Spacetime Intervals and Four-Vectors
Observers disagree about distances and times but agree on the interval between two events, which is the invariant of spacetime. Grouping quantities into four-vectors makes that invariance automatic.
What a learner can do afterwards
- Computes the interval between two events and classifies it as timelike, spacelike or lightlike
- Shows that the interval is unchanged by a Lorentz transformation
- Explains why causality survives even though the time order of some events is frame dependent
1 · Read
Observers argue about times and lengths, but they share one quantity: the interval. For two events take s squared equals c delta t squared minus the squared space gaps. Every frame computes the same s squared, since the Lorentz remix is built to preserve it.
Take two flashes 5 s apart in time and 3 light s apart in space. Then s squared is 25 minus 9, which is 16, a positive number. Positive means timelike: one slow traveller could visit both events, and all frames agree on their order.
A negative s squared is spacelike: no signal could connect the events, so frames may disagree on their time order, and neither could have caused the other. Zero is lightlike: only light itself could connect them. Causes always precede effects, so causality survives.
Bundle time with space as the four-vector c t, x, y, z. With time scaled by c in the same bundle, the invariance comes along for free.
Compute the interval, read its sign, and you know who could have caused whom.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.