Editing Lorentz transformation (section)

==Tensor formulation==

{{main|Representation theory of the Lorentz group}}
{{For|the notation used|Ricci calculus}}

=== Contravariant vectors ===
Writing the general matrix transformation of coordinates as the matrix equation
<math display="block">\begin{bmatrix} {x'}^0 \\ {x'}^1 \\ {x'}^2 \\ {x'}^3 \end{bmatrix} =
  \begin{bmatrix}
    {\Lambda^0}_0 & {\Lambda^0}_1 & {\Lambda^0}_2 & {\Lambda^0}_3 \vphantom{{x'}^0} \\
    {\Lambda^1}_0 & {\Lambda^1}_1 & {\Lambda^1}_2 & {\Lambda^1}_3 \vphantom{{x'}^0} \\
    {\Lambda^2}_0 & {\Lambda^2}_1 & {\Lambda^2}_2 & {\Lambda^2}_3 \vphantom{{x'}^0} \\
    {\Lambda^3}_0 & {\Lambda^3}_1 & {\Lambda^3}_2 & {\Lambda^3}_3 \vphantom{{x'}^0} \\
  \end{bmatrix}
  \begin{bmatrix} x^0 \vphantom{{x'}^0} \\ x^1 \vphantom{{x'}^0} \\ x^2 \vphantom{{x'}^0} \\ x^3 \vphantom{{x'}^0} \end{bmatrix}</math>
allows the transformation of other physical quantities that cannot be expressed as four-vectors; e.g., [[tensor]]s or [[spinor]]s of any order in 4-dimensional spacetime, to be defined. In the corresponding [[tensor index notation]], the above matrix expression is
<math display="block">{x'}^\nu = {\Lambda^\nu}_\mu x^\mu,</math>

where lower and upper indices label [[covariance and contravariance of vectors|covariant and contravariant components]] respectively,<ref>{{harvnb|Dennery|Krzywicki|2012|p=[https://books.google.com/books?id=ogHCAgAAQBAJ&pg=PA138 138]}}</ref> and the [[summation convention]] is applied. It is a standard convention to use [[Greek alphabet|Greek]] indices that take the value 0 for time components, and 1, 2, 3 for space components, while [[Latin alphabet|Latin]] indices simply take the values 1, 2, 3, for spatial components (the opposite for Landau and Lifshitz). Note that the first index (reading left to right) corresponds in the matrix notation to a ''row index''. The second index corresponds to the column index.

The transformation matrix is universal for all [[four-vector]]s, not just 4-dimensional spacetime coordinates. If {{mvar|A}} is any four-vector, then in [[tensor index notation]] <math display="block"> {A'}^\nu = {\Lambda^\nu}_\mu A^\mu \,.</math>

Alternatively, one writes <math display="block"> A^{\nu'} = {\Lambda^{\nu'}}_\mu A^\mu \,.</math> in which the primed indices denote the indices of A in the primed frame. For a general {{mvar|n}}-component object one may write <math display="block">{X'}^\alpha = {\Pi(\Lambda)^\alpha}_\beta X^\beta \,, </math> where {{math|Π}} is the appropriate [[Representation theory of the Lorentz group|representation of the Lorentz group]], an {{math|''n'' × ''n''}} matrix for every {{math|Λ}}. In this case, the indices should ''not'' be thought of as spacetime indices (sometimes called Lorentz indices), and they run from {{math|1}} to {{mvar|n}}. E.g., if {{mvar|X}} is a [[bispinor]], then the indices are called ''Dirac indices''.

=== Covariant vectors ===
There are also vector quantities with covariant indices. They are generally obtained from their corresponding objects with contravariant indices by the operation of ''lowering an index''; e.g.,
<math display="block">x_\nu = \eta_{\mu\nu}x^\mu,</math>
where {{mvar|η}} is the [[metric tensor]]. (The linked article also provides more information about what the operation of raising and lowering indices really is mathematically.) The inverse of this transformation is given by
<math display="block">x^\mu = \eta^{\mu\nu}x_\nu,</math>
where, when viewed as matrices, {{math|''η''{{sup|''μν''}}}} is the inverse of {{math|''η''{{sub|''μν''}}}}. As it happens, {{math|1=''η''{{sup|''μν''}} = {{math|''η''{{sub|''μν''}}}}}}. This is referred to as ''raising an index''. To transform a covariant vector {{math|''A''{{sub|''μ''}}}}, first raise its index, then transform it according to the same rule as for contravariant {{math|4}}-vectors, then finally lower the index;
<math display="block">{A'}_\nu = \eta_{\rho\nu} {\Lambda^\rho}_\sigma \eta^{\mu\sigma}A_\mu.</math>

