Here we have 3 events and 3 inertial frame moving with different velocities. I have to prove some time based inequalities [closed]

This is my homework question for relativity. I am not sure where to start. I mean we dont know what are the velocities of R, S, and Q or the difference between their velocities (stating which is more faster). The question does not state which event is more further from the origin. I know if they are not given in question then they are not required, but can you please give a hint so that I can get a coorect path to follow so that I can solve it.

QUESTION:

Consider three events A, B, and C in spacetime, and three inertial reference frames R, S, and Q. You may assume that “observers” in the three frames agree on the event=x= 0 (namely, the origin of spacetime), but maybe moving at different velocities (which are oriented in the x-direction). Denote the coordinates of event A in inertial reference frame R with the coordinate pair $$(t_A,_R,x_A,_R)$$. Show that the following scenario is mathematically forbidden: in frame R, $$t_A,_R< t_B,_R< t_C,_R$$, while in frame S, $$t_B,_S< t_C,_S< t_A,_S$$, while in frame Q,$$t_C,_Q < t_A,_Q < t_B,_Q$$. Note that these inequalities are strict!

Suppose there are three clocks on a certain rod, placed at its center and the two ends. If the rod is moving at relativistic speeds towards you, and you look at this from a frame that is at rest, the clock that is closest to you, would appear to lag behind, and the clock furthest away from you would appear to lead.

This happens because of the relativity of simultaneity.

Now imagine some events happen in the rest frame, on a straight line. Each of these events can be thought of as a clock in a straight line. Now imagine you have a moving frame, moving along the $$x$$ axis. From the perspective of this moving frame, it would appear that the three events are moving towards it instead. Remember, in its own frame the moving observer feels at rest, while everything else is moving. Just like inside a train, you feel as if the world is moving by you, while it is the exact opposite.

So, suppose this observer is moving left to right with respect to the rest frame. Then from its perspective, it would be the rest frame, and the original frame would appear to move from right to left. Hence, the clock on the extreme left would appear to lag behind, and the one on the extreme right would appear to lead. Hence, to this moving observer, the event that happens furthest away from himself, appears to happen first and then the others.

Using this line of reasoning, you can tackle these problems.

For different observers moving in different directions, the order of events would thus be different. If the observer was moving right to left, then the rest frame would appear to move left to right. The order of events according to this observer would be opposite to that of the previous observer.

I highly recommend Brian Greene's $$11$$ hour lecture on Special relativity on youtube, for these conceptual problems.

• Thanks for the in depth explanation. Commented Sep 9, 2021 at 17:08