How do the coordinates from different frames of reference moving with respect to each other relate? In this section, the relation between different coordinates systems will be studied.
Putting together the two boxed relations, the following result is obtained
When the motion between the different frames of reference is only along one of the axis, the other coordinates remain unchanged; that is, y' = y and z' = z.
The time measured by different frames of reference are different. The following discussion will help to understand the relation between the times measured by the two frames of reference.
After a time t, with respect to frame of reference O, and t', with respect to frame of reference O', the pulse is used for synchronizing secondary clocks in both frames of reference. The yellow circle correspond to the pulse of light as observed by the observer O while the orange circle correspond to the view of the same pulse from the point of view of the observer O'. The coordinates x and x' correspond to the distance traveled by the pulse in the two frames of reference. From these coordinates, the times t and t' can be calculated considering that the light travels at the same speed c for both frames of reference,
The relation between x' and x was obtained above for the Lorentz transformation for spatial dimensions.
This result is completely different than the classical mechanic result that where the time is always the same independent of the motion of the frame of reference. In this case, the times run different for different frames of reference in motion with respect to each other.
The following table resume the result encounter for the Lorentz Transformations.