Question

The coordinates of two spacetime events are given by (ct1, x1) = (2, 5) and (ct2, x2) = (3, 3) in an inertial reference frame K.

a) Calculate the squared spacetime interval between these events. Is their spacetime separation spacelike or timelike? (

b) Transform the co-ordinates of this pair of events to a new reference frame, K0 moving relative to the one used in (a) with a constant speed u = −2 × 108m/s in the x direction. Explicitly show that the squared spacetime interval between the events is the same in the new reference frame.

(c) Compare the time ordering of events in the two reference frames – you should find that the ordering has changed. In general such a change of ordering would lead to causal paradoxes, for example: the girl calls her dog, the dog obeys; versus, the dog obeys, the girl calls her dog. Explain (with the aid of a spacetime diagram) why this is not a problem in the case of the events analysed in this question

Answer 1

The ST coordinates are:

The Space Time interval is:

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Now, the new frame is moving with velocity

The boosted frame's time and space coordinates are related to the original frame as:

Taking the two results above, substituting the values, we get:

Thus, the space time interval is given by:

It is thus the same.

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In the first frame:

In the new frame,

The negative implies that the difference in the time coordinates (second event- first event) in the boosted frame is negative. This means that the second event had a time coordinate lesser than the first event.

The events with lesser time coordinates happen first. (obviously)

Thus, in the boosted frame , the second event happened first.