Belinfante–Rosenfeld stress–energy tensor

In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the stress–energy tensor that is constructed from the canonical stress–energy tensor and the spin current so as to be symmetric yet still conserved.

In a classical or quantum local field theory, the generator of Lorentz transformations can be written as an integral

${\displaystyle M_{\mu \nu }=\int \mathrm {d} ^{3}x\,{M^{0}}_{\mu \nu }}$

of a local current

${\displaystyle {M^{\mu }}_{\nu \lambda }=(x_{\nu }{T^{\mu }}_{\lambda }-x_{\lambda }{T^{\mu }}_{\nu })+{S^{\mu }}_{\nu \lambda }.}$

Here ${\displaystyle {T^{\mu }}_{\lambda }}$ is the canonical stress–energy tensor satisfying ${\displaystyle \partial _{\mu }{T^{\mu }}_{\lambda }=0}$, and ${\displaystyle {S^{\mu }}_{\nu \lambda }}$ is the contribution of the intrinsic (spin) angular momentum. The anti-symmetry

${\displaystyle {M^{\mu }}_{\nu \lambda }=-{M^{\mu }}_{\lambda \nu }}$

implies the anti-symmetry

${\displaystyle {S^{\mu }}_{\nu \lambda }=-{S^{\mu }}_{\lambda \nu }.}$

Local conservation of angular momentum

${\displaystyle \partial _{\mu }{M^{\mu }}_{\nu \lambda }=0\,}$

requires that

${\displaystyle \partial _{\mu }{S^{\mu }}_{\nu \lambda }=T_{\lambda \nu }-T_{\nu \lambda }.}$

Thus a source of spin-current implies a non-symmetric canonical stress–energy tensor.

The Belinfante–Rosenfeld tensor is a modification of the stress–energy tensor

${\displaystyle T_{B}^{\mu \nu }=T^{\mu \nu }+{\frac {1}{2}}\partial _{\lambda }(S^{\mu \nu \lambda }+S^{\nu \mu \lambda }-S^{\lambda \nu \mu })}$

that is constructed from the canonical stress–energy tensor and the spin current ${\displaystyle {S^{\mu }}_{\nu \lambda }}$ so as to be symmetric yet still conserved, i.e.,

${\displaystyle \partial _{\mu }T_{B}^{\mu \nu }=0.}$

An integration by parts shows that

${\displaystyle M^{\nu \lambda }=\int (x^{\nu }T_{B}^{0\lambda }-x^{\lambda }T_{B}^{0\nu })\,\mathrm {d} ^{3}x,}$

and so a physical interpretation of Belinfante tensor is that it includes the "bound momentum" associated with gradients of the intrinsic angular momentum. In other words, the added term is an analogue of the ${\displaystyle {\mathbf {J} }_{\text{bound}}=\nabla \times \mathbf {M} }$ "bound current" associated with a magnetization density ${\displaystyle {\mathbf {M} }}$.

The curious combination of spin-current components required to make ${\displaystyle T_{B}^{\mu \nu }}$ symmetric and yet still conserved seems totally ad hoc, but it was shown by both Rosenfeld and Belinfante that the modified tensor is precisely the symmetric Hilbert stress–energy tensor that acts as the source of gravity in general relativity. Just as it is the sum of the bound and free currents that acts as a source of the magnetic field, it is the sum of the bound and free energy–momentum that acts as a source of gravity.