All Physics as Curvature?

William Kingdon Clifford (1845-1879) was an Nineteenth Century mathematician and geometer commemorated by modern mathematicians by having Clifford algebra named after him. He was also a fellow of the Royal Society and The Metaphysical Society, wrote a children's book, and made the occasional cutting remark about the inadvisability of trusting the opinions of groups of experts (unless one knew for a fact that at least one of the group had personal first-hand knowledge of the thing that they were talking about).

Amongst relativists, Clifford is remembered as having been one of the first people to come out unambiguously in favour of the idea that physics could (and should) be modelled as a problem involving curved space.

In 1870, Clifford addressed the Cambridge Philosophical Society ("On the Space-theory of Matter" *), declaring:

"...
I hold in fact,

1. That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
2. That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
3. That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
4. That in the physical world nothing else takes place but this variation subject (possibly) to the law of continuity.
   

... "

In other words, according to Clifford, matter was simply a persistent local curvature in space. While some other well-known theorists of the time (such as Oliver Lodge) were were interested in the idea of describing matter as a sort of condensation of a presumed aetherial medium, and using ideas from fluid dynamics as a shorthand for the properties of space, Clifford considered the mathematical curvature-based descriptions as more than just a means of expressing the variation in field-effect properties associated with density-variations and distortions of an underlying medium: for Clifford, the physics was simply the geometrical curvature itself.

Clifford was one of a number of C19th mathematicians working on geometrical descriptions of physics considered as a curved-space problem, a loose association of broadly similar-minded researchers whose presentations were sometimes propagated in lectures rather than in published journal papers (and who were memorably referred to by James Clerk Maxwell as the "space-crumplers".

Clifford's view was influential, but his vision arguably wasn't quite implemented by Einstein's general theory of relativity – although GR1915 implemented curvature-based descriptions of gravitation, rotation and acceleration effects, it still fell back on an underlying flat-spacetime layer when it came to describing inertial mechanics (that layer being special relativity).

This seems to be fixable, but we're not there yet.

* "William Kingdon Clifford, Mathematical Papers", (1882) pp.21-22

Special Relativity is not Compulsory

One of the foundations of Twentieth Century relativity theory was the idea that Einstein's early "special-case" theory of relativity ("Special Relativity", or "SR") had to appear as a complete subset of any larger and more sophisticated models.

At first glance, this seemed unavoidable.

Einstein's later and more sophisticated general theory was at its heart a geometrical theory of curved spacetime... it described gravitational fields in terms of how they warp lightbeam geometry, and then used the principle of equivalence to argue that the effects associated with accelerations and rotations must also follow the same set of rules. We could then model all three classes of effect as an exercise in curved-spacetime geometry, and go on to extend the model to include more sophisticated gravitomagnetic effects.

But Einstein's general theory didn't attempt to apply these new curvature principles to simpler problems involving basic relative motion, because his earlier special theory had already dealt with those cases by assuming flat spacetime. Instead of going over the same ground a second time, Einstein simply said that, just as classically-curved surfaces reduced over sufficiently small regions to apparent flatness, so the geometry and physics of general relativity, if we zoomed in sufficiently far, ought to reduce to flat spacetime and the "flat-spacetime" version of physics described by the special theory.

There were good pragmatic reasons for Einstein's adoption of special relativity as a foundation for GR, but geometrical necessity wasn't one of them. Here's why:
... It's true that if we zoom in on a GR-type model sufficiently far, we end up with effectively-flat spacetime, but this doesn't automatically mean that we then have flat-spacetime physics. It might instead mean that we've zoomed in so far that there's no longer any meaningful classical physics to be had. We have to accept at least the logical possibility that real physical particles (and their interactions) might be unavoidably associated with spacetime curvature, and in that scenario, we can't derive their relationships by presuming absolutely flat spacetime, because that condition would only be met if our particles didn't physically exist.

Allow any form of velocity-dependent curvature at all around moving particles, and SR's flat-spacetime derivations fracture and fail. This is especially unfortunate since the experimental evidence suggests that moving particles do seem to disturb the surrounding lightbeam geometry, just as we'd expect if curvature effects were a fundamental part of physics, and if the flat-spacetime basis of special relativity was wrong.

---==---

This suggestion that "all physics is curvature" was put forward at the end of the Nineteenth Century by a mathematician called William Kingdon Clifford, who's usually remembered for having his name on Clifford Algebra. The critical thing about a "Cliffordian" model in this context is that when we implement the principle of relativity within it, we find that the resulting physics doesn't reduce to special relativity and the relationships of Minkowski spacetime. Instead of a Minkowski metric, it reduces in the presence of moving particles to something that looks more like a relativistic acoustic metric, and which appears to be much more compatible with quantum mechanics than our current classical models.

So the perfect, unbreakable geometrical proofs of SR's inevitability as physics aren't complete. In order to complete them, we have to be able to show that Cliffordian models can't work ... and that seems to be difficult, because the results of taking a Cliffordian approach seem to be pretty damned good.

To date, nobody seems to have been able to come up with a convincing disproof of this class of curvature-based solution, and until that happens we have to accept the possibility that special relativity might not be a part of our final system of physics.