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.
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.
http://en.wikipedia.org/wiki/Four-velocity