The transformation that never mentions light

The usual derivation of the Lorentz transformation begins with two postulates: the relativity principle, and the constancy of the speed of light. The second is doing a great deal of work and it is a strange thing to assume — a statement about one particular physical phenomenon, promoted to an axiom about geometry.

It is not necessary. The transformation follows from symmetry alone, with one constant left over, and light enters only when the constant has to be given a value.

The four assumptions #

Each is worth stating in the weakest form that still does the work.

Homogeneity. No place and no moment is special, so a transformation between frames maps straight lines to straight lines and uniform motion to uniform motion — which forces it to be linear. This is the assumption that does the most and is questioned the least.

Isotropy. No direction is special, so the transformation can depend on the relative velocity’s magnitude and on nothing else about the pair of frames.

The group property. Two changes of frame, one after the other, are a change of frame. That is what “frame” means if the set of them is to be usable at all.

The relativity principle. If frame B moves at $v$ relative to A, then A moves at $-v$ relative to B, and the transformation from B to A is the same function with $-v$ substituted. Neither frame is privileged.

Work those through and what survives is

$$x' = \gamma(x - vt), \qquad t' = \gamma(t - Kvx), \qquad \gamma = \frac{1}{\sqrt{1 - Kv^2}}$$

with a single constant $K$ whose dimensions are one over a speed squared. The four assumptions cannot fix its value or even its sign.

How the constant survives the algebra #

The derivation is worth sketching, because the place the constant enters is not obvious and is where all the content is.

Linearity gives $x' = A x + B t$ and $t' = C x + D t$, four coefficients depending on $v$. Requiring that the origin of the moving frame ($x' = 0$) be at $x = vt$ fixes $B = -Av$. Isotropy — reflecting both frames in a mirror must give the same family — forces $A$ and $D$ to be equal and even in $v$, while $C$ is odd.

That leaves two functions of $v$: call them $\gamma(v)$ and $C(v)$, so that the transformation is the matrix with rows $(\gamma, -\gamma v)$ and $(C, \gamma)$. The relativity principle says its inverse must be the same matrix with $v$ replaced by $-v$ — and for a two-by-two matrix that forces the determinant to be one, which is $\gamma^2 + C\gamma v = 1$. Writing $C = -Kv\gamma$ turns that into $\gamma^2(1 - Kv^2) = 1$, and therefore $\gamma = 1/\sqrt{1-Kv^2}$ with $K$ an arbitrary constant.

$K$ is a constant of integration. It appears because the relativity principle is a functional equation whose solution has a free parameter, and the group property then fixes nothing further: composition works for every $K$, which is what the third figure verifies. That is the whole reason the four assumptions cannot pick a world.

Three worlds #

$K > 0$ defines a speed $1/\sqrt{K}$ that every frame agrees about — an invariant speed, arrived at without any mention of light. $K = 0$ gives $\gamma = 1$ and $t' = t$: Galileo, with absolute time. $K < 0$ makes the transformation a rotation of the plane of space and time.

The middle panel is the one worth pausing on. With $K=0$ the preserved quantity is $t^2$ alone: every frame agrees about the time between two events and there is no statement whatever about distances. That is why a Galilean world can carry a signal infinitely fast without contradicting anything — there is no invariant speed for it to exceed.

In the left panel the hyperbolas have asymptotes, and the asymptotes are worldlines at the invariant speed. Nothing crosses them under any boost, which is what makes a maximum speed a geometric fact rather than a dynamical one. The quantity nobody argues about is the same invariant approached from the other direction.

How speeds add in each #

The composition law is $(u+v)/(1 + Kuv)$ in every case, and reading the three curves is the quickest way to see what each world is like.

With $K>0$ the result never reaches the invariant speed however large the second speed. That is speeds that refuse to add obtained without postulating anything about light, and it makes clear that the existence of a maximum speed is a consequence of the group structure rather than an extra assumption.

With $K<0$ two ordinary forward speeds compose to an infinite one and then to a backward one. That is not an inconsistency — it is what happens when a rotation angle passes a quarter turn — but it is a problem.

Why the third world goes #

Nothing in that figure is wrong. The transformations compose properly, the relativity principle holds, the geometry is Euclidean, and a chain of perfectly ordinary boosts takes any event anywhere — including to a time before the origin.

