Special Relativity Question

Question

Given 2 events, A and B, given in space-time coordinates ( ct, x ) as the following:

A: ( ct, x ) = ( 2, 3 )

B: ( ct, x ) = ( 3, 5 )

a) Find the fraction of the speed of light, β, that will cause the events A and B to appear simultaneous to a moving observer moving at that speed β.

b) Did the moving observer share the same ( ct, x ) = ( 0, 0 ) origin point as the "rest frame" (the problem then referred to "RF" as the "rest frame" from then on) observer? If not, where were they located in space, relative to the "RF" observer?

c) After what spatial location and corresponding β can a moving observer see event B happen before event A? Please explain your reasoning if this set-up is possible or not.

I will link the accompanying figure in my comments section. Thank you for any and all help!

Answer 1

Answer [a] part.

Let us assume that Events A and B appear simultaneously to an observer moving at speed βc. Therefore, space time metric ds for the observer must be same for both events.

According to Einstein’s theory of relativity

Space-time metric s is related to space-time coordinates ds^2 = (c x dt)^2 – (dx)^2

For event A: ds^2 = (c x dt)^2 – (dx)^2 = 2^2 – 3^2 = 4 – 9 = -5

For event B: ds’^2 = (β x c dt’)^2 – (dx’)^2 = β^2 x 3^2 – 5^2 = 9 β^2 – 25

For simultaneous events ds^2 = ds’^2, -5 = 9 β^2 -25 solving β we get β = 2√5/3 = 1.4907

The results show that the speed of observer should be greater than speed of light c

For observing events at separate locations at different times.

Answer [b] part. Yes, the moving observer share the same reference point origin (0, 0) as the rest frame.

Answer [c] part. Assume spatial location dx for event B observed before event A

In order to observe event B before event A

Metric for event B ds’^2 > metric for event A ds^2

9 β^2 – dx^2 > -5 implies that spatial location depends on speed of observer.

However, it is possible to observe event B before event A if observer speed is

above speed of light.

I hope this helps.

Shailesh (Sky) Kadakia, Author & Publisher Text Book

Title " Unique Physics of Light and Astronomy"