⏱️ Lorentz Transform — Minkowski Diagram

Lorentz Transformation

The Lorentz transformation relates spacetime coordinates between two inertial frames S and S′, where S′ moves at velocity v = βc relative to S:

• x′ = γ(x − βct) — space coordinate transforms (length contraction)
• ct′ = γ(ct − βx) — time coordinate transforms (time dilation)
• γ = 1/√(1−β²) — Lorentz factor, diverges as β→1

Reading the Minkowski Diagram

• The blue axes are the S frame — horizontal x, vertical ct
• The orange axes are the S′ frame — both tilted toward the light cone as β increases
• The yellow dashed 45° lines are the light cone — events on it have s² = 0 (lightlike)
• Dashed lines from event A show its coordinates in each frame
• The S′ grid lines appear physically stretched because the axes use Minkowski (not Euclidean) unit intervals

What to Explore

• Drag β close to 1 — watch the S′ axes squeeze toward the light cone
• Place event A at x = ct (on light cone) — both frames agree it is lightlike
• Place A in the "past" region (ct < 0) and increase β — ct′ stays negative (causality preserved)
• Place A with x > ct (spacelike) — the time ordering can reverse between frames, showing simultaneity is relative

Spacetime Interval

The quantity s² = c²t² − x² (or −s² = x² − c²t²) is Lorentz-invariant — the same in all inertial frames. If s² > 0 the interval is timelike (causal connection possible), if s² = 0 it is lightlike, if s² < 0 it is spacelike (no causal connection, time order can be reversed).