X-Message-Number: 19295
From: 
Date: Mon, 17 Jun 2002 03:38:16 EDT
Subject: One century of Special Relativity

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One century of Special Relativity. This is somewhat long, you may skip to the 
last paragraph to see the content.

Next year there will be one century that Einstein published its theory of 
special Relativity.  A way to mark that event could be to extend it or 
reinvent it! There is the track I suggest:

We are in euclidean space with signature: +++ (each "+" stands for the sign 
of the square of a dimension), in pseudo-euclidean spaces there is a mixing 
of + and - signs, some dimensions have a negative square. For example, the 
Minkowski space of SR is a pseudo-euclidean space with signature: +++-, the 
fourth dimension being time. The natural space of mechanics has signature: 
+++---. Why not build an extended SR on it?

Well, we know the first minus refert to time, we can think of space +++ 
tilting in the time direction, this tilt would metered by an angle, for 
example w. This angle would be real valued, not an imaginary one as common 
angle in +++. The trigonometry used would be hyperbolic, with exp (w) = ch w 
+ sh w ( ch stands for cosinus hyperbolic and sh for sinus hyperbolic 
functions). With mass m, m.exp(w) gives m.ch w = energy and m.sh w = 
impulsion. If we break m.exp (w) into a Taylor serie, we have: m + mw + 
(mw**2)/2 +... The hyperbolic tangent th = ch/sh is a function going from 0 
to 1 when w goes from 0 to the infinity. For low values of w,  we are near 
from w = th w = tangent to the curve th. The "tangent to th" is the velocity 
v, so first terms in the Taylor series are near: m + mv + (1/2)mv**2. these 
are the rest mass, the newtonian impulsion and newtonian energy. We could 
have invented postnewtonian physics with correction terms: (1/6)mv**3/c for 
impulsion and (1/24) mv**4/c**2 for energy... these are the next terms in 
exp(w) Taylor serie.

If you are not familiar with that, look at : E.F. Taylor, J.A. Wheeler, 
Spacetime Physics, W.H. Freeman 1966. It is old, but here is no better 
introductory book on the subject.

Now, what is m? the rest mass, the first element in the set? Some ten years 
ago, there has been a short communication in "Nature" looking at it as a 
second negative square dimension. We could then tilt space +++ into that 
second "minus" dimension and make a special relativity with it. Assume the 
tilt angle is u, we can expand the taylor serie  of exp(u): 1 + u + (1/2)u**2 
+ ... The simplest way to look at these terms is to think of them as angles 
(indeed u is an angle!).Or, more precisely, infinitesimal angle generators. 
So, the first term is the zero angle, the second the one angle, the third the 
two dimensional angle and so on. Such angle generators define Lie's groups 
and the Taylor serie up to the n-th term is the set of Lie's groups up to n-1 
dimensions. (see for example: N. Jacobson, Lie Algebras, ch.1, Dover, 1979. 
on that subject). Now, for each dimensionality, the simplest groups are: 
U(1): the symetry of electromagnetism, SU(2), the one of weak nuclear force, 
SU(3), that of color field giving the strong nuclear force when it leak 
outisde composite particles. We have rediscovered the gauge symetries of 
Yang-Mills theories...With a bonus: Such theories have a problem to include 
mass. Here, mass comes in a natural way because each symetry is an 
approximation of the tilt angle of space toward the mass dimension. (look Ma, 
no need for Higgs mechanism).

So the fourth dimension is time and the fiveth is mass... This is as ancien 
egyptians who metered length with one unit and height with another. We know 
the euclidean space is homogenous and so each dimension has the same 
properties as all the others. It must be the same for -- ones, so, each 
dimension here has both, the property of time and the one of mass. Here, I 
have unified newtonian impulsion and electromagnetism, newtonian energy and 
weak nuclear force and rest mass with the mysterious "1" of no tilt angle in 
the mass direction, so "1" is the mass without mass. Because basic SR unifies 
mass and energy ( if you write newtonian energy (1/2) mv**2 and not (1/2) 
mv**2/c**2 as it should, you have to multiply the first term, m, in the 
Taylor serie by c**2. So, energy without velocity reduces to : E = mc**2).To 
get back to our "1", it so turns out to be the energy of zero point energy, 
the energy of empty space. The one seen in Casimir's force for example.

We have a third "-" dimension to spent, it is the dimension of 0-point 
energy, let me call it Casimir and denote it by "C". We can think of an angle 
k who define the tilt of euclidean +++ space towards C, the hyberbolic 
trigonometry of k may be broken into the Taylor set of exp (k) = 1 + k + ... 
From here we can write the k-gauge theories... Unify them with 
electromagnetism-impulsion, newtonian energy-nuclear force,... The k-Lie's 
groups would define different approximations of the 0-point energy, and so 
different values of the true zero entropy, the reference disorder at 0 degree 
Kelvin. This would be equivalent for thermodynamics to a redefinition of the 
Boltzman's constant who tell how much entropy or disorder is ascribed to a 
given energy. This would shift many physical constants.

So, we have a 6 dimensions Special Relativity with 3 negative square 
dimensions: t for time, m for mass and C for Casimir, each one has the three 
properties because of space homogeneity. Mass is seen as gauge fields and 
these have a velocity property, this fits well with the so called 
cosmological red-shift: Mass *is* velocity. Mass-time-Casimir is too a 
thermodynamics entropy shifter, it must make many physical constants into 
variables, an explanation for the recently detected cosmological variation in 
alpha, the electromagnetic coupling constant. The k-angle of Casimir 
dimension when expanded into Taylor would give a set of inflation fields for 
cosmology.My contribution is: The homogenous negative square dimensions and 
the third one, seen  as zero point energy. Everithing else has been published 
before.

Happy Birthday to you Special Relativity.

Yvan Bozzonetti.



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