Editing Lorentz transformation (section)

===Transformation of other quantities===

In general, given four quantities {{mvar|A}} and {{math|1='''Z''' = (''Z''{{sub|''x''}}, ''Z''{{sub|''y''}}, ''Z''{{sub|''z''}})}} and their Lorentz-boosted counterparts {{mvar|A′}} and {{math|1='''Z′''' = (''Z′''{{sub|''x''}}, ''Z′''{{sub|''y''}}, ''Z′''{{sub|''z''}})}}, a relation of the form
<math display="block">A^2 - \mathbf{Z}\cdot\mathbf{Z} = {A'}^2 - \mathbf{Z}'\cdot\mathbf{Z}'</math>
implies the quantities transform under Lorentz transformations similar to the transformation of spacetime coordinates;
<math display="block">\begin{align}
           A' &= \gamma \left(A - \frac{v\mathbf{n}\cdot \mathbf{Z}}{c} \right) \,, \\
  \mathbf{Z}' &= \mathbf{Z} + (\gamma-1)(\mathbf{Z}\cdot\mathbf{n})\mathbf{n} - \frac{\gamma A v\mathbf{n}}{c} \,.
\end{align}</math>

The decomposition of {{math|'''Z'''}} (and {{math|'''Z′'''}}) into components perpendicular and parallel to {{math|'''v'''}} is exactly the same as for the position vector, as is the process of obtaining the inverse transformations (exchange {{math|(''A'', '''Z''')}} and {{math|(''A′'', '''Z′''')}} to switch observed quantities, and reverse the direction of relative motion by the substitution {{math|'''n''' ↦ −'''n'''}}).

The quantities {{math|(''A'', '''Z''')}} collectively make up a ''[[four-vector]]'', where {{mvar|A}} is the "timelike component", and {{math|'''Z'''}} the "spacelike component". Examples of {{mvar|A}} and {{math|'''Z'''}} are the following:

{| class="wikitable"
|-
! Four-vector
! {{mvar|A}}
! {{math|'''Z'''}}
|-
| Position [[four-vector]]
| [[Time]] (multiplied by {{mvar|c}}), {{math|''ct''}}
| [[Position vector]], {{math|'''r'''}}
|-
| [[Four-momentum]]
| [[Energy]] (divided by {{mvar|c}}), {{math|''E''/''c''}}
| [[Momentum]], {{math|'''p'''}}
|-
| [[Four-vector|Four-wave vector]]
| [[angular frequency]] (divided by {{mvar|c}}), {{math|''ω''/''c''}}
| [[wave vector]], {{math|'''k'''}}
|-
| [[Four-spin]]
| (No name), {{math|''s''{{sub|''t''}}}}
| [[Spin (physics)|Spin]], {{math|'''s'''}}
|-
| [[Four-current]]
| [[Charge density]] (multiplied by {{mvar|c}}), {{math|''ρc''}}
| [[Current density]], {{math|'''j'''}}
|-
| [[Electromagnetic four-potential]]
| [[Electric potential]] (divided by {{mvar|c}}), {{math|''φ''/''c''}}
| [[Magnetic vector potential]], {{math|'''A'''}}
|}

For a given object (e.g., particle, fluid, field, material), if {{mvar|A}} or {{math|'''Z'''}} correspond to properties specific to the object like its [[charge density]], [[mass density]], [[Spin (physics)|spin]], etc., its properties can be fixed in the rest frame of that object. Then the Lorentz transformations give the corresponding properties in a frame moving relative to the object with constant velocity. This breaks some notions taken for granted in non-relativistic physics. For example, the energy {{mvar|E}} of an object is a scalar in non-relativistic mechanics, but not in relativistic mechanics because energy changes under Lorentz transformations; its value is different for various inertial frames. In the rest frame of an object, it has a [[rest energy]] and zero momentum. In a boosted frame its energy is different and it appears to have a momentum. Similarly, in non-relativistic quantum mechanics the spin of a particle is a constant vector, but in [[relativistic quantum mechanics]] spin {{math|'''s'''}} depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike quantity {{math|''s''{{sub|''t''}}}}, however a boosted observer will perceive a nonzero timelike component and an altered spin.<ref>{{harvnb|Chaichian|Hagedorn|1997|page=239}}</ref>

Not all quantities are invariant in the form as shown above, for example orbital [[angular momentum]] {{math|'''L'''}} does not have a timelike quantity, and neither does the [[electric field]] {{math|'''E'''}} nor the [[magnetic field]] {{math|'''B'''}}. The definition of angular momentum is {{math|1='''L''' = '''r''' × '''p'''}}, and in a boosted frame the altered angular momentum is {{math|1='''L′''' = '''r′''' × '''p′'''}}. Applying this definition using the transformations of coordinates and momentum leads to the transformation of angular momentum. It turns out {{math|'''L'''}} transforms with another vector quantity {{math|1='''N''' = (''E''/''c''{{sup|2}})'''r''' − ''t'''''p'''}} related to boosts, see [[relativistic angular momentum]] for details. For the case of the {{math|'''E'''}} and {{math|'''B'''}} fields, the transformations cannot be obtained as directly using vector algebra. The [[Lorentz force]] is the definition of these fields, and in {{mvar|F}} it is {{math|1='''F''' = ''q''('''E''' + '''v''' × '''B''')}} while in {{mvar|F′}} it is {{math|1='''F′''' = ''q''('''E′''' + '''v′''' × '''B′''')}}. A method of deriving the EM field transformations in an efficient way which also illustrates the unit of the electromagnetic field uses tensor algebra, [[Lorentz transformation#Transformation of the electromagnetic field|given below]].