Editing Lorentz transformation (section)

===Proper transformations===

If a Lorentz covariant 4-vector is measured in one inertial frame with result <math>X</math>, and the same measurement made in another inertial frame (with the same orientation and origin) gives result <math>X'</math>, the two results will be related by
<math display="block">X' = B(\mathbf{v})X</math>
where the boost matrix <math>B(\mathbf{v})</math> represents the rotation-free Lorentz transformation between the unprimed and primed frames and <math>\mathbf{v}</math> is the velocity of the primed frame as seen from the unprimed frame. The matrix is given by<ref>{{Cite journal|last=Furry|first=W. H.|date=1955-11-01|title=Lorentz Transformation and the Thomas Precession|url=https://aapt.scitation.org/doi/10.1119/1.1934085|journal=American Journal of Physics|volume=23|issue=8|pages=517–525|doi=10.1119/1.1934085| bibcode=1955AmJPh..23..517F| issn=0002-9505}}</ref>
<math display="block">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},</math>

where <math display="inline">v=\sqrt{v_x^2+v_y^2+v_z^2}</math> is the magnitude of the velocity and <math display="inline">\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}</math> 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 <math>B(-\mathbf{v})</math>.

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

If {{math|'''u'''}} and {{math|'''v'''}} are not collinear but in different directions, the situation is considerably more complicated. Lorentz boosts along different directions do not commute: {{math|''B''('''v''')''B''('''u''')}} and {{math|''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 {{math|''R''('''ρ''')''B''('''w''')}} or {{math|''B''({{overline|'''w'''}})''R''({{overline|'''ρ'''}})}}. The {{math|'''w'''}} and {{math|{{overline|'''w'''}}}} are [[velocity addition formula|composite velocities]], while {{math|'''ρ'''}} and {{math|{{overline|'''ρ'''}}}} are rotation parameters (e.g. [[axis-angle representation|axis-angle]] variables, [[Euler angles]], etc.). The rotation in [[block matrix]] form is simply
<math display="block">\quad R(\boldsymbol{\rho}) = \begin{bmatrix} 1 & 0 \\ 0 & \mathbf{R}(\boldsymbol{\rho}) \end{bmatrix} \,, </math>
where {{math|'''R'''('''ρ''')}} is a {{math|3 × 3}} [[rotation matrix]], which rotates any 3-dimensional vector in one sense (active transformation), or equivalently the coordinate frame in the opposite sense (passive transformation). It is ''not'' simple to connect {{math|'''w'''}} and {{math|'''ρ'''}} (or {{math|{{overline|'''w'''}}}} and {{math|{{overline|'''ρ'''}}}}) to the original boost parameters {{math|'''u'''}} and {{math|'''v'''}}. In a composition of boosts, the {{mvar|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 {{math|'''w''', '''ρ''', {{overline|'''w'''}}, {{overline|'''ρ'''}}}}.

In this article the [[axis-angle representation]] is used for {{math|'''ρ'''}}. The rotation is about an axis in the direction of a [[unit vector]] {{math|'''e'''}}, through angle {{mvar|θ}} (positive anticlockwise, negative clockwise, according to the [[right-hand rule]]). The "axis-angle vector"
<math display="block">\boldsymbol{\theta} = \theta \mathbf{e}</math>
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:
* [[matrix inverse|inverse]]s: {{math|1=''B''('''v'''){{sup|−1}} = ''B''(−'''v''')}} (relative motion in the opposite direction), and {{math|1=''R''('''θ'''){{sup|−1}} = ''R''(−'''θ''')}} (rotation in the opposite sense about the same axis)
* [[identity transformation]] for no relative motion/rotation: {{math|1=''B''('''0''') = ''R''('''0''') = ''I''}}
* unit [[determinant]]: {{math|1=det(''B'') = det(''R'') = +1}}. This property makes them proper transformations.
* [[symmetric matrix|matrix symmetry]]: {{mvar|B}} is symmetric (equals [[transpose]]), while {{mvar|R}} is nonsymmetric but [[orthogonal matrix|orthogonal]] (transpose equals [[matrix inverse|inverse]], {{math|1=''R''{{sup|T}} = ''R''{{sup|−1}}}}).

The most general proper Lorentz transformation {{math|Λ('''v''', '''θ''')}} includes a boost and rotation together, and is a nonsymmetric matrix. As special cases, {{math|1=Λ('''0''', '''θ''') = ''R''('''θ''')}} and {{math|1=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 {{math|Λ('''ζ''', '''θ''')}} and {{math|''B''('''ζ''')}}.

====The Lie group SO{{sup|+}}(3,1)====

The set of transformations
<math display="block"> \{ B(\boldsymbol{\zeta}), R(\boldsymbol{\theta}), \Lambda(\boldsymbol{\zeta}, \boldsymbol{\theta}) \} </math>
with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the [[special indefinite orthogonal group]] SO{{sup|+}}(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension).

