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{{Short description|Angular momentum in special and general relativity}}
{{spacetime}}
{{main|Physics:Angular momentum}}

In [[Physics:Physics|physics]], '''relativistic angular momentum''' refers to the mathematical formalisms and physical concepts that define [[Physics:Angular momentum|angular momentum]] in [[Physics:Special relativity|special relativity]] (SR) and [[Physics:General relativity|general relativity]] (GR). The relativistic quantity is subtly different from the three-dimensional quantity in [[Physics:Classical mechanics|classical mechanics]].

Angular momentum is an important dynamical quantity derived from position and momentum. It is a measure of an object's rotational motion and resistance to changes in its rotation. Also, in the same way momentum conservation corresponds to translational symmetry, angular momentum conservation corresponds to rotational symmetry – the connection between [[Physics:Symmetry|symmetries]] and [[Physics:Conservation law|conservation law]]s is made by [[Physics:Noether's theorem|Noether's theorem]]. While these concepts were originally discovered in [[Physics:Classical mechanics|classical mechanics]], they are also true and significant in special and general relativity. In terms of abstract algebra, the invariance of angular momentum, four-momentum, and other symmetries in spacetime, are described by the [[Lorentz group]], or more generally the [[Physics:Poincaré group|Poincaré group]].

[[Physics:Physical quantity|Physical quantities]] that remain separate in classical physics are ''naturally combined'' in SR and GR by enforcing the postulates of relativity. Most notably, the space and time coordinates combine into the [[Physics:Four-vector#Four-position|four-position]], and energy and momentum combine into the [[Physics:Four-momentum|four-momentum]]. The components of these [[Physics:Four-vector|four-vector]]s depend on the [[Astronomy:Frame of reference|frame of reference]] used, and change under [[Physics:Lorentz transformation|Lorentz transformation]]s to other inertial frames or [[Physics:Non-inertial reference frame|accelerated frame]]s.

Relativistic angular momentum is less obvious. The classical definition of angular momentum is the [[Cross product|cross product]] of position '''x''' with momentum '''p''' to obtain a [[Pseudovector|pseudovector]] {{math|'''x''' × '''p'''}}, or alternatively as the exterior product to obtain a second order [[Antisymmetric tensor|antisymmetric tensor]] {{math|'''x''' ∧ '''p'''}}. What does this combine with, if anything? There is another vector quantity not often discussed – it is the time-varying moment of mass polar-vector (''not'' the [[Physics:Moment of inertia|moment of inertia]]) related to the boost of the [[Physics:Centre of mass|centre of mass]] of the system, and this combines with the classical angular momentum pseudovector to form an [[Antisymmetric tensor|antisymmetric tensor]] of second order, in exactly the same way as the electric field polar-vector combines with the magnetic field pseudovector to form the electromagnetic field antisymmetric tensor. For rotating mass–energy distributions (such as gyroscopes, [[Astronomy:Planet|planet]]s, [[Astronomy:Star|star]]s, and [[Astronomy:Black hole|black hole]]s) instead of point-like particles, the '''angular momentum tensor''' is expressed in terms of the [[Stress–energy tensor|stress–energy tensor]] of the rotating object.

In special relativity alone, in the [[Physics:Rest frame|rest frame]] of a spinning object, there is an intrinsic angular momentum analogous to the "spin" in [[Physics:Quantum mechanics|quantum mechanics]] and [[Physics:Relativistic quantum mechanics|relativistic quantum mechanics]], although for an extended body rather than a point particle. In relativistic quantum mechanics, [[Physics:Elementary particle|elementary particle]]s have ''[[Physics:Spin|spin]]'' and this is an additional contribution to the ''orbital'' angular momentum operator, yielding the ''total'' angular momentum tensor operator. In any case, the intrinsic "spin" addition to the orbital angular momentum of an object can be expressed in terms of the [[Physics:Pauli–Lubanski pseudovector|Pauli–Lubanski pseudovector]].<ref>{{cite book|title=Geometry and quantum field theory|edition=2nd| author1=D.S.A. Freed |author2=K.K.A. Uhlenbeck |year=1995|publisher=American Mathematical Society|location=Institute For Advanced Study (Princeton, N.J.)| isbn=0-8218-8683-5|url=https://books.google.com/books?id=Hs4uGmcLkmIC&q=angular+momentum+in+special+relativity&pg=PA203}}</ref>

==Definitions==
{{further|Euclidean vector|Pseudovector}}

[[File:Angular momentum bivector and pseudovector.svg|275px|thumb|The 3-angular momentum as a [[Bivector|bivector]] (plane element) and axial vector, of a particle of mass ''m'' with instantaneous 3-position '''x''' and 3-momentum '''p'''.]]

===Orbital 3d angular momentum===

For reference and background, two closely related forms of angular momentum are given.

In [[Physics:Classical mechanics|classical mechanics]], the orbital angular momentum of a particle with instantaneous three-dimensional position vector {{math|1='''x''' = (''x'', ''y'', ''z'')}} and momentum vector {{math|1='''p''' = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'')}}, is defined as the ''axial vector''
<math display="block">\mathbf{L} = \mathbf{x} \times \mathbf{p}</math>
which has three components, that are systematically given by [[Cyclic permutation|cyclic permutation]]s of Cartesian directions (e.g. change {{mvar|x}} to {{mvar|y}}, {{mvar|y}} to {{mvar|z}}, {{mvar|z}} to {{mvar|x}}, repeat)
<math display="block">\begin{align}
  L_x &= y p_z - z p_y \,, \\
  L_y &= z p_x - x p_z \,, \\
  L_z &= x p_y - y p_x \,.
\end{align}</math>

A related definition is to conceive orbital angular momentum as a ''plane element''. This can be achieved by replacing the cross product by the exterior product in the language of [[Exterior algebra|exterior algebra]], and angular momentum becomes a [[Covariance and contravariance of vectors|contravariant]] second order [[Antisymmetric tensor|antisymmetric tensor]]<ref name="Penrose p433">{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= vintage books|page=433| year=2005 | isbn=978-0-09-944068-0}} Penrose includes a factor of 2 in the wedge product, other authors may also.</ref>
<math display="block">\mathbf{L} = \mathbf{x}\wedge\mathbf{p}</math>

or writing {{math|1='''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) = (''x'', ''y'', ''z'')}} and momentum vector {{math|1='''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>, ''p''<sub>3</sub>) = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'')}}, the components can be compactly abbreviated in tensor index notation
<math display="block">L^{ij} = x^i p^j - x^j p^i</math>
where the indices {{mvar|i}} and {{mvar|j}} take the values 1, 2, 3. On the other hand, the components can be systematically displayed fully in a 3 × 3 antisymmetric matrix
<math display="block">\begin{align}
  \mathbf{L} &= \begin{pmatrix}
      L^{11} & L^{12} & L^{13} \\
      L^{21} & L^{22} & L^{23} \\
      L^{31} & L^{32} & L^{33} \\
    \end{pmatrix} = \begin{pmatrix}
      0      & L_{xy} & L_{xz} \\
      L_{yx} & 0      & L_{yz} \\
      L_{zx} & L_{zy} & 0
    \end{pmatrix} = \begin{pmatrix}
       0      & L_{xy}  & -L_{zx} \\
      -L_{xy} & 0       &  L_{yz} \\
       L_{zx} & -L_{yz} &  0
    \end{pmatrix} \\
             &= \begin{pmatrix}
       0             &  xp_y - yp_x   & -(zp_x - xp_z) \\
      -(xp_y - yp_x) &  0             &  yp_z - zp_y \\
       zp_x - xp_z   & -(yp_z - zp_y) &  0
    \end{pmatrix}
\end{align}</math>

This quantity is additive, and for an isolated system, the total angular momentum of a system is conserved.

===Dynamic mass moment===

In classical mechanics, the three-dimensional quantity for a particle of mass ''m'' moving with velocity '''u'''<ref name="Penrose p433"/><ref>{{cite book|title=Special Relativity and How it Works|author=M. Fayngold|year=2008|publisher={{wipe|John Wiley & Sons}}| page=138| isbn=978-3-527-40607-4| url=https://books.google.com/books?id=Q3egk8Ds6ogC&q=angular+momentum+in+special+relativity&pg=PA137}}</ref>
<math display="block">\mathbf{N} = m \left( \mathbf{x} - t \mathbf{u} \right) = m \mathbf{x} - t \mathbf{p}</math>
has the [[Dimensional analysis|dimensions]] of ''mass moment'' – length multiplied by mass. It is equal to the mass of the particle or system of particles multiplied by the distance from the space origin to the [[Physics:Centre of mass|centre of mass]] (COM) at the time origin ({{nowrap|1=''t'' = 0}}), as measured in the lab frame. There is no universal symbol, nor even a universal name, for this quantity. Different authors may denote it by other symbols if any (for example '''μ'''), may designate other names, and may define '''N''' to be the negative of what is used here. The above form has the advantage that it resembles the familiar [[Physics:Galilean transformation|Galilean transformation]] for position, which in turn is the non-relativistic boost transformation between inertial frames.

