Spacetime Intervals and Four-Vectors
Inertial observers can measure different time and space separations while agreeing on the spacetime interval. A four-vector transforms by the Lorentz rule, which preserves its Minkowski squared norm.
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
In flat spacetime, s squared = (c delta t) squared - (delta x) squared - (delta y) squared - (delta z) squared. Inertial frames agree on this quantity. A light-second is the distance light travels in one second. With seconds and light-seconds, c delta t has the numerical value of delta t and s squared is in light-seconds squared.
For gaps measured in seconds and light-seconds, subtract squared spatial distance from the squared time gap. A 4-second gap and 2-light-second distance give 16 - 4 = 12 light-seconds squared. Positive means timelike and permits a traveller below light speed. Negative means spacelike: no causal signal can connect the events. A permitted connection does not prove causation.
For distinct events, positive s squared means timelike, negative means spacelike, and zero means lightlike. Timelike pairs allow a signal below light speed; lightlike pairs allow one at light speed. Spacelike pairs would require a faster-than-light signal and can reverse time order between inertial frames. Future-oriented inertial frames agree on timelike and lightlike order.
In an inertial frame, a force-free object moves at constant velocity. The position four-vector X = (ct, x, y, z) has length-valued components that follow the Lorentz law; four entries alone do not make a four-vector. Differences of position four-vectors follow that law too. Their Minkowski squared norm is the spacetime interval. Unlike the positive-definite inner product for ordinary lengths, this signed norm can be negative or zero for a nonzero separation.
For an x-direction boost, ct' = gamma(ct-beta x) and x' = gamma(x-beta ct), where beta = v/c and gamma = 1/sqrt(1-beta squared). Use these equations for event separations; y and z stay unchanged. Calculate the interval before and after the boost. Expanding (ct') squared - (x') squared cancels the mixed terms and gives gamma squared times (1-beta squared) times ((ct) squared - x squared). This prefactor is one, so the interval is preserved.
A possible cause must reach its effect at speed c or below. For a future-directed timelike or lightlike separation, ct > 0 and ct >= the spatial distance. Since |beta| < 1, ct-beta x stays positive, so ct' stays positive. Only spacelike pairs can reverse order. A timelike or lightlike interval allows causal influence; it does not prove that influence occurred.
Use one sign convention and consistent units. Lorentz boosts preserve the interval; its sign limits possible causal connections.
Lesson and assessment text authored for this dataset.
Readings authored independently for this dataset.
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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.