Transform an event pair and preserve its causal class

1. Define events, frames and conserved quantities

An event specifies a place and a time; two events need not be successive positions of the same object. Begin by naming the event pair before assigning a speed, duration or length. The original coordinates use c=1: one reference length equals the distance light travels in one reference time. This is a unit convention, not a claim that meters and seconds are interchangeable.

Frame S′ moves at velocity +β relative to S. Their difference coordinates obey Δt′=γ(Δt−βΔx), Δx′=γ(Δx−βΔt), with γ=1/sqrt(1−β²). The inverse changes the sign of β. Expanding the transformed squares cancels the cross terms and gives Δt′²−Δx′²=Δt²−Δx².

A positive interval is timelike, zero is null and negative is spacelike in this metric convention. Future-directed timelike or null pairs keep their time ordering under physical subluminal boosts. A spacelike pair can reverse order or become simultaneous because no causal signal connects the two events.

The visual overlays two coordinate descriptions for comparison; it does not connect them into a new physical worldline. A spacelike ratio Δx/Δt greater than one is not a measured superluminal particle. To claim a particle history, one must supply an actual continuous causal trajectory and appropriate event selection.

2. Reconstruct the original case

For the original spacelike pair (Δt,Δx)=(3,5) and β=0.4, γ≈1.091089, Δt′≈1.091089 and Δx′≈4.146140. Both intervals are −16. At β=0.8, the transformed time is −5/3 and spatial difference 13/3, still giving −16.

3. Predict and investigate

Which event pairs can reverse time ordering without a causal contradiction?

Transform all three supplied pairs, independently reconstruct the invariant and inverse transformation, and separate causal classification from an unsupported particle-velocity interpretation.

Primary mechanism context. Checked 31 October 2026. Original derivations, records and diagrams; no physical apparatus operation or actual independent clock/particle measurement is claimed. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.