100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravitational Billiards, Dualities and Hidden Symmetries 49valid for free scalar fields in two dimensions. The main difference is that,whereas in the free field case, and more generally for p-form gauge theoriesin higher dimensions, a second dualization brings us back to the fieldfrom which we started (modulo integration constants), iterating the dualitytransformations (24) and (25) does not, as we pointed out already.It is therefore the intrinsic non-linearity of Einstein’s theory that explainsthe emergence of an infinite chain of dualizations, and consequently of aninfinite dimensional symmetry.3. Gravitational Billiards and Kac-Moody AlgebrasThe duality transformations reviewed in the previous section are invariancesof mutilated versions of Einstein’s theory only. On the other hand, what weare really after, are symmetries that would not require any such truncations.The symmetries we are about to discuss next considerably extend the onesdiscussed so far, but have not actually been shown to be symmetries ofEinstein’s theory, or some extension thereof. There are two reasons for this.First, the full gravitational field equations are far more complicated thanthe truncations discussed in the foregoing section — as evidenced by thecircumstance that no exact solutions appear to be known that would notmake use of some kind of symmetry reduction in one way or another (inthe appropriate coordinates). Consequently, any extension of the knownsymmetries to the full theory, which by necessity would be very non-local,will not be easy to identify. The second difficulty is that the Lie algebrasthat are conjectured to arise in this symmetry extension belong to theclass of indefinite Kac Moody algebras. However, after more than 35 yearsof research in the theory of Kac Moody algebras, we still do not knowmuch more about these algebras beyond their mere existence — despite thefact that they can be characterized by means of a simple set of generatorsand relations! c The main encouragement therefore derives from the factthat there exists this link between these two seemingly unrelated areas,which provides more than just a hint of an as yet undiscovered symmetryof Einstein’s theory. A key role in deriving these results was played byan analysis of Einstein’s equations near a spacelike singularity in terms ofgravitational billiards, to which we turn next.c See remarks after Table 1 to appreciate the challenge.