X-Message-Number: 19295 From: Date: Mon, 17 Jun 2002 03:38:16 EDT Subject: One century of Special Relativity --part1_83.1c4d7deb.2a3eebe8_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit 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. --part1_83.1c4d7deb.2a3eebe8_boundary Content-Type: text/html; charset="US-ASCII" [ AUTOMATICALLY SKIPPING HTML ENCODING! ] Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=19295