A.5 · IB Physics
Spacetime diagrams
A spacetime diagram for frame S with the tilted axes of frame S′ drawn on top. Place two events by dragging them or with the sliders, read their coordinates in both frames, and see which quantities every observer agrees on.
Predict first
Start from the defaults, where A and B are simultaneous in S and B is 3.0 ly further along x. If you raise v ÷ c from 0.600 to 0.800, what happens to the order of A and B in S′?
Δct′ = γ(Δct − βΔx) = −γβΔx when Δct = 0. The magnitude γβΔx grows from 2.25 ly at 0.6c to 4.00 ly at 0.8c, so B stays first and the gap widens.
Check it in the model: Move the Speed of S′ relative to S, v ÷ c slider from 0.600 to 0.800 and compare Δct′ in the A to B in S′ readout.
event A event B axes and grid of S′ · drag A or B on the diagram
x′ = γ(x − vt)
ct′ = γ(ct − vx ÷ c)
interval: (ct)² − x² = (ct′)² − x′²
c = 1 ly per year
Event A in S′:
Event B in S′:
Interval of A, (ct)² − x²:
A to B in S:
A to B in S′:
Type and order:
Try this
- Choose the preset "Simultaneous in S only". Follow the dashed green line parallel to the x′ axis from each event to the ct′ axis. Which event happens first in S′, and does that agree with Δct′ in the readout?
- Put A at x = 3.0 ly, ct = 5.0 ly and drag the v/c slider from 0 to 0.95. Watch the interval of A in S and in S′, and the angle θ of the primed axes. What stays the same, and what changes?
- Make A and B space-like separated (|Δx| larger than |Δct|). Find a speed v for which the order of the two events in S′ is the opposite of the order in S, then explain why no particle could travel from A to B.
Common error: reading S′ coordinates by dropping perpendiculars onto the tilted axes. Draw a line through the event parallel to the other primed axis, and remember the unit lengths on the primed axes are longer than on the S axes, so check the reading with x′ = γ(x − vt) and ct′ = γ(ct − vx ÷ c).