Hyperbolic

 

Yellow Submarine

As we live a life of ease
Every one of us has all we need
Sky of blue and sea of green
We all live in a yellow submarine

NO! We all live in an asymmetrical hyperboloid!

It appears as if the universe in which we reside is the turbulent lobe of an asymmetrical hyperboloid which is formed by rotating an asymmetrical  hyperbola about the about axis on which a transition occurs. This can be described  as a two-sheet hyperboloid, where there is no separation between the upper and lower sheets. That single point, the separation between order and chaos, might explain the Big Bang.

If the universe is hyperbolic, as proposed by Mabkhout, https://medcraveonline.com/PAIJ/non-dark-hyperbolic-universe.html, then there is no need to resort to Dark Matter or Dark Energy to explain the expansion of the universe. Expansion, and even inflation in its earliest stages, is instead a consequence of that hyperbolic surface. If the universe is hyperbolic, then there is no contradiction between the apparent age of the universe and the size of the observable universe. If the universe is hyperbolic, then there is consistency between the Planck length and the Planck energy. If the universe is hyperbolic, then the rotational energy of galaxies is a consequence of that shape and is not a paradox.

If the universe is hyperbolic, then Euclidean geometry does not apply for very large distances and very large speeds. (i.e. the universe is locally flat but universally hyperbolic). In a flat universe, the Euclidean equation for a hypotenuse becomes imaginary when the sum of the sides of a triangle is negative. If the universe is hyperbolic, then  the correct solution  for c²=a²+b² is NOT c=√(a²+b² ), it is cosh(c)=cosh(a)cosh(b). Einstein’s formula of Energy, E=mc² , implies a solution for the mass that is not m_0*(1/√(1-(v/c)²)), but should be ln(cosh(v/c)cosh(m₀)±sinh(v/c)sinh(m₀)). The first solution requires that the mass becomes imaginary when v>c. According to Einstein v can never exceed c. The Euclidean, flat, equation is imaginary when v>c and has a contradiction at v=c. (At the speed of light, it has to paradoxically be equal to both zero and infinity). The hyperbolic equation never becomes imaginary and has no paradox at v=c. Instead the equation only allows real solutions and is undefined if v exceeds c.

If the universe is hyperbolic, then gravity may be an apparent force, which is actually masses approaching a common center to minimize their respective energy and increase their entropy. If gravity is NOT a fundamental force, then there is no graviton to be observed, and there is no need to fit gravity into the Standard Model of Physics.

A normal equation of choice, where before the choice the value is 0 and after the choice the value is 1 (i.e. heads/tails, no/yes, off/on, false/true) has a odds of 0.5, but the outcome can only be 1 or 0.  The range, the difference between the choices, ( i.e., the median value of the Probability Density Function at the choice) is then, by definition, 0.5.  A normal distribution of discrete choice ( the logistics distribution) with a choice at µ, and range, s, of  0.5 is ½*sech²(x‑µ). The variance of this distribution is not 0 but it is 0.5*π/√3. If there is choice, then there must always be a variance and has to be greater than zero. A hyperbolic universe thus answers the debate about  a deterministic universe where variance must always be 0 and a random universe where the variance can be greater than  zero.

Variance is a consequence of reality approaching an absolute. Determinism requires a parabolic solution where reality can be equal to the absolute. Choice requires a hyperbolic solution where the reality can approach, but never equal, the absolute.

A transition occurs whenever a reality approaches an absolute. ( e.g. Speed on a road is separated into congested and congested domains at the capacity of the road; Flow in a pipe or channel is separated into laminar and turbulent domains as the flow in the pipe approaches the capacity of the pipe; The mass of a particle is hypothetically separated into subluminal and superluminal domains by the speed of light.)

This occurs because a transition ( i.e. change in behavior) has to occur whenever reality approaches an absolute. This transition arguably explains the shape of a curve in traffic flow, in Fluid Dynamics, or in general relativity. It also has a bearing on the correct computation of statistics. the formula for Standard Deviation is NOT a measure of the variance. It is a measurement of the error from the mean. Even if that is error is reduced to zero, the variance will still exist. This is why there appears to be positive moments around the mean when the moment is even, when Euclidean geometry is used. It is because the moment should actually be computed using hyperbolic, not Euclidean, geometry. Consequently, the even moment is because the Euclidian solution requires that the moments exist in an imaginary plane and every even moment in the imaginary plane becomes real, iⁿ=(√-1)ⁿ =-1 which is real not imaginary, when n is even. If statistics is computed using hyperbolic geometry, then every moment about the mean is zero when there is no error.

If the universe is hyperbolic and random, and random events follow a hyperbolic secant squared distribution and do not repeat, except in the imaginary plane, these random observations can be highly correlated with Euclidean geometry which repeats.  But correlation is not causation. Get real. We all live in a hyperbolic universe which is never imaginary.