This vector is also additive: for a system of particles, the vector sum is the resultant
<math display="block">\sum_n \mathbf{N}_n = \sum_n m_n \left(\mathbf{x}_n - t \mathbf{u}_n \right) = \left(\mathbf{x}_\mathrm{COM}\sum_n m_n - t \sum_n m_n \mathbf{u}_n \right)= M_\text{tot}(\mathbf{x}_{\mathrm{COM}}-\mathbf{u}_{\mathrm{COM}}t)</math>
where the system's [[Physics:Centre of mass|centre of mass]] position and velocity and total mass are respectively
<math display="block">\begin{align}
  \mathbf{x}_\mathrm{COM} &= \frac{\sum_n m_n \mathbf{x}_n}{\sum_n m_n}, \\[3pt]
  \mathbf{u}_\mathrm{COM} &= \frac{\sum_n m_n \mathbf{u}_n}{\sum_n m_n}, \\[3pt]
             M_\text{tot} &= \sum_n m_n.
\end{align}</math>

For an isolated system, '''N''' is conserved in time, which can be seen by differentiating with respect to time. The angular momentum '''L''' is a pseudovector, but '''N''' is an "ordinary" (polar) vector, and is therefore invariant under inversion.

The resultant '''N'''<sub>tot</sub> for a multiparticle system has the physical visualization that, whatever the complicated motion of all the particles are, they move in such a way that the system's COM moves in a straight line. This does not necessarily mean all particles "follow" the COM, nor that all particles all move in almost the same direction simultaneously, only that the collective motion of the particles is constrained in relation to the centre of mass.

In special relativity, if the particle moves with velocity '''u''' relative to the lab frame, then
<math display="block">\begin{align}
           E &= \gamma(\mathbf{u})m_0c^2, &
  \mathbf{p} &= \gamma(\mathbf{u})m_0\mathbf{u}
\end{align}</math>
where
<math display="block">\gamma (\mathbf{u}) = \frac{1}{\sqrt{1 - \frac{\mathbf{u}\cdot\mathbf{u}}{c^2}}}</math>
is the [[Lorentz factor]] and ''m'' is the mass (i.e. the rest mass) of the particle. The corresponding relativistic mass moment in terms of {{mvar|m}}, {{math|'''u'''}}, {{math|'''p'''}}, {{mvar|E}}, in the same lab frame is
<math display="block">\mathbf{N} = \frac{E}{c^2}\mathbf{x} - \mathbf{p}t = m\gamma(\mathbf{u})(\mathbf{x} - \mathbf{u}t).</math>

The Cartesian components are
<math display="block">\begin{align}
  N_x = mx - p_xt &= \frac{E}{c^2}x - p_xt = m\gamma(u)(x - u_x t) \\
  N_y = my - p_yt &= \frac{E}{c^2}y - p_yt = m\gamma(u)(y - u_y t) \\
  N_z = mz - p_zt &= \frac{E}{c^2}z - p_zt = m\gamma(u)(z - u_z t)
\end{align}</math>

==Special relativity==

===Coordinate transformations for a boost in the x direction===

Consider a coordinate frame {{math|F&prime;}} which moves with velocity {{math|1='''v''' = (''v'', 0, 0)}} relative to another frame F, along the direction of the coincident {{math|''xx&prime;''}} axes. The origins of the two coordinate frames coincide at times {{math|1=''t'' = ''t''&prime; = 0}}. The mass–energy {{math|1=''E'' = ''mc''<sup>2</sup>}} and momentum components {{math|1='''p''' = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'')}} of an object, as well as position coordinates {{math|1='''x''' = (''x'', ''y'', ''z'')}} and time {{mvar|t}} in frame {{math|F}} are transformed to {{math|1=''E''&prime; = ''m''&prime;''c''<sup>2</sup>}}, {{math|1='''p'''&prime; = (''p<sub>x</sub>''&prime;, ''p<sub>y</sub>''&prime;, ''p<sub>z</sub>''&prime;)}}, {{math|1='''x'''&prime; = (''x''&prime;, ''y''&prime;, ''z''&prime;)}}, and {{math|''t''&prime;}} in {{math|F&prime;}} according to the [[Physics:Lorentz transformation|Lorentz transformation]]s
<math display="block">\begin{align}
  t' &= \gamma(v) \left(t - \frac{vx}{c^2}\right) \,,\quad & E'   &= \gamma(v) \left(E - vp_x \right) \\
  x' &= \gamma(v) (x - vt) \,,\quad                        & p_x' &= \gamma(v) \left(p_x - \frac{vE}{c^2}\right) \\
  y' &= y \,,\quad                                         & p_y' &= p_y \\
  z' &= z \,,\quad                                         & p_z' &= p_z \\
\end{align}</math>

The Lorentz factor here applies to the velocity '''v''', the relative velocity between the frames. This is not necessarily the same as the velocity '''u''' of an object.

For the orbital 3-angular momentum '''L''' as a pseudovector, we have
<math display="block">\begin{align}
  L_x' &= y' p_z' - z' p_y' = L_x \\
  L_y' &= z' p_x' - x' p_z' = \gamma(v) ( L_y - v N_z ) \\
  L_z' &= x' p_y' - y' p_x' = \gamma(v) ( L_z + v N_y ) \\
\end{align}</math>

{{math proof
| title = Derivation
| proof =
For the x-component
<math display="block">L_x' = y'p_z' - z'p_y' = yp_z - zp_y = L_x</math>
the y-component
<math display="block">\begin{align}
L_y' &= z'p_x' - x'p_z'\\
     &= z\gamma\left(p_x - \frac{vE}{c^2}\right) - \gamma\left(x - vt\right)p_z \\
     &= \gamma\left[zp_x - z\frac{vE}{c^2} - xp_z + vtp_z\right] \\
     &= \gamma\left[\left(zp_x - xp_z\right) + v\left(p_zt - z\frac{E}{c^2}\right)\right] \\
     &= \gamma\left(L_y - vN_z\right)
\end{align}</math>
and z-component
<math display="block">\begin{align}
L_z' &= x' p_y' - y' p_x'\\
     &= \gamma\left(x - vt\right)p_y - y\gamma\left(p_x - \frac{vE}{c^2}\right)\\
     &= \gamma\left[xp_y - vtp_y - yp_x + y\frac{vE}{c^2}\right]\\
     &= \gamma\left[\left(xp_y - yp_x\right) + v\left(y\frac{E}{c^2} - tp_y\right)\right]\\
     &= \gamma\left(L_z + vN_y\right)
\end{align}</math>
}}

In the second terms of {{math|''L<sub>y</sub>''&prime;}} and {{math|''L<sub>z</sub>''&prime;}}, the {{mvar|y}} and {{mvar|z}} components of the [[Cross product|cross product]] {{math|'''v''' × '''N'''}} can be inferred by recognizing [[Cyclic permutation|cyclic permutation]]s of {{math|1=''v<sub>x</sub>'' = ''v''}} and {{math|1=''v<sub>y</sub>'' = ''v<sub>z</sub>'' = 0}} with the components of {{math|'''N'''}},
<math display="block">\begin{align}
   -v N_z &= v_z N_x - v_x N_z = \left(\mathbf{v} \times \mathbf{N}\right)_y \\
    v N_y &= v_x N_y - v_y N_x = \left(\mathbf{v} \times \mathbf{N}\right)_z \\
\end{align}</math>

Now, {{math|''L<sub>x</sub>''}} is parallel to the relative velocity {{math|'''v'''}}, and the other components {{math|''L<sub>y</sub>''}} and {{math|''L<sub>z</sub>''}} are perpendicular to {{math|'''v'''}}. The parallel–perpendicular correspondence can be facilitated by splitting the entire 3-angular momentum pseudovector into components parallel (∥) and perpendicular (⊥) to '''v''', in each frame,
<math display="block">\mathbf{L} = \mathbf{L}_\parallel + \mathbf{L}_\perp \,,\quad \mathbf{L}' = \mathbf{L}_\parallel' + \mathbf{L}_\perp'\,.</math>

Then the component equations can be collected into the pseudovector equations
<math display="block">\begin{align}
  \mathbf{L}_\parallel' &= \mathbf{L}_\parallel \\
      \mathbf{L}_\perp' &= \gamma(\mathbf{v})\left(\mathbf{L}_\perp + \mathbf{v} \times \mathbf{N} \right) \\
\end{align}</math>

Therefore, the components of angular momentum along the direction of motion do not change, while the components perpendicular do change. By contrast to the transformations of space and time, time and the spatial coordinates change along the direction of motion, while those perpendicular do not.