But <math display="block">\eta_{\rho\nu} {\Lambda^\rho}_\sigma \eta^{\mu\sigma} = {\left(\Lambda^{-1}\right)^\mu}_\nu,</math>

That is, it is the {{math|(''μ'', ''ν'')}}-component of the ''inverse'' Lorentz transformation. One defines (as a matter of notation),
<math display="block">{\Lambda_\nu}^\mu \equiv {\left(\Lambda^{-1}\right)^\mu}_\nu,</math>
and may in this notation write
<math display="block">{A'}_\nu = {\Lambda_\nu}^\mu A_\mu.</math>

Now for a subtlety. The implied summation on the right hand side of
<math display="block">{A'}_\nu = {\Lambda_\nu}^\mu A_\mu = {\left(\Lambda^{-1}\right)^\mu}_\nu A_\mu</math>
is running over ''a row index'' of the matrix representing {{math|Λ{{sup|−1}}}}. Thus, in terms of matrices, this transformation should be thought of as the ''inverse transpose'' of {{math|Λ}} acting on the column vector {{math|''A''{{sub|''μ''}}}}. That is, in pure matrix notation,
<math display="block">A' = \left(\Lambda^{-1}\right)^\mathrm{T} A.</math>

This means exactly that covariant vectors (thought of as column matrices) transform according to the [[dual representation]] of the standard representation of the Lorentz group. This notion generalizes to general representations, simply replace {{math|Λ}} with {{math|Π(Λ)}}.

=== Tensors ===
If {{mvar|A}} and {{mvar|B}} are linear operators on vector spaces {{mvar|U}} and {{mvar|V}}, then a linear operator {{math|''A'' ⊗ ''B''}} may be defined on the [[tensor product]] of {{mvar|U}} and {{mvar|V}}, denoted {{math|''U'' ⊗ ''V''}} according to<ref>{{harvnb|Hall|2003|loc=Chapter 4}}</ref>

{{Equation box 1
 |indent =:
 |equation = <math>(A \otimes B)(u \otimes v) = Au \otimes Bv, \qquad u \in U, v \in V, u \otimes v \in U \otimes V.</math> {{spaces|13}} {{EquationRef|(T1)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

From this it is immediately clear that if {{mvar|u}} and {{mvar|v}} are a four-vectors in {{mvar|V}}, then {{math|''u'' ⊗ ''v'' ∈ ''T''{{sub|2}}''V'' ≡ ''V'' ⊗ ''V''}} transforms as

{{Equation box 1
 |indent =:
 |equation =
<math>
  u \otimes v \rightarrow \Lambda u \otimes \Lambda v =
    {\Lambda^\mu}_\nu u^\nu \otimes {\Lambda^\rho}_\sigma v^\sigma =
    {\Lambda^\mu}_\nu {\Lambda^\rho}_\sigma u^\nu \otimes v^\sigma \equiv
    {\Lambda^\mu}_\nu {\Lambda^\rho}_\sigma w^{\nu\sigma}.
</math> {{spaces|13}} {{EquationRef|(T2)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

The second step uses the bilinearity of the tensor product and the last step defines a 2-tensor on component form, or rather, it just renames the tensor {{math|''u'' ⊗ ''v''}}.

These observations generalize in an obvious way to more factors, and using the fact that a general tensor on a vector space {{mvar|V}} can be written as a sum of a coefficient (component!) times tensor products of basis vectors and basis covectors, one arrives at the transformation law for any [[tensor]] quantity {{mvar|T}}. It is given by<ref>{{harvnb|Carroll|2004|page=22}}</ref>

{{Equation box 1
 |indent =:
 |equation =
<math>
  T^{\alpha'\beta' \cdots \zeta'}_{\theta'\iota' \cdots \kappa'} =
    {\Lambda^{\alpha'}}_\mu {\Lambda^{\beta'}}_\nu \cdots {\Lambda^{\zeta'}}_\rho
    {\Lambda_{\theta'}}^\sigma {\Lambda_{\iota'}}^\upsilon \cdots {\Lambda_{\kappa'}}^\zeta
    T^{\mu\nu \cdots \rho}_{\sigma\upsilon \cdots \zeta},
</math> {{spaces|13}} {{EquationRef|(T3)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

where {{math|''Λ''{{sub|''χ′''}}{{sup|''ψ''}}}} is defined above. This form can generally be reduced to the form for general {{mvar|n}}-component objects given above with a single matrix ({{math|Π(Λ)}}) operating on column vectors. This latter form is sometimes preferred; e.g., for the electromagnetic field tensor.

