The fundamental question raised by these postulates of special relativity is how different coordinate systems (reference frames) are related, i.e., how one transforms between them. (x, y, z, t) denotes the coordinates of some event in frame S, what are the coordinates (x’, y’, z’, t’) in the frame S’ moving at the velocity v relative to S? But first, a clarification on proper time and coordinate time:
Proper time is time measured between events by use of a single clock, where these events occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a shorter proper time between two events than a non-accelerated (inertial) clock between the same events.
In standard special relativity, we often want to express results in terms of a spacetime coordinate system relative to an implied observer. In this case, an event is specified by one time coordinate and three spatial coordinates. The time measured by the time coordinate is referred to as coordinate time, to distinguish it from proper time.
The answer is given by the Lorentz transformations:
x’ = y(x-vt)
y’ = y
z’ = z
t’ = y(t-vx/c2)
where y = 1/(1-v2/c2)1/2
(Notes from a lecture delivered by Bruce Gordon,
“Relativity Theory and the Nature of Time”)