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.

 

[thomate1]

Yes, you are on the right track. The general condition for two events to not be simultaneous in any frame is that the spacetime interval s is greater than zero. This means that the distance between the events in space and time is significant enough that they cannot occur at the same time in any reference frame. In other words, if the spacetime interval is greater than zero, then the events cannot be considered simultaneous in any frame of reference. This is a key concept in understanding the relationship between space and time in the theory of relativity.