==== Transformation of the electromagnetic field ====
[[File:Lorentz boost electric charge.svg|upright=1.75|thumb|Lorentz boost of an electric charge; the charge is at rest in one frame or the other.]]
{{main|Electromagnetic tensor}}
{{Further|classical electromagnetism and special relativity}}

Lorentz transformations can also be used to illustrate that the [[magnetic field]] {{math|'''B'''}} and [[electric field]] {{math|'''E'''}} are simply different aspects of the same force — the [[electromagnetic force]], as a consequence of relative motion between [[electric charge]]s and observers.<ref>{{harvnb|Grant|Phillips|2008}}</ref> The fact that the electromagnetic field shows relativistic effects becomes clear by carrying out a simple thought experiment.<ref>{{harvnb|Griffiths|2007}}</ref>
* An observer measures a charge at rest in frame {{mvar|F}}. The observer will detect a static electric field.  As the charge is stationary in this frame, there is no electric current, so the observer does not observe any magnetic field.
* The other observer in frame {{mvar|F′}} moves at velocity {{math|'''v'''}} relative to {{mvar|F}} and the charge. ''This'' observer sees a different electric field because the charge moves at velocity {{math|−'''v'''}} in their rest frame. The motion of the charge corresponds to an [[electric current]], and thus the observer in frame {{mvar|F′}} also sees a magnetic field.

The electric and magnetic fields transform differently from space and time, but exactly the same way as relativistic angular momentum and the boost vector.

The electromagnetic field strength tensor is given by
<math display="block">
  F^{\mu\nu} = \begin{bmatrix}
    0              & -\frac{1}{c}E_x & -\frac{1}{c}E_y & -\frac{1}{c}E_z \\
    \frac{1}{c}E_x &  0              & -B_z            &  B_y   \\
    \frac{1}{c}E_y &  B_z            &  0              & -B_x   \\
    \frac{1}{c}E_z & -B_y            &  B_x            &  0
  \end{bmatrix} \text{(SI units, signature }(+,-,-,-)\text{)}.
</math>
in [[SI units]]. In relativity, the [[Gaussian units|Gaussian system of units]] is often preferred over SI units, even in texts whose main choice of units is SI units, because in it the electric field {{math|'''E'''}} and the magnetic induction {{math|'''B'''}} have the same units making the appearance of the [[Electromagnetic tensor|electromagnetic field tensor]] more natural.<ref>{{harvnb|Jackson|1975|p={{page needed|date=November 2023}}}}</ref> Consider a Lorentz boost in the {{mvar|x}}-direction. It is given by<ref>{{harvnb|Misner|Thorne|Wheeler|1973}}</ref>
<math display="block">
  {\Lambda^\mu}_\nu = \begin{bmatrix}
     \gamma      & -\gamma\beta & 0 & 0\\
    -\gamma\beta &  \gamma      & 0 & 0\\
     0           &  0           & 1 & 0\\
     0           &  0           & 0 & 1\\
  \end{bmatrix}, \qquad

  F^{\mu\nu} = \begin{bmatrix}
     0   &  E_x &  E_y &  E_z \\
    -E_x &  0   &  B_z & -B_y \\
    -E_y & -B_z &  0   &  B_x \\
    -E_z &  B_y & -B_x &  0
  \end{bmatrix} \text{(Gaussian units, signature }(-,+,+,+)\text{)},
</math>
where the field tensor is displayed side by side for easiest possible reference in the manipulations below.

The general transformation law {{EquationNote|(T3)}} becomes
<math display="block">F^{\mu'\nu'} = {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}.</math>

