestrank hyperbolic KacMoody algebras, and notably , as hidden symmetries.
Contents
1 POINCARÉ and RELATIVITY
1.1 Some significant biographical dates
Starting in 1886, Poincaré holds the chair of “Physique mathématique et calcul des probabilités” at the Sorbonne. His 1888 lectures are about Maxwell’s theories and the electromagnetic theory of light. In 1889, 1892 and 1899, he lectures on the works by Helmholtz, Hertz, Larmor and, especially, Lorentz. His lectures give him the matter of four books in which he expounds all the different theories. This pedagogical work makes him fully aware of the state of the art in modern electromagnetic theories, and establishes him as a renowned expert in mathematical physics. He exchanges a correspondence with Lorentz (who keeps him informed of his work), and starts writing some research papers on electromagnetic theory.
In 1893, Poincaré becomes a member of the “Bureau des Longitudes”. It is the time where one starts to use telegraphic signals for synchronizing clocks (Bréguet, 1857; Le Verrier, 1875,…) and for measuring longitudes (see [1]).
In 1900, the “Congrès International de Physique” takes place in Paris, and Poincaré gives an invited review talk on the “Relations entre la physique expérimentale et la physique mathématique”. Four years later, on September 24, 1904, during the “Congrès international des Arts et des Sciences” of SaintLouis, Missouri (USA), Poincaré gives an invited review talk on “L’état actuel et l’avenir de la physique mathématique”.
In 1902, Poincaré publishes his popular book “La Science et l’hypothèse”; more than 16000 copies of this book have been sold. In 1905, he publishes his second popular book: “La valeur de la Science”. Two other popular books shall come out later: “Science et Méthode”, in 1908, and “Dernières Pensées”, posthumously, in 1913.
1.2 Selected citations and contributions of Poincaré to Relativity
In 1898, in a paper entitled “La Mesure du temps” [2], he writes: “Nous n’avons pas l’intuition directe de la simultanéité, pas plus que celle de l’égalité de deux durées.” He discusses in detail the fact that, in order to define time and simultaneity, one must admit, as postulates, some “rules”, e.g. that the velocity of light is constant and isotropic, and then that one must correct for the nonzero transmission times when using telegraphic signals to synchronize faraway clocks.
In the paper “La théorie de Lorentz et le principe de réaction” [3], written in 1900 at the occasion of the 25th anniversary of Lorentz’s thesis, Poincaré discusses (as emphasized in [4]) the effect of an overall translation, at some speed , on the synchronization of clocks by the exchange of electromagnetic signals. More precisely, he works only to first order in , and notes that, if moving observers synchronize their watches by exchanging optical signals, and if they correct these signals by the transmission time under the (incorrect) assumption that the signals travel at the same speed in both directions, their watches will indicate not the “real time”, but the “apparent time”, say
(1.1) 
His main point is that the “apparent time” coincides with the formal mathematical variable introduced by Lorentz in 1895 under the name of “local time” (and used by him to show the invariance of Maxwell theory under uniform translations, to first order in ).
In the book ‘‘La Science et l’hypothèse”, dated from 1902^{1}^{1}1This seems to be the only work of Henri Poincaré read by Einstein before 1905., Poincaré writes suggestive sentences such as:

“Il n’y a pas d’espace absolu et nous ne concevons que des mouvements relatifs.”

“Il n’y a pas de temps absolu.”

“Nous n’avons pas l’intuition directe de la simultanéité de deux événe
ments.” 
“ je ne crois pas, malgré Lorentz, que des observations plus précises puissent jamais mettre en évidence autre chose que les déplacements relatifs des corps matériels.” [In fact, this is reprinted from his 1900 talk at the Congrès International de Physique.]
Poincaré recalls that when experiments testing effects of first order in all came out negative, Lorentz found a general explanation at this order . When experiments testing the order again gave negative results (MichelsonMorley 1887), one found an ad hoc hypothesis (namely the LorentzFitzGerald contraction). For Poincaré, this is clearly unsatisfactory:
“Non, il faut trouver une même explication pour les uns et pour les autres et alors tout nous porte à penser que cette explication vaudra également pour tous les termes d’ordre supérieurs et que la destruction mutuelle de ces termes sera rigoureuse et absolue”.
On Mai 27, 1904, Lorentz publishes his crucial paper: “Electromagnetic phenomena in a system moving with any velocity smaller than that of light”. This paper contains the full “Lorentz transformations” linking variables associated to a moving frame, say , to the “true, absolute” time and space coordinates . [Lorentz considers only as convenient mathematical variables.]
On September 24, 1904, at the SaintLouis “Congrès International”, and later in his 1905 popular book “La Valeur de la Science”, Poincaré mentions the “Principe de Relativité” among a list of the basic principles of physics.
Then he mentions that Lorentz’s “local time” is the (apparent) time indicated by moving clocks (say, and ) when they are synchronized by exchanging light signals and by (‘wrongly’ but conventionally) assuming the isotropy of the speed of light in the moving frame, i.e. the equality between the transmission times during the two exchanges and . [However, he does not write down any equations, so that it is not clear whether he is alluding to his previous first order in result, (1.1), or to an all order result (see below)].
He also mentions the existence of a “mécanique nouvelle” where inertia goes to as , and therefore, is a limiting speed. [Note, however, that this was a feature common to all the current electron dynamics: Lorentz’s, Abraham’s, etc.]
On June 5, 1905^{2}^{2}2Notice that Einstein’s paper on Relativity was received by the Annalen der Physik on June 30, 1905., Poincaré submits to the Comptes Rendus of the Académie des Sciences the short Note: “Sur la dynamique de l’électron” [5]. This is followed by a more detailed article [6] (received on July 23, 1905). In these papers:

