View source for Noether's theorems/Mathematics/The Belinfante-Rosenfeld improvement

{{DISPLAYTITLE:The Belinfante-Rosenfeld improvement}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Moving the bookkeeping without changing the total|Moving the bookkeeping without changing the total]].''

[[File:NT-F-11.png|thumb|right|360px|alt=Two bar charts of equal total area. The left, the canonical stress-energy assignment, has uneven bars; the right, the improved assignment, has evened ones. An arrow joins them. The caption states that the total and the conservation law are unchanged and only the distribution differs.|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
<math>T^{ab}</math>, satisfying

<math>\partial_{a}T^{ab}=0\qquad\qquad\mathrm{(1)}</math>

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
[[Noether's theorems/First theorem/The Noether current and the Noether charge|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 <math>T^{ab}</math> the divergence of a three-index object:

<math>T^{ab}\ \longrightarrow\ T^{ab}+\partial_{c}S^{cab}\qquad\qquad\mathrm{(2)}</math>

subject to one condition, that <math>S</math> is antisymmetric in its first two
indices:

<math>S^{cab}=-S^{acb}\qquad\qquad\mathrm{(3)}</math>

Take the divergence of the added term:

<math>\partial_{a}\partial_{c}S^{cab}=0\qquad\qquad\mathrm{(4)}</math>

because <math>\partial_{a}\partial_{c}</math> is symmetric under interchange of
<math>a</math> and <math>c</math>, while <math>S^{cab}</math> 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
<math>\partial_{c}S^{cab}</math> 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 <math>S^{cab}</math> 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
[[Noether's theorems/Second theorem/The second theorem stated and proved|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 <math>S</math> is flagged UNVERIFIED. The connection to the
corollary of Noether's second-theorem proof is read from the public-domain German
original. See [[Noether's theorems/Sources|Sources]].''