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?
Special Relativity – Understanding Space-Time Intervals and Their Types
causalitycoordinate systemsmetric-tensorspacetimespecial-relativity
Related Solutions
Suppose you draw a spacetime diagram and measure the angle of your trajectory to the spatial and time axes. You'll find time like trajectories are more nearly parallel to the time axis - i.e. the angle the trajectory makes to the time axis is smaller than the angle it makes to the spatial axis. Conversely space like trajectories are more nearly parallel to the spatial axis. This seems to me a plausible origin for the terms.
How about this for a more "physical" definition: two points in space-time are time-like separated if and only if a massive particle starting at one could, if subjected to appropriate finite forces, reach the other.
Replace "massive" with "massless" to get the definition of light-like separation. If neither is possible the the points are space-like separated.
Best Answer
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.