he admits “sans restriction” le “Postulat de Relativité” and explores its consequences

he modifies and completes Lorentz’s 1904 paper (by giving the correct transformation laws for the electromagnetic field, and source, quantities).

he explains “dynamically” why each “electron” undergoes the Lorentz contraction by introducing a (negative) internal pressure holding the electron against its electric selfrepulsion

he proves the group structure of “Lorentz transformations” [a name that he introduces in these papers]

he proves the invariance of
(1.2) 
he introduces and makes the identification of Lorentz transformations with rotations in a 4dimensional euclidean space

he proves the addition law for velocity parameters
(1.3) 
he discusses the invariants and covariants of the Lorentz group, e.g.
(1.4) 
he demands that the Principle of Relativity apply to gravitation

he discusses possible relativistic laws of gravity (of an actionatadistance type, i.e. without assuming any explicit field content)

he mentions that relativistic “retarded” gravitational interactions propagating with velocity are in agreement with existing observational limits on (due to Laplace) because they all differ from Newton’s law only at the order

he speaks of “ondes gravifiques” both in the sense of retarded interaction and of emission of radiation

he concludes about the necessity of a more detailed discussion of deviations from Newtonian gravity.
In 19061907, Poincaré’s Sorbonne lectures [7] (published in 1953 !) are about “Les limites de la loi de Newton”:

In them Poincaré sets a limit on the ratio between the gravitational mass and the inertial mass, , by updating Laplace’s work on the polarization of the EarthMoon orbit by the Sun. [This effect has been rediscovered, in a different context, by Nordtvedt in 1968.]

he discusses observational consequences of (among many others possible modifications of Newton’s law) some selected “relativistic” laws of gravity and shows that their main observational effect is an additional advance of the perihelion of Mercury: e.g. he mentions that an electromagneticlike gravitational law (“spin1 exchange”) yields an additional perihelion advance of per century^{3}^{3}3This had been already derived by Lorentz in 1900. Poincaré knew, however, at this stage that Lorentz’s electromagneticlike gravitational law was just one possibility among many “relativistic laws”., instead of an observed value that he quotes as ^{4}^{4}4It is amusing to speculate about what would have happened if Poincaré had used the better value (obtained by the american astronomer Simon Newcomb at the end of the 19th century) of , and had noticed that the various relativistic results he was deriving were all integer submultiples of the observed value ( in the case of spin1 exchange). at the time)

