# Beyond the Einstein Addition Law and its Gyroscopic Thomas by Abraham A. Ungar

By Abraham A. Ungar

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Extra resources for Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (Fundamental Theories of Physics)

Sample text

We assume that the inner product in (also known as a scalar product) is positive definite in the sense that v • v = ||v|| ² ‡ 0 for all v ˛ and ||v||² > 0 if v „ 0. Clearly, a positive definite inner product is non-degenerate, that is, then u = 0. Indeed, if u • v = 0 for all if u,v ˛ and u • v = 0 for all v ˛ v ˛ and any given u ˛ then, in particular, for v = u we have ||u|| ² = 0 implying u = 0. 57) An isometry must be linear. 1 is a linear self-map of . VERIFYING THE COCYCLE EQUATION Some identities with lengthy, but straightforward, algebraic proof are presented in this book without proof.

Herbert Goldstein, Classical Mechanics The Einstein addition and its associated Thomas precession form an integral part of the greatest intellectual achievement of the twentieth century, that is, the understanding of spacetime geometry. However, it seems that the presence of relativistic velocities with their Einstein’s addition in spacetime geometry results in a loss of mathematical regularity since Einstein’s addition is not a group operation. Indeed, one of the goals this book is to show that this is not the case.

36) Conversely, one must also show that x = b gyr[b, a]a is indeed a solution. 18 on p. 47. 34) that contrasts our gyrogroup formalism approach is found in [Ung91c] [Ric93]. 3 on p. 36. 1 (The Loop Property). 35). 6), is a loop. 37) calling it the gyrogroup cooperation that coexists with the gyrogroup operation ¯. 39) The Einstein binary cooperation , called the Einstein coaddition, will prove useful in the algebra of Einstein’s addition. 43) that is symmetric in u and v. 44) We thus see that while Einstein’s addition ¯ possesses the left cancellation law, we need its coexisting operation, the Einstein coaddition , in order to have a right cancellation law as well.