Question

Why is this statement true?
Two events that are simultaneous in one inertial frame of reference will not necessarily be simultaneous in any other inertial frame of reference.

Answer 1

Because witnessing the events depends on reflected or emitted light reaching your eyes. Since the speed of light is constant regardless of the speed of the reference frame, the light signal from one event could reach your eyes before or after the light signal from the other depending on relative motion.

Answer 2

Let's say I have two reference frames: S and S' where the S frame is the stationary frame and S' is moving relative to S at a speed v.

The Lorentz transformation is based off of events, so lets say I have event 1 and event 2 denoted as:
Event 1:
\[\huge t_1'=\gamma(t_1-{vx\over c^2})\]

Event 2:
\[\huge t_2'=\gamma(t_2-{vx\over c^2})\]

The difference would be:
\[\huge \Delta t'=\gamma(\Delta t-{v\Delta x\over c^2})\]
Now lets say that the events were simultaneous in the S frame. This means delta t=0.
This in turn meas that
\[\huge \Delta t'=-\gamma{v\Delta x\over c^2}\]
But if the event were to also be simultaneous in the S' frame, then delta t' should also equal to 0. The only way this would be possible would be if delta x=0.

However, if delta x=0, then this means that event 1 and event 2 are in fact the same event.

Answer 3

The thing I love best about these conclusions of special relativity is that they can be proved with nothing more than the Pythagorean Theorem.