he works out the synchronization of moving clocks, by the method he had already mentioned in 19001904, to all orders in and seems to conclude (see, however, below) that the result is exactly the (all orders) “local time” introduced by Lorentz in 1904
(1.5) 
he determines how one must modify the fundamental law of dynamics so that the principle of relativity holds
In 1908, in a paper entitled “La dynamique de l’électron” [8] (which is essentially reprinted in his 1908 book “Science et Méthode”), Poincaré speaks more about gravitational waves. More precisely, he mentions that the main observable effect of the “onde gravifique” emitted at infinity by an orbiting system (“onde d’accélération”) will be, because of radiation reaction in the source, a secular acceleration of the orbital frequency, i.e. a negative value of the orbital period : . This observable effect is exactly what has been measured in binary pulsars, such as PRS 1913+16, which provided the first direct proofs that gravity propagates with the velocity of light (see, e.g., [9]).
1.3 Assessment of Poincaré’s contributions to Relativity.
The above list of statements and results is certainly an impressive list of achievements! Some people (have) claim(ed) that Poincaré should share, with Einstein, the credit for discovering the “Theory of Relativity”. When discussing this matter one should carefully distinguish various aspects of Poincaré’s contributions.
Technically, it is true that Poincaré made important new contributions related to what is now called Special Relativity. Notably, the action of the Lorentz group on electromagnetic variables ; the group structure of Lorentz transformations; the invariance of the spacetime interval ; and the proposal to consider the “Principle of Relativity” as a general principle of physics applying, for instance, to gravitation and thereby restricting possible relativistic generalizations of Newton’s law. For all those technical achievements, it would be quite reasonable to imagine that, if a Nobel prize had been given for Special Relativity before the death of Poincaré in 1912, the prize could have been shared between Einstein, Lorentz and Poincaré.
However, at the conceptual level, it seems to me (in spite of contrary claims in the literature) that Einstein was the first one to make what is the crucial step of Special Relativity, namely proposing a revolutionary^{5}^{5}5Let us note that Max Planck was the first scientist who understood the revolutionary nature of the new einsteinian conceptual setup. In 1910 he wrote that “This principle [of Relativity] has brought about a revolution in our physical picture of the world, which, in extent and depth, can only be compared to that produced by the introduction of the Copernican world system.” He also wrote that “In boldness it [Special Relativity] probably surpasses anything so far achieved in speculative natural science, and indeed in philosophical cognition theory; nonEuclidean geometry is child’s play in comparison.”[10]. change in the concept ot time. Moreover, as we shall see below, Poincaré resisted till his death such a change in the concept of time.
This conceptual revolution in the notion of time is encapsulated in the “twin paradox”, i.e. in time dilation effects, much more than in any change of synchronization conventions. Indeed, it was the idea that the variable was “time, pure and simple” which led Einstein, for the first time, to think and predict that, independently of any synchronization convention, a clock moving away and then coming back will not mark the same time when it reconvenes with its “sister clock” that remained in inertial motion. It is true that Poincaré’s discussion of synchronization in a moving frame seems close to Einstein’s synchronization process, but, when looking more carefully at what Poincaré actually wrote, one finds that there is a world of difference between the two.
First, let us mention that all the papers of Poincaré dealing with clock synchronization and published before Einstein’s 1905 work on Special Relativity either dealt only with effects (at which order there are no time dilation effects), or contained no explicit formulas [as in his SaintLouis, September 1904 lecture]. The only explicit work of Poincaré on clock synchronization which keeps all orders in is posterior to Einstein’s 1905 paper on Relativity. It is contained in his 19067 Sorbonne lectures (published only in 1953 [7]) or in his 1908 paper [8].

When looking in detail at the results actually derived by Poincaré, both in [7] and (consistently) in [8], one finds that Poincaré actually derives the following expression for what he calls the “apparent time” (“temps apparent”) marked moving clocks in the way he advocates:
(1.6) with given by (1.5). In other words, is the result of Einstein for the “time” in the moving frame (previously introduced by Lorentz as a mathematically auxiliary “local time” variable).
The crucial point is that Poincaré’s synchronized time ,(1.6), differs from Einstein’s “time, pure and simple”, , in the moving frame precisely by the timedilation factor . In other words, though moving clocks marking the Poincaré time are desynchronized among themselves with respect to the absolute time (“temps vrai”) , (because of the term in (1.6)), they do beat the same “absolute time” as a clock at rest, , and do not exhibit any “twin paradox”. [This is consistent with Poincaré’s statements, both in his 1904 SaintLouis lecture, and in “La Valeur de la Science”, that, among two moving clocks “l’une d’elles retardera sur l’autre”. The context shows that he does not speak of Einstein’s time retardation effects linked to the factor but of the constant offset between the indications of two clocks moving with the same, uniform velocity .] It is true that in a subsequent paragraph of [7] Poincaré seems to identify the result ,(1.6), of his explicit calculation with the full Lorentz “local time” . However, it seems clear to me that, in doing that, Poincaré has missed “thinking” the crucial einsteinian revolutionary step. Mathematically, Poincaré knew that the variable with good properties was (and this is the “time variable” he uses in his important technical papers of 1905 [5, 6]), but physically he never thought, nor proposed, that a moving clock will mark the time (and will therefore exhibit a “twin paradox”).
Additional evidence for this limitation of the horizon of thought of Poincaré comes from other statements of his:

Poincaré always distinguishes “le temps vrai”, , from “le temps apparent”, or , and, similarly, he always thinks in terms of “absolute space”

Poincaré kept asking for a deeper (dynamical ?) reason behind the “relativity postulate”, and the appearance of the velocity of light () in possible relativistic laws of gravity

