GRE special relativity problem: when do two events occur simultaneously in another frame?

[yxgao]

gre problem: special relativity, two events in two reference frames

34. In an inertial reference frame S, two events occur on the x-axis separated in time by $$\Delta t$$ and in space by $$\Delta x$$. in another inertial reference frame S', moving in the x-directon relative to S, the two events could occura t the same time under which, if any, of the following conditions?

a.) For any values of $$\Delta x$$ and $$\Delta t$$.
b.) Only if |$$\Delta x$$/$$\Delta t$$| < c
c.) Only if |$$\Delta x$$/$$\Delta t$$| > c
d.) Only if |$$\Delta x$$/$$\Delta t$$| = c
e.) Under no condition


Answer: c.)


Can someone explain why? It would help if you refer to specific equations that prove the answer is correct.

Thanks!

 

[Rob Woodside]

Being a dinosaur I don't write in lateX. I am condemned to Mathtype, which is not available here. So you'll get few equations from me.
Draw a space-time diagram in the frame where the two events are on the x-axis. Let the first event be at the origin and the second event at coordinates x.t (both in meters). The question asks for a frame moving along the x-axis in which the two events are simultaneous. Events that are simultaneous will live on a spatial axis. So take the origin of the moving frame at the first event and the moving x-axis on the line joining the two events. They are simultaneous in this moving frame. Since a spatial axis is space-like, so must the separation be between the two events. Back in the old frame this requires x/t>c, which is your answer c).
If you have understood this, which of your answers would you take if the question was altered so that the two events occurred at the same place in the moving frame?

 

[hhegab]

The correct answer is c.) Only if |\Delta x/\Delta t| > c.

This is because of the principle of relativity, which states that the laws of physics should be the same in all inertial reference frames. In this problem, we have two events occurring in two different reference frames, S and S'. In order for the time and space intervals to be the same in both frames, the ratio of \Delta x/\Delta t must be greater than c, the speed of light. This is because according to special relativity, time and space are not absolute, but are dependent on the observer's frame of reference. Therefore, in order for the laws of physics to remain the same in both frames, the ratio of \Delta x/\Delta t must be greater than c.

To further understand this, we can use the Lorentz transformation equations, which describe how measurements of time and space change between two inertial frames of reference. These equations show that as an object's speed approaches the speed of light, time and space intervals will appear to be different in different reference frames. So in order for the two events to occur at the same time in both frames, the ratio of \Delta x/\Delta t must be greater than c.

In summary, the correct answer is c.) Only if |\Delta x/\Delta t| > c because of the principle of relativity and the Lorentz transformation equations, which show that the ratio of time and space intervals must be greater than c in order for the laws of physics to remain the same in different reference frames.