Rod measurement and simultaneity

Two flashes that happen at the same moment for one observer happen at different moments for an observer moving past. So a moving rod, whose two ends have to be marked at the same moment to measure it, comes out shorter than the same rod measured at rest.

Section 1 defines when two distant clocks agree: light sent from A to B and reflected back must take as long going as returning. Section 2 applies that test to clocks on the ends of a rod moving at speed v, set to agree with the clocks of the resting system K. Seen from K, the light gains on the receding end B at c − v and meets the approaching end A at c + v, so t_B − t_A = r_AB/(c − v) and t′_A − t_B = r_AB/(c + v). The two times differ, so observers riding with the rod find the clocks out of step, while observers in K call them synchronous. Section 4 turns this into geometry: a sphere of radius R at rest in the moving system k, located at one time of K, is an ellipsoid with axes R√(1 − v²/c²), R and R. The instrument measures in light-seconds with c = 1. At v = 0.6c the factor √(1 − v²/c²) is 0.8, a rod 10 light-seconds long at rest in k measures 8 light-seconds in K, and two events 10 light-seconds apart at one time of K are 7.5 s apart in k.

Two frames: K, the resting system, and k, moving along K's x-axis at speed v. The instrument measures distance in light-seconds and time in seconds, so light covers one light-second each second and c = 1. Section 1's rule for two clocks at A and B: send light from A at time t_A, reflect it at B at t_B, and receive it back at A at t′_A. The clocks agree if the trip out takes as long as the trip back, t_B − t_A = t′_A − t_B. Section 2 puts clocks on the two ends of a moving rod, sets them to agree with K's clocks, and asks what riders on the rod conclude when they apply the rule. Work it out in K, where the rod has length r_AB. Going out, the light chases B, which runs ahead at v, so the gap closes at c − v and the trip takes r_AB/(c − v). Coming back, A runs toward the light, the gap closes at c + v, and the trip takes r_AB/(c + v). Put in this instrument's rod: v = 0.6 and r_AB = 8 light-seconds. Out, 8/0.4 = 20 s. Back, 8/1.6 = 5 s. Twenty seconds is not five, so by Section 1's rule the riders say their clocks disagree, while K, which set them, says they agree. Each is applying the same rule correctly in its own frame, so “at the same time” depends on the frame. The transformation of Section 3 makes this exact. Write γ = 1/√(1 − v²/c²); here that is 1/√(1 − 0.36) = 1/√0.64 = 1/0.8 = 1.25. Two events separated by Δx and Δt in K are separated in k by Δx′ = γ(Δx − vΔt) and Δt′ = γ(Δt − vΔx/c²). The default pair has Δx = 10 light-seconds and Δt = 0. Then Δx′ = 1.25 × 10 = 12.5 light-seconds, and Δt′ = 1.25 × (0 − 0.6 × 10) = −7.5 s: in k, the event farther along x happens 7.5 s earlier. Now the rod. At rest in k it is 10 light-seconds long, with its ends at x′ = 0 and x′ = 10. K measures it by finding where both ends are at one time t of K. At a fixed t, x′ = γ(x − vt) changes by γ for every light-second of x, so the ends are Δx = 10/1.25 = 8 light-seconds apart. Those two marking events are 6 s apart in k, which is why the riders do not accept 8 as their rod's length. The sphere of Section 4 works the same way in three directions: along the motion a radius of 1 light-second becomes 1 × 0.8 = 0.8, and across the motion it stays 1 and 1. A check that uses a later idea, Minkowski's of 1908: Δx² − c²Δt² is the same in both frames, 10² − 0 = 100 in K and 12.5² − 7.5² = 156.25 − 56.25 = 100 in k.

export function measureRodLength(
  e1InMeasuringFrame: Event,
  e2InMeasuringFrame: Event,
  measuringFrame: "K" | "k",
  rodRestFrame: "K" | "k",
  v: number,
  L0: number,
  c = C,
): RodMeasurementResult {
  const dt = Math.abs(e2InMeasuringFrame.t - e1InMeasuringFrame.t);
  if (dt > 1e-12) {
    const repair = selectSimultaneousEndpoints(
      rodRestFrame,
      measuringFrame,
      v,
      L0,
      e1InMeasuringFrame.t,
      c,
    );
    return {
      status: "not-applicable",
      isSimultaneous: false,
      condition: "non-simultaneous-endpoints",
      reason:
        "These endpoint events are not simultaneous in the measuring frame, so their separation is not a length measurement.",
      repairSuggestedPair: repair.status === "value" ? repair.value : undefined,
    };
  }

  const measuredLength = Math.abs(e2InMeasuringFrame.x - e1InMeasuringFrame.x);
  return {
    status: "value",
    isSimultaneous: true,
    measuredLength,
  };
}

export function causalOrder(e1: Event, e2: Event, c = C): KinematicResult<CausalOrderResult> {
  if (!finite(e1.t, e1.x, e2.t, e2.x) || !Number.isFinite(c) || c <= 0) {
    return outsideDomain("nonfinite-input", "Event coordinates and light speed must be finite.");
  }
  const dt = e2.t - e1.t;
  const dx = e2.x - e1.x;
  const dy = (e2.y ?? 0) - (e1.y ?? 0);
  const dz = (e2.z ?? 0) - (e1.z ?? 0);

  const spatialDist2 = dx * dx + dy * dy + dz * dz;
  const timeDist2 = c * c * dt * dt;
  const s2 = spatialDist2 - timeDist2;

  let classification: CausalClassification;
  let description: string;

  if (Math.abs(s2) <= 1e-12) {
    classification = "lightlike";
    description = "Events can be connected only by a light signal propagating at speed c.";
  } else if (s2 > 0) {
    classification = "spacelike";
    description =
      "No signal travelling at or below the speed of light reaches between these events; their time order can be reversed by choosing an appropriate moving reference frame.";
  } else {
    classification = "timelike";
    description =
      "A signal could travel from the first of these events to the second; their time order is invariant under all subluminal Lorentz boosts.";
  }

  return ok({
    s2,
    classification,
    description,
  });
}

export function ellipsoidAxes(
  R: number,
  beta: number,
): KinematicResult<Readonly<{ longitudinal: number; transverseY: number; transverseZ: number }>> {
  const g = gamma(beta);
  if (g.status !== "value") return g;
  if (!finite(R) || R < 0)
    return outsideDomain("invalid-length", "Radius must be finite and nonnegative.");
  return ok({
    longitudinal: R * Math.sqrt(1 - beta * beta),
    transverseY: R,
    transverseZ: R,
  });
}