Hello George,
Thanks for the Dec. 4th coment, where you wrote: "George Ellis wrote on Dec. 4, 2008 @ 10:00 GMT
Dear Dr. E (The Real McCoy)
I agree with the spirit of what you do in your essay, which as you point out is going in the same direction as mine. The main point where I differ is in the use of the imaginary time coordinate. I prefer to see it all done with a real time coordinate (see e.g. my book with Ruth Williams entitled "Flat and Curved Spacetimes"). So a key element is how proper time relates to coordinate time as we move to the future, which is what your key equation establishes; I think one can do this without introducing the imaginary quantity "i"."
Max Born writes in his book on relativity, "Born writes, "It is true that in the domain of ordinary number one cannot extract the square root of the negative quantity -c^2t^2; hence u has no elementary menaing. But mathematicians have long been accustomed to overcoming such diificulties. The imaginary quanity "i" has been firmly established in mathematics since the time of Gauss. We cannot here enter into the question of how the doctrines of imaginary numbers can be rigorously established. These numbers are essentially no more "imaginary" than a fraction such as 2/3., for numbers with which we number things or count properly comprise only the natural intergesrs 1,2,3,4. . . The number 2 is not dividsible by 3, so that 2/3 is an operation that can be carried out just as little as (-1)^(1/2). Fractions such as 2/3 signify an extension of the natural concept of numbers; however, they have become familiar through education and custom, and excite no feeling of strangeness. The introduction of imgainary numbers is a similar extension: all formulae that contain imaginary numbers have just as definite a meaning as those formed from ordinary "real" numbers, and the inferences drawn from them are just as convincing."
The imaginary number i represents very real things. It represenst the fact that while we were working in three spatial dimensions, an i popped out, representing a fourth dimension. This is analogous to working along a 1D number line, and suddenly a solution to an equation presents an i. All this meands is that you are now operating in a 2D world. MDT recognizes that the fourth dimension is expanding relative to the three spatial dimensions, at the rate of c. dx4/dt=ic . The t--on our clocks and watches--is measured in our three spatial dimensions, but the fourth dimension expands perpendicular to the three spatial dimensions, and hence the i.
MDT agress 100% with Einstein's and Minkowski's relativity. The fourth dimension is a direction that is orthogonal to the three spatial dimensions, and that's the origin of the i. MDT srealizes that the fourth dimension is expanding relative to the three spatial dimensions. In his 1912 paper Einstein just states x4 = ict. MDT begins at a more fundamental level and presents a hitherto unsung universal invariant: dx4/dt=ic, which also provides a physical model accounting for entropy, entanglement, quantum mechanics' nonlocality, and time and its arrows in all realms, in addition to relativity.
MDT contends that the fourth dimension is very much like the spatial dimensions, except that it is expanding relative to them! dx4/dt = ic.
Now, regarding i, i does not imply "imaginary" in the sense of "it doesn't really exist." But rather i implies a very real perpendicularity.
i is an imaginary number, but it can define very real entities!
For instance, in a complex plane, we can designate the x axis to be real and the y axis to be imaginary. This is a mathematical tool, but the y axis is very real! Imaginary numbers are very useful in describing oscillations and rotations. When we solve an equation and we see one, that is math's way of lettig us know--"hey there is something going on here that is perpendicular to where you started."
If we were called upon to draw i, we would draw it perpendicular to the real number line. i^2 would be -1 on the real number line i^3 would be -i--it would point "south" along the y axis. And i^4 is 1 on the real number line. So multiplying by i rotates something by 90 degrees! Multiplying by i makes something perpendicular to its former self! Now although i is "imaginary," the y-axis is a very real entity. So i has a reality to it.
So if we're solving an equation, and an i pops out, all of a sudden we need to start thinking of an orthogonal space.
Now the way I interpret x4 = ict is that x4 is perpendicular to the three spatial dimensions, and that as t advances on our clock or watch, it moves.
Consider a 2D x-y plane and an expanding 3D sphere. We could then say that the sphere will also expand in the imaginary direction, which is directed along the z axis. The expansion of the sphere would appear as an expanding circle in the 2D plane. Now, the surface of the sphere would be perpendicular to every point in the 2D plane. instead of writing the third coordinate as z, we could associate it with i--the imaginary number, which would represent the orthogonal surface at every point in our 2d plane.
Now, let us consider the above with an extra dimension, so:
consider a 3D space and an expanding 4D surface. The expansion of the 4D surface would appear as an expanding sphere in the 3D space. now, the surface of the 4D surface would be perpendicular to every point in the 3D space. instead of writing the fourth coordinate as x4, we could associate it with i--the imaginary number, which would represent the orthogonal surface at every point in our 3D space.
more clues are discussed in the paper, where towards the bottom of page 6, i write:
"Einstein definitively states x4 = ict, and time and ict are very different entities. Einstein states, "One has to keep in mind that the fourth coordinate u (which Einstein sometimes writes as x4) is always purely imaginary." It is imaginary because the expansion of the fourth dimension is
orthogonal to the three spatial dimensions in every direction, just as the radii of an expanding sphere are perpendicular to its surface at every point."
The fourth dimension is very, very real.
All motion rests upon its fundamental expansion relative to the three spatial dimensions: dx4/dt = ic. Every object moves at but one speed through space-time--c. This is because space-time moves at but one speed through every obeject--c. Catch up with the fourth expanding dimension, and you'll be going close to c relative to the three spatial dimensions. Remain stationary in the three spatial dimensions, and you'll be traveling at close to c relative to the fourth dimension. And isn't it cool that the faster an object moves, the shorter it is in the three spatial dimensions? This is because it is physically being rotated into the fourth dimension--the fundamental source of all motion by its never-ending motion, which sets the universe's maximum velcoity at c.
Relativity implies a frozen, timeless, block universe. But as Galileo said, "Yet it moves!" *Why* is this? Because dx4/dt = ic! And the spherically symmetric expansion that the expanding fourth dimension manifests itself as--this smearing of locality--jives perfectly with the motion of a photon as well as its nonlocal properties, setting its velocity to c independent of the source and rendering it timeless and ageless--stationary in the fourth expanding dimension, which would also explain entanglement with other photons with which it once shared a common origin! And we also get a *physical* model for entropy and time.
