The space-time interval is a quantity combining the time and spatial separation between two events. It is invariant because every inertial observer calculates the same interval, even though observers may measure different distances and time intervals.
For two events separated by time and position , the interval in one spatial dimension is
Multiplying time by the speed of light converts the time separation into a distance, so every term has units of distance squared.
Different inertial reference frames generally disagree about and . According to the Lorentz transformations,
and
Substituting these expressions into causes the additional terms involving and to cancel, giving
This cancellation is not an algebraic coincidence. The Lorentz transformations preserve the speed of light and mix space and time in precisely the way required for the interval to remain unchanged. The interval therefore resembles distance in ordinary geometry: coordinates can change when the coordinate system changes, but the underlying separation does not. Consequently, observers agree on the causal relationship between the events.
The sign of the squared interval classifies the separation. A positive value is timelike, so one event can influence the other. Zero is lightlike and corresponds to a light signal. A negative value is spacelike, so no signal travelling at or below light speed can connect the events. This classification is also invariant.
For example, suppose light-seconds and . Then light-seconds squared. For an observer moving at , , so light-seconds and light-seconds. Therefore,
the same interval.
A common misconception is that time and distance are individually invariant. They are not: their values change between frames, but their space-time combination does not.
This is HL-only content from A.5 Galilean and special relativity. For a Show question, write the Lorentz transformations, substitute them into the interval, and include the cancellation steps. Do not merely state that the interval is invariant.
