Assume there are two points in spacetime and . Let's say that the first one is in the origin of spacetime i.e. . The point has two possibilities
1. i.e. it's right of
2. i.e. it's right of but ahead of it as well.
Someone want to send a signal from to . Is there a way that these two points(the two ) receive the signal instantly either in space or time?
Answer & Explanation
Calculate the spacetime interval
In this sign convention,
"Someone want to send a signal from to . Is there a way that these two points(the two ) receive the signal instantly either in space or time?"
If the events are timelike-separated, there exists an inertial frame in which the events are at the same spatial location. Intuitively, if the spatial distance between them, divided by the elapsed time is less than , then it is possible for a subluminal signal to go between them at this speed . Hence you can pick an inertial frame comoving with that subluminal signal, in which the signal will be stationary... and hence those two events at the same spatial location.
Similarly, if the events are spacelike-separated, there exists an inertial frame in which the events are simultaneous. Relative to the old inertial frame, it will have speed , where is as above.
For (1), the event is co-located in time, i.e., simultaneous with event and thus the associated interval is space-like (the distance through space is greater than the distance through time in any frame).
There is no world line (time-like curve) that includes the events and so, the event cannot be the event "receive signal initiated at event ."
For (2), the temporal distance is . Since the temporal distance is larger than the spatial distance, the associated interval is time-like and thus, there is a frame of reference (the rest frame) in which the two events are co-located in space.
However, your notion of "sending a signal from to either " needs refinement. You should be thinking in terms of world lines and the event as "the event that a signal is sent" and event as "the event that the signal is received". Thinking of it this way, it's clear that (1) cannot be an .
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