These transformations are true for ''all'' {{math|'''v'''}}, not just for motion along the {{math|''xx&prime;''}} axes.

Considering {{math|'''L'''}} as a tensor, we get a similar result
<math display="block">\mathbf{L}_\perp' = \gamma(\mathbf{v})\left(\mathbf{L}_\perp + \mathbf{v} \wedge \mathbf{N} \right)</math>
where
<math display="block">\begin{align}
   v_z N_x - v_x N_z &= \left(\mathbf{v}\wedge\mathbf{N}\right)_{zx} \\
   v_x N_y - v_y N_x &= \left(\mathbf{v}\wedge\mathbf{N}\right)_{xy} \\
\end{align}</math>

The boost of the dynamic mass moment along the {{mvar|x}} direction is
<math display="block">\begin{align}
  N_x' &= m' x' - p_x' t' = N_x \\
  N_y' &= m' y' - p_y' t' = \gamma(v)\left(N_y + \frac{v L_z}{c^2}\right) \\
  N_z' &= m' z' - p_z' t' = \gamma(v)\left(N_z - \frac{v L_y}{c^2}\right) \\
\end{align}</math>

{{math proof
| title = Derivation
| proof =
For the x-component
<math display="block">\begin{align}
  N_x' &= \frac{E'}{c^2}x' - t'p_x'\\
       &= \frac{\gamma}{c^2}(E - vp_x)\gamma(x - vt) - \gamma\left(t - \frac{xv}{c^2}\right)\gamma\left(p_x - \frac{vE}{c^2}\right) \\
       &= \gamma^2\left[\frac{1}{c^2}\left(E - vp_x\right)(x - vt) - \left(t - \frac{xv}{c^2}\right)\left(p_x - \frac{vE}{c^2}\right)\right] \\
       &= \gamma^2\left[\frac{Ex}{c^2} - \frac{Evt}{c^2} - \frac{vp_x x}{c^2} + \frac{vp_xvt}{c^2} - tp_x + \frac{xv}{c^2}p_x + t\frac{vE}{c^2} - \frac{xv}{c^2}\frac{vE}{c^2}\right] \\
       &= \gamma^2\left[\frac{Ex}{c^2}\cancel{-\frac{Evt}{c^2}}\cancel{-\frac{vp_xx}{c^2}} + \frac{v^2}{c^2}p_xt - tp_x\cancel{+\frac{xv}{c^2}p_x}\cancel{+t\frac{vE}{c^2}} - \frac{v^2}{c^2}\frac{Ex}{c^2}\right] \\
       &= \gamma^2\left[\left(\frac{Ex}{c^2} - tp_x\right) + \frac{v^2}{c^2}\left(p_xt - \frac{Ex}{c^2}\right)\right] \\
       &= \gamma^2\left[1 - \frac{v^2}{c^2}\right]N_x \\
       &= \gamma^2\frac{1}{\gamma^2}N_x
\end{align}</math>
the y-component
<math display="block">\begin{align}
  N_y' &= \frac{E'}{c^2}y' - t'p_y' \\
       &= \frac{1}{c^2}\gamma(E - vp_x)y - \gamma\left(t - \frac{xv}{c^2}\right)p_y \\
       &= \gamma\left[\frac{1}{c^2}(E - vp_x)y - \left(t - \frac{xv}{c^2}\right)p_y\right] \\
       &= \gamma\left[\frac{1}{c^2}Ey - \frac{1}{c^2}vp_x y - tp_y + \frac{xv}{c^2}p_y\right] \\
       &= \gamma\left[\left(\frac{1}{c^2}Ey - tp_y\right) + \frac{v}{c^2}(xp_y - yp_x)\right] \\
       &= \gamma\left(N_y + \frac{v}{c^2}L_z\right)
\end{align}</math>
and z-component
<math display="block">\begin{align}
  N_z' &= \frac{E'}{c^2}z' - t'p_z' \\
       &= \frac{1}{c^2}\gamma(E-vp_x)z - \gamma\left(t - \frac{xv}{c^2}\right)p_z\\
       &= \gamma\left[\frac{1}{c^2}(E - vp_x)z - \left(t - \frac{xv}{c^2}\right)p_z\right]\\
       &= \gamma\left[\frac{1}{c^2}Ez - \frac{1}{c^2}vp_z z - tp_z + \frac{xv}{c^2}p_z\right]\\
       &= \gamma\left[\left(\frac{1}{c^2}Ez - tp_z\right) + \frac{v}{c^2}(xp_z - zp_x)\right]\\
       &= \gamma\left(N_z - \frac{v}{c^2}L_y\right)
\end{align}</math>
}}

Collecting parallel and perpendicular components as before
<math display="block">\begin{align}
  \mathbf{N}_\parallel' &= \mathbf{N}_\parallel \\
      \mathbf{N}_\perp' &= \gamma(\mathbf{v})\left(\mathbf{N}_\perp - \frac{1}{c^2}\mathbf{v}\times\mathbf{L}\right)\\
\end{align}</math>

Again, the components parallel to the direction of relative motion do not change, those perpendicular do change.

===Vector transformations for a boost in any direction===

So far these are only the parallel and perpendicular decompositions of the vectors. The transformations on the full vectors can be constructed from them as follows (throughout here {{math|'''L'''}} is a pseudovector for concreteness and compatibility with vector algebra).

Introduce a [[Unit vector|unit vector]] in the direction of {{math|'''v'''}}, given by {{math|1='''n''' = '''v'''/''v''}}. The parallel components are given by the [[Vector projection|vector projection]] of {{math|'''L'''}} or {{math|'''N'''}} into {{math|'''n'''}}
<math display="block">\mathbf{L}_\parallel = (\mathbf{L}\cdot\mathbf{n})\mathbf{n} \,, \quad \mathbf{N}_\parallel = (\mathbf{N}\cdot\mathbf{n})\mathbf{n}</math>
while the perpendicular component by vector rejection of '''L''' or '''N''' from '''n'''
<math display="block">\mathbf{L}_\perp = \mathbf{L} - (\mathbf{L}\cdot\mathbf{n})\mathbf{n} \,, \quad \mathbf{N}_\perp = \mathbf{N} - (\mathbf{N}\cdot\mathbf{n})\mathbf{n}</math>
and the transformations are
<math display="block">\begin{align}
  \mathbf{L}' &= \gamma(\mathbf{v})(\mathbf{L} + v\mathbf{n}\times\mathbf{N}) - (\gamma(\mathbf{v}) - 1)(\mathbf{L}\cdot\mathbf{n})\mathbf{n} \\
  \mathbf{N}' &= \gamma(\mathbf{v})\left(\mathbf{N} - \frac{v}{c^2}\mathbf{n}\times\mathbf{L}\right) - (\gamma(\mathbf{v}) - 1)(\mathbf{N}\cdot\mathbf{n})\mathbf{n} \\
\end{align}</math>
or reinstating {{math|1='''v''' = ''v'''''n'''}},
<math display="block">\begin{align}
  \mathbf{L}' &= \gamma(\mathbf{v})(\mathbf{L} + \mathbf{v}\times\mathbf{N}) - (\gamma(\mathbf{v}) - 1)\frac{(\mathbf{L}\cdot\mathbf{v})\mathbf{v}}{v^2} \\
  \mathbf{N}' &= \gamma(\mathbf{v})\left(\mathbf{N} - \frac{1}{c^2}\mathbf{v}\times\mathbf{L}\right) - (\gamma(\mathbf{v}) - 1)\frac{(\mathbf{N}\cdot\mathbf{v})\mathbf{v}}{v^2} \\
\end{align}</math>

These are very similar to the Lorentz transformations of the [[Physics:Electric field|electric field]] {{math|'''E'''}} and [[Physics:Magnetic field|magnetic field]] {{math|'''B'''}}, see [[Physics:Classical electromagnetism and special relativity|Classical electromagnetism and special relativity]].

