Spin tensor

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Not to be confused with tensor products of spin representations.

In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field theory.

The Euclidean group SE(d) of direct isometries is generated by translations and rotations. Its Lie algebra is written ${\mathfrak {se}}(d)$ .

This article uses Cartesian coordinates and tensor index notation.

Background on Noether currents

The Noether current for translations in space is momentum, while the current for increments in time is energy. These two statements combine into one in spacetime: translations in spacetime, i.e. a displacement between two events, is generated by the four-momentum P. Conservation of four-momentum is given by the continuity equation:

$\partial _{\nu }T^{{\mu \nu }}=0\,,$

where $T^{{\mu \nu }}\,$ is the stress-energy tensor, and ∂ are partial derivatives that make up the four gradient (in non-Cartesian coordinates this must be replaced by the covariant derivative). Integrating over spacetime:

$\int d^{4}xT^{{\mu 0}}({\vec {x}},t)=P^{\mu }$

gives the four-momentum vector at time t.

The Noether current for a rotation about the point y is given by a tensor of 3rd order, denoted $M_{y}^{{\alpha \beta \mu }}$ . Because of the Lie algebra relations

$M_{y}^{{\alpha \beta \mu }}(x)=M_{0}^{{\alpha \beta \mu }}(x)+y^{\alpha }T^{{\beta \mu }}(x)-y^{\beta }T^{{\alpha \mu }}(x)\,,$

where the 0 subscript indicates the origin (unlike momentum, angular momentum depends on the origin), the integral:

$\int d^{4}xM_{0}^{{\mu \nu }}({\vec {x}},t)$

gives the angular momentum tensor $M^{{\mu \nu }}\,$ at time t.

Definition

The spin tensor is defined at a point x to be the value of the Noether current at x of a rotation about x,

$S^{{\alpha \beta \mu }}({\mathbf {x}})\ {\stackrel {{\mathrm {def}}}{=}}\ M_{x}^{{\alpha \beta \mu }}({\mathbf {x}})=M_{0}^{{\alpha \beta \mu }}({\mathbf {x}})+x^{\alpha }T^{{\beta \mu }}({\mathbf {x}})-x^{\beta }T^{{\alpha \mu }}({\mathbf {x}})$

The continuity equation

$\partial _{\mu }M_{0}^{{\alpha \beta \mu }}=0\,,$

implies:

$\partial _{\mu }S^{{\alpha \beta \mu }}=T^{{\beta \alpha }}-T^{{\alpha \beta }}\neq 0$

and therefore, the stress-energy tensor is not a symmetric tensor.

The quantity S gives the spin density and M gives the angular momentum density. The angular momentum is the sum of the orbital angular momentum and spin (spin in this case is not only for a point-like particle, but also for an extended bodies).

The relation:

$T_{{ij}}-T_{{ji}}$

gives the torque density showing the rate of conversion between the orbital angular momentum and spin.

Examples

Examples of materials with a nonzero spin density are molecular fluids, the electromagnetic field and turbulent fluids. For molecular fluids, the individual molecules may be spinning. The electromagnetic field can have circularly polarized light. For turbulent fluids, we may arbitrarily make a distinction between long wavelength phenomena and short wavelength phenomena. A long wavelength vorticity may be converted via turbulence into tinier and tinier vortices transporting the angular momentum into smaller and smaller wavelengths while simultaneously reducing the vorticity. This can be approximated by the eddy viscosity.

References

A. K. Raychaudhuri, S. Banerji, A. Banerjee (2003). General Relativity, Astrophysics, and Cosmology. Astronomy and astrophysics library. Springer. pp. 66–67. ISBN 038-740-628-X.

J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 156–159, §5.11. ISBN 0-7167-0344-0.

L. M. Butcher, A. Lasenby, M. Hobson (2012). "Localizing the Angular Momentum of Linear Gravity". Phys. Rev. D. arXiv:1210.0831. doi:10.1103/PhysRevD.86.084012.

T. Banks (2008). "Modern Quantum Field Theory: A Concise Introduction". Cambridge University Press. ISBN 113-947-389-1.

S. Kopeikin, M.Efroimsky, G. Kaplan (2011). "Relativistic Celestial Mechanics of the Solar System". John Wiley & Sons. ISBN 352-763-457-6.

W. F. Maher, J. D. Zund (1968). "A spinor approach to the Lanczos spin tensor". II Nuovo Cimento A. 10 57 (4) (Springer). pp. 638–648.

External links

von Jan Steinhoff. "Canonical Formulation of Spin in General Relativity (Dissertation)". Retrieved 2013-10-27.

Tensors

Glossary of tensor theory

Scope

Mathematics	coordinate system multilinear algebra Euclidean geometry tensor algebra differential geometry exterior calculus tensor calculus

Physics Engineering	continuum mechanics electromagnetism transport phenomena general relativity computer vision

Notation

Tensor definitions

Operations

Related abstractions

dimension
basis (linear algebra)
vector, vector space
multivector
covariance and contravariance of vectors
linear transformation
matrix
spinor
Cartan formalism (physics)
differential form
exterior form
connection form
geodesic
manifold
fiber bundle

Notable tensors

Mathematics	Kronecker delta Levi-Civita symbol metric tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor

Physics	moment of inertia angular momentum tensor spin tensor Cauchy stress tensor stress–energy tensor EM tensor gluon field strength tensor Einstein tensor metric (GR)

Mathematicians

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