I am taking Introduction to Modern Physics class. There, we were studying spacetime intervals as a subsection of Lorentz's transformation. My professor said that $\Delta x^2-c^2\Delta t^2$ is invariant, and then he said us that it is a lightlike event if $\Delta x^2-c^2\Delta t^2=0$, timeline if $\Delta x^2-c^2\Delta t^2<0$ and spacelike if $\Delta x^2-c^2\Delta t^2>0$. I understand that a spacelike event is when different reference frames do not agree on the order of the events. That is because we used Lorentz's transformation to show that, if say A and B are space-like events, A and B happen simultaneously in one frame, A happens before B in one reference frame, and B can happen before A in another reference frame. These are all mathematical notions for these events. I do not understand when these events can occur in reality. Can someone give me some examples so that I can understand them?
"Events" are akin to [localized] points.
It's not the "events" that are spacelike, timelike, or lightlike.
Rather, it is the "relationship between pairs of events" that are spacelike, timelike,or lightlike.
The pair of events "A and B" is timelike related if there is a timelike path in spacetime (say the worldline of a particle with nonzero mass) from A to B.
It's possible for events "B and C" to spacelike related to each other, while each being timelike related to a third event A. So, it's about the relation of a pair of events.
In special relativity, the character of the "interval between a pair of events A and B" is given by the sign of the square-norm of the spacetime-displacement-vector from A to B.
To get a feeling for these relations, it's best to draw a spacetime diagram. Then draw in the lightcones of each event... and see which events lie inside the lightcones of others. Such events are timelike-related to each other.
Note that there is a transitivity property for future-timelike-related events. If B is in the timelike-future of A, and C is in the timelike-future of B, then C is in the timelike-future of A.
You can extend this notion to "causal-futures" (future-timelike-or-future-lightlike-related).
But this doesn't work for lightlike-relations or spacelike-relations.
An interval between two events is space like if it is not possible for light leaving one of the events to arrive at the location of the other before it happens.
Let's take an example. The Moon is about 1.3 light seconds away. Suppose I synchronise my watch with two friends, Como and Zaquette, on the moon. If I sneeze at exactly noon and Como sneezes one second later, then the interval between our two sneezes is space like, because his sneeze happened 0.3 seconds before light from my sneeze could have reached him. If Zaquette sneezes a second later, then the interval between her sneezes and mine is timeline, because light from my sneeze would have reached her 0.7 seconds before she sneezed.
Everyone will agree that my sneeze happened before Zaquette's. However, in some frames of reference moving relative to mine, Como's sneeze will seem to have happened before mine.