Abstract
Einstein built special relativity on two postulates: no experiment can distinguish one constant-velocity laboratory from another, and light travels at the same speed in every such laboratory. The first is a principle of symmetry. The second is a fact about a particular phenomenon, an observation imported into the foundations. We show this was unnecessary. The symmetry principle alone yields a family of possible universes, labelled by a single number κ. The Killing form sorts them into exactly three kinds: one with no causality, one that cannot unify space and time, and one that achieves both and demands a finite speed that is the same in all frames. Experiment is needed to measure that speed, not to establish its existence. Einstein needed one postulate, not two.
The postulate
Einstein built special relativity on two principles. The first is a symmetry requirement:
Postulate (relativity principle, 1905). The laws of physics take the same form in all inertial frames.
An inertial frame is any laboratory moving at constant velocity: a sealed room drifting through space. The postulate says that no experiment performed inside such a room can reveal how fast it is moving, or whether it is moving at all.
For Einstein this was more than a statement about experiments. It was a design principle: the mathematical framework of physics should contain no structure put in by hand. Everything should emerge from the rules themselves. Special relativity came from removing the assumption that all observers agree on which events are simultaneous. General relativity came from removing the assumption that there is a single natural way to label points in spacetime, the four-dimensional union of space and time. Each advance stripped away a background assumption that the previous framework had smuggled in.
The second postulate is different in character. It says that light travels at a finite speed c that is the same in all inertial frames. This is not a principle of symmetry. It is an observation about a particular physical phenomenon, imported into the foundations. For Einstein, who held that the laws of physics should be self-contained, this was a concession: an empirical fact doing the work of a structural argument.
What the postulate determines
How do measurements in one inertial frame relate to those in another? If you are on a train moving at speed v relative to the platform, and you measure a time t and a position x, what are the platform's values t′ and x′?
The relativity principle, together with three natural requirements (the same physics everywhere, no preferred direction, and the condition that combining two transformations yields a third, known as the group property) turns out to be extraordinarily restrictive. Since 1910, independent derivations have shown that these requirements allow exactly one family of transformation rules, governed by a single undetermined number κ:
t′ = γ(t − κv x), x′ = γ(x − v t), γ = 1 / √(1 − κv²)These deserve a moment's reading. The second equation is intuitive: the platform sees your position shifted by the train's motion. The first is where the physics lives. The term κvx mixes space into time. It means that what counts as "now" depends on where you are, and κ controls the strength of this mixing. As v approaches 1/√κ, the factor γ grows without bound, which acts as a speed limit. When κ = 0, the mixing vanishes: t′ = t, time is absolute, and we recover Newton.
Three kinds of universe arise, depending on the sign of κ:
Lorentzian (κ > 0)
Einstein's transformations. Finite maximum speed, lightcones, causal structure.
Galilean (κ = 0)
Newton's transformations. Universal time, no speed limit, space and time independent.
Euclidean (κ < 0)
All four dimensions equivalent. No lightcone, no distinction between past and future.
Every treatment since 1910 has agreed that the first postulate alone can't determine the sign of κ. To choose among the three, one must look outside: measure the speed of light, and thereby import Einstein's second postulate or something equivalent. We disagree. The symmetry rules already contain the answer.
Can the rules examine themselves?
The three universes share the same generators and combination rules. They differ only in the value of κ, which controls how strongly the elementary operations interact with each other. Can this internal structure, using only its own resources, distinguish the three cases without any appeal to experiment?
It can. Wilhelm Killing found a way to do this in 1888, seventeen years before Einstein's paper. His construction, the Killing form, applies to any set of symmetry rules.
Every set of symmetry rules is built from elementary operations called generators. For the symmetry of motion, there are six: three rotations (J₁, J₂, J₃) and three boosts (K₁, K₂, K₃). A rotation relates your laboratory to one pointing a different way. A boost relates it to one moving past you, the operation of changing velocity.
Each generator acts as a machine that reshuffles the others. The Killing form measures the total reshuffling that each pair produces, formally B(X,Y) = tr(adX ∘ adY). It produces a single number, computed entirely from the combination rules. No choices are involved and no external input is needed.
For the symmetry rules labelled by κ, the result is:
B = diag(−4 I₃, 4κ I₃)In words: the Killing form registers −4 on every rotation and 4κ on every boost. So we have the self-test.
Three verdicts
κ < 0: No causal structure. The Killing form is non-zero on both rotations and boosts. But the geometry treats all four dimensions equivalently. No lightcone, no causal ordering. A universe with κ < 0 has no notion of cause and effect.
κ = 0: The algebra goes blind. The Killing form returns zero on every boost generator. The self-test comes back blank for exactly the operations that connect different inertial frames. Velocity space has no natural ruler. Time is absolute. The algebra preserves this structure but doesn't generate it. Absolute time is a background structure, assumed rather than derived. This is what the relativity postulate was meant to forbid.
κ > 0: Everything is determined. The Killing form returns −4 on rotations and 4κ on boosts. Every generator is visible. The invariant speed is V = 1/√κ, finite, real, and the same in all frames. Spacetime acquires a definite geometry. The only quantity all observers agree on is the spacetime interval, combining space and time inseparably. This geometry has lightcones. Past and future are distinguished. Causal structure exists.
| κ < 0 | κ = 0 | κ > 0 | |
|---|---|---|---|
| Killing form on boosts | 4κ < 0 | 0 | 4κ > 0 |
| Invariant speed | imaginary | undefined | finite, real |
| Spacetime metric | Euclidean | dt² only | Lorentzian |
| Causal structure | none | none | lightcones |
| Space–time unification | all alike | impossible | complete |
| Background structure needed | none | yes | none |
Structure and scale
A finite invariant speed must exist, and spacetime must be Lorentzian. What the argument doesn't determine is the numerical value of that speed. V depends on κ, which sets the conversion factor between units of space and units of time. Measuring c ≈ 3 × 10⁸ m/s fixes κ = 1/c². But this is calibration: fitting a number to a framework whose architecture is already determined.
Why does this matter? Because Einstein's second postulate was calibration dressed up as foundation. The symmetry rules themselves demand a finite invariant speed. Experiment tells us how fast. The algebra tells us that.
Einstein was right that the relativity principle was the deeper of his two postulates, and right that background structure should be eliminated. The tools to finish the job existed in his time. But the conclusion is the one he reached: there is a finite invariant speed, spacetime is Lorentzian, and space and time are unified.
The speed's value is empirical. Its existence is not.
Einstein needed one postulate, not two.
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