Spacelike, timelike and lightlike

Quoted from StackExchange.

Suppose we have two events \((x_1,y_1,z_1,t_1)\) and \((x_2,y_2,z_2,t_2)\). Then we can define

\[\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2, \]

which is called the spacetime interval. The first event occurs at the point with coordinates \((x_1,y_1,z_1)\) and the second at the point with coordinates \((x_2,y_2,z_2)\) which implies that the quantity

\[r^2 = \Delta x^2+\Delta y^2+\Delta z^2 \]

is the square of the separation between the points where the events occur. In that case the spacetime interval becomes \(\Delta s^2 = r^2 - c^2\Delta t^2\). The first event occurs at time \(t_1\) and the second at time \(t_2\) so that \(c\Delta t\) is the distance light travels on that interval of time.

In that case, \(\Delta s^2\) seems to be comparing the distance light travels between the occurrence of the events with their spatial separation. We now have the following definitions:

• If \(\Delta s^2 <0\), then \(r^2 < c^2\Delta t^2\) and the spatial separation is less than the distance light travels and the interval is called timelike.

• If \(\Delta s^2 = 0\), then \(r^2 = c^2\Delta t^2\) and the spatial separation is equal to the distance light travels and the interval is called lightlike.

• If \(\Delta s^2 >0\), then \(r^2 > c^2\Delta t^2\) and the spatial separation is greater than the distance light travels and the interval is called spacelike.