#spacetime-interval

Two observers watch the same pair of events. They agree that the events happened, but they may assign them different distances and different elapsed times.

That can sound like a problem. If one observer measures one distance and another measures a different distance, how can both still agree that the events are timelike-, lightlike-, or spacelike-separated?

The answer is that relativity does not preserve space and time as separate quantities. It preserves a particular combination of them: the spacetime interval.

If the geometric picture is unfamiliar, Light Cones: What Can Affect What in Spacetime? provides the useful starting point. A light cone places events inside, on, or outside its boundary. The spacetime interval is the calculation that makes the same classification without relying on a diagram.

Space and Time Change Together

Consider two events, \(A\) and \(B\). In one inertial frame, their coordinate differences are

$$ \Delta t = t_B-t_A $$

and

$$ \Delta x = x_B-x_A. $$

Another observer moving at a steady velocity relative to the first may obtain different values for both \(\Delta t\) and \(\Delta x\). Neither value, taken alone, is shared by every inertial observer.

But the changes are not arbitrary. When an observer's motion changes the measured distance between the events, it changes the measured elapsed time in a coordinated way. The spacetime interval captures that coordination.