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

52 H. Nicolaidegrees of freedom, this action is further simplified to⎡S[β a ] = 1 ∫ ∫d∑d d x dx 0 Ñ −1 (⎣) ( d∑2 ˙βa −4a=1 a=1≡ 1 ∫ ∫d d x dx 0 Ñ −1 G ab ˙βa ˙βb4˙β a ) 2⎤⎦(36)where G ab is the restriction of the superspace metric (à la Wheeler-DeWitt)to the space of scale factors. A remarkable, and well known property of thismetric is its indefinite signature (− + · · · +), with the negative sign correspondingto variations of the conformal factor. This indefiniteness will becrucial here, because it directly relates to the indefiniteness of the generalizedCartan-Killing metric on the associated Kac Moody algebra. In theHamiltonian description the velocities ˙β a are replaced by their associatedmomenta π a ; variation of the lapse Ñ yields the Hamiltonian constraintH = ∑ aπ 2 a − 1d − 1( ∑aπ a) 2≡ G ab π a π b ≈ 0 (37)Here G ab is the inverse of the superspace metric, i.e. G ac G bc = δ a c . Theconstraint (37) is supposed to hold at each spatial point, but let us concentrateat one particular spatial point for the moment. It is easy to checkthat (37) is solved by the well known conditions on the Kasner exponents.In this approximation, one thus has a Kasner-like metric at each spatialpoint, with the Kasner exponents depending on the spatial coordinate. Interms of the β-space description, we thus have the following picture of thedynamics of the scale factors at each spatial point. The solution to theconstraint (37) corresponds to the motion of a relativistic massless particle(often referred to as the ‘billiard ball’ in the remainder) moving in theforward lightcone in β-space along a lightlike line w.r.t. the ‘superspacemetric’ G ab . The Hamiltonian constraint (37) is then re-interpreted as arelativistic dispersion relation for the ‘billiard ball’.Of course, the above approximation does not solve the Einstein equations,unless the Kasner exponents are taken to be constant (yielding thewell known Kasner solution). Therefore, in a second step one must nowtake into account the spatial dependence and the effects of non-vanishingspatial curvature, and, eventually, the effect of matter couplings. At firstsight this would seem to bring back the full complications of Einstein’sequations. Surprisingly, this is not the case. Namely, one can show (at leastheuristically) that 65