For the magnetic field one obtains
<math display="block">\begin{align}
  B_{x'} &= F^{2'3'}
            = {\Lambda^2}_\mu {\Lambda^3}_\nu F^{\mu\nu}
            = {\Lambda^2}_2 {\Lambda^3}_3 F^{23}
            = 1 \times 1 \times B_x \\
         &= B_x, \\
  B_{y'} &= F^{3'1'}
            = {\Lambda^3}_\mu {\Lambda^1}_\nu F^{\mu \nu}
            = {\Lambda^3}_3 {\Lambda^1}_\nu F^{3\nu}
            = {\Lambda^3}_3 {\Lambda^1}_0 F^{30} + {\Lambda^3}_3 {\Lambda^1}_1 F^{31} \\
         &= 1 \times (-\beta\gamma) (-E_z) + 1 \times \gamma B_y
            = \gamma B_y + \beta\gamma E_z \\
         &= \gamma\left(\mathbf{B} - \boldsymbol{\beta} \times \mathbf{E}\right)_y \\
  B_{z'} &= F^{1'2'}
            = {\Lambda^1}_\mu {\Lambda^2}_\nu F^{\mu\nu}
            = {\Lambda^1}_\mu {\Lambda^2}_2 F^{\mu 2}
            = {\Lambda^1}_0 {\Lambda^2}_2 F^{02} + {\Lambda^1}_1 {\Lambda^2}_2 F^{12} \\
         &= (-\gamma\beta) \times 1\times E_y + \gamma \times 1 \times B_z
            = \gamma B_z - \beta\gamma E_y \\
         &= \gamma\left(\mathbf{B} - \boldsymbol{\beta} \times \mathbf{E}\right)_z
\end{align}</math>

For the electric field results
<math display="block">\begin{align}
  E_{x'} &= F^{0'1'}
            = {\Lambda^0}_\mu {\Lambda^1}_\nu F^{\mu\nu}
            = {\Lambda^0}_1 {\Lambda^1}_0 F^{10} + {\Lambda^0}_0 {\Lambda^1}_1 F^{01} \\
         &= (-\gamma\beta)(-\gamma\beta)(-E_x) + \gamma\gamma E_x
            = -\gamma^2\beta^2(E_x) + \gamma^2 E_x
            = E_x(1 - \beta^2)\gamma^2 \\
         &= E_x, \\
  E_{y'} &= F^{0'2'}
            = {\Lambda^0}_\mu {\Lambda^2}_\nu F^{\mu\nu}
            = {\Lambda^0}_\mu {\Lambda^2}_2 F^{\mu 2}
            = {\Lambda^0}_0 {\Lambda^2}_2 F^{02} + {\Lambda^0}_1 {\Lambda^2}_2 F^{12} \\
         &= \gamma \times 1 \times E_y + (-\beta\gamma) \times 1 \times B_z
            = \gamma E_y - \beta\gamma B_z \\
         &= \gamma\left(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B}\right)_y \\
  E_{z'} &= F^{0'3'}
            = {\Lambda^0}_\mu {\Lambda^3}_\nu F^{\mu\nu}
            = {\Lambda^0}_\mu {\Lambda^3}_3 F^{\mu 3}
            = {\Lambda^0}_0 {\Lambda^3}_3 F^{03} + {\Lambda^0}_1 {\Lambda^3}_3 F^{13} \\
         &= \gamma \times 1 \times E_z - \beta\gamma \times 1 \times (-B_y)
            = \gamma E_z + \beta\gamma B_y \\
         &= \gamma\left(\mathbf{E} + \boldsymbol{\beta} \times \mathbf{B}\right)_z.
\end{align}</math>

Here, {{math|1='''''β''''' = (''β'', 0, 0)}} is used. These results can be summarized by
<math display="block">\begin{align}
  \mathbf{E}_{\parallel'} &= \mathbf{E}_\parallel \\
  \mathbf{B}_{\parallel'} &= \mathbf{B}_\parallel \\
       \mathbf{E}_{\bot'} &= \gamma \left( \mathbf{E}_\bot + \boldsymbol{\beta} \times \mathbf{B}_\bot \right) = \gamma \left( \mathbf{E} + \boldsymbol{\beta} \times \mathbf{B} \right)_\bot,\\
       \mathbf{B}_{\bot'} &= \gamma \left( \mathbf{B}_\bot - \boldsymbol{\beta} \times \mathbf{E}_\bot \right) = \gamma \left( \mathbf{B} - \boldsymbol{\beta} \times \mathbf{E} \right)_\bot,
\end{align}</math>
and are independent of the metric signature. For SI units, substitute {{math|''E'' → {{frac|''E''|''c''}}}}. {{harvtxt|Misner|Thorne|Wheeler|1973}} refer to this last form as the {{math|3 + 1}} view as opposed to the ''geometric view'' represented by the tensor expression
<math display="block">F^{\mu'\nu'} = {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu},</math>
and make a strong point of the ease with which results that are difficult to achieve using the {{math|3 + 1}} view can be obtained and understood. Only objects that have well defined Lorentz transformation properties (in fact under ''any'' smooth coordinate transformation) are geometric objects. In the geometric view, the electromagnetic field is a six-dimensional geometric object in ''spacetime'' as opposed to two interdependent, but separate, 3-vector fields in ''space'' and ''time''. The fields {{math|'''E'''}} (alone) and {{math|'''B'''}} (alone) do not have well defined Lorentz transformation properties. The mathematical underpinnings are equations {{EquationNote|(T1)}} and {{EquationNote|(T2)}} that immediately yield {{EquationNote|(T3)}}. One should note that the primed and unprimed tensors refer to the ''same event in spacetime''. Thus the complete equation with spacetime dependence is
<math display="block">
  F^{\mu' \nu'}\left(x'\right) =
    {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}\left(\Lambda^{-1} x'\right) =
    {\Lambda^{\mu'}}_\mu {\Lambda^{\nu'}}_\nu F^{\mu\nu}(x).
</math>