Poincaré had no firm theoretical a priori conviction in the “relativity principle”; e.g.
 In his 1908 paper [8] (and in his 1908 book on ‘‘Science et Méthode”^{6}^{6}6Though he added a last minute footnote stating that the more recent experiments of Bucherer agreed with relativistic dynamics.), commenting Kaufmann’s early experiments (that did not seem to quite agree with relativistic dynamics), he expresses doubts about the exact validity of the relativity principle: He writes that the latest experiments of Kaufmann
“ont donné raison à la théorie d’Abraham. Le Principe de Relativité n’aurait donc pas la valeur rigoureuse qu’on était tenté de lui attribuer; ”
By contrast, Einstein, commenting in 1907 the same experimental results, states that the agreement with relativity is rather good in first approximation, and that the deviations are probably due to systematic errors. Indeed, Einstein writes that it is a priori more probable that relativistic dynamics be correct, rather than Abraham’s dynamics, because the former is based on a general principle having wide ranging consequences for physics as a whole.
 in his 19061907 lectures, Poincaré concludes that the most probable explanation for the anomaly in Mercury’s perihelion advance is the existence of an inframercurial ring of matter. He has not enough trust in any of the possible relativistic gravitational theories he had introduced in 1905 to propose their effects as a likely explanation for it. [As we said above, had he noticed that they all gave an integer submultiple of the observed anomaly, he might have suspected that one of them might give the correct explanation.]
 in 1904 (SaintLouis), 1905 (“La valeur de la science”), and in 1908 (i.e. several years after Einstein’s famous September 1905 paper on ), Poincaré speaks of the recoil in reaction to the emission of electromagnetic waves and says : “Ce que nous avons envoyé au loin, ce n’est plus un projectile matériel, c’est de l’énergie et l’énergie n’a pas de masse”, i.e. “energy has no mass” ! (“Science et méthode”, livre III, chapter II, 1908)
 in 1912, a few months before his death, Poincaré writes [11] some sentences that have been quoted as evidence for Poincaré’s role in conceptualizing (or at least accepting the EinsteinMinkowski) spacetime as a physical structure. For instance, he writes: “Tout se passe comme si le temps était une quatrième dimension de l’espace; et comme si l’espace à quatre dimensions résultant de la combinaison de l’espace ordinaire et du temps pouvait tourner non seulement autour d’un axe de l’espace ordinaire [] mais autour d’un axe quelconque. [] dans la nouvelle conception l’espace et le temps ne sont plus deux entités entièrement distinctes et que l’on puisse envisager séparément, mais deux parties d’un même tout [] qui sont comme étroitement enlaçées ”. However, if one reads the full text, one realizes that Poincaré explains here a conception proposed by “some physicists”, and that he is not at all ready to accept this new conception (or “convention”). Indeed, he ends his text by writing:
“Aujourd’hui certains physiciens veulent adopter une convention nouvelle [] ceux qui ne sont pas de cet avis peuvent légitimement conserver l’ancienne pour ne pas troubler leurs vieilles habitudes. Je crois, entre nous, que c’est ce qu’ils feront encore longtemps.”
This last sentence, which constitutes the last words written by Poincaré on Special Relativity, shows clearly that Poincaré never believed in the physical relevance of the conceptual revolution brought by Einstein in the concept of time (and extended by Minkowski to a revolutionary view of the physical meaning of spacetime).
1.4 Poincaré on Einstein, concerning Relativity

Poincaré never mentioned Einstein’s work on relativity (neither in his papers or books, nor, as far as I know, in his letters). Poincaré seemed to be unaware of Einstein’s work during the years 19061909. His attention was probably brought to the work of Einstein and Minkowski only in the spring of 1909. [In April 1909 Poincaré gave some lectures in Göttingen, notably on “la mécanique nouvelle”, without mentioning the names of Einstein or Minkowski.]

Maybe he thought that
 technically, there was nothing new in Einstein’s work on Relativity
 conceptually, Einstein was ‘‘cheating”^{7}^{7}7I thank David Gross for a useful discussion on this point. because he simply assumed (kinematically) what had to be proven (dynamically) [as Lorentz thought, and as Poincaré’s electronpressure model contributed to proving]
 Einstein’s real contribution remained physically obscure to him, because Poincaré always thought that “apparent time” should be different from “real time” (while Einstein summarized his main contribution as being “the realization that an auxiliary term introduced by H. A. Lorentz and called by him ‘local time’ could be defined as ‘time pure and simple’.”).

