Solutions of Shape Dynamics

This chapter deals with the most important results in SD, namely, the classical solutions of the theory in which the equivalence with (GR) breaks down. Firstly, I study the case of homogeneous but not isotropic cosmologies, known as ‘Bianchi IX’ universes in detail. In this case, each solution that reaches the big bang singularity can be continued uniquely through it, just by requiring continuity of the conformally- and scale-invariant degrees of freedom. The result is a couple of cosmological solutions with opposite orientation glued at the big bang. This result is more general than the homogeneous case, and can be extended to a large class of solutions if the BKL conjecture is valid. In the case of spherically symmetric solutions one has to couple gravity to some form of matter in order to have dynamically non-trivial degrees of freedom. The simplest case is a series of concentric infinitely thin shells of dust in a universe with the topology of a three-sphere. In this case too a departure from the dynamics of (GR) is seen, that manifests itself in a failure of the CMC slicing when one of the shells collapses (no spacetime corresponding to that solution of SD exists). The conformally invariant degrees of freedom, again, seem to still be regular when this happens. In the last part of the chapter I will discuss the sense in which one can talk about asymptotically flat solutions of SD, and past results in this regime.