Thread by @WKCosmo

A few words about this paper, which was a lot of fun to write. 1/N https://t.co/PRg3YHPvDr

https://twitter.com/WKCosmo/status/1682195003731329026

Most of the heavy lifting on this project was done on a five-week visit to @iitmadras in Chennai in January, working with my awesome collaborators L. Sriramkumar and @suvashis_maity.

This is why we do these research visits! You simply cannot do this kind of thing remotely. You have to be in the place, talking at the blackboard together. There is no substitute.

What we did in the paper is to revisit the famous Borde-Guth-Vilenkin theorem, which showed that inflationary spacetimes are geodesically incomplete.

"Geodesic completeness" concerns itself with the question of whether or not spacetime of the universe is smooth and well-behaved, or ends at a singularity in the past, or the future, or both.

The big bang is an example of a past singularity: 13.8 billion years ago, everything smunched together to infinite density, and our equations describing it all blow up.

Inflation changes this a bit: you get rid of the initial singularity and replace it with the end of an earlier epoch of inflation.

This raises the question: can you get rid of singularities in altogether, and make the universe infinitely old?

BGV showed that the answer is NO, as long as the universe is expanding! You can — as in the case of inflation — push that singularity as far back in time as you like, but you can't get rid of it.

My grad student Nina Stein and I recently extended this result to certain kinds of bouncing cosmologies as well, with cycles of expansion and contraction, showing that these too have a singularity lurking in their past.

arxiv.org/abs/2110.15380

In this new paper, we tried to look at the BGV Theorem in a very general way — not just smooth cosmological spacetimes — with as few assumptions as possible about the symmetry of the spacetime.

We do this by following the path (called a "world line") of an observer through space and time, and asking what that observer sees.

So we measure time as the time measured on a clock carried by that observer, and we measure space as a set of nested surfaces perpendicular to the observer's world line.

So we slice 4-dimensional spacetime up into a stack of 3-dimensional surfaces, like the pages of a book, called a "foliation".

You can define a generalized notion of expansion in terms of how fast neighboring world lines diverge, or pull apart, as they extend forward in time.

The important point is that this does not depend on the global properties of the spacetime, but only how it behaves locally, or nearby the observer you're following.

What we showed is that you can re-write this expansion in terms of how much the perpendicular spatial surfaces — the pages of the book — bend, called their extrinsic curvature.

What this means is that the total history of the expansion measured along any interval on that observer's world line, from the infinite past to the infinite future, depends only on the properties of the perpendicular spatial surfaces at the boundary.

This is like being able to infer what happens in the whole book just by reading its first and last pages. It is similar to holography in that regard.

This does not depend on any assumptions about the spacetime. It's just a general fact about geometry, which makes it very powerful.

Now imagine following the path of an observer who sees time extend infinitely into both the past and the future, with one caveat: that observer sees a net positive amount of expansion.

Now consider a second observer, who is moving relative to the first observer.

The BGV Theorem then says that, as long as the first observer sees his local neighborhood expanding on average, the moving observer will encounter a singularity at FINITE time into the past.

His world line (or geodesic) will be incomplete.

We went further and looked at sets of such observers, and found the rmarkable result that the stationary and moving observers see themselves in different, but overlapping universes!

If the stationary observer sees a geometrically flat spacetime, extending infinitely into the past and the future, the moving observer sees a geometry with negative curvature, which is inevitably singular at some point in the finite past! (This is super-weird.)

The two sets of observers see DIFFERENT METRICS.

From the viewpoint of the non-singular metric, the singular metric looks like the nucleation of a spherical bubble, expanding outward at the speed of light. INSIDE the bubble is an infinite universe, with negative curvature.

The singularity is at the bubble wall, which the stationary observer sees as a spatial boundary, but the moving observer sees as being in his PAST.

Note that this is the same spacetime, just seen from two different viewpoint!

Now we have a fully geometric picture of the BGV Theorem, which lets us do new things.

For example, it is now easy to see that if you turn this around and assume that, instead of net expansion, there is net contraction, the expanding bubble turns into a collapsing "antibubble", which has a singularity in the FUTURE.

This is where Penrose's Conformal Cyclic Cosmology comes in.

Penrose's construction of a cyclic universe is done by stitching together the infinite future of one spacetime and the infinite past of another, and then the infinite future of that one onto the infinite past of yet another, and so on to infinity.

Exactly HOW one does this is left a little vague, so we looked at the case which is relevant for realistic cosmologies, in which the universe at infinite future time is dominated by a cosmological constant.

In this case, the geometric formulation of the BGV Theorem immediately tells you that the bubble solution in the first universe (which Penrose calls an "aeon"), which is singular in the past, maps onto an antibubble solution in the next aeon.

The antibubble solution is singular on its boundary in the FUTURE.

So BGV tells us — at least in this case — that the CCC construction doesn’t get rid of the singularities, it just repeats them in an endless cycle of past-singular expanding bubbles and future-singular collapsing antibubbles.

A big open question here is, what is the nature of these singularities? Are they physical things, or just an illusion, an artifact of the choice of coordinate system?

We don't answer that question in our paper. But unbeknownst to us, while we were completing our work, @ghazalgesh Eric Ling, and Jerome Quintin were working on exactly that, and put a beautiful paper out in May.

arxiv.org/abs/2305.01676

@ghazalgesh This is a crazy-hard mathematical paper, but the result is that the bubble singularities are (probably) physically real, and not coordinate artifacts, in any realistic universe not protected by very strong symmetries. Wow!

The bottom line here is that expanding universes always have singularities hiding in their past, and even if you stitch a bunch of them together at infinity, a la Penrose, you just end up with an ever-multiplying tower of more singularities.

There is no free lunch.

/FIN