Simultaneity, the notion of two
events happening at different
locations in
space, but at identical
times, is an idea which is all but meaningless to
modern physics. To see this, it is unnecessary to know any
math or
physics at all; one just needs to accept the main fundamental premise of
Einstein's Special Theory of Relativity: The
speed of light is the same in all
reference frames.
Like most of the new concepts in special relativity, the best way to describe the breakdown of simultaneity is with a thought experiment.
Imagine two world leaders are planning to sign a peace treaty, but neither leader wants to be the first one to sign the treaty. A system is devised to ensure that both parties sign the treaty simultaneously: each leader sits at the opposite end of a long table with her copy of the treaty. In the center of the table, there is a light bulb, currently switched off. When the appropriate moment arrives, the light bulb is switched on, which is the signal for each leader to sign. Since the leaders are an equal distance from the light bulb, the light should reach each of them at the same moment, and their signatures should be simultaneous. It seems like an infallible plan.
Now, this signing was (in)conveniently scheduled to take place on a moving train. Observers from one of the nations, let's call it Backzia, are watching the signing from the ground outside the train. They witness something problematic. Since the train is moving with respect to their reference frame, it forces the light to travel a greater distance to reach the front end of the car, and therefore it takes a longer amount of time, and the leader at the back end of the train car is observed to sign the treaty first. This enrages and infuriates the Backzians, who feel they have been cheated.
But that's not all.
The people of Frontasia are witnessing the same scene from a second traincar moving alongside the first but at twice the speed. From their frame of reference, the beam of light hits the front end of the conference table first, and the leader at the front end of the car is observed to sign the treaty first.
War breaks out until Einstein explains to them that simultaneity is relative, because the speed of light is not.
The physical interpretation of all this is that each observer uses her own clocks (and metersticks) to determine the times and locations of events in spacetime. Each observer is equally "right" about what actually transpired; there is no universal reference frame. From the point of view of someone on the train, objects on the train are stationary and objects outside are moving, but for observers outside, the opposite is true.
"Wait a minute," you might say if you were still paying attention, "If no reference frame is 'right' about which event happened first, then doesn't that totally freak with our view of causality?" No; causality is preserved. For two events to have this ambiguity about which happened first, there is a certain requirement: their spatial separation must be sufficiently large, and their temporal separation must be sufficiently small (i.e. the table must be very big and the length of time between signings must be very short). Specifically, they need to be far enough apart that a beam of light could not traverse the distance in the given separation of time. So, since nothing* can travel faster than light, if this ambiguity is to arise, the events must be too far apart for any information to be transferred from one location to the other in time. In this case, one event can't affect the other's outcome, meaning they can't be causally related. The two events happen completely independently of one another, and thus it doesn't physically matter which one happens first. I state all of this in a rather cavalier manner, but it is a very nontrivial point, and it took some time for physicists to come to terms with this dilemma.
The relativity of simultaneity is just one of the many changes forced into our reasoning of the universe by the Special Theory of Relativity. The other significant changes include time dilation and length contraction.
*Nothing real, anyway. This is a subject I'd rather not get into, but here's a link if you're interested. I suggest not listening to most of these noders, although Norton_I and a few others bring up some good points.