 So let's say we have two coordinate systems like this, and two events happen, which we assign x, y, and z coordinates with respect to each coordinate system. Now we can calculate what we call the Euclidean distance between these points. We call this d, and just using the Pythagoras theorem, d squared is equal to distance in x squared plus distance in y squared plus distance in z squared. And before we knew about relativity, this distance was thought to be invariant between frames. It didn't matter if someone was moving, if someone was rotated, the distance that two people would measure would be the same. However, with things like length contraction, we now know that this is no longer the case. So what can we use instead as our idea of distance? So the answer turns out to be something that we call the invariant interval, ds. And this is calculated by ds squared is change in x squared plus y squared plus z squared minus c squared times the time spacing between the two events. Now this is the same between any two frames, which you can work out by just plugging in the Lorentz transformations. Now in our transformations, we've made assumptions that all the motion is along the x-axis, that the coordinate systems are not rotated relative to each other, and then at time t equals zero, the origins coincide. Now you don't have to make any of those assumptions. So you can go ahead and do the most general kind of transformation, which would be a lot more complicated, but you'd still find that this interval is invariant. Now we never actually deal with ds. We always talk about the interval squared, ds squared. And the reason for this is that the interval can be negative because of this minus c squared delta t squared term. So for simplicity, let's assume the two events occurred along the x-axis. So there's no separation in y, there's no separation in z. And let's look at the three cases for if ds squared is greater than zero equal to zero or less than zero. So if ds squared is greater than zero, that tells us that delta x squared is greater than c squared delta t squared. Or in other words, delta x on delta t is greater than c. What this means is if an object wants to travel between these two events, it would have to travel at a speed, distance over time, greater than the speed of light. So in other words, it's impossible. Nothing, no signal, can travel between these two events. And we say that the events are space-like separated. If ds squared is equal to zero, then we have delta x is equal to c delta t. Or delta x on delta t is equal to c. So if something wanted to travel between these two events, it would have to travel at the speed of light. And for that reason, we call these events light-like separated. Lastly, that's the case where ds squared is less than zero. So these are the two events where signals can travel between them. One of them can influence the other. And these are the events that we call time-like. So the reason for calling these events space-like and time-like is you can show if two events are space-like separated, you can find a reference frame where they happen at the same time, but at different points in space. Meanwhile, if two events are time-like separated, you can find a reference frame where they happen at the same place, but at different points in time, one after the other.