Think
I ain't no psychiatrist, I ain't no
doctor with degrees
But, it don't take too much high IQ's
To see what you're doing to me.
I also ain’t no doctor with degrees, but
Dr. Nash meet Pythagoras
and Dr. Einstein
The two equations with which most people are familiar are probably:
- c2=a2+b2, Pythagoras’ Theorem for the hypotenuse
of a right triangle; and
- E=mc2, Einstein’s formula for energy.
Most people are also aware that the c s in these equations are variables which
represent different things:
- the hypotenuse in Pythagoras’ Theorem, and
- the speed
of light in Einstein’s formula.
What most people do not
realize, is that BOTH equations are actually for triangles: Pythagoras’ Theorem
by definition, and Einstein’s formula as an outcome of its Triangle of Energy, E= (mc2)2
= (mvc)2 + (m0c2)2.
Further, if m=m0/ϒ, and the terms are simplified, then Einstein’s
Triangle becomes ϒ2=1 - v2/c2. Both equations ignore any consideration of the imaginary plane. If
they are treated as complex numbers and the imaginary plane has a coefficient of
zero, they should be c2=a2+b2+02i
and ϒ2=1 - v2/c2+ 02i . Their solution on an elliptical or flat surface is the conventional c=√(a2+b2)
and ϒ=√(1 - v2/c2). However on a hyperbolic surface there
would be two solutions.
c=ln(cosh( √(a2+b2)) ± sinh( √(a2+b2))), or ln(0 ± 2cosh( √(a2+b2)))
and
c=ln(cosh(-√(a2+b2)) ± sinh(-√(a2+b2))), or ln(0 ± 2cosh(-√(a2+b2)))
for Pythagoras’s Theorem, and
ϒ=ln(cosh( √(1 - v2/c2))
± sinh( √(1 - v2/c2))), or ln(0 ± 2cosh( √(1 - v2/c2)))
and
ϒ=ln(cosh(-√(1 - v2/c2))
± sinh(-√(1 - v2/c2))), or ln(0 ± 2cosh(-√(1 - v2/c2)))
for the factor of relativistic dilation, which is the Loretz Transform on a flat
Euclidean Surface.
Since cosh(x) is an odd function, that is cosh(x)=cosh(-x),
both solutions are the same and there is thus effectively only one solution. If
at the origin of (0,0,0) there is also a rotation of the coefficient of the imaginary
axis in the transformation of coordinates as a complex number, then according
to Euler’s’ formula, the coefficients of the imaginary axis, sin(0) and sin(π)
are both zero, but the coefficients of the real axis will be cos(0)=1 and cos(π)=-1. Thus only with a rotation of the imaginary axis
by π radians can a function pass though the origin from one sheet of a two-sheeted
hyperboloid to the other sheet if this volume is formed by rotating a hyperbolic surface about the imaginary axis. It is proposed that any function passing through
the origin from one sheet to the other requires a rotation by π radians
at the origin. An input of zero for the natural logarithm, ln, has a value of 1.
Thus the range of the solution for in the positive sheet of a two-sheet hyperboloid for Pythagoras’ Theorem is 1<c<1+ln(2cosh(√(a2+b2)))
and for the relativistic dilation factor
is 1<ϒ<1+ln(2cosh(√(1-v2/c2))). In the negative sheet for Pythagoras’
Theorem it is -1>c>-1-ln(2cosh(√(a2+b2))); and for the relativistic dilation factor
it is -1>ϒ-1-ln(2cosh(√(1 - v2/c2))).
The midpoint, average, of these ranges is in each sheet is, respectively, √(a2+b2)
and √(1 - v2/c2) preceded by a negative sign in the negative sheet and a positive
sign in the positive sheet. From the perspective within each sheet, all values would
be positive, including this midpoint average.
Assuming that the
universe is a hyperbolic surface as proposed by Mabkhout (Mabkhout,
2012)
appears reasonable and consistent with this analysis. Mabkhout proposes that if Einstein’s
tensors are re-solved for a hyperbolic surface,
then there is no need to resort to Dark Energy or Dark Matter, and the inflation after the Big Bang at the
origin appears to be a consequence of the hyperbolic surface. He also proposes
that the hyperbolic solutions are consistent with the age and size of the universe
at large scales, and the Planck Energy and Planck Length at quantum scales.
It is noted that a
range, uncertainty (as in Heisenberg’s Uncertainty Principle), also appears to be an outcome for the solution on a hyperbolic surface. Pythagoras’ Theorem and the Lorentz’s Transform are merely AVERAGES
of the ranges of uncertainty required by the hyperbolic solutions. When the uncertainty in the solutions is small, then the values
introduced by imagination can be ignored. However at a scale of 2/3 of the absolute,
uncertainty becomes too large to be ignored. At
5/6 of the absolute, the average approximations of the midpoint of the ranges are
no longer useful.
The values of 2/3
and 5/6 are consistent with n individuals acting as a single system,
called a Nash Equilibrium after mathematician John Nash. For a normal random function
of the absolute, for example the hyperbolic secant squared, or logistics, distribution,
the mean/median, µ, is one half of the absolute. Zero is that mean minus
3 Standard Deviations, σ. Between
0 and µ-2σ there is a small value for the approximation and a large uncertainty.
Between µ-2σ and µ+2σ are almost all of the solutions and the uncertainty
causing an error from this approximation is small. Only between µ+2σ
and µ+3σ, which if µ is ∞/2
and µ-3σ is zero, 1/∞, then µ+3σ is ∞, the absolute,
and σ=∞/6, is the approximation large and its error also large. A Nash Equilibrium
again is n individuals acting as a system and almost all of those individuals,
over 90 percent, will be between µ-2σ
and µ+2σ, in other words 2/3 of the absolute. While almost 5% of the individuals
will be between zero and µ-2σ, the error will be large, but the approximation
will be very small. Only for the almost 5% of
the individuals between µ+2σ and the absolute, i.e. 5/6 of the absolute,
will the error be large, and the approximation also be large. Thus according to
Nash, Pythagoras’s Theorem is
correct over 90% of the time and when the values are less that 5/6 of the absolute,
this approximation can be used. Pythagoras’ Theorem, like all models, may be wrong, but according to Nash it is
useful.
Works Cited
Mabkhout, S. (2012). The infinite distance horizon
and the hyperbolic inflation in the hyperbolic universe. Phys. Essays,
25(1), p.112.