Alternatively, starting from the vector Lorentz transformations of time, space, energy, and momentum, for a boost with velocity {{math|'''v'''}},
<math display="block">\begin{align}
           t' &= \gamma(\mathbf{v})\left(t - \frac{\mathbf{v}\cdot\mathbf{r}}{c^2}\right)\,,\\
  \mathbf{r}' &= \mathbf{r} + \frac{\gamma(\mathbf{v}) - 1}{v^2}(\mathbf{r}\cdot\mathbf{v})\mathbf{v} - \gamma(\mathbf{v}) t\mathbf{v}\,,\\
  \mathbf{p}' &= \mathbf{p} + \frac{\gamma(\mathbf{v}) - 1}{v^2}(\mathbf{p}\cdot\mathbf{v})\mathbf{v} - \gamma(\mathbf{v})\frac{E}{c^2}\mathbf{v}\,,\\
           E' &= \gamma(\mathbf{v})\left(E - \mathbf{v}\cdot\mathbf{p}\right)\,,\\
\end{align}</math>
inserting these into the definitions
<math display="block">\begin{align}
  \mathbf{L}' &= \mathbf{r}'\times\mathbf{p}' \,, &
  \mathbf{N}' &= \frac{E'}{c^2}\mathbf{r}' - t'\mathbf{p}'
\end{align}</math>
gives the transformations.

{{math proof
| title = Derivation of vector transformations directly
| proof =
The orbital angular momentum in each frame are
<math display="block">\mathbf{L}'=\mathbf{r}'\times\mathbf{p}' \,,\quad \mathbf{L}=\mathbf{r}\times\mathbf{p}</math>
so taking the cross product of the transformations
<math display="block">\begin{align}
  \mathbf{L}' &= \left[\mathbf{r} + (\gamma - 1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma tv\mathbf{n}\right] \times \left[\mathbf{p} + (\gamma - 1)(\mathbf{p}\cdot\mathbf{n})\mathbf{n}-\gamma\frac{E}{c^2}v\mathbf{n}\right]\\
              &= [\mathbf{r} + (\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma tv\mathbf{n}]\times\mathbf{p} + (\gamma - 1)(\mathbf{p}\cdot\mathbf{n})[\mathbf{r} + (\gamma - 1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma tv\mathbf{n}]\times\mathbf{n} - \gamma\frac{E}{c^2}v[\mathbf{r} + (\gamma - 1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma tv\mathbf{n}]\times\mathbf{n}\\
              &= \mathbf{r}\times\mathbf{p} + (\gamma - 1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}\times\mathbf{p} - \gamma tv\mathbf{n}\times\mathbf{p} + (\gamma - 1)(\mathbf{p}\cdot\mathbf{n})\mathbf{r}\times\mathbf{n} - \gamma\frac{E}{c^2}v\mathbf{r}\times\mathbf{n}\\
              &= \mathbf{L} + \left[\frac{\gamma - 1}{v^2}(\mathbf{r}\cdot\mathbf{v}) - \gamma t\right]\mathbf{v}\times\mathbf{p} + \left[\frac{\gamma - 1}{v^2}(\mathbf{p}\cdot\mathbf{v}) - \gamma\frac{E}{c^2}\right]\mathbf{r}\times\mathbf{v}\\
              &= \mathbf{L} + \mathbf{v}\times\left[\left(\frac{\gamma - 1}{v^2}(\mathbf{r}\cdot\mathbf{v}) - \gamma t\right)\mathbf{p} - \left(\frac{\gamma - 1}{v^2}(\mathbf{p}\cdot\mathbf{v}) - \gamma\frac{E}{c^2}\right)\mathbf{r}\right]\\
              &= \mathbf{L} + \mathbf{v}\times\left[\frac{\gamma - 1}{v^2}\left((\mathbf{r}\cdot\mathbf{v})\mathbf{p} - (\mathbf{p}\cdot\mathbf{v})\mathbf{r}\right) + \gamma\left(\frac{E}{c^2}\mathbf{r} - t\mathbf{p}\right)\right]
\end{align}</math>

Using the [[Triple product|triple product]] rule
<math display="block">\begin{align}
  \mathbf{a}\times(\mathbf{b}\times\mathbf{c}) &= \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b}) \\
  (\mathbf{a}\times\mathbf{b})\times\mathbf{c} &= (\mathbf{c}\cdot\mathbf{a})\mathbf{b} - (\mathbf{c}\cdot\mathbf{b})\mathbf{a} \\
\end{align}</math>
gives
<math display="block">\begin{align}
(\mathbf{r}\times\mathbf{p})\times\mathbf{v} &= (\mathbf{v}\cdot\mathbf{r})\mathbf{p} - (\mathbf{v}\cdot\mathbf{p})\mathbf{r}\\
(\mathbf{v}\cdot\mathbf{r})\mathbf{p} - (\mathbf{v}\cdot\mathbf{p})\mathbf{r} &= \mathbf{L}\times\mathbf{v}\\
\end{align}</math>
and along with the definition of {{math|'''N'''}} we have
<math display="block">\mathbf{L}' = \mathbf{L} + \mathbf{v} \times \left[\frac{\gamma - 1}{v^2}\mathbf{L}\times\mathbf{v} + \gamma\mathbf{N}\right]</math>

Reinstating the unit vector {{math|'''n'''}},
<math display="block">\mathbf{L}' = \mathbf{L} + \mathbf{n}\times\left[(\gamma - 1)\mathbf{L}\times\mathbf{n} + v\gamma\mathbf{N}\right]</math>

Since in the transformation there is a cross product on the left with {{math|'''n'''}},
<math display="block">\mathbf{n}\times(\mathbf{L}\times\mathbf{n}) = \mathbf{L}(\mathbf{n}\cdot\mathbf{n})-\mathbf{n}(\mathbf{n}\cdot\mathbf{L}) = \mathbf{L} - \mathbf{n}(\mathbf{n}\cdot\mathbf{L})</math>
then
<math display="block">\mathbf{L}' = \mathbf{L} + (\gamma-1)(\mathbf{L} - \mathbf{n}(\mathbf{n}\cdot\mathbf{L})) + \beta\gamma c\mathbf{n}\times\mathbf{N} = \gamma(\mathbf{L} + v\mathbf{n}\times\mathbf{N}) - (\gamma - 1)\mathbf{n}(\mathbf{n}\cdot\mathbf{L})</math>
}}

===4d angular momentum as a bivector===

In relativistic mechanics, the COM boost and orbital 3-space angular momentum of a rotating object are combined into a four-dimensional [[Bivector|bivector]] in terms of the [[Physics:Four-vector|four-position]] '''X''' and the [[Physics:Four-momentum|four-momentum]] '''P''' of the object<ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= vintage books|pages=437–438, 566–569| year=2005 | isbn=978-0-09-944068-0}} '''Note:''' Some authors, including Penrose, use ''Latin'' letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.</ref><ref>{{cite book|title=Special Relativity and How it Works|author=M. Fayngold|year=2008|publisher=John Wiley & Sons| pages=137–139|isbn=978-3-527-40607-4|url=https://books.google.com/books?id=Q3egk8Ds6ogC&q=angular+momentum+in+special+relativity&pg=PA137}}</ref>
<math display="block">\mathbf{M} = \mathbf{X}\wedge\mathbf{P}</math>

In components
<math display="block">M^{\alpha\beta} = X^\alpha P^\beta - X^\beta P^\alpha</math>
which are six independent quantities altogether. Since the components of {{math|'''X'''}} and {{math|'''P'''}} are frame-dependent, so is {{math|'''M'''}}. Three components
<math display="block">M^{ij} = x^i p^j - x^j p^i = L^{ij}</math>
are those of the familiar classical 3-space orbital angular momentum, and the other three
<math display="block">M^{0i} = x^0 p^i - x^i p^0 = c\,\left(t p^i - x^i \frac{E}{c^2} \right) = - c N^i</math>
are the relativistic mass moment, multiplied by {{math|−''c''}}. The tensor is antisymmetric;
<math display="block">M^{\alpha\beta} = -M^{\beta\alpha}</math>

The components of the tensor can be systematically displayed as a [[Matrix (mathematics)|matrix]]
<math display="block">\begin{align}
  \mathbf{M}
  &= \begin{pmatrix}
    M^{00} & M^{01} & M^{02} & M^{03} \\
    M^{10} & M^{11} & M^{12} & M^{13} \\
    M^{20} & M^{21} & M^{22} & M^{23} \\
    M^{30} & M^{31} & M^{32} & M^{33}
  \end{pmatrix} \\[3pt]
  &= \left(\begin{array}{c|ccc}
    0     & -N^1 c  & -N^2 c  & -N^3 c \\ \hline
    N^1 c &  0      &  L^{12} & -L^{31} \\
    N^2 c & -L^{12} &  0      &  L^{23} \\
    N^3 c &  L^{31} & -L^{23} &  0
  \end{array}\right) \\[3pt]
  &= \left(\begin{array}{c|c}
    0                       & -\mathbf{N} c \\ \hline
    \mathbf{N}^\mathrm{T} c &  \mathbf{x}\wedge\mathbf{p} \\
  \end{array}\right)
\end{align}</math>
in which the last array is a [[Block matrix|block matrix]] formed by treating '''N''' as a row vector which matrix transposes to the column vector '''N'''<sup>T</sup>, and {{math|'''x''' ∧ '''p'''}} as a 3 × 3 antisymmetric matrix. The lines are merely inserted to show where the blocks are.

