Editing Angular momentum (section)

== Angular momentum in general relativity ==
[[File:Angular momentum bivector and pseudovector.svg|thumb|The 3-angular momentum as a [[bivector]] (plane element) and [[axial vector]], of a particle of mass ''m'' with instantaneous 3-position '''x''' and 3-momentum '''p'''.]]{{Main|Relativistic angular momentum}}
In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see [[#Angular momentum in quantum mechanics|below]]) is described using a different formalism, instead of a classical [[pseudovector]]. In this formalism, angular momentum is the [[2-form]] [[Noether charge]] associated with rotational invariance. As a result, angular momentum is generally not conserved locally for general [[curved space]]times, unless they have rotational symmetry;<ref name="Hawking">
{{cite book
|last1 = Hawking
|first1 = S. W.
|last2 = Ellis
|first2 = G. F. R.
|title = The Large Scale Structure of Space-Time
|publisher = Cambridge University Press
|date=1973
|url=https://doi.org/10.1017/CBO9780511524646
|pages= 62–63
|doi = 10.1017/CBO9780511524646
|isbn = 978-0-521-09906-6
}}</ref> whereas globally the notion of angular momentum itself only makes sense if the spacetime is asymptotically flat.<ref name="Misner">
{{cite book
|last1 = Misner
|first1 = C. W.
|last2 = Thorne
|first2 = K. S.
|last3 = Wheeler
|first3 = J. A.
|title = Gravitation
|publisher = W. H. Freeman and Company
|date=1973
|url=https://archive.org/details/GravitationMisnerThorneWheeler
|chapter= 20: Conservation laws for 4-momentum and angular momentum
}}</ref> If the spacetime is only axially symmetric like for the [[Kerr metric]], the total angular momentum is not conserved but <math>p_{\phi}</math> is conserved which is related to the invariance of rotating around the symmetry-axis, where note that <math>p_{\phi}=g_{\mu \phi}p^{\phi}=mg_{\mu \phi} dX^{\mu}/d\tau</math> where <math>g_{\mu\nu}</math> is the [[Metric tensor (general relativity)|metric]], <math>m=\sqrt{|p_\mu p^{\mu}|}</math> is the [[rest mass]], <math>dX^{\mu}/d\tau</math> is the [[four-velocity]], and <math>X^{\mu}=(t,r,\theta,\phi)</math> is the four-position in spherical coordinates.

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element:
<math display="block">\mathbf{L} = \mathbf{r} \wedge \mathbf{p} \,,</math>
in which the [[exterior product]] (∧) replaces the [[cross product]] (×) (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined using the vectors '''x''' and '''p''', and the expression is true in any number of dimensions. In Cartesian coordinates:
<math display="block">\begin{align}
  \mathbf{L} &= \left(xp_y - yp_x\right)\mathbf{e}_x \wedge \mathbf{e}_y + \left(yp_z - zp_y\right)\mathbf{e}_y \wedge \mathbf{e}_z + \left(zp_x - xp_z\right)\mathbf{e}_z \wedge \mathbf{e}_x\\
             &= L_{xy}\mathbf{e}_x \wedge \mathbf{e}_y + L_{yz}\mathbf{e}_y \wedge \mathbf{e}_z + L_{zx}\mathbf{e}_z \wedge \mathbf{e}_x \,,
\end{align}</math>
or more compactly in index notation:
<math display="block">L_{ij} = x_i p_j - x_j p_i\,.</math>

The angular velocity can also be defined as an anti-symmetric second order tensor, with components ''ω<sub>ij</sub>''. The relation between the two anti-symmetric tensors is given by the moment of inertia which must now be a fourth order tensor:<ref>Synge and Schild, Tensor calculus, Dover publications, 1978 edition, p. 161. {{ISBN|978-0-486-63612-2}}.</ref>
<math display="block">L_{ij} = I_{ijk\ell} \omega_{k\ell} \,. </math>

Again, this equation in '''L''' and '''''ω''''' as tensors is true in any number of dimensions. This equation also appears in the [[geometric algebra]] formalism, in which '''L''' and '''''ω''''' are bivectors, and the moment of inertia is a mapping between them.

In [[relativistic mechanics]], the [[relativistic angular momentum]] of a particle is expressed as an [[antisymmetric tensor|anti-symmetric tensor]] of second order:
<math display="block">M_{\alpha\beta} = X_\alpha P_\beta - X_\beta P_\alpha</math>
in terms of [[four-vector]]s, namely the [[four-position]] ''X'' and the [[four-momentum]] ''P'', and absorbs the above '''L''' together with the [[moment of mass]], i.e., the product of the relativistic mass of the particle and its [[center of mass]], which can be thought of as describing the motion of its center of mass, since mass–energy is conserved.

In each of the above cases, for a system of particles the total angular momentum is just the sum of the individual particle angular momenta, and the center of mass is for the system.