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. The rotation was discovered by Émile Borel in 1913, rediscovered and proved by Ludwik Silberstein in his 1914 book The Theory of Relativity, rediscovered by Llewellyn Thomas in 1926, and rederived by Eugene Wigner in 1939. 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. Goldstein:
The spatial rotation resulting from the successive application of two non-collinear Lorentz transformations have been declared every bit as paradoxical as the more frequently discussed apparent violations of common sense, such as the twin paradox. Einstein's principle of velocity reciprocity (EPVR) reads
We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non-preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of v to −v With less careful interpretation, the EPVR is seemingly violated in some situations, but on closer analysis there is no such violation. 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 coordinates of the relative velocities of these two bodies will not be opposite even though the actual velocity vectors themselves are indeed opposites (with the fact that the coordinates are not opposites being due to the fact that the two travellers are not using the same coordinate basis vectors). If A and B both started in the lab system with coordinates matching those of the lab and subsequently use coordinate systems that result from their respective boosts from that system, then the velocity that A will measure on B will be given in terms of A's new coordinate system by:
v A B = 1 1 + u ⋅ v c 2 [ ( 1 + 1 c 2 γ u 1 + γ u u ⋅ v ) u + 1 γ u v ] , {\displaystyle \mathbf {v} _{AB}={\frac {1}{1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\left(1+{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {u} }}{1+\gamma _{\mathbf {u} }}}\mathbf {u} \cdot \mathbf {v} \right)\mathbf {u} +{\frac {1}{\gamma _{\mathbf {u} }}}\mathbf {v} \right],}
And the velocity that B will measure on A will be given in terms of B's coordinate system by:
v B A = 1 1 + v ⋅ u c 2 [ ( 1 + 1 c 2 γ v 1 + γ v v ⋅ u ) v + 1 γ v u ] , {\displaystyle \mathbf {v} _{BA}={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}}}\left[\left(1+{\frac {1}{c^{2}}}{\frac {\gamma _{\mathbf {v} }}{1+\gamma _{\mathbf {v} }}}\mathbf {v} \cdot \mathbf {u} \right)\mathbf {v} +{\frac {1}{\gamma _{\mathbf {v} }}}\mathbf {u} \right],}
The Lorentz factor for the velocities that either A sees on B or B sees on A are the same:
γ = γ u ⊕ v = γ v ⊕ u = γ u γ v ( 1 + u ⋅ v c 2 ) , {\displaystyle \gamma =\gamma _{\mathbf {u} \oplus \mathbf {v} }=\gamma _{\mathbf {v} \oplus \mathbf {u} }=\gamma _{\mathbf {u} }\gamma _{\mathbf {v} }\left(1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}\right)\,,}
but the components are not opposites - i.e. v A B ≠ − v B A {\textstyle \mathbf {v} _{AB}\neq -\mathbf {v} _{BA}}
However this does not mean that the velocities are not opposites as the components in each case are multiplied by different basis vectors (and all observers agree that the difference is by a rotation of coordinates such that the actual velocity vectors are indeed exact opposites). The angle of rotation can be calculated in two ways:
cos ϵ = ( 1 + γ + γ u + γ v ) 2 ( 1 + γ ) ( 1 + γ u ) ( 1 + γ v ) − 1 , {\displaystyle \cos \epsilon ={\frac {(1+\gamma +\gamma _{\mathbf {u} }+\gamma _{\mathbf {v} })^{2}}{(1+\gamma )(1+\gamma _{\mathbf {u} })(1+\gamma _{\mathbf {v} })}}-1~,}
Or:
cos ϵ = − ( v A B ⋅ v B A ) ( v A B 2 ) {\displaystyle \cos \epsilon =-{\frac {(\mathbf {v} _{AB}\cdot \mathbf {v} _{BA})}{(v_{AB}^{2})}}}
And the axis of rotation is:
e = − u × v | u × v | . {\displaystyle \mathbf {e} =-{\frac {\mathbf {u} \times \mathbf {v} }{|\mathbf {u} \times \mathbf {v} |}}~.}
Setup of frames and relative velocities between them
Two general boosts 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 rotation. 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
and
γ u = 1 1 − | u | 2 c 2 {\displaystyle \gamma _{\mathbf {u} }={\frac {1}{\sqrt {1-{\frac {|\mathbf {u} |^{2}}{c^{2}}}}}}}
is the Lorentz factor of the velocity u (the vertical bars |u| indicate the magnitude of the vector). The velocity u can be thought of the velocity of a frame Σ′ relative to a frame Σ, and v is the velocity of an object, say a particle or another frame Σ′′ relative to Σ′. In the present context, all velocities are best thought of as relative velocities of frames unless otherwise specified. The result w = u ⊕ v is then the relative velocity of frame Σ′′ relative to a frame Σ. Although velocity addition is nonlinear, non-associative, and non-commutative, the result of the operation correctly obtains a velocity with a magnitude less than c. If ordinary vector addition was used, it would be possible to obtain a velocity with a magnitude larger than c. The Lorentz factor γ of both composite velocities are equal,
γ = γ u ⊕ v = γ v ⊕ u = γ u γ v ( 1 + u ⋅ v c 2 ) , {\displaystyle \gamma =\gamma _{\mathbf {u} \oplus \mathbf {v} }=\gamma _{\mathbf {v} \oplus \mathbf {u} }=\gamma _{\mathbf {u} }\gamma _{\mathbf {v} }\left(1+{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}\right)\,,}
and the norms are equal under interchange of velocity vectors
| u ⊕ v | = | v ⊕ u | = c γ γ 2 − 1 . {\displaystyle |\mathbf {u} \oplus \mathbf {v} |=|\mathbf {v} \oplus \mathbf {u} |={\frac {c}{\gamma }}{\sqrt {\gamma ^{2}-1}}\,.}
Since the two possible composite velocities have equal magnitude, but different directions, one must be a rotated copy of the other. More detail and other properties of no direct concern here can be found in the main article.