As a final vista on the conceptual difference between Poincaré and Einstein, let us mention the following revealing anecdote. Poincaré and Einstein met only once, at the Solvay meeting of 1911. Maurice de Broglie (who was one of the secretaries of this first Solvay meeting) wrote (in 1954 [12]) the following:
“Je me rappelle qu’un jour à Bruxelles, comme Einstein exposait ses idées [sur la “mécanique nouvelle” c’estàdire relativiste], Poincaré lui demanda: ‘ Quelle mécanique adoptezvous dans vos raisonnements ?’ Einstein lui répondit :‘Aucune mécanique’, ce qui parut surprendre son interlocuteur.”
This conversation on “relativistic mechanics” (which contrasts the “dynamical approach” of Poincaré to the “kinematical” one of Einstein) is not reported in the official proceedings of the 1911 Solvay meeting which concerned the (old) theory of quanta.
2 RELATIVITY, BILLIARDS and SYMMETRY
2.1 Introduction and overview
A remarkable connection between the asymptotic behavior of certain Einsteinmatter systems near a cosmological singularity and billiard motions in the Weyl chambers of some corresponding Lorentzian Kac–Moody algebras was uncovered in a series of works [13, 14, 15, 16, 17, 18, 19]. This simultaneous appearance of billiards (with chaotic properties in important physical cases) and of an underlying symmetry structure (infinitedimensional Lie algebra) is an interesting fact, which deserves to be studied in depth. This topic would have pleased Poincaré because it involves several mathematical structures dear to his heart: notably, discrete reflection groups (and their fundamental polytope), Lobachevskii space and chaotic dynamics. Before explaining the techniques that have been used to uncover this connection, we will start by reviewing previous related works, and by stating the main results of this billiard/symmetry connection.
The simplest example of this connection concerns the pure Einstein system in dimensional spacetime. The Einstein’s equations requiring the vanishing of the Ricci tensor ( are nonlinear PDE’s for the metric components. Near a cosmological spacelike singularity, here chosen as , the spatial gradients are expected to become negligible compared to time derivatives (20, 21, 22] of the asymptotic behavior of the general solution of the Einstein’s equations, close to the singularity, and showed that it can be described as a chaotic [23, 24] sequence of generalized Kasner solutions. The Kasner metric is of the type ); this then suggests the decoupling of spatial points and allows for an approximate treatment in which one replaces the above partial differential equations by (a threedimensional family) of ordinary differential equations. Within this simplified context, Belinskii, Khalatnikov and Lifshitz (BKL) gave a description [
(2.7) 
where the constants obey^{8}^{8}8In the gauge, they also obey .
(2.8) 
An exact Kasner solution, with a given set of ’s and ’s, can be represented by a null line in a 3dimensional auxiliary Lorentz space with coordinates equipped with the metric given by the quadratic form above. The auxiliary Lorentz space can be radially projected on the unit hyperboloid or further on the Poincaré disk (i.e. on the hyperbolic plane ): the projection of a null line is a geodesic on the hyperbolic plane. See Figure 1.
BKL showed that, because of nonlinearities in Einstein’s equations, the generic solution behaves as a succession of Kasner epochs, i.e., as a broken null line in the auxiliary Lorentz space, or (after projection) a broken geodesic on the Poincaré disk. This broken geodesic motion is a “billiard motion” (seen either in Lorentzian space or in hyperbolic space). See Figure 2.
The billiard picture naturally follows from the Hamiltonian approach to cosmological behavior and was first obtained in the homogeneous (Bianchi IX)
fourdimensional case [25, 26] and then extended to higher spacetime dimensions with forms and dilatons [27, 28, 29, 30, 31, 15, 18, 32, 19]. Recent work
[19] has improved the derivation of the billiard picture by using the
Iwasawa decomposition of the spatial metric.
Combining this decomposition with the ArnowittDeserMisner [33] Hamiltonian formalism highlights
the mechanism by which all variables except the scale factors and the dilatons get asymptotically frozen.
The nonfrozen variables (logarithms of scale factors and dilatons) then undergo a billiard motion. This billiard
motion can be seen either in a dimensional Lorentzian space, or, after radial projection,
on dimensional hyperbolic space. Here, is the spacetime dimension and the number
of dilaton fields (see below for details). The Figures 1 and 2 correspond to the case , .
A remarkable connection was also established [13, 14, 15, 16, 17, 18, 19] between certain specific Einsteinmatter systems and Lorentzian KacMoody (KM) algebras [34]. In the leading asymptotic approximation, this connection is simply that the Lorentzian billiard table within which the motion is confined can be identified with the Weyl chamber of some corresponding Lorentzian KM algebra. This can happen only when many conditions are met: in particular, (i) the billiard table must be a Coxeter polyhedron (the dihedral angles between adacent walls must be integer submultiples of ) and ii) the billiard must be a simplex. Surprisingly, this occurs in many physically interesting Einsteinmatter systems. For instance, pure Einstein gravity in dimensional spacetime corresponds to the Lorentzian KM algebra [16] which is the overextension of the finite Lie algebra : for , the algebra is the Cartan matrix of which is given by
