Path-time Diagram in Newtonian Physics
The black axes labelled x and ct on the adjoining diagram are the coordinate system of an observer which we will refer to as 'at rest', and who is positioned at x=0. His world line is identical with the time axis. Each parallel line to this axis would correspond also to an object at rest but at another position. The blue line, however, describes an object moving with constant speed v to the right, such as a moving observer.
This blue line labelled ct' may be interpreted as the time axis for the second observer. Together with the path axis (labeled x, which is identical for both observers) it represents his coordinate system. Both observers agree on the location of the origin of their coordinate systems. The axes for the moving observer are not perpendicular to each other and the scale on his time axis is stretched. To determine the coordinates of a certain event, two lines parallel to the two axes must be constructed passing through the event, and their intersections with the axes read off.
Determining position and time of the event A as an example in the diagram leads to the same time for both observers, as expected. Only for the position different values result, because the moving observer has approached the position of the event A since t=0. Generally stated, all events on a line parallel to the path axis (x axis) happen simultaneously for both observers. There is only one universal time t=t' which corresponds with the existence of only one common path axis. On the other hand due to two different time axes the observers usually measure different path coordinates for the same event. This graphical translation from x and t to x' and t' and vice versa is described mathematically by the so called Galilean transformation.
Read more about this topic: Minkowski Diagram
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