Mathematican William Kingdon Clifford (1845-1879) is usually best-remembered for his development of Clifford algebra, and for his promotion of the idea that "all physics is curvature".
The concept of a Cliffordian universe (a universe in which in which the existence of particles and the physics of their interactions is inextricably associated with curvature effects) is important to gravitational theory, as, as a "proof of concept", it violates and possibly disproves an important principle commonly relied on by C20th gravitational theorists, that gravitational theories must be perfect supersets of special relativity.
|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.
|— William Kingdon Clifford, "On the Space-Theory of Matter", 1876
The "reduction to SR" argument
- " If we consider any gravitational theory which obeys field laws that are "classical" (subject to the law of continuity, ruling out discontinuous QM-style behaviour) and "simple" (ruling out continuous fractal geometries), then, just as an arbitrarily-small magnified section of a circle becomes indistinguishable from a section of straight line, so, if we "zoom in" sufficiently far on a region of curved-spacetime physics, we must eventually arrive at an arbitrarily-flat region of spacetime in which the operating physics must be flat-spacetime physics. If we also require physics to obey the principle of relativity, then the only system of equations that supports perfectly flat spacetime are those of Einstein's 1905 special theory. Therefore, we know for a geometrical fact, that any simple classical geometrical theory of curved spacetime MUST reduce perfectly to the physics of special relativity over a sufficiently small region. The equations of special relativity MUST be contained within the physical laws of the larger theory, as a limiting case, or else it is wrong. "
The "Cliffordian universe" counter-argument
- " If all physics is reducable to geometry, then moveable particles must represent transmissable geometrical features in spacetime, and particles and their interactions must be associated with intrinsic curvature - the physics of how particles interact must be describable as the geometry of how their associated curvatures interact. In such a universe there is no such thing as flat-spacetime physics, and the concept is a contradiction in terms. Although we can zoom in arbitrarily far on a patch of spacetime to obtain a region that is arbitrarily flat, that region will then by definition not contain any particles or significant interactions – the patch will not represent the limit at which curved-spacetime physics reduces to special relativity, it will represent the empty limit at which no meaningful physics can be said to be taking place. While we require physics to obey the principle of relativity the PoR is not required to also apply to invented, fictitious, hypothetical, unphysical, purely mathematical observers outside physics, whose presence we cannot detect, and whose existence does not correspond to any known physical laws. If such a mathematical ghost were to observe a violation of the PoR, they would have no way of communicating it to us, and we would be none the wiser – the logic of these ghostly observers may be valid mathematics, but the physical universe is not compelled to obey laws designed for non-participants. "
Implications of a Cliffordian universe
In a Cliffordian universe, the relationships between relatively-moving particles are not described geometrically by Minkowski spacetime. Since Minkowski spacetime is constructed from the relationships of SR, and SR can be considered to be the physical embodiment of Minkowski spacetime, changing the lightbeam geometry away from flat Minkowski spacetime to a different type of curved system necessarily involves changing the relationships away from those of special relativity, to Something Else.
If our universe is Cliffordian, then "ordinary general relativity, and also every other gravitational theory that depends on a perfect reduction to special relativity – essentially, every textbook curved-spacetime theory that we currently consider to be worth studying – is wrong.
This makes the question of whether or not our universe really is Cliffordian a matter of some interest.
- W.K. Clifford, On the space-Theory of Matter (abstract) Proceedings of the Cambridge Philosphical Society Vol II, 1864-76 (archive.org)
- Until we can demonstrate that we do not live in a Cliffordian universe, the argument that reduction to SR physics is compulsory cannot be treated as a geometrical result, it is merely an assertion.
- SR-style arguments applied within a Cliffordian universe can still be used to derive special relativity, as a "null theory" with no domain of applicability that only appears in the special circumstance that no observers are actually present. But in that universe, the equations of motion for the "null" theory will not correspond to the equations of motion of actual particles, which will be subject to different geometrical constraints.
- The principle of relativity, applied within a Cliffordian universe, appears to generate a relativistic acoustic metric.