Projective Cosmology, and the topological failure of Einstein's General Theory
The graphic above is from my old, defunct, 1990s website, and I also borrowed it for chapter 12 of the book.
It shows a rather fun observerspace projection: if we assume that the universe is (hyper-) spherical, but we colour it in as it's seen to be rather than how we deduce it to be, expansion and Hubble shift result in a description in which things are more redshifted towards the universe's farside. Free-falling objects recede from us faster towards the apparent farside-point, as if they were falling towards some hugely massive object at the opposite end of the universe, and as if there was a corresponding gravitational field centred on the farside. At a certain distance between us and where this (apparent) gravitational field would be expected to go singular, there's a horizon (the cosmological horizon) censoring the extrapolated Big Bang singularity from view, and that looks gravitational, too.
And, funnily, enough, this "warped" worldview turns out to be defensible (as an observer-specific description) using the available optical evidence. Since we reckon that the universe is expanding, and we're seeing older epochs of the universe's history as we look further away, we're seeing those distant objects as they were in the distant past, when the universe was smaller and denser and the background gravitational field-density was greater than it is now.
Our perspective view is showing us an angled slice through space and time that really does include a gravitational gradient – between "there-and-then" and "here-and-now". The apparent gravitational differential is physically real within our observerspace projection, and viewed end-on, the projection describes a globular universe with a great big black hole at the opposite end to wherever the observer happens to be.
This projection is fascinating: it means that we end up describing cosmological-curvature effects with gravitational-curvature language, and it cuts down on the number of separate things that our universe model has to contain. If we take this topological projection seriously, some physics descriptions need to be unified. If we can agree on a single definition of relative velocity, the projection means that cosmological shifts (as a function of cosmological recession velocity) have to follow the same law as gravitational shifts (as a function of gravitational terminal velocity) ... and then, since gravitational shifts can be calculated from their associated terminal velocities as conventional motion shifts, we have have three different effects (cosmological, gravitational and velocity shifts) all demanding to be topologically transformed into one another, and all needing to obey the same laws. This all sounds great, and at this point someone who hasn't done advanced gravitational physics will probably be anticipating the punchline – that when we work out what this unified set of laws would have to be, we find that they're the set given by Einstein's special and general theories, QED.
Except that they aren't. We don't believe that cosmological shifts obey the relationship between recession velocity and redshift supplied by special relativity.
We dealt with this by ignoring the offending geometry. Since cosmological horizons had to be leaky, and GR1915 told us (wrongly) that gravitational horizons had to give off zero radiation, we figured that these had to be two physically-irreconcilable cases, and that any approach that unified the two descriptions was therefore misguided. Since a topological re-projection couldn't be "wrong", it had to be "inappropriate". Instead of listening to the geometry and going for unification, we stuck with the current implementation of general relativity, and suspended the usual rules of topology to force a fit.
But then Stephen Hawking used quantum mechanics to argue that gravitational horizons should emit indirect radiation after all, as the projection predicts. So we'd broken geometrical laws (in a geometrical theory!) to protect an unverified physical outcome that turned out to be wrong. Where we should have been able to predict Hawking radiation across a gravitational horizon from simple topological arguments in maybe the 1930's, by using the closed-universe model and topology, we instead stuck with existing theory and had to wait until the 1970's for QM to tap us on the shoulder and point out that statistical mechanics said that we'd screwed up somewhere.
If we look at this projection, and consider the consequences, it suggests that the structure of current general relativity theory, when applied to a closed universe, doesn't give a geometrically consistent theory ... or at least, that the current theory is only "consistent" if we use the condition of internal consistency to demand that any logical or geometrical arguments that would otherwise crash the theory be suspended (making the concept almost worthless). It basically tells us that current classical theory is a screw-up. And that's why you probably won't see this projection given in a C20th textbook on general relativity.