(2.9) 
Chaotic billiard tables have finite volume in hyperbolic space, while nonchaotic ones have infinite volume; as a consequence, chaotic billiards are associated with hyperbolic KM algebras; this happens to be the case for pure gravity when .
Another connection between physically interesting Einsteinmatter systems and KM algebras concerns the lowenergy bosonic effective actions arising in string and theories. Bosonic string theory in any spacetime dimension is related to the Lorentzian KM algebra [15, 17]. The latter algebra is the canonical Lorentzian extension of the finitedimensional algebra . The various superstring theories (in the critical dimension ) and theory have been found [15] to be related either to (when there are two supersymmetries in , i.e. for type IIA, type IIB and theory) or to (when there is only one supersymmetry in , i.e. for type I and II heterotic theories), see the table below. A construction of the Einsteinmatter systems related to the canonical Lorentzian extensions of all finitedimensional Lie algebras , , , , , , , and (in the above “billiard” sense) is presented in Ref. [17]. See also Ref. [35] for the identification of all hyperbolic KM algebras whose Weyl chambers are Einstein billiards.
The correspondence between the specific Einstein–threeform system (including a Chern–Simons term) describing the bosonic sector of 11dimensional supergravity (also known as the “lowenergy limit of theory”) and the hyperbolic KM group was studied in more detail in [18]. Reference [18] introduces a formal expansion of the field equations in terms of positive roots, i.e. combinations of simple roots of , , , where the ’s are integers . It is then useful to order this expansion according to the height of the positive root , defined as . The correspondence discussed above between the leading asymptotic evolution near a cosmological singularity (described by a billiard) and Weyl chambers of KM algebras involves only the terms in the field equation whose height is . By contrast, the authors of Ref. [18] managed to show, by explicit calculation, that there exists a way to define, at each spatial point , a correspondence between the field variables , (and their gradients), and a (finite) subset of the parameters defining an element of the (infinitedimensional) coset space where denotes the maximal compact subgroup of , such that the (PDE) field equations of supergravity get mapped onto the (ODE) equations describing a null geodesic in up to terms of height 30. A complementary check of the correspondence between dimensional supergravity and the model has been obtained in [37]. This result was further extended to the correspondence between the model and, both, massive 10dimensional IIA supergravity [38], and 10dimensional IIB supergravity [39].
These tantalizing results suggest that the infinitedimensional hyperbolic Kac–Moody group may be a “hidden symmetry” of supergravity in the sense of mapping solutions onto solutions (the idea that might be a symmetry of supergravity was first raised by Julia long ago [36, 40]). Note that the conjecture here is that the continuous group be a hidden symmetry group of classical supergravity. At the quantum level, i.e. for M theory, one expects only a discrete version of , say , to be a quantum symmetry. See [41] for recent work on extending the identification of [18] between roots of and symmetries of supergravity/Mtheory beyond height 30, and for references about previous suggestions of a possible role for . For earlier appearances of the Weyl groups of the series in the context of duality see [42, 43, 44]. A series of recent papers [45, 46, 47, 48, 49] has also explored the possible role of (a nonhyperbolic extension of ) as a hidden symmetry of M theory.
It is also tempting to assume that the KM groups underlying the other (special) Einsteinmatter systems discussed above might be hidden (solutiongenerating) symmetries. For instance, in the case of pure Einstein gravity in spacetime, the conjecture is that be such a symmetry of Einstein gravity. This case, and the correspondence between the field variables and the coset ones is further discussed in [19].
Note that rigorous mathematical proofs [31, 50, 32, 51]
concerning the PDE/billiard connection are only available
for ‘non chaotic’ billiards.
In the remainder of this paper, we will outline various arguments explaining the above results; a more complete derivation can be found in [19].
2.2 General Models
The general systems considered here are of the following form
(2.10) 
Units are chosen such that , is Newton’s constant and the spacetime dimension is left unspecified. Besides the standard EinsteinHilbert term the above Lagrangian contains a dilaton^{9}^{9}9The generalization to any number of dilatons is straightforward. field and a number of form fields (for ). The form field strengths are normalized as
(2.11) 
As a
convenient common formulation we adopt the Einstein conformal
frame and normalize the kinetic term of the dilaton with
weight one with respect to the Ricci scalar. The Einstein metric
has Lorentz signature and is used
to lower or raise the indices; its determinant is denoted by . The dots in the action (2.10) above
indicate possible modifications of the field strength by
additional Yang–Mills or Chapline–Mantontype couplings
[52, 53]. The real parameter measures the strength
of the coupling of to the dilaton. When , we assume
that so that there is only one dilaton.
2.3 Dynamics in the vicinity of a spacelike singularity
The main technical points that will be reviewed here are the following

