Imagine that time is just space at an angle. (This isn't far from the truth, actually).
Now put that aside, and consider "real" space at an angle... that is, north is at 90 degrees to east, and you can lay out a grid and measure the "northness" and "eastness" of every house on the block. This works as long as we know what north is, and calculate east from it. But what if we disagree on what north is? Use the north star? Use the compass? The compass points at a (non-zero) angle to the north star. It's "in a different direction". So, all the answers come out
different. But they come out different in a special way. In both cases, if you square the north and east figures and then add them, the answer is identical, even though the input numbers are different. Pretty cool, huh? It's just the distance formula, or pythagorean theorem, in disguise.
When you walk in a straight line, you are walking some amount north and some amount east at the same time, and that relationship is fixed by the same formula. You can measure house positions by how much "in front" and how much "to the side" they are, and the same relationship governs the results. The angle you walk acts like a "portable north" for you.
Ok, now popj time being space at an angle. You are traveling through time
but not through space (from your own point of view). You are always "right here", but "now" keeps slipping into the past. Other people may see you moving through space, but only because you are moving with respect to them.
So, now consider what happens when you snap your fingers twice, one second apart. To you, the snaps happened at the same place ("right here"), but at different times (one second by your watch). To your friend, screaming by in a blitzrocket, those two snaps happened in two different places
: "right in front of me" and "way behind me".
But, they are the same two snaps. So, in spacetime, they have to have the same relationship to each other. They are the same "spaceetime" distance apart. (This is called the "interval")
It turns out that not only are you imagining time is space at an angle, time actually is
"imaginary space" at an angle. The equivalent relationship turns out to be that you can square the time difference between the two snaps, square the space difference between the two snaps, subtract them
, and the result is the same for you and for your friend. (Yes, there's a conversion factor involved between space units and time units). We say the interval is "invariant".
The conversion factor is c, the speed of light. (Hmmmmm....)
Further, if you use i
c as the conversion factor (i
being the square root of -1), you can use the ordinary pythagorean theorem again, and add instead of subtract. The i
, when squared, becomes the requisite -1.
This needs to be visualized in four dimensional complex space, but a cartoon of it looks like this: Use the vertical axis for time, the horizontal axis for space, and that's "you, at rest", moving up the graph in time. You always see yourself that way because you are still with respect to yourself.
Somebody whizzing by will be moving through this spacetime diagram at some angle from the vertical from your point of view. It will lean towards the 45 degree diagonal that represents the speed of light. But your view of their "space axis" will also lean towards that same diagonal. As you view it, their space and their time will not be at 90 degrees, but will scrunch closer together. At the speed of light, your view of their space and your view of their time will be superimposed. And that's as far as you can go - where space and time meet.
Still, from their POV, they see their own space and time at 90 degrees, and it's your axes that seem scrunched.
For an excellent and accessible book on the subject, check out "Spacetime Physics" by Taylor and Wheeler.