Understanding the Spacetime Interval and the Condition for Simultaneity

[theneedtoknow]

The spacetime interval s between two events is s^2 = c^2*t^2 - x^2 where t is the time between the 2 events and x is the distance between the 2 events in a given frame of reference.


What is the general condition on s such that two events cannot be simultaneous in any
frame?

I don't really understand the question..
What am i supposed to do?
I mean...the shortest possible time between 2 events is 0, so picking a reference frame in which they are simultaneous, the spacetime interval between them would simply be the distance x between them
or
s = root (-x^2)
which is not a real number...(even though I think the spacetie interval can be imaginary)
So if the time between any 2 events is more than 0, then the spacetime interval would be greater than root (-x^2) since we'd be substracting -x^2 from a number larger than zero...so is the restriction that if s is less than root (-x^2), then 2 events can' tbe simultaneous in any frame?
Can someone please tell me if I am on the right track at all??

 

[ak1948]

Does this make sense?

The spacetime interval between two events doesn't change when you change from one inertial frame to another, although x and t do. If s^2 = 0, the measured time between events (t) is equal to the time required for light to travel the measured distance between the events (x/c). So only if x=0 could the events be simultaneous.

If s^2 < 0, the measured time between events is less than the time required for light to travel the measured distance between events; so a frame can be found in which the events are simultaneous. In this frame x^2 = -s^2.

If s^2 > 0 , |t| > |x/c|. Since the distance between the events can't be negative in any frame, t must be greater than 0. There is no frame in which the events are simultaneous.