Noether's theorems/Mathematics/The Belinfante-Rosenfeld improvement

Plain-language counterpart: Moving the bookkeeping without changing the total.

The Belinfante improvement as a redistribution

The stress-energy tensor the first theorem produces has two defects, and one standard repair. This page derives why the repair is allowed, and flags what it does not derive.

The defect

Apply the first theorem to translations in space and time. The currents assemble into a two-index object, the canonical stress-energy tensor $T^{ab}$, satisfying

${\partial }_a{T}^{ab}=0\mathrm{(}\mathrm{1}\mathrm{)}$

on solutions. Two things are wrong with it for the purposes of general relativity.

It is not symmetric in general. And it is not unique, since the current page showed that any current admits additions with identically vanishing divergence.

General relativity needs a symmetric tensor, because the object appearing on the right of Einstein's equations is symmetric by construction. So the canonical tensor as delivered cannot be the one that couples to gravity.

Why an improvement is allowed, derived

Add to $T^{ab}$ the divergence of a three-index object:

$T^{ab}⟶T^{ab}+{\partial }_c{S}^{cab}\mathrm{(}\mathrm{2}\mathrm{)}$

subject to one condition, that $S$ is antisymmetric in its first two indices:

$S^{cab}=-S^{acb}\mathrm{(}\mathrm{3}\mathrm{)}$

Take the divergence of the added term:

${\partial }_a{\partial }_c{S}^{cab}=0\mathrm{(}\mathrm{4}\mathrm{)}$

because ${\partial }_a{\partial }_c$ is symmetric under interchange of $a$ and $c$, while $S^{cab}$ is antisymmetric under it by (3), and the contraction of a symmetric with an antisymmetric object vanishes.

So (1) still holds after (2). The conservation law survives the change identically, not merely on solutions.

The charges survive too. The added contribution to a charge is an integral of ${\partial }_c{S}^{cab}$ over a slice, which reduces to a surface term at infinity and vanishes for fields falling off there.

Same conservation law, same total, different distribution. The improvement moves energy and momentum around within the system without changing what flows or how much there is.

The construction, not derived

UNVERIFIED. That an $S^{cab}$ satisfying (3) can always be chosen so that the improved tensor is symmetric — the Belinfante-Rosenfeld construction, built from the spin current supplied by the first theorem applied to rotations — is stated in the literature and is not derived here.

Deriving it requires the rotation currents for fields carrying spin, which this volume does not set up.

The point that is usually missed

The non-uniqueness is a property of the theorem, not a defect of a derivation.

Noether's proof determines the current up to exactly this freedom, and says so: the corollary at the end of the second theorem's proof produces a non-vanishing function system whose divergence vanishes identically, which is the freedom (2) arriving as an output.

So the improvement is not a repair of Noether's result. It is a choice made within a latitude her result already exhibited, and the criterion for the choice — symmetry — comes from general relativity rather than from the theorem.

Provenance: (1) to (4) are standard and (4) is derived above. The construction of a symmetrising $S$ is flagged UNVERIFIED. The connection to the corollary of Noether's second-theorem proof is read from the public-domain German original. See Sources.