Reversed configuration Consider the reversed configuration, namely, frame Σ moves with velocity −u relative to frame Σ′, and frame Σ′, in turn, moves with velocity −v relative to frame Σ′′. In short, u → − u and v → −v by EPVR. Then the velocity of Σ relative to Σ′′ is (−v) ⊕ (−u) ≡ −v ⊕ u. By EPVR again, the velocity of Σ′′ relative to Σ is then wi = v ⊕ u. (A) One finds wd ≠ wi. While they are equal in magnitude, there is an angle between them. For a single boost between two inertial frames, there is only one unambiguous relative velocity (or its negative). For two boosts, the peculiar result of two inequivalent relative velocities instead of one seems to contradict the symmetry of relative motion between any two frames. Which is the correct velocity of Σ′′ relative to Σ? Since this inequality may be somewhat unexpected and potentially breaking EPVR, this question is warranted.
Formulation in terms of Lorentz transformations
Two boosts equals a boost and rotation The answer to the question lies in the Thomas rotation, and that one must be careful in specifying which coordinate system is involved at each step. When viewed from Σ, the coordinate axes of Σ and Σ′′ are not parallel. While this can be hard to imagine since both pairs (Σ, Σ′) and (Σ′, Σ′′) have parallel coordinate axes, it is easy to explain mathematically. Velocity addition does not provide a complete description of the relation between the frames. One must formulate the complete description in terms of Lorentz transformations corresponding to the velocities. A Lorentz boost with any velocity v (magnitude less than c) is given symbolically by
X ′ = B ( v ) X {\displaystyle X'=B(\mathbf {v} )X}
where the coordinates and transformation matrix are compactly expressed in block matrix form
X ′ = [ c t ′ r ′ ] B ( v ) = [ γ v − γ v c v T − γ v c v I + γ v 2 γ v + 1 v v T c 2 ] X = [ c t r ] {\displaystyle X'={\begin{bmatrix}ct'\\\mathbf {r} '\end{bmatrix}}\quad B(\mathbf {v} )={\begin{bmatrix}\gamma _{\mathbf {v} }&-{\dfrac {\gamma _{\mathbf {v} }}{c}}\mathbf {v} ^{\mathrm {T} }\\-{\dfrac {\gamma _{\mathbf {v} }}{c}}\mathbf {v} &\mathbf {I} +{\dfrac {\gamma _{\mathbf {v} }^{2}}{\gamma _{\mathbf {v} }+1}}{\dfrac {\mathbf {vv} ^{\mathrm {T} }}{c^{2}}}\\\end{bmatrix}}\quad X={\begin{bmatrix}ct\\\mathbf {r} \end{bmatrix}}}
and, in turn, r, r′, v are column vectors (the matrix transpose of these are row vectors), and γv is the Lorentz factor of velocity v. The boost matrix is a symmetric matrix. The inverse transformation is given by
B ( v ) − 1 = B ( − v ) ⇒ X = B ( − v ) X ′ . {\displaystyle B(\mathbf {v} )^{-1}=B(-\mathbf {v} )\quad \Rightarrow \quad X=B(-\mathbf {v} )X'.}
It is clear that to each admissible velocity v there corresponds a pure Lorentz boost,
v ↔ B ( v ) . {\displaystyle \mathbf {v} \leftrightarrow B(\mathbf {v} ).}
Velocity addition u⊕v corresponds to the composition of boosts B(v)B(u) in that order. The B(u) acts on X first, then B(v) acts on B(u)X. Notice succeeding operators act on the left in any composition of operators, so B(v)B(u) should be interpreted as a boost with velocities u then v, not v then u. Performing the Lorentz transformations by block matrix multiplication,
X ″ = B ( v ) X ′ , X ′ = B ( u ) X ⇒ X ″ = Λ X {\displaystyle X''=B(\mathbf {v} )X'\,,\quad X'=B(\mathbf {u} )X\quad \Rightarrow \quad X''=\Lambda X}
the composite transformation matrix is
Λ = B ( v ) B ( u ) = [ γ − a T − b M ] {\displaystyle \Lambda =B(\mathbf {v} )B(\mathbf {u} )={\begin{bmatrix}\gamma &-\mathbf {a} ^{\mathrm {T} }\\-\mathbf {b} &\mathbf {M} \end{bmatrix}}}
and, in turn,
