Physics HL · Chapter 6: Relativity
6.3 Velocity Addition and Simultaneity
Use relativistic velocity composition and explain why simultaneity depends on frame for separated events.
Estimated time: 22 minutes
Relativistic Velocity Addition
These replace u = u' + v at high speed and guarantee that composed speeds stay below c for material objects.
The chapter demonstrates that even when two velocities are each close to c, their composition remains below c. This resolves the contradiction that appears if you force Galilean addition onto electromagnetic signals or near-light projectiles.
For light itself, setting u-prime equal to c returns u equal to c directly from the formula. That algebraic result is the practical signature that the transformation framework is internally consistent with Einstein's second postulate.
Relativity of Simultaneity
If two events are simultaneous in one frame and occur at different positions, another frame in relative motion generally assigns different times to them. This is not an optical delay artifact. It is a coordinate-time difference produced by Lorentz transformation.
When delta t-prime is zero and delta x-prime is nonzero, delta t is usually nonzero.
The train-lightning style setup in the source chapter is useful because it separates two ideas: emission order and reception order. Events that meet at one location can be simultaneous for all frames, while spatially separated emissions need not be. Distinguishing those cases prevents many paradox-style mistakes.
Frame Logic for Simultaneity Experiments
The chapter's train example can be read in a strict sequence. First define the two emission events A and B and their coordinates in one frame. Next transform those event coordinates, not detector-arrival times. Only after that should you discuss which observer receives which signal first. Mixing these stages is the main source of simultaneity errors.
If S' says events are simultaneous but spatially separated, S predicts a nonzero time offset with sign set by v and the event ordering used in delta x-prime.
Sign discipline matters. If event B is ahead in the +x' direction so that delta x-prime = x'B - x'A is positive, and S' moves with +v relative to S, then delta t = tB - tA is positive: A happened earlier in S. This is exactly the direction logic shown in the text's quantitative train-light example.
Simulation: Moderate-Speed Frame Comparison
Use a mid-relativistic speed to see that simultaneity offsets exist before extreme gamma growth dominates intuition.
Relativity Clock + Spacetime Lab
γ
1.2500
v
0.600 c
v (m/s)
1.799e+8
t from τ
20.00 s
Minkowski diagram (x, ct)
This diagram is coordinate-based: the same event pair gets different simultaneity slices in S and S′, while light-cone boundaries preserve causal structure.
Interpretation guidance: hold the two sliders for proper time and coordinate time equal, then vary beta. At beta = 0.6 the gamma value is only 1.25, but the model already shows clear separation between local clock interval and transformed frame interval. This is a good regime for conceptual debugging because numbers stay simple while relativity is undeniably active.
Test Yourself
In frame S', two flashes at different x' positions are simultaneous (delta t' = 0) and x'B > x'A. If S' moves at +v relative to S, which statement is correct for S?
Hint: Start from the reduced simultaneity equation with delta t-prime set to zero.