The success of the special theory of relativity in explaining many things that Newtonian mechanics could not while still being consistent with Newtonian mechanics at low speeds gives us a great deal of confidence.
Any two inertial observers will agree on the speed of light – the constancy of the speed of light being one of the postulates of the special theory. In fact there is more on which they can both agree. All physical events take place in 'spacetime'. All events can be given a set of coordinates,
where
are space coordinates and
is time. These four coordinates together determine uniquely the coordinates of an event.
Because of the phenomena of length contraction and time dilation predicted by special relativity, two observers in uniform relative motion will not necessarily agree on any of the coordinates (x,y,z,t) for an event. They will agree on a quantity quite similar in concept to distance in ordinary three dimensional space, called the 'spacetime interval'. If two observers synchronize meet at some point, labelled
and synchronize their clocks at
then any later event E, which has coordinates
for one observer and
will satisfy
To make the constancy of this quantity for any two observers clear, we write
and in general any two observers will agree on the spacetime interval squared,
between any two events.
If
then
is an ordinary three dimensional distance squared,
in space, so we can write
Then
The events must be such that a light ray could have travelled between them, and in fact may be for example, an emission of a photon by an atom at one point, and absorption of a photon by some other atom at some other point a distance
away and timelater.
If
then
so the distance between the two events is too great for any particle or light ray to have travelled between them. One event cannot be the cause of the other.
If
then
so the distance between the two events is small enough for a particle or light ray to have travelled between them. The earlier event may be the cause of the latter.