A few words about intervals:

In class we talked a little about the interval between events. We said that that interval can be timelike, which is, for example, the case when two events occur in the same place but at different times. When that is so, there is no reference frame in which the order of the two events is reversed. Certainly, there are reference frames in which the time between the events is increased or decreased, but none in which the second event occurs first. We agreed, I think, that this was important for the question of causality: if A causes B, then we feel much better if A happens before B, and if everyone, in every reference frame, agrees that it does.

By contrast, we can consider two events that (in some reference frame) happen at the same time but in different places. We say that the interval between them is spacelike. Then, in no reference frame can these two events occur in the same place. Once again, there will be reference frames in which the events occur closer together or farther apart, but none in which they occur at the same place. We believe that if the interval between A and B is spacelike, then neither event could possibly have caused the other. After all, no cause (such as a bullet or a burst of light) could have propagated from one event to the other; if so, the reference frame in which that bullet was at rest would be one in which the two events occur at the same place.

Let's get quantitative:

We can define the interval between events A and B very simply: Square the distance between the events, then subtract from that the square of the time between them, multiplied by the square of the speed of light. Both terms have dimensions of length-squared. If the interval you calculate is positive, then the interval is spacelike. (After all, if two events occur at the same time in different places, then the time difference is zero and the interval is just the square of the distance between them, and that has to be positive.)

If the interval is negative, then it is timelike. (After all, if two events occur in the same place at different times, then the distance between them is zero and the interval is just minus c-sqared times the square of the time difference, and that has to be negative.)

Finally, if the interval is exactly zero, then we say it is lightlike. This is the case for such pairs of events as A (bulb emits pulse of light.) and B (pulse reaches a detector.).

This definition of the interval allows you to work with any two events, even if they occur at different times and places, and to determine whether, for example, one event could have caused the other.

Useful consequence:

The interval between two events is the same in any inertial reference frame. You can easily verify this by applying the Lorentz transformations to the x's and t's for two events. You'll find that the interval between the primed values is the same as the interval between the unprimed values.

Facts to remember:

If two events are separated by a timelike interval, then there is a reference frame in which both events occur in the same place. Also, in every reference frame, the order in which the two events occur is the same. Causally-related events, therefore, are separated by a timelike (or lightlike) interval.

If two events are separated by a spacelike interval, then there is a reference frame in which both events occur at the same time. Also, there is no reference frame in which the events occur in the same place. Neither event could have caused the other event.