near the singularity, , due to the decoupling of space points, Einstein’s PDE equations become ODE’s with respect to time.

The study of these ODE’s near , shows that the diagonal spatial metric components “” and the dilaton move on a billiard in an auxiliary dimensional Lorentz space.

All the other field variables ( freeze as .

In many interesting cases, the billiard tables can be identified with the fundamental Weyl chamber of an hyperbolic KM algebra.

For SUGRA, the KM algebra is . Moreover, the PDE’s are equivalent to the equations of a null geodesic on the coset space , up to height 30.
2.3.1 ArnowittDeserMisner Hamiltonian formalism
To focus on the features relevant to the billiard picture, we assume here that there are no Chern–Simons and no Chapline–Manton terms and that the curvatures are abelian, . That such additional terms do not alter the analysis has been proven in [19]. In any pseudoGaussian gauge and in the temporal gauge ( and , ), the ArnowittDeserMisner Hamiltonian action [33] reads
(2.12) 
where the Hamiltonian density is
(2.13)  
(2.14)  
(2.15)  
(2.16) 
and is the spatial curvature scalar. is the rescaled lapse. The dynamical equations of motion are obtained by varying the above action with respect to the spatial metric components, the dilaton, the spatial form components and their conjugate momenta. In addition, there are constraints on the dynamical variables,
(2.17)  
(2.18)  
(2.19) 
with
(2.20)  
(2.21) 
where the subscript stands
for spatially covariant derivative.
2.3.2 Iwasawa decomposition of the spatial metric
We systematically use the Iwasawa decomposition of the spatial metric and write
(2.22) 
where is an upper triangular matrix with ’s on the diagonal. We will also need the Iwasawa coframe ,
(2.23) 
as well as the vectorial frame dual to the coframe ,
(2.24) 
where the matrix is the inverse of , i.e., . It is again an upper triangular matrix with 1’s on the diagonal. Let us now examine how the Hamiltonian action gets transformed when one performs, at each spatial point, the Iwasawa decomposition 2.22 of the spatial metric. The kinetic terms of the metric and of the dilaton in the Lagrangian (2.10) are given by the quadratic form
(2.25) 
The change of variables corresponds to a point transformation and can be extended to the momenta as a canonical transformation in the standard way via
(2.26) 
Note that the momenta
(2.27) 
conjugate to the
nonconstant offdiagonal Iwasawa components are only
defined for ; hence the second sum in (2.26) receives
only contributions from .
2.3.3 Splitting of the Hamiltonian
We next split the Hamiltonian density (2.13) in two parts: , which is the kinetic term for the local scale factors and the dilaton , and , a “potential density” (of weight 2) , which contains everything else. Our analysis below will show why it makes sense to group the kinetic terms of both the offdiagonal metric components and the forms with the usual potential terms, i.e. the term in (2.14). Thus, we write
(2.28) 
with the kinetic term of the variables
(2.29) 
where denotes the inverse of the metric of Eq. (2.25). In other words, the right hand side of Eq. (2.29) is defined by
(2.30) 
where are the momenta conjugate to and , respectively, i.e.
(2.31) 
The total (weight 2) potential density,
(2.32) 
is naturally split into a “centrifugal” part linked to the kinetic energy of the offdiagonal components (the index referring to “symmetry,”), a “gravitational” part , a term from the forms, , which is a sum of an “electric” and a “magnetic” contribution and also a contribution to the potential coming from the spatial gradients of the dilaton .

“centrifugal” potential
(2.33) 
“gravitational” (or “curvature”) potential
(2.34) where
(2.35) and
(2.36) while is a polynomial of degree two in the first derivatives and of degree one in the second derivatives .

form potential
(2.37) which is a sum of an “electric” and a “magnetic” contribution. The “electric” contribution can be written as
(2.38) where and are the ”electric wall” forms,
(2.39) And the “magnetic” contribution reads,
(2.40) where and the are the magnetic linear forms
(2.41) 
dilaton potential
(2.42) (2.43) where
(2.44)
2.3.4 Appearance of sharp walls in the BKL limit
In the decomposition of the hamiltonian as , is the kinetic term for the ’s while all other variables now only appear through the potential which is schematically of the form
(2.45) 
where . Here are the linear wall forms already introduced above:
symmetry walls  
gravitational walls  
electric walls  
magnetic walls 
In order to take the limit which corresponds to tending to future timelike infinity, we decompose into hyperbolic polar coordinates , i.e.
(2.46) 
where are coordinates on the future sheet of the unit hyperboloid which are constrained by
(2.47) 
and is the timelike variable defined by
(2.48) 
which behaves like at the BKL limit. In terms of these variables, the potential term looks like
(2.49) 
The essential point now is that, since , each term becomes a sharp wall potential, i.e. a function of which is zero when , and when . To formalize this behavior we define the sharp wall function ^{10}^{10}10 One should more properly write , but since this is the only step function encountered here, we use the simpler notation . as
(2.50) 
A basic formal property of this function is its invariance under multiplication by a positive quantity. Because all the relevant prefactors are generically positive near each leading wall, we can formally write
(2.51)  
valid in spite of the increasing of the spatial gradients [19]. Therefore, the limiting dynamics is equivalent to a free motion in the space interrupted by reflections against hyperplanes in this space given by which correspond to a potential described by infinitely high step functions
(2.52) 
The other dynamical variables (all variables but the ’s) completely disappear from this limiting Hamiltonian and therefore they all get frozen as .
2.3.5 Cosmological singularities and Kac–Moody algebras
Two kinds of motion are possible according to the volume of the billiard table on which it takes place, i.e. the volume of the region where for , also characterized by the conditions,
(2.53) 
Depending on the fields present in the lagrangian, on their dilatoncouplings and on the spacetime dimension, the (projected) billiard volume is either of finite or infinite. The finite volume case corresponds to neverending, chaotic oscillations for the ’s while in the infinite volume case, after a finite number of reflections off the walls, they tend to an asymptotically monotonic Kasnerlike behavior, see Figure 3:
In Figure 3 the upper panels
are drawn in the Lorentzian space spanned by . The billiard tables are represented as
“wedges” in –dimensional (or dimensional, if there are
no dilatons) space, bounded by
hyperplanar walls on which the
billiard ball undergoes specular reflections. The upper left
panel is a (critical) “chaotic” billiard table (contained
within the space future light cone), while the upper
right one is a (subcritical) “nonchaotic” one (extending beyond
the light cone). The lower panels represent the corresponding
billiard tables (and billiard motions) after projection onto
hyperbolic space ( if there are no dilatons). The latter
projection is defined in the text by central projection onto
space (i.e. the unit hyperboloid , see the upper panels), and is
represented in the lower panels by its image in the Poincaré
ball (disk).
In fact, not all the walls are relevant for determining the billiard table. Some of the walls stay behind the others and are not met by the billiard ball. Only a subset of the walls , called dominant walls and here denoted are needed to delimit the hyperbolic domain. Once the dominant walls are found, one can compute the following matrix
(2.54) 
where
. By definition, the diagonal elements are all equal to 2. Moreover, in many interesting cases, the offdiagonal elements happen to be non positive integers. These are precisely the characteristics of a generalized Cartan matrix, namely that of an infinite KM algebra (see appendix A). As
recalled in the introduction, for pure gravity in spacetime dimensions, there are dominant
walls and the
matrix is exactly the generalized Cartan matrix of the hyperbolic KM
algebra which is hyperbolic for . More generally, bosonic string theory in
spacetime dimensions is related to the Lorentzian
KM algebra [15, 17] which is the canonical
Lorentzian extension of the finitedimensional Lie algebra .
The various superstring theories, in the critical dimension
, and theory have been found [15] to be related
either to (when there are two supersymmetries,
i.e. for type IIA, type IIB and theory) or to (when
there is only one supersymmetry, i.e. for type I and II
heterotic theories), see the table.
The hyperbolic KM algebras are those relevant for chaotic billiards since their fundamental Weyl chamber has a finite volume.
Theory  Corresponding Hyperbolic KM algebra 

