From: tgcapizzi@gmail.com   
      
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   On Thursday, September 16, 2004 at 8:09:24 AM UTC-4, DRLunsford wrote:   
   > This occured to me at lunch, and is sort of fun...   
   > We know that the Lorentz transformation can be derived from a   
   > group-theoretic analysis of space and time, based on simple   
   > assumptions of isotropy, linearity etc. There exist both synthetic and   
   > analytic versions of this derivation that have been posted here. The   
   > end result is, there is a parameter with the dimensions of a velocity   
   > that is either finite, or infinite, that characterizes the geometry.   
   > Of course, it is C.   
   > One might ask, is Euclidean geometry so characterized? The answer is   
   > yes!   
   > Metric geometry sits inside projective geometry by positing a   
   > fundamental quadric. In relativity it is of course the light cone - in   
   > 1+1 dimensions   
   > x^2 - (ct)^2 = 0   
   > which can be factored   
   > (x - ct)(x + ct) = 0   
   > so x/t = +-c - this shows how the fundamental quadric is related to   
   > the characteristic parameter.   
   > What about Euclidean plane geometry? The fundamental quadric is   
   > x^2 + y^2 = 0   
   > which seems like an empty statement, but makes sense in the context of   
   > projective geometry as the "circular points at infinity". We can now   
   > factor this as   
   > (x - iy)(x + iy) = 0   
   > The characteristic parameter of Euclidean geometry is the imaginary   
   > unit! So "i" plays the role of the "speed of imaginary light" in   
   > Euclidean geometry :)   
   > This has a beautiful interpretation. Intuitively, one knows that, on   
   > the Euclidean plane, one can imagine a thing called "infinity" which   
   > can never be got closer to, from which all regular points are   
   > "infinitely" distant. No matter how far you go, you're always   
   > "infinitely" far away from "infinity". This is the *exactly analogous*   
   > result to the impossibility of attaining the speed C in relativity.   
   > This may be the most basic way complex numbers enter into physics.   
   > -drl   
      
   This information needs to be combined with eigenvector decomposition based   
   on the Lorentz matrix. In the first place, calling it the Lorentz matrix makers   
   it appear to be a physics thing. Wrong. It was a simple, hyperbolic rotation   
   long, long before Lorentz's name was attached to it.   
      
   "> We know that the Lorentz transformation can be derived from a   
   > group-theoretic analysis of space and time, based on simple   
   > assumptions of isotropy, linearity etc."   
      
   Nothing more sophisticated than Euclidean geometry is needed. Indeed,   
   the hyperbolic geometry that underpins relativity was first used by the   
   map-maker, Mercator, centuries before it was claimed by physics.   
   Centuries before group math and relativity.   
      
   The details of hyperbolic trigonometry were published in the mid-1700's.   
   The "Lorentz" Transformation is nothing more than a restatement of the   
   hyperbolic identities for the cosh and sinh of the sum of two hyperbolic   
   angles. Given a hyperbolic angle, w, that is the sum of two independent   
   variables, A and B, those identities are cosh(w) = cos(A+B) =   
   cosh(A)cosh(B)+sinh(A)sinh(B), and sinh(w) =   
   sinh(A)cosh(B)+cosh(A)sinh(B). A point, (x,y), on a unit hyperbola with   
   horizontal symmetry satisfies the equation x²-y² = 1. The coordinates   
   of this point are equal to (cosh(A),sin(A)).  Since the addition of   
   hyperbolic angles is commutative and linear, it doesn't matter which of   
   the two angles refers to the initial state. The final state is just the   
   sum of an initial state and an increment. There is a simple proof, using   
   geometric algebra, that the sum of hyperbolic rotation angles is linear.   
   So, we can state that the point (cosh(w),sinh(w)) = (cosh(B+A),   
   sinh(B+A)). Using the above identities, cosh(B+A)=   
   cosh(B)cosh(A)+sinh(B)sinh(A) sinh(B+A) = sinh(B)cosh(A)+cosh(B)sinh(A)   
   The astute observer will notice that this is nothing more than an   
   expansion of the linear algebra equation: â_cosh(w)â_  â_cosh(B)   
   sinh(B)â_â_cosh(A)â_ â_ sinh(w)â_=â_sinh(B) cosh(B)â_â_ sinh(A)â_ In   
   Greek symbols and more specific coordinates: â_ct'â_  â_ γ   
   -βγâ_â_ctâ_ â_ r' â_=â_-βγ  γâ_â_ r â_ Using the same identities,   
   we can derive the hyperbolic identity of the tanh of the sum of two   
   hyperbolic angles: v3 = c tanh(w) = c tanh(A+B)  =   
   c(sinh(A)cosh(B)+cosh(A)sinh(B))/(cosh(A)cosh(B)+sinh(A)sinh(B)) =   
   c(sinh(A)/cosh(A)+sinh(B)/cosh(B))/(1+sinh(A)/cosh(A)*sinh(B)/cosh(B)) =   
   c(tanh(A)+tanh(B))/(1+tanh(A)*tanh(B)) In its more familiar form, v1 = c   
   tanh(A) and v2 = c tanh(B): (v1+v2)/(1+v1/c*v2/c), the velocity addition   
   "rule" of special relativity. It's just a hyperbolic identity. Physics   
   did not invent it, even if they take credit for it.   
      
   "> Metric geometry sits inside projective geometry by positing a   
   > fundamental quadric. In relativity it is of course the light cone - in   
   > 1+1 dimensions   
   > x^2 - (ct)^2 = 0   
   > which can be factored   
   > (x - ct)(x + ct) = 0"   
      
   Minkowski's initial foray into relativity identified time as an   
   imaginary component, so that the quadric becomes  x^2 + (ict)^2 =  x^2 -   
   (ct)^2 = 0. This equation generates the rectangular hyperbola. In   
   general, x^2 - (ct)^2 = ±s^2, where the choice of sign depends on an   
   arbitrary convention. The two values, s and w, are coordinates in a   
   Cartesian grid. They represent hyperbolic coordinates in which they are   
   invariant in the absence of external forces. A shift of either   
   coordinate has no effect on the other coordinate, a standard property of   
   orthogonal coordinates. Since w can represent arbitrary combinations of   
   rotation angles, it is convenient to divide it into 2 components, one   
   being its initial value, w0, at some arbitrary time, t=0, and the other,   
   its value at some future time, w1. Then w = w1-w0. It makes no   
   difference if it is a reference to one frame which changes its   
   hyperbolic angle, or a given frame that is observed by a non co-moving   
   frame.   
      
      
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