We know that time acts like a fourth dimension akin to the three

spatial dimensions because it appears in all the equations of physics

virtually interchangeably with the three spatial dimensions. Since the

equations are valid representations of reality ( at least within our

current measurement errors), we can use them as a touchstone for the

validity of our physical concepts. If it waddles like a duck and it

quacks like a duck...

Now once we have at least three dimensions, we can perform

rotations. In the usual textbook example, the x and y axes are rotated

about the z axis. But if time is also a dimension, we ought to be able

to rotate the x and t axes about either the y axis or z axis. So what

happens when we do this? The answer is that when we do this, the slope

of the rotated axis, as seen from our unrotated axis, is x/t. But x/t is

the speed with something is moving! Why does it seem like a speed?

Because we're moving down the time axis. So in this respect, the time

axis is unique and is NOT interchangeable with the spatial axes, (since

we're moving down the time axis and not down the spatial axes).

Another way of looking at it would be this.

Suppose you and I are walking down a country road until we come to a

fork where one branch slants off to the right and the other branch

slants off to the left. You take the right fork and I take the left

fork. We'll now start to separate from each other with a speed of

separation that depends on the angle at which my fork deviates from your

fork, and upon how fast we're walking. From my perspective, it will seem

as though I'm going straight ahead and you're walking off to the right.

>From your perspective, it will seem as though you're going staight ahead

and I'm walking off to the left. It will also seem to me as though

you're falling behind me, in the sense that I have to look back over my

RIGHT shoulder to see you on your path. Conversely, it will seem to you

as though I'm falling behind you, in the sense that you have to look

back over your LEFT shoulder to see me on my path. Neither one of us is

more correct than the other, but these are precisely the kinds of

effects over which we scratch our heads in special relativity.

The most significant difference, when we consider rotating the time

axis and one spatial axis instead of two spatial axes, is that when

we're think we're standing still, we're actually moving down the time

axis. We're aware of this in the sense that we know that time is

passing, but we can't look around and see what's behind us or ahead of

us along the time axis. And when we come to a "fork in the road", you're

able to go off to the right, down your slanted time axis, by

accelerating to the right--in other words, by acquiring a rightward

velocity (slope) of x/t. I'm able to make my time axis fork to the left

by accelerating to the left, thereby acquiring a slope (velocity) of

(-x)/t. Now we're moving apart at a speed of (2x)/t. We can't

geometrically see our motion down the time axis. But to someone who can

see four-dimensional structures, or who can simply plot this out on a

sheet of paper, it looks exactly like the fork-in-the-road situation in

the preceding paragraph. We move down the time axis together and then

you veer to the right by accelerating to the right and I veer to the

left by accelerating to the left.

And here again, if we could see in four dimensions, when I looked

back over my right shoulder at you, it would look as though you had

fallen behind me because you would be moving off diagonally from me. And

when you looked back at me over your laft shoulder, it would look as

though I had fallen behind you. To say this another way, both of us

would say that 30 seconds had elapsed. But you wouldn't have progressed

as far down MY time axis as I had progressed down MY time axis, so it

would appear to me as though your time intervals--your seconds--were

shorter than mine. After all, if we've both gone for 30 seconds and

you're not as far along my time axis as I am, then your seconds must be

shorter than mine. But since your clock speed is proportional to one

over the length of your second, that would mean that your clock is going

faster than mine. Now at this point, you're going to rightfully say to

me, "Wait a minute! In special relativity, moving clocks go SLOWER than

stationary clocks." And you're absolutely right. In order to keep this

discussion intuitive, I have pretended that the rotations of the x and t

axes are ordinary spatial rotations rather than hyperbolic rotations.

But the types of effects are the same. All that the fact that the

rotations are hyperbolic does is to in"invert" the numbers. Let me give

an example.

Suppose that you and I are walking down a road that has a 60 fork

to the left. You go straight ahead and I take the left fork for some

distance, following the first leg of an equilateral triangle, until my

fork suddenly switches back 120 to the right. In other words, I find

myself on the second leg of an equilateral triangle, heading back toward

the main road that we were on and that you're still following. After a

while, I'll get back to the main road and of course, I'll be behind you,

since I've taken a detour. In fact, I will have gone twice as far as you

had to go to get to the place where the two roads rejoined. But if,

instead of traveling down the x-axis, we were traveling down the t-axis,

then the rotation I would have made in order to take the left fork would

have been a hyperbolic rotation of 40.89. Similarly, when the road made

its 120 rotation to bring me back onto the main road, that would also

have been a hyperbolic rotation and the angle would have been soemwhere

between 40.89 and 45. (At the moment, I'm too lazy to calculate it )

And when I got back, to the main road, I would have travelled only half

as far (the reciprocal of twice as far) as you had to travel to get to

the place where the roads rejoined. In other words, if we're travelling

down the t-axis instead of the x-axis, it's shorter taking the detour

than it is following the main road! This is counter-intuitive. But

that's the way hyperbolic rotations work.

 

What we've just considered is the "twins paradox" of special

relativity. The travelling twin, who takes the 40.89 space-time detour,

will seem to have travelled only half as far down his time axis as we

terrestrial observers will have travelled down our time axis.

In practice, the travelling twin, before he accelerates, will

perceive the distance to Alpha Centauri to be 4.29 light-years. Then if,

within a few hours, he accelerates to 7/8ths of the speed of light, he

will now perceive the distance to Alpha Centauri as being only half what

it was, or 2.145 light years. If he travels toward Alpha Centauri for

2.145 years at 7/8ths of the speed of light, decelerating within a few

hours so that he's at rest with respect to Alpha Centauri, the distance

between the earth and Alpha Centauri will once again appear to be 4.29

light years. Now he turns his rocket ship around and rapidly accelerates

back toward the earth until he's travelling in the opposite direction at

7/8ths of the speed of light. Now the earth will only appear to him to

be 2.145 light years away. After 2.145 additional years, he will return

home and decelerate to match the earth's velocity. And once again, the

distance to Alpha Centauri will appear to be 4.29 light-years. The

travelling twin will have aged 4.29 years, plus what ever little bit of

time he spent accelerating, decelerating, and visiting Alpha Centauri.

We will have aged 8.58 years, plus whatever little bit of time he spent

accelerating, decelerating, and visiting Alpha Centauri.

So to recap: why do we rotate the time axis? Because that's what

happens when we accelerate something to some velocity relative to us.

Its time axis becomes rotated away from ours. If we could see in four

dimensions, we would be able to see this directly. But since we can't

see in four dimensions, we have to resort to drawings (or potentially,

to computer simulations) to see what's going on.

An acceleration is a rotation of a moving object's time axis away from

ours.