Minkowski unified space with time in terms of a fourth dimension as related to the speed of light.
But his 4th dimension does not enter into the system in the same manner as the other three dimensions. Minkowski’s “time dimension” is not written as “t”, but as “-ict”, where “-i” is the negative square root of minus one, c is light velocity, and t is time. (This in itself holds that time is a complex number system, which implies an orthorotation into a higher physical dimensionality, which seems to be the situation in every case where the square root of -1 comes up in an equation, as in electric engineering.)
When we examine the distance traveled on a journey through this “space-time”, we can calculate the distance traveled on each axis in terms of a graph which measures the distances traveled along each of the several axes. In normal terms, without time involved, we can measure the distance traveled along the axes in of units along the axes, then calculate the actual distance traveled by use of the Pythagorean theorum. However, when the time axis is included, this calculation based on Pythagoras must be altered to include time. Thus, if we are interested in the space-time distance traveled, to measure along a line AB, relative to two of the axes, say x and y, our equation must look like:
AB^2 = x^2 + (ict)^2
or AB^2 = x^2 – (c^2 t^2) since i squared is -1
According to this, when x = ct the distance along AB = 0 which says that we have not traveled any distance at all! (A conundrum, since we were moving with the velocity of light in the first place.)
This is one of the famous “null lines” of relativity.
So when we are moving at speed v, the spatial distance x, traveled in t seconds is simply vt. Substituting vt for x in the previous equation, we have:
SqRt(v^2 t^2 – c^2 t^2) = SqRt(t^2 [v^2 – c^2] ) = distance
This equation indicates that when v = c the space-time length vanishes. By this, no time will have elapsed on our journey between points A and B because there is effectively no distance between A and B. Another conundrum.
Now, on to the actual theme of the title:
When we are looking at the projective non-linear solutions to the Maxwell equations known as the Moebius transforms, we find that the propagation velocity of such discontinuities is allowed to be ANY velocity, from zero velocity to infinite velocity.
Let us examine what happens to the above situation when considering the Moebius transform E/M propagations. In place of our term “c” then, let us place a term which can vary from zero to infinity, as do the propagation velocities of the Moebius transform E/M propagations. We shall use the term (0 => v => U), where v is some measurable velocity and U is an unlimited velocity.
Then we have, substituting into the first equation above,
AB^2 = x^2 + ( i [ 0 => v => U ] t )^2
Here the solutions involved with zero velocity and infinite velocity are interesting of themselves, while any numerical velocity between zero and infinity is readily tractable. And also revealing.
What we see from this is that the distance traveled along the space-time from A to B is countable and further, is never zero. That is to say, that for Moebius transform E/M, there are no “null lines”. Further, we can clearly see that time is always measurable, as well, with regard to the Moebius transform E/M, except in the case of an infinite velocity, whereupon time ceases to flow
for entity aboard the Moebius transform propagation. At the same time, it is obvious that superluminal velocities are obtained by the Moebius E/M and its passengers, while time flows according to the equation above.
This looks something like
SqRt(t^2 [v^2 – [ 0 => v => U ]^2] ) = distance
What this means is that there are two different kinds of time, depending on whether we are contemplating normal light or the Moebius E/M. In terms of consciousness, an entity traveling along with a Moebius transform E/M propagation will have an awareness of motion, and the passage of time, until the point that an infinite velocity is reached, whereupon time stops. Lorentz contraction does not apply here in the same manner as it does in standard relativity theory. Rather, it proceeds as the function [ 0 => v => U ] , as the propagation velocity approaches infinity.
Pardon my awkward math, but I think these understandings need to be incorporated into relativity theory, as well as QM. There are other implications as well, if you understand what all this means. For example, I think this understanding could be important for interstellar travel.