If two events have Δs²<0 we say they are “timelike separated.”

There is always an observer who describes them as happening at the same place but different times.

Crucially, according to special relativity everyone agrees on the order of two events that are timelike separated.

Finally, a pair of events with Δs²=0 are “null” or “lightlike separated.” Everyone agrees on their order, as well as the fact that a signal from the first would have to move at the speed of light to reach the second.

The meaning of Δs² doesn't exactly fit with our intuitive ideas about distance. (More on that later.)

But the fact that it can be +,-, or 0 is how we sort out essential questions like “Can observers disagree on the order of events?"

If two events have Δs² > 0 we say they are "spacelike separated.” There is always an observer who describes that pair of events as happening at the same time but at different places. Different observers may disagree on the order in which two spacelike events occur!

Earlier I referred to a “peculiar notion of distance.” Since Δs² has units of length², you might think its square root is probably some sort of distance.

Except, Δs² can be positive, zero, or even *negative*. In the last case its square root wouldn't give a sensible distance.

This quantity is called the “spacetime interval.” We usually denote it by Δs².

Δs² = - c² Δt² + Δx² + Δy² + Δz²

Minkowski built the geometry of his four-dimensional spacetime around this invariant notion.

"Invariant" in this context means that it stays the same if we switch from one observer's description to that of another observer moving relative to the first at constant speed along a straight line. I could be a little more precise here, but let's keep going.

Einstein explained how to reconcile these descriptions. I won’t give the full formulas here, but it's a set of rules analogous to (but not the same as) how you account for the extra 12m of relative motion in our earlier example.

His explanation used rules first conceived by Dutch physicist Hendrik Lorentz. An important consequence of the rules is that different descriptions still agree on one thing. If c is the speed of light, then

- c² Δt² + Δx² + Δy² + Δz² = - c² Δt'² + Δx'² + Δy'² + Δz’²

How do we make sense of things if we disagree on these descriptions? How do we do physics?

Especially the “when” part – can we disagree on really important things like which event happens first?

Wouldn’t that screw up cause-and-effect? This all sounds very bad!

Take any two events. I say time Δt passes between them, and they are separated in space by Δx, Δy, and Δz.

If you are moving relative to me (and I am moving relative to you) you would say time Δt’ passes between the events, with separation Δx’, Δy’, and Δz’.

We would record different values for the time between those events, and for the part of the “where” involving the direction of our relative motion.

According to Einstein’s theory of special relativity, this isn’t right.

Two people (“observers”) who are moving with respect to each other will in general give different accounts of both the where and when of an event, even if they gave the same description for a previous event.

If we see each other as moving it’s a bit trickier, but we still know how to reconcile those descriptions.

As you move away from me you might say x’=10m and x’=27m for those events.

But once you account for 12m of relative motion you understand that from my point of view they were separated by only 5m.

So we always agree on when things happen. And if we see each other as not moving, we immediately agree on distances between events.

I might say two events happen at x=0m and x=5m, so Δx=5m. You might be further away and record x'=10m and x'=15m, but you'd still say Δx’=5m.

(I'm using "Newtonian" to mean someone operating according to the ideas about space and time central to Newton's formulation of physics and generally accepted as true by other physicists until the early part of the 20th century. An equally good term here would be "Galilean.")

That’s true, the Newtonian would say, whether or not you and I see each other as being at rest or in motion.

If we’re moving away from each other you’d keep revising the “where” of my watch — x=10m, then x=15m, then x=20m, and so on. But we’d still log the same times for events.

A "Newtonian" would say that part makes perfect sense, but they would expect us to agree on *when* the watch ticked.

After accounting for the light from my watch taking a split second to reach your eyes 10m away, you would also log that in your records as an event at 12:04pm.

(You record your "where" description as x', y', and z', so we don’t mix it up with my x, y, z.)

Every one of us has our own personal way of describing where and when things happen. At the very least, we tend to put ourselves at the center of things. So when the minute hand on my watch (located on me!) hits 12:04pm, I'd probably label the “where” of that event as x=0, y=0, z=0.

On the other hand, you'd probably describe that event differently. If I was 10m away from you in what you call the x direction, you would say the event takes place at x'=10m, y'=0, z'=0.

Likewise, I’ll use Δx, Δy, and Δz to refer to the differences in where the events take place.

If the two events happen in the same place then all three quantities would be zero.

If one of them happens 20m further away in the direction I’m calling “x,” then Δx=20m, and so on.

Let's refer to the time that passes between the two events as Δt.

If the first event happens at 12:04pm and the second event happens at 12:05pm, Δt would be 60 seconds.

We use Δ as shorthand for "the change in," so Δt just means "how much the clock changed" between the events.

I'll record the info for these events so I have it all in one place, like so:

Event 1: t1, x1, y1, z1
Event 2: t2, x2, y2, z2

Then I can unambiguously describe the where and when to anyone who asks. If they recorded the same info, we could even compare our descriptions.

Suppose I observe two events. From my point of view, each one happens at a particular time t and a specific place labeled by three coordinates x, y, z. Maybe they occur at the same time but different places, or vice-versa. Maybe they differ in both place and time.

As opening lines go, these are pretty good. Just comes right out and swings for the fences, boldly claiming that two concepts hardwired into your understanding of physical reality are "doomed to fade away into shadows."

Today we refer to this 4-dimensional arena for special relativity, equipped with a peculiar notion of "distance," as Minkowski spacetime.

Let me explain one of the ways in which it is different than the Newtonian picture of space and time that preceded it.