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Proper transformations

If a Lorentz covariant 4-vector is measured in one inertial frame with result X {\displaystyle X}, and the same measurement made in another inertial frame (with the same orientation and origin) gives result X ′ {\displaystyle X'}, the two results will be related by X ′ = B (v) X {\displaystyle X'=B(\mathbf {v})X} where the boost matrix B (v) {\displaystyle B(\mathbf {v})} represents the rotation-free Lorentz transformation between the unprimed and primed frames and v {\displaystyle \mathbf {v} } is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by B (v) = =, {\displaystyle B(\mathbf {v})={\begin{bmatrix}\gamma &-\gamma v_{x}/c&-\gamma v_{y}/c&-\gamma v_{z}/c\\-\gamma v_{x}/c&1+(\gamma -1){\dfrac {v_{x}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{y}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{z}}{v^{2}}}\\-\gamma v_{y}/c&(\gamma -1){\dfrac {v_{y}v_{x}}{v^{2}}}&1+(\gamma -1){\dfrac {v_{y}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{y}v_{z}}{v^{2}}}\\-\gamma v_{z}/c&(\gamma -1){\dfrac {v_{z}v_{x}}{v^{2}}}&(\gamma -1){\dfrac {v_{z}v_{y}}{v^{2}}}&1+(\gamma -1){\dfrac {v_{z}^{2}}{v^{2}}}\end{bmatrix}}={\begin{bmatrix}\gamma &-\gamma {\vec {\beta }}^{T}\\-\gamma {\vec {\beta }}&I+(\gamma -1){\dfrac {{\vec {\beta }}{\vec {\beta }}^{T}}{\beta ^{2}}}\end{bmatrix}},}

where v = v x 2 + v y 2 + v z 2 {\textstyle v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}} is the magnitude of the velocity and γ = 1 1 − v 2 c 2 {\textstyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} is the Lorentz factor. This formula represents a passive transformation, as it describes how the coordinates of the measured quantity changes from the unprimed frame to the primed frame. The active transformation is given by B (− v) {\displaystyle B(-\mathbf {v})}.

If a frame F ′ is boosted with velocity u relative to frame F, and another frame F ′′ is boosted with velocity v relative to F ′, the separate boosts are X ″ = B (v) X ′, X ′ = B (u) X {\displaystyle X''=B(\mathbf {v})X'\,,\quad X'=B(\mathbf {u})X} and the composition of the two boosts connects the coordinates in F ′′ and F, X ″ = B (v) B (u) X. {\displaystyle X''=B(\mathbf {v})B(\mathbf {u})X\,.} Successive transformations act on the left. If u and v are collinear (parallel or antiparallel along the same line of relative motion), the boost matrices commute: B (v) B (u) = B (u) B (v). This composite transformation happens to be another boost, B (w), where w is collinear with u and v.

If u and v are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: B (v) B (u) and B (u) B (v) are not equal. Although each of these compositions is not a single boost, each composition is still a Lorentz transformation as it preserves the spacetime interval. It turns out the composition of any two Lorentz boosts is equivalent to a boost followed or preceded by a rotation on the spatial coordinates, in the form of R (ρ) B (w) or B (w) R (ρ). The w and w are composite velocities, while ρ and ρ are rotation parameters (e.g. axis-angle variables, Euler angles, etc.). The rotation in block matrix form is simply R (ρ) =, {\displaystyle \quad R({\boldsymbol {\rho }})={\begin{bmatrix}1&0\\0&\mathbf {R} ({\boldsymbol {\rho }})\end{bmatrix}}\,,} where R (ρ) is a 3d rotation matrix, which rotates any 3d vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is not simple to connect w and ρ (or w and ρ) to the original boost parameters u and v. In a composition of boosts, the R matrix is named the Wigner rotation, and gives rise to the Thomas precession. These articles give the explicit formulae for the composite transformation matrices, including expressions for w, ρ, w, ρ.

In this article the axis-angle representation is used for ρ. The rotation is about an axis in the direction of a unit vector e, through angle θ (positive anticlockwise, negative clockwise, according to the right-hand rule). The "axis-angle vector" θ = θ e {\displaystyle {\boldsymbol {\theta }}=\theta \mathbf {e} } will serve as a useful abbreviation.

Spatial rotations alone are also Lorentz transformations since they leave the spacetime interval invariant. Like boosts, successive rotations about different axes do not commute. Unlike boosts, the composition of any two rotations is equivalent to a single rotation. Some other similarities and differences between the boost and rotation matrices include:

The most general proper Lorentz transformation Λ(v, θ) includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, Λ(0, θ) = R (θ) and Λ(v, 0) = B (v). An explicit form of the general Lorentz transformation is cumbersome to write down and will not be given here. Nevertheless, closed form expressions for the transformation matrices will be given below using group theoretical arguments. It will be easier to use the rapidity parametrization for boosts, in which case one writes Λ(ζ, θ) and B (ζ).