Again, this tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each constituent of the system:
<math display="block">\mathbf{M}_\text{tot} = \sum_n \mathbf{M}_n = \sum_n \mathbf{X}_n \wedge \mathbf{P}_n \,.</math>

Each of the six components forms a conserved quantity when aggregated with the corresponding components for other objects and fields.

The angular momentum tensor '''M''' is indeed a tensor, the components change according to a [[Physics:Lorentz transformation|Lorentz transformation]] matrix Λ, as illustrated in the usual way by tensor index notation
<math display="block">\begin{align}
  {M'}^{\alpha\beta} &= {X'}^\alpha {P'}^\beta - {X'}^\beta {P'}^\alpha \\
                     &= {\Lambda^\alpha}_\gamma X^\gamma {\Lambda^\beta}_\delta P^\delta - {\Lambda^\beta}_\delta X^\delta {\Lambda^\alpha}_\gamma P^\gamma \\
                     &= {\Lambda^\alpha}_\gamma {\Lambda^\beta}_\delta \left( X^\gamma P^\delta - X^\delta P^\gamma \right) \\
                     &= {\Lambda^\alpha}_\gamma {\Lambda^\beta}_\delta M^{\gamma \delta}  \\
\end{align} ,</math>
where, for a boost (without rotations) with normalized velocity {{math|1='''β''' = '''v'''/''c''}}, the Lorentz transformation matrix elements are
<math display="block">\begin{align}
  {\Lambda^0}_0 &= \gamma \\
  {\Lambda^i}_0 &= {\Lambda^0}_i = -\gamma \beta^i \\
  {\Lambda^i}_j &= {\delta^i}_j + \frac{\gamma - 1}{\beta^2} \beta^i\beta_j
\end{align}</math>
and the covariant ''β<sub>i</sub>'' and contravariant ''β<sup>i</sup>'' components of '''β''' are the same since these are just parameters.

In other words, one can Lorentz-transform the four position and four momentum separately, and then antisymmetrize those newly found components to obtain the angular momentum tensor in the new frame.

{{math proof
| title = Vector transformations derived from the tensor transformations
| proof =
The transformation of boost components are

<math display="block">\begin{align}
  M'^{k0} &= {\Lambda^k}_\mu {\Lambda^0}_\nu M^{\mu\nu}\\
          &= {\Lambda^k}_0{\Lambda^0}_0 M^{00} + {\Lambda^k}_i {\Lambda^0}_0 M^{i0} + {\Lambda^k}_0 {\Lambda^0}_j M^{0j} + {\Lambda^k}_i {\Lambda^0}_j M^{ij} \\
          &= \left({\Lambda^k}_i {\Lambda^0}_0 - {\Lambda^k}_0 {\Lambda^0}_i\right) M^{i0} + {\Lambda^k}_i {\Lambda^0}_j M^{ij} \\
\end{align}</math>
as for the orbital angular momentum
<math display="block">\begin{align}
  {M'}^{k\ell} &= {\Lambda^k}_\mu {\Lambda^\ell}_\nu M^{\mu\nu} \\
               &= {\Lambda^k}_0 {\Lambda^\ell}_0 M^{00} + {\Lambda^k}_i {\Lambda^\ell}_0 M^{i0} + {\Lambda^k}_0 {\Lambda^\ell}_j M^{0j} + {\Lambda^k}_i {\Lambda^\ell}_j M^{ij} \\
               &= \left({\Lambda^k}_i {\Lambda^\ell}_0 - {\Lambda^k}_0 {\Lambda^\ell}_i\right) M^{i0} + {\Lambda^k}_i {\Lambda^\ell}_j M^{ij}
\end{align}</math>

The expressions in the Lorentz transformation entries are
<math display="block">\begin{align}
  {\Lambda^k}_i {\Lambda^\ell}_0 - {\Lambda^k}_0 {\Lambda^\ell}_i
    &= \left[{\delta^k}_i + \frac{\gamma - 1}{\beta^2}\beta^k\beta_i\right]\left(-\gamma\beta^\ell\right) -
         \left(-\gamma\beta^k\right)\left[{\delta^\ell}_i + \frac{\gamma - 1}{\beta^2} \beta^\ell \beta_i\right] \\
    &= \gamma\left[\beta^k {\delta^\ell}_i - \beta^\ell {\delta^k}_i\right] \\

  {\Lambda^k}_i {\Lambda^0}_0 - {\Lambda^k}_0 {\Lambda^0}_i
    &= \left[{\delta^k}_i + \frac{\gamma - 1}{\beta^2} \beta^k \beta_i\right]\gamma - (-\gamma\beta^k)(-\gamma\beta^i) \\
    &= \gamma\left[{\delta^k}_i + \frac{\gamma - 1}{\beta^2}\beta^k \beta_i - \gamma\beta^k \beta^i\right] \\
    &= \gamma\left[{\delta^k}_i + \left(\frac{\gamma - 1}{\beta^2} - \gamma\right)\beta^k \beta_i\right] \\
    &= \gamma\left[{\delta^k}_i + \left(\frac{\gamma - \gamma\beta^2 - 1}{\beta^2}\right)\beta^k \beta_i\right] \\
    &= \gamma\left[{\delta^k}_i + \left(\frac{\gamma^{-1} - 1}{\beta^2}\right)\beta^k \beta_i\right] \\
    &= \gamma{\delta^k}_i - \left[\frac{\gamma - 1}{\beta^2}\right]\beta^k \beta_i \\

  {\Lambda^k}_i {\Lambda^\ell}_j
    &= \left[{\delta^k}_i + \frac{\gamma - 1}{\beta^2}\beta^k \beta_i\right]\left[{\delta^\ell}_j + \frac{\gamma - 1}{\beta^2}\beta^\ell \beta_j\right] \\
    &= {\delta^k}_i {\delta^\ell}_j + \frac{\gamma - 1}{\beta^2}{\delta^k}_i \beta^\ell \beta_j + \frac{\gamma - 1}{\beta^2}\beta^k \beta_i {\delta^\ell}_j +
         \frac{\gamma - 1}{\beta^2}\frac{\gamma - 1}{\beta^2}\beta^\ell \beta_j \beta^k \beta_i
\end{align}</math>
gives
<math display="block">\begin{align}
  cN'^k &= \left({\Lambda^k}_i {\Lambda^0}_0 - {\Lambda^k}_0 {\Lambda^0}_i\right)cN^i + {\Lambda^k}_i {\Lambda^0}_j \varepsilon^{ijn} L_n \\
        &= \left[\gamma{\delta^k}_i - \left(\frac{\gamma - 1}{\beta^2}\right)\beta^k \beta_i\right] cN^i +
             -\gamma\beta^j \left[{\delta^k}_i + \frac{\gamma - 1}{\beta^2}\beta^k \beta_i\right] \varepsilon^{ijn} L_n \\
        &= \gamma cN^k -\left(\frac{\gamma - 1}{\beta^2}\right)\beta^k \left(\beta_i cN^i\right) -
              \gamma\beta^j {\delta^k}_i \varepsilon^{ijn} L_n - \gamma\frac{\gamma - 1}{\beta^2}\beta^j \beta^k \beta_i \varepsilon^{ijn} L_n \\
        &= \gamma cN^k -\left(\frac{\gamma - 1}{\beta^2}\right)\beta^k \left(\beta_i cN^i\right) - \gamma\beta^j \varepsilon^{kjn} L_n \\
\end{align}</math>
or in vector form, dividing by {{math|''c''}}
<math display="block">\mathbf{N}' = \gamma \mathbf{N} -\left(\frac{\gamma - 1}{\beta^2}\right)\boldsymbol{\beta} \left(\boldsymbol{\beta}\cdot \mathbf{N}\right) - \frac{1}{c}\gamma \boldsymbol{\beta}\times\mathbf{L}</math>
or reinstating {{math|1='''β''' = '''v'''/''c''}},
<math display="block">\mathbf{N}' = \gamma \mathbf{N} - \left(\frac{\gamma - 1}{v^2}\right)\mathbf{v} \left(\mathbf{v} \cdot \mathbf{N}\right) - \gamma\mathbf{v}\times\mathbf{L}</math>
and
<math display="block">\begin{align}
  L'^{k\ell} &= \left({\Lambda^k}_i {\Lambda^\ell}_0 - {\Lambda^k}_0 {\Lambda^\ell}_i\right)cN^i + {\Lambda^k}_i {\Lambda^\ell}_j L^{ij} \\
             &= \gamma c \left(\beta^k {\delta^\ell}_i - \beta^\ell {\delta^k}_i\right) N^i +
                  \left[{\delta^k}_i {\delta^\ell}_j + \frac{\gamma - 1}{\beta^2}{\delta^k}_i \beta^\ell \beta_j +
                  \frac{\gamma - 1}{\beta^2}\beta^k \beta_i {\delta^\ell}_j +
                  \frac{\gamma - 1}{\beta^2}\frac{\gamma - 1}{\beta^2}\beta^\ell \beta_j \beta^k \beta_i\right] L^{ij} \\
             &= \gamma c \left(\beta^k N^\ell -\beta^\ell N^k\right) + L^{k\ell} + \frac{\gamma - 1}{\beta^2} \beta^\ell \beta_j L^{kj} + \frac{\gamma - 1}{\beta^2}\beta^k \beta_i L^{i\ell} \\
\end{align}</math>
or converting to pseudovector form
<math display="block">\begin{align}
  \varepsilon^{k\ell n}L'_n &= \gamma c \left(\beta^k N^\ell - \beta^\ell N^k\right) + \varepsilon^{k\ell n}L_n + \frac{\gamma - 1}{\beta^2} \left(\beta^\ell \beta_j \varepsilon^{kjn} L_n - \beta^k \beta_i \varepsilon^{\ell i n} L_n \right) \\
\end{align}</math>
in vector notation
<math display="block">\mathbf{L}' = \gamma c \boldsymbol{\beta}\times\mathbf{N} + \mathbf{L} + \frac{\gamma - 1}{\beta^2} \boldsymbol{\beta}\times(\boldsymbol{\beta}\times\mathbf{L})</math>
or reinstating {{math|1='''β''' = '''v'''/''c''}},
<math display="block">\mathbf{L}' = \gamma \mathbf{v}\times\mathbf{N} + \mathbf{L} + \frac{\gamma - 1}{v^2} \mathbf{v}\times\left(\mathbf{v}\times\mathbf{L}\right)</math>
}}

