#timelike-separation

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.

Suppose an instrument at one location emits a signal. Two seconds later, a station far away records an alarm.

Could the first event have caused the second?

The answer does not come from the words before and after. It depends on how far apart the events are, how much time separates them, and whether any influence could have crossed that distance in time. Relativity gives us a diagram for answering exactly that question: the light cone.

A light cone is often introduced as a picture of the rule that nothing travels faster than light. That is true, but it puts the emphasis in the wrong place. The deeper use of the diagram is to show which events can be causally connected. The speed of light matters because it marks the boundary of possible influence.

1. Begin With One Event

In relativity, an event is something that happens at a particular place and time: a detector clicks, a lamp turns on, a message arrives. A light-cone diagram begins by choosing one such event as its reference point.

Now imagine that the event emits a flash. As time passes, the light moves outward in every spatial direction. In ordinary three-dimensional space, the expanding wavefront would be a sphere.

A basic spacetime diagram suppresses two spatial dimensions and shows only one horizontal direction, with time running vertically. The expanding light therefore appears as two diagonal lines. If we include both the future and the past of the chosen event, those lines form the familiar double cone.