Is the Universe in the complex domain?

Whether or not the Universe is in the so called real or complex mathematical domain could be the basis of a consistent description. You have to get in the right ballpark before you can even start the game that counts. So far all descriptions are founded on the former, with the result of serious foundational questions at the large and small scales.

The complex domain is used regularly throughout science and engineering to come up with real-world solutions -- without a second thought. But with a second though ... . Yet, it is posed as all encompassing --

There it is in blue. The cover illustration is a little more restricted, classified as pure "imaginary" in dark blue in the light blue field. The complex domain is unavoidable.

No familiar number when squared (positive or negative) yields a negative number. To get a negative number like -1 after you square it, as in the cover illustration, you have to give it a new name; historically that name is "imaginary" shortened to i, which is unfortunate, because it implies unreality. Then i = (-1)^1/2. It is intended to suggest i is quite "real" in the everyday physical sense in addition to the mathematical sense.

When large-scale space is brought up general relativity might automatically come to mind. But the acceleration of the Universe is proposed to be near the order of 10^-13 m/s^2, as discussed, which might be that of grass growth. Then special relativity can be considered in certain circumstances. According to the Hubble Law, the velocity at which a galaxy moves away from us is directly dependent on the distance that galaxy is from us -- the farther away, the faster it moves. This means that a sufficiently distant galaxy may exceed light speed, particularly in our accelerating Universe, which is not permitted according to special relativity (expanding space to be discussed) -- or is it? Commonly, for relativistic mass, m'

(1) m' = m(1- v^2/c^2)^-1/2, ......(v<c)

while the Hubble Law reads

(2) v = Hs ....................................(v<c) or (v>c) and (v≠c)

where v is the velocity of a distant galaxy, s is the distance between galaxies considered and H is the Hubble Constant (actually H sub-0). Notably, in the Hubble Law greater than light speeds are permitted. But note the contradiction for velocity in Equation (2). A cluster of galaxies cannot go from v<c to v>c without passing though c. Therefore either v<c or v>c must be discounted in an accelerating Universe (any galaxy will eventually approach c). Since galaxies are free to move at v>c (but not relative to space) the v<c condition must be discounted. For consistency Equation (1) cannot be valid universally, as commonly suspected.

Further, in that galaxies are not moving with respect to space in the Hubble expansion, they can be described as being embedded in space. If at this scale, what of lower scales? If there is embedding, galaxies should be imbedded at any scale. Otherwise the question, At what point does the embedding begin? This question proceeds down to the elementary particle.*

If special relativity is not applicable beyond the scale of clusters of galaxies, the same might be said of general relativity, where the original logic does not allow for or predict accelerated space expansion beyond clusters of galaxies. Why should general relativity be permitted a place holder in the equation (the cosmological constant) and special relativity not be permitted an appropriate modification? But special relativity does not need any modification if it is expressed this way

(3) m' = -im(v^2/c^2 -1)^-1/2. ...... (v>c)

However with this expression

• mass is "imaginary" (in the mathematical sense)
• mass is negative
• velocities less than light speed are not permitted for normal particles.

a) Imaginary mass. We know that distant galaxies are not imaginary in the sense of being unreal or non-physical, so that the quantity i does not negate reality here.

b) Negative mass. Clusters of galaxies, are moving away from one another as though they were composed of negative mass -- perhaps a more apt description than at present. And if all mass/energy in the Universe is negative (represented by all mass/energy within such clusters), it may be defined as positive.

c) Velocities less than light speed are not permitted. Who is to say that all non-zero rest mass particles in the Universe are not exceeding light speed, and that c is not the lower limit? Then Equation (2) for the Hubble Law is consistent when

(2') v = Hs ............................ (v>c)

only if Equation (3) is universal so that Paragraphs a, b and c hold, and the entire visible Universe is exceeding light speed, i.e., space(time) and that which is imbedded in space. Then the Universe might be consistently described mathematically if it is in the complex domain, perhaps from smallest to largest scales.*

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* Letters, Dec 5, 2021 https://www.dhirubhai.net/pulse/where-boundary-between-small-large-scale-warren-frisina/