Take the origin of a spacetime diagram as here and now, and send light out from it to the left and right. The world lines of the light are two lines at degrees; drawn with two dimensions of space they form a cone, and so they are called the light cone.
Inside the upward cone are the events that light, or anything slower, can reach from the origin: the future. Inside the downward cone are the events from which the origin can be reached: the past.
The rest, outside the cone, are events too far away for even light to bridge in time. Nothing from the origin reaches them, and nothing from them reaches the origin.
Describe the separation of two events by the difference in position and the difference in time . Both take different values in different frames.
The following combination, however, comes out the same whatever frame measures it. It is called the spacetime interval.
The two events in the figure have and , so .
The figure reads the separation between the origin and the event again along tilted axes. In the frame with it becomes , and with it becomes : both values change.
Yet is whether it is computed as , or .
Put in the Lorentz transformation , , and the terms in cancel, leaving
Since , the right side equals the original .
An ordinary squared distance is a sum, and points of equal value lie on a circle. The interval is a difference, and points of equal value lie on a hyperbola.
When the time term is the larger. One event lies inside the other's light cone and the two can be joined by something slower than light. This is a timelike separation, and the order of the events is the same in every frame.
When the two events are joined exactly by light. This is a lightlike separation.
When the space term is the larger and even light cannot bridge the gap. This is a spacelike separation, and depending on the frame the order of the events can be reversed.
For the three points in the figure, , and .
For two events with a timelike separation, a single clock moving at constant velocity can pass through both. In that clock's frame , so .
This is simply what the clock reads. It is called proper time and written . Because , proper time comes out the same whichever frame it is calculated in.
The clock in the figure moves at ; in the ground frame it covers while . Then and , matching from time dilation.