Causal Ordering & the Spacetime Interval

Source

Lecture: 18 — Faster than Light
Simulation: Simulation 2 — guided-inquiry causality activity
Concept: The spacetime interval between two events determines whether their time ordering is frame-invariant (timelike) or frame-dependent (spacelike).
Serves: Analyse causality using lightcone structure · Explain why the speed of light is an absolute speed limit · Connect superluminal propagation, acausality and instabilities.
Inquiry: Where does E₂ fall relative to E₁'s lightcone? Can a boost reverse the time ordering? What rule connects the classification to reversibility?
Misconception: "Time ordering is absolute" — that if E₁ precedes E₂ in one frame it must in every frame.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	Single concept	Every element serves one question: does the interval fix the ordering? No dynamics, energy or momentum.
A3/A4	Formal layer off by default	The Lorentz transform Δt′=γ(Δt−vΔx) lives in the ∑ Formal panel, off on load. The lightcone + readouts are enough alone.
B3	Define symbols	A "Symbol key" panel defines s², Δt, Δx, γ, v, v_sim and distinguishes Δt vs Δt′.
C1	No orphan numbers	Each readout (Δt′, Δx′, s², class, v) is named in at least one step prompt.
C2	Frame-labeled quantities	"Δt′ (frame S′, v=0.40c)" and "s² (invariant, all frames)".
D1	Viz is primary	The Minkowski diagram fills the hero; readouts are a compact strip beneath it.
D3	Labeled entities	E₁ (cyan) and E₂ (amber) circles with labels; lightcone and regions labeled.
D5	No clipping	Fixed ±5 range holds all presets and their boosted images; fixed-width numbers prevent jitter.
D6	Incremental emphasis	Steps brighten the boost slider and add ghost markers without ever disabling them.
D7	No gating	Drag, presets, slider and prediction all work from first render.
D8	Labels, not prose, on canvas	Canvas carries axis/event/ordering labels only; explanations stay in panels.
F2	Conceptual prediction	"Does the ordering change?" is a qualitative yes/no.
F9–F14	Prompt phasing	Step 1 open-ended; Step 2 holds tension; Step 4 asks for a warranted rule before the summary.
G2	Invariant labeled	s² is permanently annotated "(invariant, all frames)".
G3	Real re-render	Boosting Lorentz-moves E₂ on the diagram; Δt′, Δx′ change while s² holds.
G4	Boost = that rest frame	The slider observes from a frame moving at v; axes relabel S→S′.
G5	Live-bound text	The rebuttal inserts the computed v_sim from the current E₂ position.
G7	Events as points	E₁, E₂ are coordinate events, not bodies on worldlines.
H1	Structural misconception	The wrong idea is never a standing card; it is only a prediction option.
H2	Misconception as option	Option (A) "ordering same in all frames" triggers a rebuttal only when chosen for a spacelike pair.
J2	Display honesty	s² comes from the un-boosted coordinates and stays constant; the hyperbolic locus is drawn accurately.
J5	Stable axes	Diagram axes are fixed at ±5 regardless of boost — no per-frame rescale.

Causal Ordering & the Interval · Minkowski diagram (c = 1)

E₁ first

Active frameS (rest)

Δt (this frame)+1.00

Δx (this frame)+2.50

s² = Δt²−Δx² (invariant)−5.25

Classificationspacelike

Guided Inquiry

1 · Identify the events

Drag E₂ or pick a preset pair. Where does E₂ fall relative to E₁'s lightcone — inside (timelike), outside (spacelike), or on it (lightlike)? Read s² and the classification in the strip below the diagram.

2 · Predict the ordering

For the current event pair, is the time ordering (which event happens first) the same in every inertial frame?

Your guess is evaluated against the current pair.

3 · Boost & check

Slide Frame boost v (in Controls). Watch E₂ glide along its hyperbola while the lightcone stays fixed at 45°. Does the ordering label flip? For a spacelike pair the green v_sim marker shows exactly where Δt′ = 0.

4 · Generalize

Try all three presets. What rule connects the classification (timelike / spacelike / lightlike) to whether the order can reverse? State it in your own words, then open The rule notes below.

5 · Synthesise

Timelike (s² > 0) pairs keep their order in every frame — a cause can precede its effect for all observers. Spacelike (s² < 0) pairs have frame-dependent order, so no signal can causally link them without paradox. This is why c is an absolute speed limit. See The rule and ∑ Formal, then hit Finish to explore freely.

Controls

▾

Preset pair

— or drag E₂ anywhere on the diagram.

Frame boost v

v = +0.00c

The rule (notes)

▾

Timelike (s² > 0, inside the lightcone): a causal signal could connect E₁ and E₂. The causal order is frame-invariant — no subluminal boost reverses it.

Spacelike (s² < 0, outside the lightcone): the spatial gap exceeds c × the time gap, so no signal can connect them. "Before" and "after" are frame-dependent — a boost at v = Δt/Δx flips the order.

Lightlike (s² = 0, on the cone): only a light signal connects them; the order is invariant (flipping would need v = c).

Symbol key

▾

Δt, Δxtime / space separation of E₂ from E₁ in the rest frame S (c = 1). Δt′, Δx′the same separation seen in the boosted frame S′. s²Δt² − Δx², the invariant interval — same in all frames. vboost: speed of frame S′ relative to S, in units of c. γLorentz factor 1/√(1−v²). v_simboost that makes Δt′ = 0 (the simultaneity frame), = Δt/Δx.

Lorentz factor

Lorentz boost (c = 1)

Invariant interval — sign convention

Classification

Time ordering under boost

Order-flip (simultaneity) velocity