Length contraction has an effect on [[charge density]] {{mvar|ρ}} and [[current density]] {{math|'''J'''}}, and time dilation has an effect on the rate of flow of charge (current), so charge and current distributions must transform in a related way under a boost. It turns out they transform exactly like the space-time and energy-momentum four-vectors,
<math display="block">\begin{align}
  \mathbf{j}' &= \mathbf{j} - \gamma\rho v\mathbf{n} + \left( \gamma - 1 \right)(\mathbf{j} \cdot \mathbf{n})\mathbf{n} \\
        \rho' &= \gamma \left(\rho - \mathbf{j} \cdot \frac{v\mathbf{n}}{c^2}\right),
\end{align}</math>

or, in the simpler geometric view,
<math display="block">j^{\mu'} = {\Lambda^{\mu'}}_\mu j^\mu.</math>

Charge density transforms as the time component of a four-vector. It is a rotational scalar. The current density is a 3-vector.

The [[Maxwell equations]] are invariant under Lorentz transformations.

=== Spinors ===
Equation {{EquationNote|(T1)}} hold unmodified for any representation of the Lorentz group, including the [[bispinor]] representation. In {{EquationNote|(T2)}} one simply replaces all occurrences of {{math|Λ}} by the bispinor representation {{math|Π(Λ)}},

{{Equation box 1
 |indent =:
 |equation =
<math>\begin{align}
u \otimes v \rightarrow \Pi(\Lambda) u \otimes \Pi(\Lambda) v &=
  {\Pi(\Lambda)^\alpha}_\beta u^\beta \otimes {\Pi(\Lambda)^\rho}_\sigma v^\sigma\\ &=
  {\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma u^\beta \otimes v^\sigma\\ &\equiv
  {\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma w^{\beta\sigma}
\end{align}</math> {{spaces|13}} {{EquationRef|(T4)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

The above equation could, for instance, be the transformation of a state in [[Fock space]] describing two free electrons.

==== Transformation of general fields ====
A general ''noninteracting'' multi-particle state (Fock space state) in [[quantum field theory]] transforms according to the rule<ref>{{harvnb|Weinberg|2002|loc=Chapter 3}}</ref>
{{NumBlk||<math display="block">\begin{align}
         &U(\Lambda, a) \Psi_{p_1\sigma_1 n_1; p_2\sigma_2 n_2; \cdots} \\
    = {} &e^{-ia_\mu \left[(\Lambda p_1)^\mu + (\Lambda p_2)^\mu + \cdots\right]}
            \sqrt{\frac{(\Lambda p_1)^0(\Lambda p_2)^0\cdots}{p_1^0 p_2^0 \cdots}}
            \left( \sum_{\sigma_1'\sigma_2' \cdots} D_{\sigma_1'\sigma_1}^{(j_1)}\left[W(\Lambda, p_1)\right] D_{\sigma_2'\sigma_2}^{(j_2)}\left[W(\Lambda, p_2)\right] \cdots \right)
            \Psi_{\Lambda p_1 \sigma_1' n_1; \Lambda p_2 \sigma_2' n_2; \cdots},
  \end{align}</math>
 | {{EquationRef|1}}
}}
where {{math|''W''(Λ, ''p'')}} is the [[Wigner's classification|Wigner's little group]]<ref>{{Cite web |title=INSPIRE |url=https://inspirehep.net/literature/26312 |access-date=2024-09-04 |website=inspirehep.net}}</ref> and {{math|''D''{{sup|(''j'')}}}} is the {{nowrap|{{math|(2''j'' + 1)}}-dimensional}} representation of {{math|SO(3)}}.