===Rigid body rotation===


For a particle moving in a curve, the [[Cross product|cross product]] of its [[Physics:Angular velocity|angular velocity]] {{math|'''''ω'''''}} (a pseudovector) and position {{math|'''x'''}} give its tangential velocity
<math display="block">\mathbf{u} = \boldsymbol{\omega} \times \mathbf{x}</math>

which cannot exceed a magnitude of {{math|''c''}}, since in SR the translational velocity of any massive object cannot exceed the [[Physics:Speed of light|speed of light]] ''c''. Mathematically this constraint is {{math|0 ≤ {{abs|'''u'''}} < ''c''}}, the vertical bars denote the [[Norm (mathematics)|magnitude]] of the vector. If the angle between {{math|'''''ω'''''}} and {{math|'''x'''}} is {{mvar|θ}} (assumed to be nonzero, otherwise '''u''' would be zero corresponding to no motion at all), then {{math|1={{abs|'''u'''}} = {{abs|'''''ω'''''}} {{abs|'''x'''}} sin ''θ''}} and the angular velocity is restricted by
<math display="block">0 \leq |\boldsymbol{\omega}| < \frac{c}{|\mathbf{x}|\sin\theta}</math>

The maximum angular velocity of any massive object therefore depends on the size of the object. For a given |'''x'''|, the minimum upper limit occurs when {{math|'''''ω'''''}} and {{math|'''x'''}} are perpendicular, so that {{math|1=''θ'' = ''π''/2}} and {{math|1=sin ''θ'' = 1}}.

For a rotating [[Rigid body|rigid body]] rotating with an angular velocity {{math|'''''ω'''''}}, the {{math|'''u'''}} is tangential velocity at a point {{math|'''x'''}} inside the object. For every point in the object, there is a maximum angular velocity.

The angular velocity (pseudovector) is related to the angular momentum (pseudovector) through the [[Physics:Moment of inertia|moment of inertia]] tensor {{math|'''I'''}}
<math display="block">\mathbf{L} = \mathbf{I}\cdot\boldsymbol{\omega} \quad \rightleftharpoons \quad L_i = I_{ij} \omega_j</math>
(the dot {{math|·}} denotes [[Tensor contraction|tensor contraction]] on one index). The relativistic angular momentum is also limited by the size of the object.

==Spin in special relativity==

=== Four-spin ===

A particle may have a "built-in" angular momentum independent of its motion, called [[Physics:Spin|spin]] and denoted '''s'''. It is a 3d pseudovector like orbital angular momentum '''L'''.

The spin has a corresponding [[Physics:Spin magnetic moment|spin magnetic moment]], so if the particle is subject to interactions (like [[Physics:Electromagnetic field|electromagnetic field]]s or [[Physics:Spin-orbit coupling|spin-orbit coupling]]), the direction of the particle's spin vector will change, but its magnitude will be constant.

The extension to special relativity is straightforward.<ref>{{cite book|first=J. D.|last=Jackson| title=Classical Electrodynamics|pages=[https://archive.org/details/classicalelectro00jack_0/page/556 556–557]|edition=2nd|year=1975| orig-year=1962| isbn=0-471-43132-X|publisher={{wipe|John Wiley & Sons}}|chapter=Chapter 11|chapter-url-access=registration| chapter-url=https://archive.org/details/classicalelectro00jack_0/page/556}} Jackson's notation: '''S''' (spin in F, lab frame), '''s''' (spin in F&prime;, rest frame of particle), S<sup>0</sup> (timelike component in lab frame), S&prime;<sup>0</sup> = 0 (timelike component in rest frame of particle), no symbol for 4-spin as a 4-vector</ref> For some lab frame F, let F&prime; be the rest frame of the particle and suppose the particle moves with constant 3-velocity '''u'''. Then F&prime; is boosted with the same velocity and the Lorentz transformations apply as usual; it is more convenient to use {{math|1='''β''' = '''u'''/''c''}}. As a [[Physics:Four-vector|four-vector]] in special relativity, the four-spin '''S''' generally takes the usual form of a four-vector with a timelike component ''s<sub>t</sub>'' and spatial components '''s''', in the lab frame
<math display="block">\mathbf{S} \equiv \left(S^0, S^1, S^2, S^3\right) = (s_t, s_x, s_y, s_z)</math>
although in the rest frame of the particle, it is defined so the timelike component is zero and the spatial components are those of particle's actual spin vector, in the notation here '''s'''&prime;, so in the particle's frame
<math display="block">\mathbf{S}' \equiv \left({S'}^0, {S'}^1, {S'}^2, {S'}^3\right) = \left(0, s_x', s_y', s_z'\right)</math>

Equating norms leads to the invariant relation
<math display="block">s_t^2 - \mathbf{s}\cdot\mathbf{s} = -\mathbf{s}'\cdot\mathbf{s}'</math>
so if the magnitude of spin is given in the rest frame of the particle and lab frame of an observer, the magnitude of the timelike component ''s<sub>t</sub>'' is given in the lab frame also.

{{math proof
| title = Vector transformations derived from the tensor transformations
| proof =
The boosted components of the four spin relative to the lab frame are
<math display="block">\begin{align}
  {S'}^0 &= {\Lambda^0}_\alpha S^\alpha = {\Lambda^0}_0 S^0 + {\Lambda^0}_i S^i = \gamma \left(S^0 - \beta_i S^i\right) \\
         &= \gamma \left(\frac{c}{c} S^0 - \frac{u_i}{c} S^i\right) = \frac{1}{c} U_0 S^0 - \frac{1}{c} U_i S^i \\[3pt]
  {S'}^i &= {\Lambda^i}_\alpha S^\alpha = {\Lambda^i}_0 S^0 + {\Lambda^i}_j S^j \\
         &= -\gamma\beta^i S^0 + \left[\delta_{ij} + \frac{\gamma - 1}{\beta^2} \beta_i\beta_j \right] S^j \\
         &= S^i + \frac{\gamma^2}{\gamma + 1}\beta_i \beta_j S^j - \gamma\beta^i S^0
\end{align}</math>

Here {{math|1=''γ'' = ''γ''('''u''')}}. ''S''&prime; is in the rest frame of the particle, so its timelike component is zero, {{math|1=''S''&prime;<sup>0</sup> = 0}}, not {{math|''S''<sup>0</sup>}}. Also, the first is equivalent to the inner product of the four-velocity (divided by {{math|''c''}}) and the four-spin. Combining these facts leads to
<math display="block">{S'}^0 = \frac{1}{c} U_\alpha S^\alpha = 0</math>
which is an invariant. Then this combined with the transformation on the timelike component leads to the perceived component in the lab frame;
<math display="block">S^0 = \beta_i S^i</math>

