I'm somewhat disturbed by the fact that, although Google presents the linked article as the top result (at least to me), the rest of the first page is an assortment of unrelated theory. How such an obviously important isomorphism (that between special relativity's law of velocity addition and the law of probability addition) could go so unnoticed in an era of increasing awareness of the relevance of "it from bit" is beyond me, but I do know that web pages tend to go away at a high frequency. If that were to happen to this web page, it would relegate the isomorphism far less accessible to the interested amateur who might be willing to risk his non-existent reputation on a different paradigm. For that reason I'm pasting the web page here:
*Probabilities and Velocities* If two events, denoted by A and B, are mutually exclusive and have the individual probabilities P(A) and P(B), then the probability of "A or B" is just the sum of the individual probabilities, i.e., P(A ÈB) = P(A) + P(B) However, if the events A and B are not mutually exclusive, meaning that there is some non-zero probability P(AÇB) of *both* events occurring, then the above formula represents an over-estimate of their combined probability, because it counts the intersection P(AÇB) twice. To correct for this, the general expression for the probability of the union of two arbitrary events is P(A ÈB) = P(A) + P(B) - P(AÇB) In particular, if A and B are independent, the probability of them both occurring equals the product of their individual probabilities, i.e., P(AÇB) = P(A) P(B), so the formula for the probability of independent events is P(A ÈB) = P(A) + P(B) - P(A) P(B) It's interesting to compare these formulas with the composition formulas for speeds with respect to different frames of reference. Of course, in ordinary units a speed can have a magnitude greater than 1, which makes it incommensurable with probabilities, which always have magnitude less than or equal to 1. However, if we agree to express all speeds as fractions of the speed of light, then the speed of every physical entity with respect to any inertial system of coordinates will be dimensionless and less than or equal to 1. Another potential difficulty is that speeds can be negative. In fact, we might even imagine complex speeds, similar to the complex values of the Schrödinger wave equation in quantum mechanics. Taking the same approach that Max Born proposed for the interpretation of the wave function, we could identify the squared norm of the speed v with the "probability". The squared norm of a complex number is simply the product of the number and its complex conjugate, i.e., . For ordinary real-valued speeds, this implies that we associate the "probability" with the value v2. Two speeds may be called independent if they are in orthogonal directions. For example, suppose a particle is moving with a speed vy in the positive y direction of an inertial coordinate system S, and suppose the spatial origin of S is moving with a speed vx in the positive x direction of another inertial coordinate system S' whose axes are aligned with those of S. What is the combined speed V = vx È vy of the particle with respect to the S' system? According to Galilean kinematics, these orthogonal squared speeds are simply additive, so P(vx È vy) = P(vx) + P(vy) which expresses the Pythagorean vector addition law V2 = vx2 + vy2 However, if the squared norm of a speed is to actually represent a probability, and if speeds in orthogonal directions are to be regarded as independent, then we can argue that the true law for the composition of orthogonal speeds should be P(vx È vy) = P(vx) + P(vy) - P(vx) P(vy) In terms of the actual speeds this represents the formula V2 = vx2 + vy2 - vx2vy2 Since these speeds are all expressed as fractions of the speed of light, it's clear that the term vx2vy2would be negligible unless at least one of the speeds involved was near the speed of light. Interestingly, this is precisely the correct formula for the composition of orthogonal speeds according to Einstein's theory of special relativity. To see this, let t,x,y,z denote the inertial coordinates of system S, and let T,X,Y,Z denote the (aligned) inertial coordinates of system S'. In S the particle is moving with speed vy in the positive y direction so its coordinates are The Lorentz transformation for a coordinate system S' whose spatial origin is moving with the speed v in the positive x (and X) direction with respect to system S is so the coordinates of the particle with respect to the S' system are The first of these equations implies t = T(1 - vx2)1/2, so we can substitute for t in the expressions for X and Y to give The total squared speed V2 with respect to these coordinates is given by Incidentally, subtracting 1 from both sides and factoring the right hand side, this relativistic composition rule for orthogonal speeds can be written in the form This is essentially DeMorgan's Rule, which says that the negation of the logical OR of two events is the logical AND of the negations of the two events. Symbolically, this can be expressed as The composition of three orthogonal speeds (such as in the x, y, and z directions) is likewise given by the probabilistic rule for independent events Notice that the relativistic energy of a particle of rest mass m0 is m0/(1-V 2)1/2, so this gives a decomposition of the kinetic components of the relativistic energy into orthogonal factors. Moreover, we can decompose the vibrational modes of a complex configuration into an infinite family of orthogonal components, and the relativistic energies combine in accordance with this rule. Incidentally, the above formulas for the composition of orthogonal velocities can be generalized to cover any two velocities *u* and *v*. In terms of the dot product, the magnitude of the mutual velocity*V* between two particles that are moving with the velocities *u* and *v* relative to any given inertial coordinate system is The denominator of the right hand side is unity if *u* and *v* are perpendicular, in which case this formula reduces to the previous expression. On Tue, Mar 4, 2014 at 12:42 PM, James Bowery <jabow...@gmail.com> wrote: > More to the point -- or perhaps I should say, to the bit -- is that it > makes no more sense to talk about speeds greater than light than it does > probabilities greater than 1: > > http://www.mathpages.com/home/kmath216/kmath216.htm > > > On Tue, Mar 4, 2014 at 12:35 PM, D R Lunsford <antimatter3...@gmail.com>wrote: > >> No one will ever take cold fusion seriously if they come here and read >> nonsense about how relativity is wrong. All of these specious arguments >> focus on the constancy of the speed of light. >> >> What is never understood is that C isn't the speed of anything in >> particular. It is a parameter that characterizes the geometry of spacetime, >> which is no longer Euclidean. The structure of this geometry emerges from a >> very simple (group theoretic) analysis. The parameter C emerges out of the >> analysis and is either finite, or not. Experience shows that it is finite. >> The derivation is here, I gave it some years ago and this person has added >> commentary, most of which is helpful. Only simple algebra is required. >> >> That light goes at C is incidental to the existence of a universal >> constant with the dimensions of speed. It does so because the corresponding >> field is massless. The most important point to be grasped is that one does >> not assume C=constant - this comes right out of the symmetry and >> homogeneity analysis. Euclidean geometry is also characterized by a >> constant - however it is imaginary, and corresponds to the "circular points >> at infinity" in projective geometry. >> >> http://membrane.com/sidd/wundrelat.txt >> >> -drl >> >> >> -- >> "Time flies like an arrow, but fruit flies like a banana." - Marx >> > >
