That's a fantastic explanation. I did a Physics undergrad and masters, and somehow I never heard this particular metaphor for it, but using a velocity vector through spacetime makes for a very clear image of what's going on. Is that from general relativity? (I only did special relativity)
This is just special relativity (space need not be curved for this description to work).
Brian Greene's "Fabric of the Cosmos" gives a good layman's exposition of this idea, using a loaf-of-bread metaphor (this was the first time I saw it explained this way and it made a huge impact on my way of thinking about SR)
He does the same in "The Elegant Universe", using the metaphor of a car driving at a fixed speed (c) across field that is space in one direction and time in another. That was my big "aha" moment with special relativity: understanding that "space" and "time" are dimensions in essentially the same way that "length" and "width" are dimensions -- with the caveat that we always have this tremendous velocity (c) across one or the other.
Then you contemplate the kinetic energy that must be associated with c, and e=Mc^2 pops right out at you. Absolutely blew my mind when I first grokked that intuitively.
In the end, Greene failed to convince me that string theory was particularly interesting, but his descriptions of relativity are absolutely first-rate.
These are actually sometime the worst things to read if you actually want to understand stuff. The publishers of these kind of popular science books enforce the rule for authors that no equations should ever appear. If authors get really upset with that they allow printing one equation, usually, E=Mc^2. This leads even talented authors to water down everything with faulty and many time absurd metaphors. There is no real substitute to reading real physics books. On a lighter side, check this out: http://www.youtube.com/watch?v=w5VVEw4ZSRI
Well, having an intuitive understanding goes a long way too. Being able to derive formulae rather than just having them memorized is good. Of course, I don't want a book that doesn't include the formulae, but I want to know the "why" of it as well.
If you measure time in seconds, and speed in units of C, then vx^2+vy^2+vz^2+vt^2 is always a constant (in flat spacetime). So you have no choice over the matter. If you are stationary, time passes at a rate of 1 second / stationary second. If you move in space, rate of passage changes exactly by the amount required to compensate. So in a way, you always move at the same speed through spacetime, speed of light, and can only choose the direction.
The reason is that time dilation factor is the lorenz factor, (1-u^2/c^2). This is how much time will pass in your watch per one second of a stationary clock.
the minus sign in front of the vt is very important and gives the SR (hyperbolic) structure of flat spacetime. The rotating a vector thing is just an analogy to euclidean space rotations.
Yes it does. Gravity changes spacetime, so the locally flat spacetime at the bottom of a gravity well isn't the same as the locally flat spacetime at the top of a well, but it's still flat in both cases.
