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Events and Reference Frames

The vocabulary for relativity without hidden meaning: events, frames, simultaneity at three-fifths light speed, and what is frame-dependent.

Two events, $A$ and $B$, happen at the same time in a frame $S$: $A$ at $(t,x)=(0,0)$ and $B$ at $(0,1)$. Are they simultaneous?

The question has no answer that does not depend on the frame. Take a second frame $S'$ moving at $v=3/5$ relative to $S$ (so $\gamma=5/4$, in units where $c=1$). Executed with exact rationals:

$$ t'_A = 0, \qquad t'_B = \gamma(0 - v\cdot1) = \tfrac54\cdot\left(-\tfrac35\right) = -\tfrac34 . $$
In $S'$, $B$ happens three quarters of a time unit *before* $A$. Both frames are right. Simultaneity belongs to a frame's way of assigning time coordinates, not to the events. What both frames agree on is the interval:
$$ S:\ 0^2 - 1^2 = -1, \qquad S':\ \left(-\tfrac34\right)^2 - \left(\tfrac54\right)^2 = \tfrac{9}{16} - \tfrac{25}{16} = -1 . $$
Everything in this part of the book is a variation on that observation: some quantities depend on the frame, and the interval does not.

This chapter fixes the vocabulary for talking about relativity without importing hidden meaning: events, frames, and what depends on the frame. It builds on claims (Chapter 1) and assumptions (Chapter 2), and it is the base for Chapters 18 to 22.

Events

An event is a point of spacetime: something that happens somewhere at some moment, with no duration and no extent. A collision, a clock tick, a flash. In one space and one time dimension, an event has two coordinates, $(t,x)$. Two descriptions of the same event in different frames have different coordinates and are still one event.

The engine’s claims already follow this convention. The claims lorentz-interval-invariance-1d-* have variables $t, x$ (one event’s coordinates), a boost that produces $t', x'$ (the same event in another frame), and a conclusion about $t^2-x^2$.

Reference frames

A frame is a rule for giving coordinates to events. Operationally it is built from rods and clocks:

  • rods lay out a spatial coordinate $x$;
  • clocks at every position give a time coordinate $t$;
  • the clocks must be synchronized, and synchronization is a convention. The standard one, Einstein’s, uses light signals and assumes that light has the same speed in both directions. That is partly stipulation, not purely measurement. (Background from the physics literature, not a repository record.)

An inertial frame is one in which free bodies move in straight lines at constant speed. Which frames are inertial is an empirical matter (Chapter 18). The Lorentz transformation relates two inertial frames.

This matters for the engine because it marks what a claim would have to state. A “physical” Lorentz claim silently assumes inertial frames, Einstein synchronization, ideally behaved rods and clocks, and $c=1$. The current claim contract records $c=1$ and the real domain. It records none of the others; they live in the model, not the contract (Chapter 4).

What depends on the frame, and what does not

Quantity Depends on the frame? Example here
Coordinates $(t,x)$ of an event Yes $B$ is at $t=0$ in $S$ and $t'=-3/4$ in $S'$
Whether two events are simultaneous Yes The opening example
Time between two events ($\Delta t$) Yes Chapter 21
Distance between two events ($\Delta x$) Yes Chapter 21
The interval $\Delta t^2 - \Delta x^2$ No $-1$ in both frames
Whether the interval is positive, zero or negative No Chapter 22

The last two rows are what the machine-checked theorem of Chapter 20 provides. The rest follow from the boost formula.

The boost between two frames

Chapter 19 derives the transformation. Here, take it as given. Frame $S'$ moves at velocity $v$ along $x$ relative to $S$, and the origins coincide at $t=0$:

$$ t' = \gamma(t - vx), \qquad x' = \gamma(x - vt), \qquad \gamma = \frac{1}{\sqrt{1-v^2}}, \quad \lvert v\rvert < 1 . $$
For $v=3/5$ the numbers stay exact, because $\sqrt{1-9/25} = 4/5$ and $\gamma = 5/4$. Executed: $\gamma^2(1-v^2)=1$, the hypothesis of the algebraic theorem.

What this establishes

No claim is qualified in this chapter. The statements it uses, in the form a future claim contract would need:

Candidate claim Statement Engine standing
(illustrative) relativity of simultaneity There exist events simultaneous in $S$ and not simultaneous in $S'$ Executed for one instance ($v=3/5$); not a claim
(qualified) lorentz-interval-invariance-1d-physical@1.1.0 The boost preserves $t^2-x^2$ Kernel-accepted in Lean and Rocq (Chapter 20)

The interval invariance is the qualified evidence. The relativity of simultaneity is a single exact computation. That proves an existential claim (one pair of events works) and nothing about frames in general. The fully general statement, that for every $v\neq0$ some pair of events simultaneous in $S$ is not simultaneous in $S'$, is a short theorem, but nobody has put it through a kernel here. The one-line computation above is checkable by hand; no review is recorded.

This chapter fixes the objects. It says nothing about whether nature’s frames behave like these. In particular, the engine cannot say that clocks synchronized by Einstein’s convention exist in practice, that rods and clocks transform as the model assumes, or that the frames in a real laboratory are inertial. Those are modeling and empirical questions (Chapter 4 and Chapter 18).

Try it yourself

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

The “Ch 17” section of part3_kinematics_demo.py prints the $S'$ times $0$ and $-3/4$ and the two intervals.

Seeded error. In the demo, change v to Rational(1, 2). The line assert gamma == Rational(5, 4) fails, because $\gamma = 2/\sqrt3$ is irrational and the exact-rational presentation breaks. That is why these chapters use $3/5$.

Exercises.

  1. With $v=3/5$, at what $S$-time must an event at $x=1$ happen to be simultaneous with $A$ in $S'$? (Answer: $t'=0$ requires $t=vx=3/5$.)
  2. Compute the $S'$ position of $B$ and check that the interval is still $-1$. (Answer: $x'_B = \gamma(1-0) = 5/4$, and $(-3/4)^2-(5/4)^2 = -1$.)

Open questions.

  1. Einstein synchronization is a convention. Should the claim contract record it as an assumption with its own role, or leave it in the model?
  2. Is a “frame” a primitive object in the formal development (a coordinate chart), or only the boost formula between coordinate pairs? Physlib models the Lorentz group as matrices that preserve the Minkowski metric (according to the project’s survey of formalizations); how a frame would be represented there is open.
  3. What would an operational claim about frames look like, in terms of rods, clocks and synchronization procedures, and which part of the engine could hold it?

Evidence record

The qualified evidence for interval invariance is in Chapter 20: Lean and Rocq kernel acceptance, SymPy and Z3, with statuses from the accepted R2 report and the R4 bundle. The simultaneity computation is illustrative.

Sources

  • Albert Einstein, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik (1905), https://doi.org/10.1002/andp.19053221004, cited by the physical claim.
  • Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics: Introduction to Special Relativity (W. H. Freeman, 1992), as a standard modern treatment.

Neither has been read closely for this edition; they are background, not evidence.