Thanks SpeedFreek, that’s an excellent article! Not easy reading by any means, but at least it’s not encumbered by horrible tensor notation. (Dear God of the Cosmos, deliver me from tensor bundles, and leadeth me not into super Grassman manifolds …)
They seem to be assuming only one initial singularity. That must be either an oversight on the authors’ part, or else maybe there is some well-known (except to me) reason why Lorentzian geometry prohibits multiple initial singularities. Or maybe the results in the paper are actually still valid even with many Big Bangs, and they just didn’t mention it.
The paper does uniquely define a cosmological time function, but it only has nice properties if it satisfies the regularity conditions. Basically, that time since the BB is finite everywhere, and goes to 0 as you go back in time along a timelike geodesic.
So, it is still in question whether our Universe has only one BB, and if so, whether the time function defined in terms of it is regular.
Also, even though the time function is well-defined, how can we Earthlings know what it is? How can we say that, e.g., we are looking at light from a star that was emitted when the Universe was 5 billion years old?
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“That’s no moon.”
-Obi-Wan Kenobi