Derive velocity addition from transformed differentials

1. Define events, frames and conserved quantities

Velocity is a ratio of spatial and time increments in one frame. A transformation must change both increments before the ratio is taken. Frame S′ moves at +β relative to S, and a particle has primed speed u′. The inverse Lorentz map gives dt=γ(dt′+βdx′) and dx=γ(dx′+βdt′).

Dividing by dt′ and then dividing dx by dt yields u=(u′+β)/(1+βu′). The common γ cancels, but the time-transformation denominator remains. The Galilean sum u′+β neglects that denominator and is accurate only in an appropriate low-speed limit.

For u′=+1 or −1, the expression returns the same light speed in S. Those endpoints describe light signals, not permissible speeds of a massive inertial frame. For two massive collinear frame speeds, the same composition rule can be applied sequentially; the result stays subluminal.

The original ledger sets dt′=1 and dx′=u′ to make the coordinate transformation explicit. It also transforms the resulting velocity back and checks dt²−dx²=1−u′². A conserved interval and recovered input provide independent consistency tests beyond merely obtaining a plausible number below one.

2. Reconstruct the original case

At β=0.6 and u′=0.5, dt=1.625 and dx=1.375, so u=11/13≈0.846154 rather than 1.1. Both coordinate-increment intervals are 0.75, and the inverse formula recovers u′=0.5.

3. Predict and investigate

What goes wrong if space is transformed but the original time increment is retained?

Transform the explicit differential pair, calculate the ratio, reverse it and check the interval. Compare counter-moving and light cases before treating a sum as a massive-frame speed.

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.