Pure gravity in 

Mtheory, IIA and IIB Strings 

type I and heterotic Strings 

closed bosonic string in 

This table displays the Coxeter–Dynkin diagrams which encode the geometry of the
billiard tables describing the asymptotic cosmological behavior of General Relativity and of
three blocks of string theories: theory,
type IIA and type IIB superstring theories, type I and the two heterotic superstring theories, and
closed bosonic string theory in .
Each node of the diagrams represents a dominant wall of the
cosmological billiard. Each Coxeter diagram of a
billiard table corresponds to the Dynkin diagram of a (hyperbolic) KM algebra: ,
and .
The precise links between a chaotic billiard and its corresponding Kac–Moody algebra can be summarized as follows

the scale factors parametrize a Cartan element ,

the dominant walls correspond to the simple roots of the KM algebra,

the group of reflections in the cosmological billiard is the Weyl group of the KM algebra, and

the billiard table can be identified with the Weyl chamber of the KM algebra.
2.3.6 and a “small tension” limit of SUGRA
The main feature of the gravitational billiards that can be
associated with the KM algebras is that there exists a group
theoretical interpretation of the billiard motion: the asymptotic
BKL dynamics is equivalent (in a sense to be made precise below),
at each spatial point, to the asymptotic dynamics of a
onedimensional nonlinear model based on a certain
infinitedimensional coset space , where the KM group and
its maximal compact subgroup depend on the specific model. As
we have seen, the walls that determine the billiards are the dominant walls. For the KM billiards, they correspond to the simple roots of the KM algebra. As we discuss below, some of the
subdominant walls also have an algebraic interpretation in terms
of higherheight positive roots. This enables one to go beyond the
BKL limit and to see the beginnings of a possible identification
of the dynamics of the scale factors and of all the
remaining variables with that of a nonlinear model
defined on the cosets of the KM group divided by its
maximal compact subgroup [18, 19].
For concreteness, we will only consider one specific example here: the relation between the cosmological evolution of supergravity and a null geodesic on [18] where is the maximally compact subgroup of . The model is formulated in terms of a oneparameter dependent group element and its Lie algebra value derivative
(2.55) 
The action is
(2.56) 
with a lapse function whose variation gives rise to the Hamiltonian constraint ensuring that the trajectory is a null geodesic. The symmetric projection
(2.57) 
is introduced in order to define an evolution on the coset space. Here is the standard invariant bilinear form on ; is the “transpose” of defined with the Chevalley involution^{11}^{11}11The Chevalley involution is defined by as . This action is invariant under :
(2.58) 
Making use of the explicit Iwasawa parametrization of the generic group element together with the gauge choice (Borel gauge), one can write
with and
Using the commutation relations in form (see [38, 39] for other decompositions of the algebra) together with the bilinear form for , one obtains up to height 30 ^{12}^{12}12We keep only the generators and corresponding to the roots with height ( are simple roots and integers) ,
(2.59)  
where with
and all “contravariant indices” have been raised by . The “covariant” time derivatives are defined by (with )