Calibration of Minkowski Diagram: Explaining Invariance of S^2

[Pushoam]

Can anyone please illustrate highlighted part?
Can anyone please explain me how invariance of s^2 is useful in characterization of events?

I didn't understand the highlighted part i.e. how to determine velocity of S' relative to S?

 

[Ibix]

##s^2## is useful for characterising the separation between events. If it's zero in one frame it's zero in all, and the distance between the events must be equal to the distance light can travel in the time between them in all frames.

If it's positive in one frame (using the -+++ sign convention your book has, which is not universal) then it's positive in all, so not even light can get from one event to another. Note that the x difference cannot change sign in this case - because it cannot do so without passing through zero which must yield a non-positive ##s^2##.

Likewise a negative value is negative in all frames; this time it's the t difference between the events that cannot change sign. This means that all frames must agree on the time order of the events - which is good because influences traveling below lightspeed can get from one event to the other, and we don't like theories that let effect precede cause.

The velocity of the frame is easy. How do you get the velocity of an object from a displacement-time graph? What's the relationship between the time axis of S' and the graph of an object at rest in S'?