Friday, 1 May 2009

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


Anonymous said...

You seem to think that, if particles of matter and energy are considered to be “crumpled space” as Clifford suggested, then spacetime is not flat in the limit of small regions. That is incorrect. The idea that particles of matter are really just curved configurations of the space(time) manifold does not in any way conflict with the fact that a curved manifold approaches flattness on a sufficiently small scale. If there is a high degree of crumpling (curvature) on a very small scale, it merely means that we must examine the manifold on an even smaller scale in order to find that it approaches flattness. Regardless, the property of approaching flattness on a sufficiently small scale is inseparable from the concept of a Riemannian manifold. So your claim that the Cliffordian view of matter implies that space(time) does not reduce to flattness on sufficiently small scales is self-evidently false.

Furthermore, such a view of matter does not conflict with the fact that the physics of particles reduces to special relativity on the scale of ordinary particle interactions. This is essentially the content of the equivalence principle, which is the empirical foundation of general relativity. Even gravity gravitates, i.e., a small crumpled region of spacetime representing a material particle possesses a calcuable energy and momentum, and, as Einstein and Infeld showed in 1938, such a configuration propagates along geodesics of the surrounding (uncrumpled) spacetime.

The Cliffordian concept of matter has not been overlooked or neglected. It has actually been explored extensively in the context of general relativity. Of course, every particle of matter and/or energy possesses macroscopic gravity, so there is a degree of curvature associated with any particle in that sense, but the strength of gravity is 1E-40 times smaller than the strength of electromagnetism, so the macroscopic curvature is negligible when focusing on a few electrons zooming around and interacting electromagnetically in otherwise empty space, i.e., physics reduces to special relativity, but this is not to deny that on a much smaller scale, the detailed structure of an electron might involve spacetime curvature. In fact, many physicists have examined the consequences of regarding particles of matter as manifestations (sometimes even singularities) of spacetime curvature in the context of general relativity – but none of this changes the self-evident fact that a curved Riemannian manifold approaches flattness on a sufficiently small scale, unless you admit singularities, but even in that case Einstein showed that such singularities would follow geodesic paths through the surrounding non-singular spacetime, which approaches the flat spacetime of special relativity in sufficiently small regions. Hence your beliefs on this subject are mistaken.

ErkDemon (Eric Baird) said...

The flaw in the usual "geometrical reduction to SR" argument is that even if there's a geometrical reduction to flat spacetime /in principle/, it doesn't logically follow that there has to be a reduction to the physical laws of special relativity /in practice/.

Suppose that the existence of a pair of particles with significant relative velocity is always associated with a distortion of the local lightbeam geometry. In that scenario, it might be that the curvature IS the physics.

Suppose that it is. We'd still be able to follow your suggestion that we merely need to zoom in further on part of this curved region to find an "effectively-flat" patch, and then argue that the internal physics of that patch must correspond to the physics of special relativity, by invoking (a) flatness, and (b) the principle of relativity.

But in this scenario, _there_is_no_meaningful_internal_physics_for_the_patch_. There are no objects or observers inside the patch for "the physics of flat spacetime" to refer to. The patch is only able to be "effectively flat" because it doesn't contain any interacting particles or other interesting behaviour. Its contents are "null physics" - you can derive abstract things for it mathematically, but they needn't correspond to how real particles interact, because as soon as you introduce a particle or two into the region to see if they really do obey the flat-spacetime laws that you've just constructed, you've destroyed the condition of flatness, and all bets are off.

In this hypothetical situation, the real physics wouldn't involve the relativity of notional paths through flat Minkowski spacetime, it'd involve the relativity of the appearance of curved surfaces at the particle scale.

You don't have to buy into this curved-spacetime worldview that doesn't include SR as "physical" physics, but you do have to accept it as a logical possibility that needs to be disposed of before the "geometrical reduction" argument can be given the status of a real proof.

Until someone manages to find a way to eliminate the alternative, "single-stage, entirely curvature-based" approach to tackling relativity theory, there's still a chance that the principle of relativity is correct, but that special relativity was an inappropriate implementation of the idea.

Anonymous said...

There is no “single-stage, entirely curvature-based” approach. You typed that phrase, but it doesn’t correspond to anything that you (or anyone else) has ever articulated. In fact, everything you say is inconsistent with your professed longing for a “curvature-based approach”. For example, you advocate taking Newtonian Mechanics rather than special relativity as the basis for physics, and yet Newtonian Mechanics entails perfectly flat Euclidean space. Likewise your verbiage containing phrases like “acoustic metric” doesn’t correspond to any Cliffordian concept of curved space either. Most studies of acoustic metrics are carried out in flat Galilean space and time, and the only curvature is not of space, but of the abstract manifold of acoustic distances. So neither of the mechanistic frameworks that you have mentioned embodies the principles that you claim to espouse.

Also, you continue to ignore the fact that general relativity does exactly what you claim some “new approach” is needed to do, i.e., whenever there are particles present, spacetime is curved, and this affects the motions and interactions of those particles. Now, the idea that you seem to be trying (and failing) to articulate is that we should seek a unified field theory, in which not just gravity but also electromagnetism and all other kinds of interactions are subsumed into the geometry of spacetime, and that even the appearance of particles would really be just configurations of curved spacetime. Well, duh... this is what Einstein (and Eddington, and Weyl, and Schrodinger, and Kaluza, and...) spent decades trying to do, although they soon realized that curvature alone is not sufficient, because all the curvature degrees of freedom are already used for the gravitational field – unless you postulate extra dimensions – so they invoked other things like torsion and generalized metrics (higher than second order) and complex dimensions, etc. But none of them misunderstood things badly enough to suggest that Lorentz invariance is violated, because Lorentz invariance is abundantly confirmed by experience.