So the third case is excluded by an argument about causality, and that argument is an extra assumption. The four symmetry requirements do not contain it. If one insists that the order of causally connected events cannot depend on the observer, $K<0$ goes; if one does not, it stays, and describes a consistent world that is not this one.

Being clear about which conclusions come from which assumption is the whole reason for doing the derivation this way. The maximum speed comes from the group structure. The exclusion of rotations comes from causality. Only the value of the constant comes from experiment.

Measuring the constant #

The reasoning behind the bound is short. If the composition law is $(u+v)/(1+Kuv)$ and an experiment finds that two speeds of order $v$ add to within a fraction $\epsilon$ of their sum, then $Kv^2$ must be smaller than about $2\epsilon$, so the invariant speed is at least $v/\sqrt{2\epsilon}$. Both the speed and the precision help, and the speed helps directly while the precision helps only as a square root.

The bound scales with the speed used, which is why the question could not be settled mechanically. Two carts on a bench bound the invariant speed to something smaller than walking pace times a hundred, however carefully the collision is timed. A rifle bullet fired from a train does better and nowhere near well enough.

The measurement needs either something moving very fast or a precision beyond anything mechanical, and the first thing available in either category was light. That is the honest role of the light postulate: not a foundation, but the first experimental handle on a constant the symmetry argument leaves free.

Once the sign and the value are known, everything else follows — and everything else was already determined before they were known, which is the point.

The two limits, and the two names #

There is a symmetry between the two physically acceptable cases that is easy to miss and worth naming.

Galilean relativity is often described as the limit of the Lorentz case for small speeds, and that is right as far as it goes. It is also a perfectly good theory in its own right, satisfying all four assumptions exactly rather than approximately, and its composition law is exact rather than a first term.

The two are related as $K \to 0$, and the limit is not smooth in one respect that matters: the causal structure changes discontinuously. At any $K>0$, however small, there is a light cone and a set of events that cannot influence one another. At $K=0$ exactly, every event can influence every other, and the cone has opened to the whole plane. A structural feature has appeared out of nothing at the limit, which is why questions about causality cannot be settled by taking limits.

The same discontinuity is why the Galilean theory is not merely an approximation to be used at low speeds but a genuinely different description with different things in it, and why the transition between the two is where most of the confusion in learning the subject lives.

What this buys #

Deriving the transformation this way changes what the theory looks like, and three consequences are worth having.

Relativity is a statement about symmetry, not about electromagnetism. The historical route runs through Maxwell’s equations and their strange invariance, which makes it look as though light is somehow constitutive. It is not. Any theory in a homogeneous, isotropic world with a relativity principle has this structure, and Maxwell’s equations happen to be one whose invariant speed is $c$.

The invariant speed need not be the speed of anything. It is a conversion factor between time and distance, and it would exist even if nothing travelled at it. That photons do is a separate fact — a consequence of their masslessness, not of geometry.

A test of relativity is a measurement of $K$. Every experiment that constrains Lorentz violation is, in this language, a measurement of whether the constant is the same for every particle and in every direction — which is a much clearer statement of what such experiments do than “testing whether the speed of light is constant”.

Who did it, and when #

The derivation is not modern and is not obscure. Vladimir Ignatowski published it in 1910, five years after Einstein’s paper, and Frank and Rothe gave a more careful version in 1911. It has been rediscovered every couple of decades since, most influentially by Lévy-Leblond in 1976, and is standard in the literature on the foundations of the subject.

That it has to be rediscovered is the interesting part. The light postulate is historically prior — Einstein reached the transformation through it, and through Maxwell’s equations before that — and a route that arrives at the same place without it does not change any prediction, so there is nothing to be gained by it except understanding.

Understanding is the thing being gained, and it is not nothing. A theory whose foundations are two postulates, one of them about a specific phenomenon, invites the question of what happens if that phenomenon behaves differently. A theory whose foundations are four symmetry statements and one measured constant invites a different question, which is what else the constant governs — and the answer is everything.

What the constant does elsewhere #

Once the constant is measured, it turns up in places that have nothing to do with changing frames, and the list is a good check that it is a property of spacetime rather than of light.

