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Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation — the so-called Thomas (or Wigner) rotation — and a single boost. Using spacetime algebra, the geometric algebra for Minkowski spacetime, we show in this paper how to relate the Thomas rotation to the Doppler shift experienced between two observers in relative motion. Both observers are moving away from a third observer, set in the laboratory, with relative non-parallel velocities.