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The Postulates and Their Roles

Six premises behind the Lorentz derivation, sorted into empirical, mathematical, and conventional: what each contributes and what the repository never formalized.

“Special relativity follows from two postulates.” That sentence is a slogan, and a derivation that takes it literally will not go through. A closer count of what a standard derivation uses looks like this:

  1. The principle of relativity: the laws of physics take the same form in all inertial frames.
  2. Constancy of the speed of light: light travels at the same speed $c$ in all inertial frames.
  3. Homogeneity of space and time, so that the transformation between frames is linear.
  4. Isotropy, so that there is no preferred direction along the line.
  5. A synchronization convention (Chapter 17).
  6. Reciprocity: if $S'$ moves at $v$ relative to $S$, then $S$ moves at $-v$ relative to $S'$, and the transformation depends only on $v$.

The first two are the famous ones. The others are routinely used in textbook derivations without being called postulates. Which of them a given proof uses, and which it quietly assumes, is exactly what a claim contract exists to record. (The list is general background, not a repository record.)

This chapter maps which premises are empirical, which are mathematical, and where each enters. It builds on the model-versus-nature distinction of Chapter 4 and the objects of Chapter 17.

Three kinds of premise

The evidence policy separates “a mathematical inconsistency, a disputed modeling assumption, and an experimental disagreement.” Applied to the list above:

Premise Kind What could challenge it
Principle of relativity Empirical, then adopted as a model principle Experiments that distinguish frames
Constancy of $c$ Empirical, then adopted Measurements of light-speed anisotropy
Homogeneity, isotropy Modeling, with empirical backing A preferred-frame model
Synchronization convention Convention Nothing empirical; only a choice, with consequences for what counts as “simultaneous”
Linearity from homogeneity Mathematical consequence of the above A mistake in the argument
Real-number structure of coordinates Mathematical modeling A different number system

The engine can speak only to the bottom two rows. Those are the premises it can turn into claims and check. The upper rows would need an empirical layer (the MATH06 milestone), which has not started.

What the repository records, and what it does not

The physical claim cites einstein-1905 for four assumptions: the real domain, subluminal speed, the gamma definition and $c=1$. These are model assumptions; they encode a choice of units and a parameterization. They do not record the postulates.

No claim in the repository has the form “if the postulates hold, then the Lorentz transformation follows”. What exists is the other direction: given the boost formula, the interval is preserved (Chapter 20). The difference matters. The engine can say “this formula preserves this quantity”. It cannot yet say “this formula is forced by premises you may accept or reject”. Chapter 19 sketches the second argument using light-cone coordinates, but as a derivation on paper, not a theorem in a kernel.

A claim of the needed form would look like this:

Candidate Shape Standing
(not in repository) postulates ⇒ Lorentz “For every family of transformations satisfying linearity, reciprocity and light-speed invariance, the transformation is the boost with $\gamma = 1/\sqrt{1-v^2}$” Not formalized; a derivation sketch is in Chapter 19

Where the second postulate does work

If light moves at speed $1$ in $S$, its worldline is $x=t$. The boost sends that line to another light line. Executed, with exact scale factors at $v=3/5$:

$$ t' - x' = 2\,(t-x), \qquad t' + x' = \tfrac12\,(t+x). $$
So $t=x$, that is $t-x=0$, maps to $t'=x'$: light still moves at speed $1$. The factors are $\gamma(1+v) = \tfrac54\cdot\tfrac85 = 2$ and $\gamma(1-v) = \tfrac54\cdot\tfrac25 = \tfrac12$, and their product is $1$.

Contrast the false shear from Chapter 5: $t'=t$, $x'=x+vt$. A light ray $x=t$ goes to $x' = (1+v)\,t'$. At $v=1/2$ that is speed $3/2$. The shear is not a relativistic transformation, because it does not preserve the speed of light. Here the second postulate works as a filter: it excludes the shear, just as the interval test does. (The shear computation is by hand.)

What this establishes

There is no evidence here for the postulates themselves. The executed computation, that at $v=3/5$ the boost maps light lines to light lines with scale factors $2$ and $1/2$, supports the consequence (the boost keeps the speed of light), not the postulate.

The postulates are what turn a piece of mathematics into a theory about the world, and that is also why they are the least checkable. A proof of “if the postulates, then Lorentz” can happen inside a kernel. Establishing the premises cannot. The engine’s honest position is that it records the mathematics of the implication and leaves the premises to experiment. A chapter that said “relativity is confirmed” would be relying on empirical (MATH06-class) evidence that does not exist here.

The evidence policy treats an unconventional claim the same way. A claim that rejects a postulate gets a contract and an evidence bar like any other, and the same limits on what it may conclude.

Try it yourself

uv run python content/examples/math/part3/part3_kinematics_demo.py

The “Ch 18/19” section of part3_kinematics_demo.py prints the light-cone scale factors.

Seeded error. Change the boost to $t' = \gamma(t-vx)$, $x' = \gamma(x+vt)$, a sign slip. At the event $(t,x)=(5,3)$ the correct interval is $16$ (the factors $t'-x'$ and $t'+x'$ are both $4$); with the slip it is $-161/4$ (factors $-7/2$ and $23/2$). Executed on 2026-11-09 with SymPy 1.14.0, using ch05_10_18_22_24_checks.py. The event $(0,1)$ from Chapter 17 is not a useful test: with $t=0$ the sign slip changes nothing.

Exercise. Check that the shear maps the left-moving light ray $x=-t$ to a ray of speed $1-v$ (in the negative direction). What is that speed at $v=1/2$? (Answer: $x' = -t + vt = -(1-v)\,t'$, so the speed is $1-v = 1/2$.)

Open questions.

  1. Is the list of six premises complete and right? Different textbooks use different minimal sets, and a reader who knows the literature should challenge it.
  2. Could the engine hold an empirical claim of the form “the speed of light is isotropic to within $\varepsilon$”, with a dataset and an uncertainty model? What would empirically_supported then require beyond the evidence policy’s wording?
  3. If a criticism of relativity rejected premise 3 or premise 6, which claim contracts would need a new version, and which would stay valid?

Sources

  • Albert Einstein, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik (1905), https://doi.org/10.1002/andp.19053221004 (a historical source)
  • H. A. Lorentz, “Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than that of Light”, Proceedings of the Royal Netherlands Academy of Arts and Sciences (1904)
  • Hermann Minkowski, “Raum und Zeit”, Physikalische Zeitschrift (1909)

None has been read closely for this edition. Experimental references on light-speed isotropy would be cited as scientific_experiment sources under the citation conventions; none is cited yet. In the repository: experiments/relativity/special/lorentz-physical-1.1.0.claim.json (the assumptions and their einstein-1905 source) and docs/methodology/evidence-policy.md.