It sets the ratio of a body’s rest energy to its mass, so the energy released by a nuclear reaction is a statement about the same number. It sets the relation between electric and magnetic fields under a boost, which is why magnetism is electricity seen sideways is a statement about geometry. It appears in the relation between a body’s momentum and its velocity, so it is measured every time a fast particle is bent in a magnetic field. And it fixes the rate at which a moving clock runs slow, which is the clock that has to slow.

None of those is an optical measurement. Each of them determines the same constant, and the agreement among them is far stronger evidence for the structure than any one determination — because a constant that appeared in one phenomenon would be a property of that phenomenon, and one that appears in all of them is a property of the arena.

That is the practical version of the argument this essay makes. The constant is not the speed of light; it is a conversion between time and distance, which light happens to travel at because it is massless.

Where the model stops #

Homogeneity is doing more work than the others. It is what forces linearity, and it is an assumption about spacetime having no structure of its own — which is false in the presence of gravity. General relativity keeps the argument locally and abandons it globally, so everything here is a statement about a small enough patch.

Isotropy has been assumed in three dimensions and used in one. The full derivation in three spatial dimensions needs rotations to be included in the group and produces the same answer, but the one-dimensional version presented here is a shortcut that would hide an inconsistency if there were one. There is not; the check is standard and long.

Continuity has been assumed silently. The derivation supposes the transformations vary smoothly with the velocity, which rules out pathological solutions that no one wants but that the stated assumptions permit. Adding the requirement explicitly is the usual fix.

The derivation says nothing about what transforms. It fixes how coordinates relate and leaves entirely open how a field, a wavefunction or a tensor behaves under the same change of frame — which is the content of the representation theory that the subject is really built on, and which is where the spin of a particle comes from, as the turn that has to be made twice works out for the rotation part.

And the causality argument is doing real work. It is not a symmetry requirement and it cannot be derived from the four. A reader who finds the exclusion of $K<0$ unsatisfying is not confused: it is genuinely the weakest step in the chain.

Where a modern test actually looks #

Reading the theory this way changes what an experimental programme looks like, and the change is not cosmetic.

If the constant is a property of spacetime, then every particle must have the same one, and it must be the same in every direction and at every energy. Each of those is a separate thing to check, and each has a name: a difference between species is a violation of universality, a difference between directions is anisotropy, a difference with energy is dispersion in vacuum.

The sharpest limits come from places where a tiny difference accumulates. A photon from a distant burst arriving with its high-energy part delayed against its low-energy part would show vacuum dispersion, and the billions of years in flight amplify a difference no laboratory could see. A pair of atomic clocks with different orientations, watched as the Earth turns, tests anisotropy at the level of parts in $10^{18}$. Neither is a measurement of the speed of light in any useful sense; both are measurements of whether one constant is doing all the work.

A test framed as “is the speed of light constant?” is hard to make precise and easy to misreport. Framed as “does one constant appear in every one of these relations, identically?”, it becomes a list of separate, well-posed measurements — which is what the field actually does.

What the pictures cannot show #

The three-panel figures place the three worlds side by side as though they were alternatives to be chosen between by inspection. They are not distinguishable at any speed a person has experienced: at walking pace all three panels are the same picture to a part in $10^{16}$. The figures are drawn with the constant set to one so that the differences are visible, which means every one of them is drawn at a speed of half the invariant speed.

The causality figure draws boosts as discrete steps, and a boost is a continuous parameter. Drawing a chain of them as dots on a circle makes the rotation visible and makes it look as though something is being done repeatedly; what is really being shown is a single one-parameter family of frames, all of them equally valid, one of which disagrees with the first about the order of two events.

Where the ladder goes next #

The spacetime-diagram ladder began with two axes, one speed, where the diagram does the arguing that algebra would otherwise have to, went on to the quantity nobody argues about and the interval, and then to the diagram a ruler cannot read, where the scale on the axes is not the scale on the page. This rung asks what the diagram would look like if the second postulate were withheld, and finds that almost all of it survives.

The rung after it is the same argument with rotations included, where the group is derived rather than assumed and the result is that the invariant speed and the rotation group fit together in only one way. The habit worth carrying is the one this rung is built on: when a derivation uses a physical postulate, ask how much of the result survives without it — because a postulate that turns out to fix only a constant was never a foundation.