Absolute relativity is the idea that careful adherence to the laws of relativity, represented in geometric algebra, provides unique insights into computationally simulating physical systems [1].
An optical clock drifts from several sources at once, none of them directly measurable. An RF package misses its specification where heat, strain and field interact. Both are hard for the same reason: the physics is coupled, and the usual tools simulate the pieces separately and combine the results. Velar simulates them together. Below we lay out the approach that makes this possible, and how it fits into the physics we were taught in school.
We tend to credit Einstein with the idea that gravity is the shape of space. But the geometry came first, and it came as pure mathematics, worked out by people with no physics in mind at all.
More than two thousand years ago, in 300 BC, Euclid set out the rules of geometry in his book Elements. Geometry was flat, three-dimensional, with parallel lines never touching.
But mathematicians were bothered by his rule that parallel lines can never touch. It did not feel as fundamental as the other rules in his text. So for two thousand years they tried to derive the parallel rule from the others, and failed, until Johann Heinrich Lambert [9] tried something new in 1766. He worked out what a geometry would look like if you could draw more than one parallel line through the same point. The idea sounds absurd, esoteric, and against the whole point of parallel lines, but the mathematics was elegant, and his solution turned out to look like the formulae of a sphere. Of what sort of sphere, he could only say this:
…we should almost conclude that the third hypothesis occurs on an imaginary radius.
An imaginary radius puts a minus sign into the measure of distance. A century and a half before physics needed that geometry, Lambert had glimpsed what a curved world would require.
Lambert took the idea no further. The break came in the 1820s and 1830s, when Nikolai Lobachevsky in Russia and János Bolyai in Hungary, working separately, stopped trying to rescue Euclid and built the new geometry in full. Drop the parallel postulate, they found, and a complete and consistent world opens up, one where a single point has not one parallel line through it but infinitely many. Bolyai, barely into his twenties, wrote to his father [10]:
Out of nothing I have created a strange new universe.
What they had changed was not the parallel lines. It was the ruler. In their world, distance is measured by a non-intuitive kind of measuring stick, one that appears different depending how you look at it, but one you can rely on nonetheless. (It turns out the minus sign Lambert had seen on his imaginary sphere is key to how that ruler works.)
In 1854 Bernhard Riemann [11] carried the idea all the way. Space, he showed, can be curved in any number of dimensions, and it need not stretch on forever. It can close gently back on itself, finite yet with no edge anywhere, the way a globe’s surface is finite but has no boundary.
In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent.
Then William Kingdon Clifford [12], who would put Riemann’s lecture into English three years later, took the boldest step of all. If space can curve, he reasoned, then perhaps matter is nothing but curved space. He read the paper out in 1870, more than forty years before general relativity existed.
(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.
…
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter.
It was a guess, which Clifford did not live to develop; he died at 33. But the instinct was right: mass could be represented as geometry.
The world would not know it until relativity asked what geometry a universe must have if the speed of light is the same for every observer. The answer was a geometry with exactly Lambert’s minus sign, now carried by time. It had been waiting seventy years. Einstein then gave the geometry its meaning: relativity was about clocks and simultaneity and a world with no state of absolute rest. Vladimir Varićak [13] later showed that velocities in relativity combine by Lobachevsky’s rules, and that a change of speed is a rotation through a hyperbolic angle.
Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
Minkowski had given relativity a geometry. But there is a difference between using geometry to picture the physics and making geometry the language the physics is written in. You can say that spacetime behaves relativistically and write down the rules for how measurements change from one observer to the next. It is a harder and deeper thing to build a language in which the objects themselves already move that way, so that relativity is not a rule laid on from outside but something the symbols do on their own.
Yet building such a language becomes surprisingly simple if one pictures spacetime as a material, something with stiffness that can be stretched and twisted. The idea is old and respectable. In 1839 James MacCullagh [15] built a working theory of light from a medium whose only springiness was a resistance to twisting, what we would now call torsion, and it succeeded where every rival failed: it gave light as a twist travelling through the medium and no pressure wave alongside it, which is what light turns out to be. Maxwell [16] pictured space as a sea of tiny spinning cells and developed modern electromagnetic theory straight out of it.
The medium fell out of fashion after 1887, when Michelson and Morley [17] failed to find the Earth’s motion through it. But we typically remember the wrong conclusion from that work [18]:
According to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether… But this ether may not be thought of as endowed with the quality characteristic of ponderable media… The idea of motion may not be applied to it.
