S16. Minkowskian Spacetime

In the previous section we briefly discussed the general feature of spacetime. From here on, we will focus on the geometrical aspect of the spacetime. So far, we have only assumed a frame $S$ being a collection of clock and a ruler. But there is another view point that is perfectly consistent and is more useful in our study of special relativity. In this view point(Minkowskian viewpoint) we assume time as a fourth dimensional axis which is 'at right angle' to the $X$,$Y$ & $Z$ axis.

The figure shows the diagram of the world line.  

World line: The curve traced out in the in the diagram is the world line of the particle.

We must note that the slope of the world line must be greater than the slope of the world line of a photon since all material particle moves with the speed less than that the speed of light. This is the reason we haven't drawn the brown arrow whose slope is less than that of the slope of world line of the photon. 

We have the Lorentz transformation as the following:

Now, to find the coordinate axes for the $S'$ frame, we simply put $x'$=0 for constant $t'$ and $t'$=0 for constant $x'$. We then have the familiar spacetime diagram.

It is now clearly evident why the coordinate axes of the $S'$ frame were oblique in our earlier discussions. It is important to note that with increasing speed of the $S'$ frame relative to the $S$ frame, these axes close in on the world line of the photon passing through the common origin. 

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