For simplicity, look at the infinitesimal Lorentz boost in the {{mvar|x}} direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the [[Taylor expansion]] of the boost matrix to first order about {{math|1=''ζ'' = 0}},
<math display="block"> B_x = I + \zeta \left. \frac{\partial B_x}{\partial \zeta } \right|_{\zeta=0} + \cdots </math>
where the higher order terms not shown are negligible because {{mvar|ζ}} is small, and {{math|''B''{{sub|''x''}}}} is simply the boost matrix in the ''x'' direction. The [[matrix calculus|derivative of the matrix]] is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at {{math|1=''ζ'' = 0}},
<math display="block"> \left. \frac{\partial B_x }{\partial \zeta } \right|_{\zeta=0} = - K_x \,. </math>

For now, {{math|''K''{{sub|''x''}}}} is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a [[matrix exponential]] is obtained
<math display="block"> B_x =\lim_{N\to\infty}\left(I-\frac{\zeta }{N}K_x\right)^{N} = e^{-\zeta K_x} </math>
where the [[Exponential function#Formal definition|limit definition of the exponential]] has been used (see also [[characterizations of the exponential function]]). More generally<ref group="nb">Explicitly,
<math display="block"> \boldsymbol{\zeta} \cdot\mathbf{K} = \zeta_x K_x + \zeta_y K_y + \zeta_z K_z </math>
<math display="block"> \boldsymbol{\theta} \cdot\mathbf{J} = \theta_x J_x + \theta_y J_y + \theta_z J_z </math>
</ref>

<math display="block">B(\boldsymbol{\zeta}) = e^{-\boldsymbol{\zeta}\cdot\mathbf{K}} \, , \quad R(\boldsymbol{\theta}) = e^{\boldsymbol{\theta}\cdot\mathbf{J}} \,. </math>

The axis-angle vector {{math|'''θ'''}} and rapidity vector {{math|'''ζ'''}} are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are {{math|1='''K''' = (''K''{{sub|''x''}}, ''K''{{sub|''y''}}, ''K''{{sub|''z''}})}} and {{math|1='''J''' = (''J''{{sub|''x''}}, ''J''{{sub|''y''}}, ''J''{{sub|''z''}})}}, each vectors of matrices with the explicit forms<ref group=nb>In [[quantum mechanics]], [[relativistic quantum mechanics]], and [[quantum field theory]], a different convention is used for these matrices; the right hand sides are all multiplied by a factor of the imaginary unit {{math|''i'' {{=}} {{sqrt|−1}}}}.</ref>

<math display="block">\begin{alignat}{3}

K_x &= \begin{bmatrix}
  0 & 1 & 0 & 0 \\
  1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  \end{bmatrix}\,,\quad &
K_y &= \begin{bmatrix}
  0 & 0 & 1 & 0\\
  0 & 0 & 0 & 0\\
  1 & 0 & 0 & 0\\
  0 & 0 & 0 & 0
  \end{bmatrix}\,,\quad &
K_z &= \begin{bmatrix}
  0 & 0 & 0 & 1\\
  0 & 0 & 0 & 0\\
  0 & 0 & 0 & 0\\
  1 & 0 & 0 & 0
\end{bmatrix}

\\[10mu]

J_x &= \begin{bmatrix}
  0 & 0 & 0 &  0 \\
  0 & 0 & 0 &  0 \\
  0 & 0 & 0 & -1 \\
  0 & 0 & 1 &  0 \\
  \end{bmatrix}\,,\quad &
J_y &= \begin{bmatrix}
  0 &  0 & 0 & 0 \\
  0 &  0 & 0 & 1 \\
  0 &  0 & 0 & 0 \\
  0 & -1 & 0 & 0
  \end{bmatrix}\,,\quad &
J_z &= \begin{bmatrix}
  0 & 0 &  0 & 0 \\
  0 & 0 & -1 & 0 \\
  0 & 1 &  0 & 0 \\
  0 & 0 &  0 & 0
  \end{bmatrix}

\end{alignat}</math>

These are all defined in an analogous way to {{math|''K''{{sub|''x''}}}} above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: {{math|'''J'''}} are the ''rotation generators'' which correspond to [[angular momentum]], and {{math|'''K'''}} are the ''boost generators'' which correspond to the motion of the system in spacetime. The derivative of any smooth curve {{math|''C''(''t'')}} with {{math|1=''C''(0) = ''I''}} in the group depending on some group parameter {{mvar|t}} with respect to that group parameter, evaluated at {{math|1=''t'' = 0}}, serves as a definition of a corresponding group generator {{mvar|G}}, and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map {{mvar|G}} smoothly back into the group via {{math|''t'' → exp(''tG'')}} for all {{mvar|t}}; this curve will yield {{mvar|G}} again when differentiated at {{math|1=''t'' = 0}}.

Expanding the exponentials in their Taylor series obtains
<math display="block"> B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^2</math>
<math display="block">R(\boldsymbol {\theta })=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^2\,.</math>
which compactly reproduce the boost and rotation matrices as given in the previous section.