The inverse relations are
<math display="block">\begin{align}
  S^0 &= \gamma \left({S'}^0 + \beta_i {S'}^i\right) \\
  S^i &= {S'}^i + \frac{\gamma^2}{\gamma+1}\beta_i\beta_j {S'}^j + \gamma\beta^i {S'}^0
\end{align}</math>
}}

The covariant constraint on the spin is orthogonality to the velocity vector,
<math display="block">U_\alpha S^\alpha = 0</math>

In 3-vector notation for explicitness, the transformations are
<math display="block">\begin{align}
          s_t &= \boldsymbol{\beta}\cdot\mathbf{s} \\
  \mathbf{s}' &= \mathbf{s} + \frac{\gamma^2}{\gamma + 1}\boldsymbol{\beta} \left(\boldsymbol{\beta}\cdot\mathbf{s}\right) - \gamma\boldsymbol{\beta} s_t
\end{align}</math>

The inverse relations
<math display="block">\begin{align}
         s_t &= \gamma \boldsymbol{\beta}\cdot\mathbf{s}' \\
  \mathbf{s} &= \mathbf{s}' + \frac{\gamma^2}{\gamma + 1} \boldsymbol{\beta}\left(\boldsymbol{\beta}\cdot\mathbf{s}'\right)
\end{align}</math>
are the components of spin the lab frame, calculated from those in the particle's rest frame. Although the spin of the particle is constant for a given particle, it appears to be different in the lab frame.

===The Pauli–Lubanski pseudovector===

The [[Physics:Pauli–Lubanski pseudovector|Pauli–Lubanski pseudovector]]
<math display="block">S_\mu = \frac{1}{2}\varepsilon_{\mu\nu\rho\sigma} J^{\nu\rho} P^\sigma ,</math>
applies to both massive and [[Physics:Massless particle|massless particle]]s.

==Spin–orbital decomposition==
In general, the total angular momentum tensor splits into an orbital component and a [[Physics:Representation theory of the Lorentz group|spin component]],
<math display="block">J^{\mu\nu} = M^{\mu\nu} + S^{\mu\nu} ~.</math>
This applies to a particle, a mass–energy–momentum distribution, or field.

==Angular momentum of a mass–energy–momentum distribution==

{{main|Spin tensor|Physics:Pauli–Lubanski pseudovector}}

===Angular momentum from the mass–energy–momentum tensor===

The following is a summary from MTW.<ref>{{cite book |pages=156–159, §5.11|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne | title=Gravitation| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref> Throughout for simplicity, Cartesian coordinates are assumed.
In special and general relativity, a distribution of mass–energy–momentum, e.g. a fluid, or a star, is described by the [[Stress–energy tensor|stress–energy tensor]] ''T<sup>βγ</sup>'' (a second order [[Tensor field|tensor field]] depending on space and time). Since ''T''<sup>00</sup> is the energy density, ''T''<sup>''j''0</sup> for ''j'' = 1, 2, 3 is the ''j''th component of the object's 3d momentum per unit volume, and ''T<sup>ij</sup>'' form components of the [[Physics:Cauchy stress tensor|stress tensor]] including shear and normal stresses, the '''orbital angular momentum density''' about the position 4-vector ''{{overline|X}}<sup>β</sup>'' is given by a 3rd order tensor
<math display="block">\mathcal{M}^{\alpha\beta\gamma} = \left(X^\alpha - \bar{X}^\alpha\right) T^{\beta\gamma} - \left(X^\beta - \bar{X}^\beta\right) T^{\alpha\gamma}</math>

This is antisymmetric in ''α'' and ''β''. In special and general relativity, ''T'' is a symmetric tensor, but in other contexts (e.g., quantum field theory), it may not be.

Let Ω be a region of 4d spacetime. The [[Boundary (topology)|boundary]] is a 3d spacetime hypersurface ("spacetime surface volume" as opposed to "spatial surface area"), denoted ∂Ω where "∂" means "boundary". Integrating the angular momentum density over a 3d spacetime hypersurface yields the angular momentum tensor about {{overline|''X''}},
<math display="block">M^{\alpha\beta}\left(\bar{X}\right) = \oint_{\partial \Omega} \mathcal{M}^{\alpha\beta\gamma} d \Sigma_\gamma</math>
where dΣ<sub>γ</sub> is the volume [[One-form|1-form]] playing the role of a [[Unit vector|unit vector]] normal to a 2d surface in ordinary 3d Euclidean space. The integral is taken over the coordinates ''X'', not {{overline|''X''}}. The integral within a spacelike surface of constant time is
<math display="block">M^{ij} = \oint_{\partial \Omega} \mathcal{M}^{ij0} d \Sigma_0 = \oint_{\partial \Omega} \left[\left(X^i - Y^i\right)T^{j0} - \left(X^j - Y^j\right) T^{i0}\right] dx \, dy \, dz</math>
which collectively form the angular momentum tensor.

===Angular momentum about the centre of mass===

There is an intrinsic angular momentum in the centre-of-mass frame, in other words, the angular momentum about any event
<math display="block">\mathbf{X}_\text{COM} = \left(X^0_\text{COM}, X^1_\text{COM}, X^2_\text{COM}, X^3_\text{COM}\right)</math>
''on'' the wordline of the object's center of mass. Since ''T''<sup>00</sup> is the energy density of the object, the spatial coordinates of the [[Physics:Center of mass|center of mass]] are given by
<math display="block">X^i_\text{COM} = \frac{1}{m_0} \int_{\partial \Omega} X^i T^{00} dxdydz</math>

Setting ''Y'' = '''X'''<sub>COM</sub> obtains the orbital angular momentum density about the centre-of-mass of the object.

===Angular momentum conservation ===

The [[Physics:Conservation law|conservation]] of energy–momentum is given in differential form by the [[Continuity equation|continuity equation]]
<math display="block"> \partial_\gamma T^{\beta\gamma} = 0</math>
where ∂<sub>''γ''</sub> is the [[Physics:Four-gradient|four-gradient]]. (In non-Cartesian coordinates and general relativity this would be replaced by the [[Covariant derivative|covariant derivative]]). The total angular momentum conservation is given by another continuity equation
<math display="block">\partial_\gamma \mathcal{J}^{\alpha\beta\gamma} = 0</math>

The integral equations use Gauss' theorem in spacetime
<math display="block">\begin{align}
                  \int_\mathcal{V} \partial_\gamma T^{\beta\gamma} \, c dt \, dx \, dy \, dz &= \oint_{\partial \mathcal{V}} T^{\beta\gamma} d^3 \Sigma_\gamma = 0 \\
  \int_\mathcal{V} \partial_\gamma \mathcal{J}^{\alpha\beta\gamma} \, c dt \, dx \, dy \, dz &= \oint_{\partial \mathcal{V}} \mathcal{J}^{\alpha\beta\gamma} d^3 \Sigma_\gamma = 0
\end{align}</math>

==Torque in special relativity==

The torque acting on a point-like particle is defined as the derivative of the angular momentum tensor given above with respect to proper time:<ref>{{cite journal|title=Torque and angular momentum on a system at equilibrium in special relativity|author=S. Aranoff|journal=American Journal of Physics|volume=37|year=1969|issue=4|pages=453–454|url=https://www.researchgate.net/publication/237053370|doi=10.1119/1.1975612| bibcode=1969AmJPh..37..453A}} This author uses '''T''' for torque, here we use capital Gamma '''Γ''' since ''T'' is most often reserved for the [[Stress–energy tensor|stress–energy tensor]].</ref><ref>{{cite journal|title=Equilibrium in special relativity|author=S. Aranoff|journal=Nuovo Cimento|volume=10| year=1972| issue=1|url=http://www.analysis-knowledge.com/Physics/Equilibrium%20in%20Special%20Relativity.pdf|page=159| doi=10.1007/BF02911417| bibcode=1972NCimB..10..155A|s2cid=117291369|access-date=2013-10-27|archive-date=2012-03-28|archive-url=https://web.archive.org/web/20120328060625/http://www.analysis-knowledge.com/Physics/Equilibrium%20in%20Special%20Relativity.pdf|url-status=dead}}</ref>
<math display="block">\boldsymbol{\Gamma} = \frac{d \mathbf{M}}{d\tau} = \mathbf{X}\wedge \mathbf{F}</math>
or in tensor components:
<math display="block">\Gamma_{\alpha\beta} = X_\alpha F_\beta - X_\beta F_\alpha</math>
where '''F''' is the 4d force acting on the particle at the event '''X'''. As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distribution of mass.

