Space and Time in Special Relativity (continued)
I had promised to make clearer the differences between space and time according to SR. The similarities I mentioned already are the following three facts:
a) Just as there is no unique, absolute "Forwards" direction in space (what is forwards for one observer is a combination of forwards and sideways to another one rotated with respect to him) there is no unique, absolute "Time" direction in spacetime (what is time for one observer is a combination of time and space for another one that moves with respect to him). As a consequence there is no unique, absolute state of rest.
b) Just like observers that are rotated which respect to each other give different values for forward distance dx and sideway distance dy between points, but agree in the total distance ds, observers moving with respect to each other will give different values to spatial distance ds between events and temporal interval dt between them, while agreeing in the total spacetime distance DS.
c) The spatial distance ds is related to the three distances in the spatial directions by Pythagoras' Theorem ds^2 = dx^2 + dy^2 + dz^2; the spacetime distance DS is related to the four distances in the four spacetime directions by DS^2 = dx^2 + dy^2 + dz^2 – dt^2 (a formula called "Minkowski's Metric"). In both these formulas the right hand side is an absolute which all observers agree in* but each term in the left hand side is relative to the orientation and state of motion of the observer.
So space and time are not identical: Minkowski's Metric** has a minus where a four-dimensional Pythagorean Theorem would have a plus. What concrete differences follow from that sign difference? I will explain them with an analogy: I will describe a set of observers rotated at different angles with respect to each other, in a two-dimensional plane, as they would be if the two directions in that plane behaved like space and time. In the following, "Forwards" is an analogy for the future direction in time and "Sideways" is an analogy for the space dimension.
So imagine you are standing, facing forwards, in a room with many other people. The people have their bodies rotated at different angles with respect to yours, but none at more than 45 degrees in either direction. That is, you see people turned a bit to your right, and a bit to your left, perhaps some at (almost) 45 degrees to your right or to your left, but nobody more than that. The analogy with space and time is that the "rotation" in spacetime, which is to be moving at a certain velocity, cannot be more than a certain amount: it is impossible to move faster than the speed of light. (And yes. This limitation in "spacetime rotations" as opposed to spatial rotations is a direct consequence of the minus sign. But I can't explain that without maths.)
But now you must be thinking that what I said contradicted fact a) above. After all, if you see someone rotated (say) 44 degrees to the right and someone else rotated 44 degrees to the left, then the first of them "must" see the second rotated at 88 degrees to the left. [Translation: if you see someone moving to the right at almost the speed of light, and someone else moving to the left at almost the speed of light, then the first one "must" see the second one moving at almost twice the speed of light to the left.] The only way out of this seems to be saying that you are special: you, only you, are looking to the true forwards [translation: only you are truly at rest] and so the rule of "no rotation of more than 45 degrees" [no speed greater than light] applies only to you. Contrary to what I wrote in a) there is an absolute, unique "Forwards" direction [an absolute, unique "Time" direction], which is the one valid for you. In short: an absolute, unsurpassable angle of rotation [speed] seems incompatible with the relativity, non-absoluteness, of spatial [spacetime] directions.
This conclusion is intuitive, but false. In our analogy, what would happen is that if you see John rotated 44 degrees to the right and Mary rotated 44 degrees to the left, then John would see you rotated 44 degrees to the left, and… Mary rotated 44.99 degrees to the left! Which means in reality, that if you see John moving at 99% of the speed of light to the right, and Mary. moving at 99% of it to the left, then John sees Mary moving at about 99.995% of it to the left. In reality, giving up our analogy, angles are additive: if you are rotated A degrees with respect to John, and Mary is rotated B degrees with respect to you, Mary is rotated A+B degrees with respect to John. Velocities, on the other hand, are not additive. If you are moving at A km/h with respect to John, and Mary moves at B km/h with respect to you, then Mary does not move at A+B km/h with respect to John.
(If you insist on knowing the real formula for the velocity of Mary with respect to John, it is (A+B) / (1+ AB/c^2), where c is the speed of light. You can check by yourself that this is very close to the commonsense answer A+B if A and B are much smaller than c, which is the reason we usually don't notice this effect, but that if A and B are close to c the answer is very close to c but not larger than it. And if both A and B are exactly c, then their "addittion" with this law is c again!)
So here there is a big difference between space and time: rotations in space are additive, rotations in spacetime not (at least not in the same way). And this is also a direct consequence of the minus sign, although once more I can't get into the details***. From this it is clear how the speed of light can be an absolute and unsurpassable constant while still keeping fully the "Principle of Relativity" that there is no unique and absolute "Time direction", or what is the same, that there is no unique and absolute "state of rest" (motion is relative). Historically, Einstein thought first that the absoluteness of the speed of light and the relativity of motion should both be true principles (both had experimental support, specially from the Michelson-Morley experiment****) and noticing that they seemed to be contradictory, discovered that they were actually compatible if velocities were not additive. A few years later Minkowski came and showed that all of Einstein's theory followed naturally if one started from the expression for DS^2 that has the famous extra minus sign.
Another difference between space and time that follows from this: with rotations in space you can go "all the way round", including facing backwards when you have rotated 180 degrees; but with rotations in spacetime (movement) you cannot "rotate more than 45 degrees" (go faster than light) and so you cannot go backwards in time. (Maybe you saw in some sci-fi movie the assertion that going faster than light amounts to time traveling. It is true, but both are impossible, at least in the sense intended.)
And I think that will be all for today. I'll be most happy to answer questions that might be unclear.
* Well, in the first one, all observers agree about ds^2 as long as they are not moving.
** Actually it is called Minkowski's metric, with no capital M in "metric", but it looked better like that.
*** For those of you who are curious and know some calculus, this has to do with the difference between trigonometric functions (which parametrize circles) and hyperbolic functions (which parametrize hyperbolas). I can also mention that the analogy between rotation in space and motion in spacetime becomes mathematically an equivalence if time is measured with imaginary numbers
**** Recently voted the greatest physics experiment ever, though I gave my vote to Galileo (who ended up in fourth place).