By Abraham A. Ungar
"I can't outline twist of fate [in mathematics]. yet 1 shall argue that accident can continually be increased or geared up right into a superstructure which perfonns a unification alongside the coincidental parts. The lifestyles of a accident is robust proof for the life of a masking conception. " -Philip 1. Davis [Dav81] Alluding to the Thomas gyration, this ebook offers the speculation of gy rogroups and gyrovector areas, taking the reader to the immensity of hyper bolic geometry that lies past the Einstein targeted idea of relativity. quickly after its creation via Einstein in 1905 [Ein05], exact relativity concept (as named by way of Einstein ten years later) grew to become overshadowed by way of the ap pearance of common relativity. to that end, the exposition of exact relativity the strains laid down by means of Minkowski, within which the position of hyperbolic ge ometry isn't emphasised. this may potentially be defined by means of the strangeness and unfamiliarity of hyperbolic geometry [Bar98]. the purpose of this ebook is to opposite the fad of neglecting the function of hy perbolic geometry within the targeted conception of relativity, initiated through Minkowski, by means of emphasizing the vital position that hyperbolic geometry performs within the theory.
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Extra info for Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces
This rotation, denoted gyr[u, v], is known as the Thomas precession generated by u and v, giving rise to the gyrocommutative law of Einstein's addition uEI\ v = gyr[u, v](vesu). 5) The Thomas precession gyr[u, v] will be studied in Section 3. We use the notation EI\ = e when no confusion arises. 7) that appears as a factor in the Einstein velocity addition law. This functional equation, called the cocycle equation, will prove useful in the extension of Einstein's addition between 3-vectors to the Lorentz transformation of 4-vectors between inertial frames.
In Chapter 2 we will extend these properties by abstraction, obtaining the formal definition of a mathematical grouplike object called a gyrogroup. Despite not being groups, gyrogroups with their gyrations possess rich structure. As we hinted earlier in this chapter, this structure turns out to reveal striking analogies shared by Euclidean and hyperbolic geometry, analogies which are obscured by traditional approaches. 2), in the open c-ball ~ of the Euclidean 3-space :lR3 can be written as u~v = 1+ 1 U·V c2 1 1 'Yu (u·v)u } .
For the discussion of the composite velocity reciprocity principle in Section 4, an object is shown moving uniformly with velocity y (velocity x) relative to I: (to r}'). 3 THOMAS PRECESSION AND GYROGROUPS The Thomas precession is illustrated in Fig. 1. Let~, ~, and ~" be three inertial frames of reference that were coincident at time t = O. The three inertial frames ~, ~, and ~", equipped with spacetime coordinates (only two space coordinates are shown in the Figure for clarity), are in relative velocities u and v.