It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the ''infinitesimal'' level the product
<math display="block"> \begin{align}
\Lambda
&= (I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots )(I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots ) \\
&= (I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots )(I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots ) \\
&= I - \boldsymbol {\zeta } \cdot \mathbf {K}  + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots  
\end{align} </math>
is commutative because only linear terms are required (products like {{math|('''θ'''·'''J''')('''ζ'''·'''K''')}} and {{math|('''ζ'''·'''K''')('''θ'''·'''J''')}} count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential
<math display="block">\Lambda (\boldsymbol{\zeta}, \boldsymbol{\theta}) = e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} }.</math>

The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
<math display="block">e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} } \ne e^{-\boldsymbol{\zeta} \cdot\mathbf{K}} e^{\boldsymbol{\theta} \cdot\mathbf{J}},</math>
because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation ''in principle'' (this usually does not yield an intelligible expression in terms of generators {{math|'''J'''}} and {{math|'''K'''}}), see [[Wigner rotation]]. If, on the other hand, ''the decomposition is given'' in terms of the generators, and one wants to find the product in terms of the generators, then the [[Baker–Campbell–Hausdorff formula]] applies.

====The Lie algebra so(3,1)====

Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the [[set (mathematics)|set]] of all Lorentz generators
<math display="block">V = \{ \boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J}  \} </math>
together with the operations of ordinary [[matrix addition]] and [[matrix multiplication#Scalar multiplication|multiplication of a matrix by a number]], forms a [[vector space]] over the real numbers.<ref group=nb>Until now the term "vector" has exclusively referred to "[[Euclidean vector]]", examples are position {{math|'''r'''}}, velocity {{math|'''v'''}}, etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see [[linear algebra]] and [[vector space]] for details. The generators of a Lie group also form a vector space over a [[field (mathematics)|field]] of numbers (e.g. [[real number]]s, [[complex number]]s), since a [[linear combination]] of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3-dimensional space.</ref> The generators {{math|''J''{{sub|''x''}}, ''J''{{sub|''y''}}, ''J''{{sub|''z''}}, ''K''{{sub|''x''}}, ''K''{{sub|''y''}}, ''K''{{sub|''z''}}}} form a [[basis (linear algebra)|basis]] set of ''V'', and the components of the axis-angle and rapidity vectors, {{math|''θ''{{sub|''x''}}, ''θ''{{sub|''y''}}, ''θ''{{sub|''z''}}, ''ζ''{{sub|''x''}}, ''ζ''{{sub|''y''}}, ''ζ''{{sub|''z''}}}}, are the [[coordinate vector|coordinate]]s of a Lorentz generator with respect to this basis.<ref group=nb>In ordinary 3-dimensional [[position space]], the position vector {{math|'''r''' {{=}} ''x'''''e'''{{sub|''x''}} + ''y'''''e'''{{sub|''y''}} + ''z'''''e'''{{sub|''z''}}}} is expressed as a linear combination of the Cartesian unit vectors {{math|'''e'''{{sub|''x''}}, '''e'''{{sub|''y''}}, '''e'''{{sub|''z''}}}} which form a basis, and the Cartesian coordinates {{math|''x, y, z''}} are coordinates with respect to this basis.</ref>

Three of the [[commutation relation]]s of the Lorentz generators are
<math display="block">[ J_x, J_y ] =  J_z \,,\quad [ K_x, K_y ] = -J_z \,,\quad [ J_x, K_y ] =  K_z \,, </math>
where the bracket {{math|1=[''A'', ''B''] = ''AB'' − ''BA''}} is known as the ''[[commutator]]'', and the other relations can be found by taking [[cyclic permutation]]s of {{mvar|x}}, {{mvar|y}}, {{mvar|z}} components (i.e. change {{mvar|x}} to {{mvar|y}}, {{mvar|y}} to {{mvar|z}}, and {{mvar|z}} to {{mvar|x}}, repeat).

These commutation relations, and the vector space of generators, fulfill the definition of the [[Lie algebra]] <math>\mathfrak{so}(3, 1)</math>. In summary, a Lie algebra is defined as a [[vector space]] ''V'' over a [[field (mathematics)|field]] of numbers, and with a [[binary operation]] [ , ] (called a [[Lie bracket]] in this context) on the elements of the vector space, satisfying the axioms of [[Bilinear map|bilinearity]], [[alternatization]], and the [[Jacobi identity]]. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators ''V'' as given previously, and the field is the set of real numbers.

Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.

The [[exponential map (Lie theory)|exponential map]] from the Lie algebra to the Lie group,
<math display="block">\exp \, : \, \mathfrak{so}(3,1) \to \mathrm{SO}(3,1),</math>
provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. In the case of the Lorentz group, the exponential map is just the [[matrix exponential]]. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is [[surjective function|surjective]] (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.