Doug--you write "The term "ic" makes no sense. The concept of the imaginary number is an ad hoc invention concocted to deal with the fact that (-1)^2 = +1."
How dare you stipulate that ic makes no sense?!?
Instead, one must ask, "what is the *physical* meaning of ic?" as MDT does. And the answer to that leads us to a brand new physical invariant--the fourth dimension is expanding relative to the three spatial dimensions at the rate of c, or dx4/dt=ic.
Einstein wasn't kiding when he stated, "Curiosity is mor eimportant than knoweldge." Curiosity is also more important than arrogant snark wild handwaving.
You will find that humble curiosity is a far greater force in advancing physics than the snarky imposition of one's lack of curiosity and humility on the beauty of physical reality--simple curiosity is far greater than the methodology of the antitheory regimes who have built billion-dollar, televised groupthink regimes of censorship, snark, and intimidation while killing progress in physics.
MDT explains the vast and profound significance of ic in x4 = ict. It means the fourth dimension is expanindg relative to the three spatial dimensions, in a manner perpendicular to the three spatial dimensions!
dx4/dt = ic is what MDT states, which comes straight from Einstein's work.
x4 = fourth dimension
i = imaginary number
c = velocity of light
t = time
Suppose I told you x4 = ict and asked you to draw x4 at t=1, t=2, t=3 . . . etc.
Would you not draw x4 in different places for different t's?
Then, since Einstein and Minkoswki agree that the fourth dimension x4 = ict, does it not make sense that the fourth dimension moves over time?
Towards the bottom of page 6 of my paper, 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,"
i represents an orthogonality, so I would infer that i represents a perpendicularity--the fourth dimension is perpenidular to the three spatial dimensions. This makes sense!! For is this not what a dimension ought be--perpendicular to the other dimensions, providing an independent degree of freedom? x2 is perpendicular to x3 and x1, and x4 is perpendicular to x1, x2, and x3! That is why the i showed up! For when one goes into calculations, and an i pops out, it means that we need to think of a brand new perpendicularity in our system!! i has so very much significance, and ic has ven more! x4=ict means that the fourth dimension is expanding relative to the three spatial dimensions at the rate of c!
now einstein writes x4 = ict, so that dx4/dt = ic.
let me illustrate the meaning of this.
consider a 2D x-y plane and an expanding 3D sphere. 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.
now, 4D is a tough entity to envision, and i don't know if MDT improves on previous treatments.
but, it does do the following:
1. recognizes that time is not the fourth dimension, but x4 = ict, as Einstein and Minkowski agreed.
2. shows all of time's arrows from various realms derive from a common physical reality--the fourth dimension is expanding relative to the three spatial dimensions: dx4/dt = ic
3. shows all of time's assymetries from various realms derive from a common physical reality--the fourth dimension is expanding relative to the three spatial dimensions: dx4/dt = ic
4. derives relativity from a simple postulate/equation: dx4/dt = ic
5. presents a physical model for QM's nonlocality & entanglement, relativity's silumtaneity, and entropy
6. unifies the dualities with a common postulate: wave-particle duality, space-time duality, and matter-energy duality are all natural results of a fourth expanding dimension, which distributes nonlocality via its expansion. note that entanglement always occurs between two particles that were formerly in contact, and thus all nonlocality emerges from a common point or locality--another clue.
7. provides a physical model for time's fundamental assymetry in this universe
8. shows that time, as measured on our watches and witnessed in the world around us, emerges from a deeper reality dx4/dt = ic.
MDT agress 100% with Einstein's and Minkowski's relativity. The fourth dimension is a dimension that is orthogonal to the three spatial dimensions. All that MDT states is 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: dx4/dt=ic, which also provides a physical model accounting for entropy, entnaglement, 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.
The block universe froze eternity,
'til MDT rose to set us all free!
Best,
Dr. E (The Real McCoy)
