Absolute relativity writes the laws of relativity in geometric algebra, a mathematics in which a direction, a rotation and a plane of twist are single objects you calculate with [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. Doing that takes a particular kind of mathematics, and that mathematics is older than the physics it now describes.
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. Euclid’s other rules are one line each and obvious on sight: any two points can be joined by a straight line. The parallel rule takes several lines to state, and it reads like something that ought to be provable from the rest. So for two thousand years they tried to prove it from the rest, and failed, until Johann Heinrich Lambert [9] tried the reverse in 1766. He assumed the rule was false, allowing more than one parallel line through the same point, and followed the consequences out expecting to reach an absurdity. He reached none. The geometry held together, and its formulae turned out to be the formulae of a sphere, a surface where the flat rules fail too. 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 number is one that squares to a negative. A sphere of imaginary radius therefore has a negative radius squared, and that minus sign lands in 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 infinitely many parallels run through a point that Euclid allowed only one line through. Bolyai, barely into his twenties, wrote to his father [10]:
Out of nothing I have created a strange new universe.
Underneath the parallel lines, what they had changed was the ruler. Euclid’s rule for distance adds squares together: go three paces along and four across, and you are five away. The new geometry subtracts one of those squares. That is the minus sign Lambert had seen on his imaginary sphere, and it is what opens the space up.
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 raised the question. In 1905 Einstein started from a speed of light that is the same for every observer and worked out the physics: clocks, simultaneity, and a world with no state of absolute rest. Three years later Hermann Minkowski showed that the whole of it is a geometry. Space and time are one four-dimensional thing, and its measure of distance subtracts the time part from the space part. That is Lambert’s minus sign, now carried by time, and it had been sitting in the mathematics since Lobachevsky and Bolyai.
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.
Vladimir Varićak [13] closed the circle in 1910. Velocities in relativity combine by Lobachevsky’s rules, and a change of speed, called a boost, is a rotation through a hyperbolic angle. The geometry the mathematicians had built with no physics in mind is the geometry of relativity.
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 deeper thing to build a language whose objects already move that way, so that relativity becomes something the symbols do on their own.
One picture makes such a language straightforward to build: spacetime as a material, something with stiffness that can be stretched and twisted. The idea is old and respectable.
In the 1800s light was known to be a wave, and every wave anyone could point to travelled through something: sound through air, ripples through water. So physicists asked what light travels through, and modelled it as an elastic solid. But squeeze an ordinary elastic solid and it pushes back, so it carries a push-pull wave like sound alongside the side-to-side wave that light is. Nobody could find light’s push-pull wave.
In 1839 James MacCullagh [15] worked out the mathematics of a medium with no springiness against squeezing at all, one that resisted only twisting, what we would now call torsion. Such a medium carries the side-to-side wave and nothing else, which is what light turns out to be. Maxwell [16] pictured space as a sea of tiny spinning cells and developed modern electromagnetic theory out of it.
If light travels through a medium, and the Earth travels through the same medium, then the Earth’s motion should show up in the speed of light. A swimmer in a river meets the same effect. A round trip downstream and back takes longer than the same trip across the current and back: the upstream leg costs more time than the downstream leg saves. In 1887 Michelson and Morley [17] raced two beams of light that way, one along the Earth’s motion and one across it. The beams tied, and the idea of spacetime as a medium you could move through, the way a swimmer moves through a river, fell out of fashion. But what if the medium cannot flow? That possibility survived the experiment untouched, and it is the one we typically forget [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.
What Michelson and Morley ruled out was a medium you could measure your speed against. A medium you cannot measure your speed against gives the same answer for the speed of light however you are moving. That fixed speed of light is where relativity starts. So what survives the experiment is a medium with physical properties and no state of motion. Its geometry is Minkowski’s, carrying 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. Torsion is the name for that twist, and in this picture it is a state of the material, present even where there is no matter, because empty space is the material.
If spacetime is such a material, then what materials science knows about ordinary materials should apply to it, starting with their flaws. A crystal can have a row of atoms missing, or a wedge of material forced in where it does not belong. Bilby, Bullough and Smith showed in 1955 [19] that a spread of missing rows is, in the mathematics, the torsion of the material’s geometry. A spread of forced-in wedges gives its curvature. Curvature and torsion, the geometry of spacetime, are also the mathematics of a solid with flaws in it.
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 writes physics in the natural language of such a medium. In that language a direction, a turn, and a plane of twist are single objects you add and multiply, the way you would numbers. Curvature and torsion are then things the algebra already contains. That language is Clifford algebra, also called geometric algebra, named for the same William Clifford we met a moment ago. The algebra starts by saying what it means to multiply two vectors together, something ordinary vector arithmetic leaves undefined. The one rule is that a vector multiplied by itself gives the square of its length. From that 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, square to ±1. That is Clifford’s rule for a unit vector. 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 square to ±1 and anticommute, meaning that swapping two of them flips the sign. They are 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, measuring how ψ changes from one point to the next. The imaginary unit is gone. The object that squares to −1 and drives the phase is γ2γ1, a bivector, which is a plane in spacetime with a direction of turn marked on it. It 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 equation’s geometry visible.
Which planes those rotations turn in is a further question. John Williamson’s answer comes from the confined-photon electron model [7] and from 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 acting on it from both sides at once. Turn a rotor once all the way around and it reads −1; a second full turn brings it back. The electron’s wavefunction behaves the same way. In the matrix form that is a puzzle, since turning anything all the way around should leave it as it was. For a rotor it is ordinary [5]. Because the rotor acts from both sides, R and −R leave the frame in the same place. ψ carries the rotor, 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 came from John Williamson and Martin van der Mark in 1997 [7]. The electron is light: a photon confined into a small closed loop, wound with the topology of a torus. Its circulation rate is the Compton frequency, about 1020 turns a second, 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, German for trembling motion, 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 algebra is what the Velar engine is built on. It is for problems the existing tools get wrong, or take too long to get right.
It runs today in CliffordClock, which predicts how stray electric fields shift an optical clock. A full-wave simulator for RF materials and advanced packages opens to beta users in late 2026.
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.