==Angular momentum as the generator of spacetime boosts and rotations==


The angular momentum tensor is the generator of boosts and rotations for the [[Lorentz group]].<ref>{{cite book|title=Quantum Mechanics|author=E. Abers|publisher=Addison Wesley|year=2004|pages=11, 104, 105, 410–411|isbn=978-0-13-146100-0}}</ref><ref>{{cite journal|title=The Proper Homogeneous Lorentz Transformation Operator ''e<sup>L</sup>'' = ''e''<sup>− ''ω''·''S'' − ''ξ''·''K''</sup>, Where's It Going, What's the Twist| author1=H.L. Berk |author2=K. Chaicherdsakul |author3=T. Udagawa |url=http://w3fusion.ph.utexas.edu/ifs/ifsreports/907_berk.pdf|journal=American Journal of Physics|doi=10.1119/1.1371919|year=2001|volume=69|issue=996}}</ref> Lorentz boosts can be parametrized by [[Physics:Rapidity|rapidity]], and a 3d [[Unit vector|unit vector]] {{math|'''n'''}} pointing in the direction of the boost, which combine into the "rapidity vector"
<math display="block">\boldsymbol{\zeta} = \zeta\mathbf{n} = \mathbf{n}\tanh^{-1}\beta</math>
where {{math|1=''β'' = ''v''/''c''}} is the speed of the relative motion divided by the speed of light. Spatial rotations can be parametrized by the [[Axis–angle representation|axis–angle representation]], the angle {{math|''θ''}} and a unit vector {{math|'''a'''}} pointing in the direction of the axis, which combine into an "axis-angle vector"
<math display="block">\boldsymbol{\theta} = \theta\mathbf{a}</math>

Each unit vector only has two independent components, the third is determined from the unit magnitude. Altogether there are six parameters of the Lorentz group; three for rotations and three for boosts. The (homogeneous) Lorentz group is 6-dimensional.

The boost generators {{math|'''K'''}} and rotation generators {{math|'''J'''}} can be combined into one generator for Lorentz transformations; {{math|'''M'''}} the antisymmetric angular momentum tensor, with components
<math display="block">M^{0i} = -M^{i0} = K_i \,,\quad M^{ij} = \varepsilon_{ijk} J_k \,.</math>
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix {{math|'''ω'''}}, with entries:
<math display="block">\omega_{0i} = - \omega_{i0} = \zeta_i \,,\quad \omega_{ij} = \varepsilon_{ijk} \theta_k \,,</math>
where the summation convention over the repeated indices ''i, j, k'' has been used to prevent clumsy summation signs. The general [[Physics:Lorentz transformation|Lorentz transformation]] is then given by the [[Matrix exponential|matrix exponential]]
<math display="block">\Lambda(\boldsymbol{\zeta},\boldsymbol{\theta}) = \exp\left(\frac{1}{2}\omega_{\alpha\beta}M^{\alpha\beta}\right) = \exp \left(\boldsymbol{\zeta} \cdot \mathbf{K} + \boldsymbol{\theta}\cdot\mathbf{J}\right)</math>
and the summation convention has been applied to the repeated matrix indices ''α'' and ''β''.

The general Lorentz transformation Λ is the transformation law for any [[Physics:Four vector|four vector]] '''A''' = (''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>), giving the components of this same 4-vector in another inertial frame of reference
<math display="block">\mathbf{A}' = \Lambda(\boldsymbol{\zeta},\boldsymbol{\theta}) \mathbf{A}</math>

The angular momentum tensor forms 6 of the 10 generators of the [[Physics:Poincaré group|Poincaré group]], the other four are the components of the four-momentum for spacetime translations.

==Angular momentum in general relativity==

The angular momentum of test particles in a gently curved background is more complicated in GR but can be generalized in a straightforward manner.  If the [[Physics:Lagrangian (field theory)|Lagrangian]] is expressed with respect to angular variables as the [[Generalized coordinates|generalized coordinates]], then the angular momenta are the [[Functional derivative|functional derivative]]s of the Lagrangian with respect to the [[Physics:Angular velocity|angular velocities]].  Referred to Cartesian coordinates, these are typically given by the off-diagonal shear terms of the spacelike part of the [[Stress–energy tensor|stress–energy tensor]].  If the spacetime supports a [[Killing vector field]] tangent to a circle, then the angular momentum about the axis is conserved.

One also wishes to study the effect of a compact, rotating mass on its surrounding spacetime.  The prototype solution is of the [[Astronomy:Kerr metric|Kerr metric]], which describes the spacetime around an axially symmetric [[Astronomy:Black hole|black hole]].  It is obviously impossible to draw a point on the event horizon of a Kerr black hole and watch it circle around.  However, the solution does support a constant of the system that acts mathematically similarly to an angular momentum.

==See also==
* {{annotated link|Physics:Thomas precession}}
* {{annotated link|Physics:Angular momentum of light}}
* {{annotated link|Physics:Two-body problem in general relativity}}
* {{annotated link|Physics:Relativistic mechanics}}
* {{annotated link|Mathisson–Papapetrou–Dixon equations}}

==References==
{{reflist}}
{{refbegin}}
* {{cite journal|title=Center of mass in special and general relativity and its role in an effective description of spacetime|author1=C. Chryssomalakos |author2=H. Hernandez-Coronado |author3=E. Okon |year=2009|location=Mexico|journal= J. Phys. Conf. Ser.|arxiv=0901.3349| doi=10.1088/1742-6596/174/1/012026 |volume=174 |issue=1 |page=012026|bibcode=2009JPhCS.174a2026C |s2cid=17734387 }}
*{{cite book|title=Special relativity|author=U.E. Schroder|page=139|publisher=[[Company:World Scientific|World Scientific]]|year=1990|volume=33|series=Lecture Notes in Physics Series|isbn=981-02-0132-X|url=https://books.google.com/books?id=sLQ1rSNUjYAC&q=angular+momentum+in+special+relativity&pg=PA139}}
{{refend}}

==Further reading==

===Special relativity===
*{{cite book|title=Relativity and Geometry|publisher=Courier Dover Publications|year=1996|author=R. Torretti|series=Dover Books on Physics Series|isbn=0-486-69046-6|url=https://books.google.com/books?id=vpW_sBxwr88C&q=angular+momentum+in+special+relativity&pg=PA355}}

===General relativity===
*{{cite book|title=Mass and motion in general relativity|author1=L. Blanchet |author2=A. Spallicci |author3=B. Whiting |page=87| publisher=Springer |year=2011|isbn=978-90-481-3015-3|volume=162|series=Fundamental theories of physics|url=https://books.google.com/books?id=DU81SMmL4GYC&q=angular+momentum+in+general+relativity&pg=PA87}}
*{{cite book|title=General Relativity: A Geometric Approach|author=M. Ludvigsen|publisher={{wipe|Cambridge University Press}}|year=1999 |page=77| isbn=0-521-63976-X|url=https://books.google.com/books?id=hQdh3SVgZ8MC&q=angular+momentum+in+general+relativity&pg=PA77}}
*{{cite book|title=General Relativity and Gravitation 1989: Proceedings of the 12th International Conference on General Relativity and Gravitation |year=1990|publisher=Cambridge University Press|isbn=0-521-38428-1|author=N. Ashby, D.F. Bartlett, W.Wyss| url=https://books.google.com/books?id=hQbKO5vT-9cC&q=angular+momentum+in+general+relativity&pg=PA77}}
*{{cite book|title=Directions in General Relativity: Volume 1: Proceedings of the 1993|publisher=Cambridge University Press|year=2005 |volume=1 |series=Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner |isbn=0-521-02139-1|author1=B.L. Hu |author2=M.P. Ryan |author3=M.P. Ryan |author4=C.V. Vishveshwara |page=347 |url=https://books.google.com/books?id=zpGZwmTJZIUC&q=angular+momentum+in+general+relativity&pg=PA347}}
*{{cite book|title=Lectures on General Relativity|author=A. Papapetrou|year=1974|isbn=90-277-0514-3|publisher=Springer |url=https://books.google.com/books?id=SWeOggyp1ZsC&q=angular+momentum+in+special+relativity&pg=PA132}}

==External links==
*{{cite news|title=Relativistic Angular Momentum|author=N. Menicucci|year=2001|url=http://panda.unm.edu/Courses/Finley/p495/TermPapers/relangmom.pdf }}
*{{cite web|url=http://www.physics.usu.edu/Wheeler/EM/EMRelativity.pdf|title=Special Relativity|access-date=2013-10-30|url-status=dead| archive-url=https://web.archive.org/web/20131104054917/http://www.physics.usu.edu/Wheeler/EM/EMRelativity.pdf|archive-date=2013-11-04}}

[[Category:Dynamics (mechanics)]]
[[Category:General relativity|Angular momentum]]
[[Category:Rotation]]
[[Category:Special relativity|Angular momentum]]

{{Sourceattribution|Relativistic angular momentum}}