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Posted by Mark on October 10, 2001 17:38:37 UTC

Mathematics showing space and time to be interchangeable:

Space measured in m (meters)
Time measured in s (seconds)

Let c be a constant, understood to be the maximal spatial displacement (m) allowed in a given time duration (s). In other words, c is the upper bound to any possible speed while 0 is the lower bound.

Under Galilean rule of reference change, velocities of objects are added and subtracted according to the frame of motion one wishes to make measurements. This means that from a stationary reference frame (relative to me), if I throw a baseball with velocity V1 head-on toward a moving vehicle with velocity V2, a passenger of the vehicle will measure the velocity of me to be -V2 (same speed, opposite direction), and that of the ball to be -V2 + -V2 (simple vector addition). (We assume the ball travels in a straight line, so as to keep all frames inertial... If you'd like, we can move away from Earth's G field and launch this argument into space). That's Galilean relativity in a nutshell... motion is not absolute but rather relative.

Here comes the kicker...
Newtonian mechanics were founded upon the principle of Galilean invariance, however they neglect to account for a maximum speed. Not only that, but they neglect to take into consideration Maxwell's constancy of the speed of electromagnetic propagation (c), (understandably overlooked due to the chronology of the theories). Because of this kicker, Newtonian mechanics will fall apart for high velocities. Newton only told "half the story" to an accurate approximation.

What happens as VERY HIGH velocities are approached (considerable fractions of c)? It can be shown that space and time are interchanged for different reference frames. Why is that?... Because ct = s (speed of light x time = space). A revised principle of invariance could not simply add vectors as performed above. Somehow, the metric used to measure vectors must stretch and contract under transformation to different referential coordinate systems; only then will a constant speed be conserved as required by good ol' Maxwell. But wait... the metric used to measure the vectors is none other than space and time themselves! Can it be that space and time can contract when subjected to motion (relative of course, as measured by an observer in motion relative to another).

Well, everybody knows the answer to that...
It may have been profound and shocking a hundred years ago, but we've all been familiarized with it these days.

Lorentz transformation coefficient: ? = (1/(1-v2/c2))1/2

Because the Lorentz transformation coefficient is directly dependent upon velocity, one is lead to say that space contracts for an object in motion as measured by a relative observer. What does it mean for space to contract? Well the term "squashed length" gives a good visual description. What happens to time? Well durations of time are "stretched", so as to give the appearance of slower ticking watches. Since space contracts in the exact amount to compensate for stretched time, one could say that space-time as a whole never changes. It's just the orientation of our light cones that change, relative to eachother's motion. So if my light cone can tip into yours, and just as Alex said, a 45 degree angle separates space from time, then what appears to be one persons space (outside the their light cone) can be somebody else's time (tipped into their light cone). Space and time are inexorably stitched together, in an eternal 4-dimensional rug of interwoven light cones. We simply live it in steps... three dimensional frame by frame. At least that's how Einstein saw it.....

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