The Bach equation and the matching of spacetimes in the Conformal Cyclic Cosmology models

Abstract

We consider the problem of matching two spacetimes, the previous and present aeons, in the Conformal Cyclic Cosmology model. The common boundary between them inherits two sets of constraints -one for each solution of the Einstein field equations extended to the conformal boundaries. The previous aeon is assumed to be an asymptotically de Sitter spacetime, so the standard conformal formulation of the Einstein field equations suffice to derive the constraints on the future null infinity. For the future aeon, which is supposed to evolve from an initial singularity, they are obtained with the use of the Bach equation. This equation is regular at the past conformal infinity for conformally flat and conformally Einstein spacetimes, so we will mostly focus on them here. An example of the electrovacuum spacetime which does not fall into this class and has regular conformal Bach tensor will be discussed in the appendix.