Ricci Time in Lemaˆıtre-Tolman Model and Block Universe

Abstract It is common to think of our universe according to the “block universe” idea, which says that spacetime consists of many “stacked” 3-surfaces varied as a function of some kind of proper time τ . Standard ideas do not distinguish past and future, but Ellis’ “evolving block universe” tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. The main idea of this work is to investigate the shape of the {τ= constant} surfaces relative to the the null surfaces, and determine what makes these surfaces timelike or spacelike. We use the Lemaˆıtre-Tolman metric as our inhomogeneous spacetime model, and we find the necessary and sufficient conditions for these {τ= constant} surfaces to be spacelike or timelike. Furthermore, we indicate whether or not timelike surfaces appear inside black holes and other strong gravity domains, by determining the location of the timelike regions relative to the apparent horizon. Based on this idea, we find that the regions where these surfaces become timelike are often close to the apparent horizons, but always outside them, and in particular timelike regions occur outside black holes. They are always spacelike near the big bang, and at late times (near the crunch or the extreme far future), they are only timelike under special circumstances.