A medium with real properties and no state of motion is not in conflict with Michelson and Morley's experiment. That medium turns out to be what a fixed speed of light describes with the same minus sign that first appeared on Lambert’s imaginary sphere. And a medium can do things empty space cannot: it can be held under tension, and it can carry a twist. Treat spacetime as an elastic solid and its two kinds of flaw, as Bilby, Bullough and Smith showed in 1955 [19], turn out to be exactly curvature and torsion, the raw ingredients of gravity. That twist is not something matter puts in and takes away. It is a state of the material, present even where there is no matter, because empty space is the material.
Give a medium tension and the freedom to twist, and its natural motions, its stretches and its turns, are the boosts and rotations of relativity. The transformations stop being imposed. They become the things the medium does.
Absolute relativity represents spacetime curvature and torsion natively by writing physics in the natural language of such a medium, a language in which a direction, a turn, and a plane of twist are all things you calculate with as directly as numbers. That language is Clifford algebra, also called geometric algebra, named for the same William Clifford we met a moment ago. The algebra grows from a single idea, that a vector multiplied by itself gives the square of its length, and from that one rule the whole geometry of a tensioned, twisting spacetime unfolds. The introduction to Clifford algebra builds it from the ground up. Writing physics this way is what we call the Mathematics of Absolute Relativity (MART).
And a physicist has met this language already, without being told its name. The gamma matrices of the Dirac equation, the ones that encode the electron’s spin, obey exactly Clifford’s rule. The algebra of a spacetime with tension and torsion was hiding inside the electron all along.
The Dirac equation was developed by Paul Dirac in 1928. It predicts the electron: its spin, its magnetic moment, the existence of antimatter. In its familiar form it carries four gamma matrices and the imaginary unit i.
The gamma matrices arrive as four objects that anticommute and square to ±1 — an algebraic device, chosen so the equation behaves as needed [4]. The i is the scalar imaginary of the complex numbers: it turns the phase. Both work, so the question of what they are is often overlooked.
Those gamma matrices are a representation of an orthonormal frame: four vectors, one along each direction of spacetime [2]. Read them that way and the equation becomes a statement in geometric algebra. In this algebra the Dirac equation reads:
Same equation. Same predictions, new language. What changed is that every symbol now has a precise physical meaning. The operator ∇ is the spacetime gradient. The imaginary unit is gone: the object that squares to −1 and drives the phase is γ2γ1, a bivector — an oriented plane in spacetime, which squares to −1 because a quarter turn taken twice reverses direction. And the wavefunction gets represented as a function of three physical concepts [3]:
This reading of the Dirac equation is David Hestenes’ work, published from 1967 onward in peer-reviewed physics journals [2], [4]. It makes the geometry that was always inside the equation visible.
What it does not say is which planes those rotations turn in. That physical reading is John Williamson’s, developed in the confined-photon electron model [7] and in the generalised field equation written dG = 0 [8].
The imaginary unit is a rotation. In the matrix equation, i is a scalar you multiply by. Here its stand-in is γ2γ1, an oriented plane, and multiplying by it turns a quantity a quarter of the way around that plane. In MART the electron’s phase is a precise rotation within spacetime: a definite plane, with an orientation and a rate [2], [3].
Spin-½ is a rotor. The wavefunction contains a rotor — the geometric-algebra form of a rotation, an even element that turns a frame by a two-sided product. Turn a rotor through a full 2π and it reads −1, not +1; only 4π brings it back. The sign change of a wavefunction under a 360° rotation, a paradox in the matrix picture, is an ordinary property of rotors [5]. Because a frame is turned two-sidedly, R and −R give the same frame. ψ carries the rotor itself, so ψ registers the difference.
The rotor picture invites a physical question. If the electron’s wavefunction is a rotation turning at a steady rate, what is turning?
One answer, put forward by John Williamson and Martin van der Mark in 1997, is that the electron is light — a photon confined into a small closed loop, wound with the topology of a torus [7]. Its circulation rate is the Compton frequency, the same rate that sets the electron’s phase in the Dirac equation. Charge, spin, and mass become properties of the winding: how the loop closes, and how many times the field turns before it rejoins itself.
Hestenes had already read an internal circulation into the Dirac theory — the zitterbewegung, a rapid internal motion the equation carries in its own solutions [6]. The confined-light model gives that circulation a definite shape and a topology.
This geometric algebra approach enables precision simulation of some of our hardest problems in engineering, where the accuracy and the cost of their simulation have both been out of reach.
Where it is already running is in simulating atomic clocks.
The geometric-algebra reformulation of the Dirac equation is peer-reviewed physics, in mainstream journals and standard references. The confined-light electron model appears in a specialist journal and a conference proceedings. References [9] to [19] are the historical and continuum-mechanics sources behind the account above.