gre problem

[yxgao]

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 occured at the same place in the moving frame?