[nkoren]

Here's a quick analogy to help (loosely adapted from Brian Greene):

You're on a grass field, sitting on one of those riding lawnmower thingies, with a broken throttle. It's moving at a fixed speed of 1mph. You can't ever change its velocity. But you can steer it. If you are going precisely east-west, then it means you're not going north-south. The more you go north-south, the less you'll be going east-west. If you're going precisely north-south, it means you're not going east-west at all. One direction is traded against the other.

Pretty straightforward, right?

So here's the analogy: that grass field is a "dimension" in the same way that "spacetime" is a "dimension". The two "directions" of spacetime aren't "east-west" and "north-south", but "space" and "time". These are inherently traded against each other. The more you're moving through one, the less you're moving through the other.

So what about that constant-velocity rideable lawnmower? That's "c" -- the speed of light. You're always traveling at this velocity. If you are sitting still in space, then you are nonetheless moving through time. Your rate of movement through time is "c". But as soon as you start moving through space, it means you are moving less through time. This is exactly the same tradeoff as moving north-south vs. east-west. If you devote 100% of your "c" to moving in the direction of the "space" axis, then it means you're not moving on the "time" axis at all.

(This is basically all it means for something to be a "dimension": different axes that are traded against one another.)

This analogy can be used to understand quite precisely how movement relates to time dilation. (It also helped me understand e=mc^2. Why is "c" there? What does the speed of light have to do with the embodied energy of matter at rest? Answer: nothing is ever at rest; all static matter is moving through through time at the velocity of "c", and obviously that movement must have kinetic energy.) But it's not a completely perfect analogy. Weirder relativistic effects like length contraction and frame dragging need much weirder analogies.

[tsimionescu]

I like this analogy, but I think an even bigger problem than distance contraction is that in fact the lawnmower can go east-west on one axis, but only towards the north on the other axis, towards the future, at least as far as we have observed in macroscopic systems.

This may not be a problem for some definitions of time, but for the notion of time which goes from past to future, I don't think the analogy holds very well.

[nkoren]

Actually the analogy is a bit poorer than that, but also more consistent. You can only go one direction on either axis. Any movement in 3D space advances you along the "+space" axis, to the the detriment of your default movement along the "+time" axis. But just as there's no "-time" axis, there's also no "-space" axis.

[tsimionescu]

Let me put it another way: from any point in space, we know you can move to any other point in space, in a finite amount of time (disregarding the accelerating expansion of the universe).

As far as we have observed, the same is not true with (the common-language definition of) time - I can't go back to the moment I was born, for example.

[nkoren]

I think your comment about disregarding the accelerating expansion of the universe is a key point. We can't actually move anywhere in space; reacheable space is constrained by our reference frame relative to the reference frame of some other part of the universe that we're accelerating away from. At a certain point, the parts of the universe become unreacheable, because we would have to accelerate faster than the speed of light in order to reach them.

If my understanding is correct, this same condition exists inside the photon limit of a black hole. Technically you're still in navigable space -- not inside the singularity yet -- and can move in any direction. But to actually escape the black hole would require accelerating faster than the speed of light.

Again, if my understanding is correct -- and I'm definitely not a phycisist by any stretch of the imagination -- our movement through time is exactly the same phenomena. We can slow our velocity through time (by moving through space instead), but we can't escape our local reference frame without moving faster than the speed of light. If we could exceed the speed of light, then we would be moving into spatial regions which are otherwise causally inaccessible to us; in other words, we'd be going backwards in time.

So: if we could go FTL, we could escape from black holes, visit parts of the universe beyond the locally-observable limit, and go backwards in time. I think (IANAP) that these are all describing precisely the same thing.

Dunno if this helps. The lawnmower analogy has definitely broken down by this point.

Also, it makes me think: if our experience of time is navigationally equivalent to the experience of space for someone getting sucked into a black hole, does that imply the existence of a higher-dimensional universe where ordinary, non-accelerating time is as fully navigable as our ordinary "non-accelerating" space? And in that higher-dimension universe, are we living near the surface of some kind of singularity? Do the inhabitants of that universe wonder about how sad it must be for poor creatures like us, forced to live on a time gradient which inexorably slopes in just one direction, the way we might commiserate the fate of those sucked into a black hole?