γ = γ v γ u ( 1 + v T u c 2 ) a = γ c u ⊕ v , b = γ c v ⊕ u M = γ u γ v v u T c 2 + ( I + γ v 2 γ v + 1 v v T c 2 ) ( I + γ u 2 γ u + 1 u u T c 2 ) {\displaystyle {\begin{aligned}\gamma &=\gamma _{\mathbf {v} }\gamma _{\mathbf {u} }\left(1+{\frac {\mathbf {v} ^{\mathrm {T} }\mathbf {u} }{c^{2}}}\right)\\\mathbf {a} &={\frac {\gamma }{c}}\mathbf {u} \oplus \mathbf {v} \,,\quad \mathbf {b} ={\frac {\gamma }{c}}\mathbf {v} \oplus \mathbf {u} \\\mathbf {M} &=\gamma _{\mathbf {u} }\gamma _{\mathbf {v} }{\frac {\mathbf {vu} ^{\mathrm {T} }}{c^{2}}}+\left(\mathbf {I} +{\frac {\gamma _{\mathbf {v} }^{2}}{\gamma _{\mathbf {v} }+1}}{\frac {\mathbf {vv} ^{\mathrm {T} }}{c^{2}}}\right)\left(\mathbf {I} +{\frac {\gamma _{\mathbf {u} }^{2}}{\gamma _{\mathbf {u} }+1}}{\frac {\mathbf {uu} ^{\mathrm {T} }}{c^{2}}}\right)\end{aligned}}}
Here γ is the composite Lorentz factor, and a and b are 3×1 column vectors proportional to the composite velocities. The 3×3 matrix M will turn out to have geometric significance. The inverse transformations are
X = B ( − u ) X ′ , X ′ = B ( − v ) X ″ ⇒ X = Λ − 1 X ″ {\displaystyle X=B(-\mathbf {u} )X'\,,\quad X'=B(-\mathbf {v} )X''\quad \Rightarrow \quad X=\Lambda ^{-1}X''}
and the composition amounts to a negation and exchange of velocities,
Λ − 1 = B ( − u ) B ( − v ) = [ γ b T a M T ] {\displaystyle \Lambda ^{-1}=B(-\mathbf {u} )B(-\mathbf {v} )={\begin{bmatrix}\gamma &\mathbf {b} ^{\mathrm {T} }\\\mathbf {a} &\mathbf {M} ^{\mathrm {T} }\end{bmatrix}}}
If the relative velocities are exchanged, looking at the blocks of Λ, one observes the composite transformation to be the matrix transpose of Λ. This is not the same as the original matrix, so the composite Lorentz transformation matrix is not symmetric, and thus not a single boost. This, in turn, translates to the incompleteness of velocity composition from the result of two boosts; symbolically,
B ( u ⊕ v ) ≠ B ( v ) B ( u ) . {\displaystyle B(\mathbf {u} \oplus \mathbf {v} )\neq B(\mathbf {v} )B(\mathbf {u} )\,.}
To make the description complete, it is necessary to introduce a rotation, before or after the boost. This rotation is the Thomas rotation. A rotation is given by
X ′ = R ( θ ) X {\displaystyle X'=R({\boldsymbol {\theta }})X}
where the 4×4 rotation matrix is
R ( θ ) = [ 1 0 0 R ( θ ) ] {\displaystyle R({\boldsymbol {\theta }})={\begin{bmatrix}1&0\\0&\mathbf {R} ({\boldsymbol {\theta }})\end{bmatrix}}}
and R is a 3×3 rotation matrix. In this article the axis-angle representation is used, and θ = θe is the "axis-angle vector", the angle θ multiplied by a unit vector e parallel to the axis. Also, the right-handed convention for the spatial coordinates is used (see orientation (vector space)), so that rotations are positive in the anticlockwise sense according to the right-hand rule, and negative in the clockwise sense. With these conventions; the rotation matrix rotates any 3d vector about the axis e through angle θ anticlockwise (an active transformation), which has the equivalent effect of rotating the coordinate frame clockwise about the same axis through the same angle (a passive transformation). The rotation matrix is an orthogonal matrix, its transpose equals its inverse, and negating either the angle or axis in the rotation matrix corresponds to a rotation in the opposite sense, so the inverse transformation is readily obtained by
R ( θ ) − 1 = R ( θ ) T = R ( − θ ) ⇒ X = R ( − θ ) X ′ . {\displaystyle R({\boldsymbol {\theta }})^{-1}=R({\boldsymbol {\theta }})^{\mathrm {T} }=R(-{\boldsymbol {\theta }})\quad \Rightarrow \quad X=R(-{\boldsymbol {\theta }})X'\,.}
A boost followed or preceded by a rotation is also a Lorentz transformation, since these op