Lorentz Transformation

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SPACETIME TRANSFORMATIONS OF THE ORDERED VACUUM George B. Smith orderedvacuum@gmail.com July, 2014 Every model of spacetime behavior transforms in a certain way as the frame of reference is changed from one inertial observer to another. Deduce the transformation properties of the ordered vacuum model developed on this webpage to see if it matches reality.

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For an observer at rest in the particle’s rest frame, the particle appears to be jumping around at the zitterbewegung frequency executing an un-biased random walk. An observer moving to the left with uniform speed would see the particle moving to the right. The observers would disagree on right-left status of some subset of vacuum exchange scattering events. The observer sees the particle encountering the same number of vacuum exchange scattering events on the left as on the right. This observer would have to see more vacuum exchange scattering events moving the particle to the right than to the left.

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To reconcile this, what is sought is a (linear) transformation between time and the coordinate in the direction of the motion (taken to be the x-direction) which accounts successfully for the velocity (averaged over a large number of vacuum exchange scattering events) in both the particle's inertial rest frame and the other inertial reference frame moving with uniform velocity v in the x direction. In order to approach this mathematically, consider the coordinate vectors x 1 , x 2 , . . . , x N of the N points to which vacuum exchange scattering brings the particle in time D t = t N - t 1 in its rest frame. The total distance covered can be written as D x = | x N - x 1 | . Similarly, D x' and D t' can be written for the moving (primed) frame , and these frames can be related with the linear transformation L:

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Since the unprimed frame is being considered as the rest frame, then it must be that D x = 0 for N sufficiently large, so that In the moving (primed) frame, the particle's speed must be v = - D x'/ D t', so that dividing these equations gives With the definition g = D t'/ D t, it follows that and .

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The transformation matrix becomes: The inverse transformation is the forward transformation with v replaced by –v, so that gives and . The transformation matrix now becomes:

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Consider two transformations, L 1 and L 2 , associated with velocities v 1 and v 2 , respectively. Closure implies the existence of a transformation, L, such that L = L 1 L 2 . Since the transformation matrix generally has equal diagonal elements, then forming the product, L(v 1 )L(v 2 ) and equating diagonal elements gives the equation: This has three solutions. Two are ruled out on physical grounds since they produce a non-unitary determinant for L. The remaining one is: Because of the independence of v 1 and v 2 , (with c being a constant with velocity units) it follows that Dividing by v 1 v 2 , g (v 1 ) g (v 2 ) gives:

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The transformation matrix now becomes: with, and with c being the limiting velocity. Thus, the set of ideas that have been put forth in this work regarding the nature of particles and how they interact with the ordered vacuum is sufficient to uniquely determine the Lorentz transformation as the only spacetime transformation that is compatible with these ideas. Indeed, it has been shown that the Lorentz transformation is precisely what is needed to enforce the proposition that vacuum exchange scattering is able to account for the particle's motion in any and all inertial reference frames, and therefore, that all motion can be regarded as based on vacuum exchange scattering in an ordered vacuum.