A New Geometry for Einstein's Theory of Relativity

As a one-time mathematician, this was a really fascinating article. The similarities seem to be entirely coincidental, but what would have been my doctoral dissertation was also about generalizing some concepts from smooth manifolds to a "non-smooth" setting, and the crux of my work also hinged on optimal transport.

Actually I feel optimal transport is a pretty underrated concept in both pure and applied math, and I would have loved to explore it had I continued in academia. But oh well, one must make choices in life...

This is all really cool, but is getting new singularity theorems really a positive sign? Like, my understanding was that it was generally hoped that an improved, quantum theory of gravity would eliminate such singularities -- that such singularities were generally considered to be non-physical artifacts that occur in GR due to its deficiencies at the most extreme scales (where quantum gravity would be relevant), not that they are in fact real and physical. So I'd consider it a better sign if these predicted black holes, which we see, but without singularities!

Small world. My graduate research was precisely on this topic as well. I was going in a more algebraic direction, though. My master's thesis was essentially about different discrete analogues of curvature using cooked-up cohomological constructions.

I really wish academia consistently provided as much security as industry. Would have loved to continue this line of research.

I can't simply help but think that optimal transport is intricately linked to the principle of least action (and as we know POLA is everywhere in nature). At the end, natural interactions seem to be one big optimization problem.

It is a positive sign, and here are three reasons why.

First, even if space is smooth, it is sometimes well-approximated by a singularity. In which case understanding that approximation has value for real universe predictions. As https://www.scientificamerican.com/article/naked-singulariti... points out, models strongly suggest that it is possible for naked singularities to form in GR. If we understand better how GR with singularities behaves, we may be able to make testable predictions about what astronomers should look for to verify them.

Second, it may be that the right quantum theory of gravity, contains singularities after all. QM is filled with smooth fields that are quantized particles. For example smooth electromagnetic waves give rise to discrete photons. Shouldn't we expect that a graviton, in the right quantized particle, also looks like a discrete particle? In that case, shouldn't it be some kind of singularity? If so, then a better understanding of singularities in GR may help us find a unified theory.

And third, extending from a smooth model to one with singularities, may result in developing better mathematical tools. For a historical example, consider the development of distributions such as the Dirac delta as an extension of theories built using Calculus on smooth functions. There is a chance that history will repeat. But we won't know until we try to develop these new tools.