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In theoretical physics, the composition of two non-collinear Lorentz boosts results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. This rotation is called Thomas rotation, Thomas–Wigner rotation or Wigner rotation. If a sequence of non-collinear boosts returns an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.[1]
The rotation was discovered by Émile Borel in 1913,[2][3][4] rediscovered and proved by Ludwik Silberstein in his 1914 book 'Relativity', rediscovered by Llewellyn Thomas in 1926,[5] and rederived by Wigner in 1939.[6] Wigner acknowledged Silberstein.
There are still ongoing discussions about the correct form of equations for the Thomas rotation in different reference systems with contradicting results.[7] Goldstein:[8]
Einstein's principle of velocity reciprocity (EPVR) reads[9]
With less careful interpretation, the EPVR is seemingly violated in some models.[10] There is, of course, no true paradox present.
Let it be u the velocity in which the lab reference frame moves respect an object called A and let it be v the velocity in which another object called B is moving, measured from the lab reference frame. If u and v are not aligned the relative velocities of these two bodies will not be opposite, that is since there is a rotation between them
The velocity that A will measure on B will be:
The Lorentz factor for the velocities that either A sees on B or B sees on A:
The angle of rotation can be calculated in two ways:
Or:
And the axis of rotation is:
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