A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry by Carlile Lavor & Sebastià Xambó-Descamps & Isiah Zaplana

Author:Carlile Lavor & Sebastià Xambó-Descamps & Isiah Zaplana
Language: eng
Format: epub
ISBN: 9783319906652
Publisher: Springer International Publishing

The first equation and the condition δ > 0 allow us to state that there is a unique such that , . From the second equation we infer that there exists such that and . Substituting these values into δξ′− δ′ξ = 1, we get λ = 1, whereby the third equation is automatically satisfied. Thus the matrix of f has the form , with and . Given that |u| < 1 and γ = (1 − u 2)−1∕2, it is clear that f is the Lorentz boost of velocity u (or rapidity α). □

In sum, the intrinsic notion corresponding to the group of Lorentz transformations (denoted G c in [74]) is the group of the proper orthochronous isometries of E 1,3. It is a normal subgroup of the isometry group O1,3 of E 1,3 and of the subgroup SO1,3 ⊂O1,3 of the proper isometries.

The essence of special relativity is the study of concepts and relations that are invariant by the action of . It is, therefore, a particular case of Klein’s geometry, but its very origin explains its extraordinary potential for expressing statements of geometric and physical content. It is what we try to show in the pages that follow.

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