Wigner rotation

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. The rotation was discovered by Llewellyn Thomas in 1926, and derived by Wigner in 1939. 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. u ⊕ v = 1 1 + u ⋅ v c 2 [ ( 1 + 1 c 2 γ u 1 + γ u u ⋅ v ) u + 1 γ u v ] , {displaystyle mathbf {u} oplus mathbf {v} ={frac {1}{1+{frac {mathbf {u} cdot mathbf {v} }{c^{2}}}}}left,}     (VA 2) Λ = B ( v ) B ( u ) = [ γ − a T − b M ] {displaystyle Lambda =B(mathbf {v} )B(mathbf {u} )={egin{bmatrix}gamma &-mathbf {a} ^{mathrm {T} }\-mathbf {b} &mathbf {M} end{bmatrix}}} The decomposition process described (below) can be carried through on the product of two pure Lorentz transformations to obtain explicitly the rotation of the coordinate axes resulting from the two successive 'boosts'. In general, the algebra involved is quite forbidding, more than enough, usually, to discourage any actual demonstration of the rotation matrix 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. The rotation was discovered by Llewellyn Thomas in 1926, and derived by Wigner in 1939. 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. There are still ongoing discussions about the correct form of equations for the Thomas rotation in different reference systems with contradicting results. Goldstein: Einstein's principle of velocity reciprocity (EPVR) reads With less careful interpretation, the EPVR is seemingly violated in some models. There is, of course, no true paradox present. When studying the Thomas rotation at the fundamental level, one typically uses a setup with three coordinate frames, Σ, Σ′ Σ′′. Frame Σ′ has velocity u relative to frame Σ, and frame Σ′′ has velocity v relative to frame Σ′. The axes are, by construction, oriented as follows. Viewed from Σ′, the axes of Σ′ and Σ are parallel (the same holds true for the pair of frames when viewed from Σ.) Also viewed from Σ′, the spatial axes of Σ′ and Σ′′ are parallel (and the same holds true for the pair of frames when viewed from Σ′′.) This is an application of EVPR: If u is the velocity of Σ′ relative to Σ, then u′ = −u is the velocity of Σ relative to Σ′. The velocity 3-vector u makes the same angles with respect to coordinate axes in both the primed and unprimed systems. This does not represent a snapshot taken in any of the two frames of the combined system at any particular time, as should be clear from the detailed description below. This is possible, since a boost in, say, the positive z-direction, preserves orthogonality of the coordinate axes. A general boost B(w) can be expressed as L = R−1(ez, w)Bz(w)R(ez, w), where R(ez, w) is a rotation taking the z-axis into the direction of w and Bz is a boost in the new z-direction. Each rotation retains the property that the spatial coordinate axes are orthogonal. The boost will stretch the (intermediate) z-axis by a factor γ, while leaving the intermediate x-axis and y-axis in place. The fact that coordinate axes are non-parallel in this construction after two consecutive non-collinear boosts is a precise expression of the phenomenon of Thomas precession. The velocity of Σ′′ as seen in Σ is denoted wd = u ⊕ v, where ⊕ refers to the relativistic addition of velocity (and not ordinary vector addition), given by