Noether's theorems/Second theorem/Proper and improper conservation laws

Plain-language counterpart: A conservation law that is only wearing the costume.

This is her sixth section and the reason the paper was written. It answers a claim of Hilbert's, and it answers it with a biconditional.

The definition

Let $I$ admit a ${𝔊}_{\mathrm{\infty }\varrho }$, and let ${𝔊}_{\sigma }$ be any finite group arising by specialising the arbitrary functions — a subgroup of the infinite one. The infinite group gives dependencies (N16); the finite subgroup gives divergence relations (N13).

The dependencies can be written

$\sum {\psi }_i{a}_i^{(\lambda )}=\mathrm{D}\mathrm{i}\mathrm{v}{\chi }^{(\lambda )}\mathrm{(}\mathrm{1}\mathrm{)}$

where the ${\chi }^{(\lambda )}$ are linear combinations of the Lagrange expressions and their derivatives. Because the $\psi$ enter linearly in both (N13) and (N16), the divergence relations must be linear combinations of the dependencies, so

$\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}=\mathrm{D}\mathrm{i}\mathrm{v}(\sum \alpha \cdot {\chi }^{(\lambda )})\mathrm{(}\mathrm{2}\mathrm{)}$

and the $B^{(\lambda )}$ themselves compose linearly out of the Lagrange expressions and their derivatives, together with functions whose divergence vanishes identically — such as the $B-\Gamma$ obtained at the end of the previous proof.

Divergence relations whose $B^{(\lambda )}$ can be composed in that way Noether calls uneigentlich, improper. All others are eigentlich, proper.

An improper conservation law is an identity wearing the costume of a conservation law. Its conserved quantity is built from the field equations themselves, so the statement that it is conserved carries no information the field equations did not already carry.

The criterion, which is a biconditional

Conversely, if the divergence relations are linear combinations of the dependencies — that is, improper — then invariance under ${𝔊}_{\sigma }$ follows from invariance under ${𝔊}_{\mathrm{\infty }\varrho }$, and $𝔊$ becomes a subgroup of ${𝔊}_{\mathrm{\infty }\varrho }$.

Hence: the divergence relations belonging to a finite group ${𝔊}_{\sigma }$ are improper if and only if ${𝔊}_{\sigma }$ is a subgroup of an infinite group with respect to which $I$ is invariant.

The "if and only if" is Noether's and it is the whole content. A one-directional statement of this criterion is not a weaker version of it; it is a different and much less useful claim.

Specialising to translations

Let the translation group be

$y_i=x_i+{\epsilon }_i;v_i(y)=u_i(x)\mathrm{(}\mathrm{3}\mathrm{)}$

so that

$\Delta x_i={\epsilon }_i,\Delta u_i=0,\overline{\delta }{u}_i=-\sum _{\lambda }\frac{\partial u_i}{\partial x_{\lambda }}{\epsilon }_{\lambda }\mathrm{(}\mathrm{4}\mathrm{)}$

Invariance under this group says that the $x$ do not occur explicitly in $f$. The corresponding $n$ divergence relations

$\sum {\psi }_i\frac{\partial u_i}{\partial x_{\lambda }}=\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}(\lambda =1,2,\dots n)\mathrm{(}\mathrm{5}\mathrm{)}$

Noether calls the energy relations, since the conservation laws $\mathrm{D}\mathrm{i}\mathrm{v}{B}^{(\lambda )}=0$ belonging to the variational problem correspond to the energy theorems and the $B^{(\lambda )}$ to the energy components.

The criterion then reads: if $I$ admits the translation group, the energy relations are improper if and only if $I$ is invariant under an infinite group containing the translation group as a subgroup.

Noether adds, in a footnote, that the energy theorems of classical mechanics and of the old relativity theory are proper, since no infinite groups occur there. That footnote is what makes the criterion a criterion rather than an observation: it exhibits the other side of the biconditional.

Her worked example

Not general relativity — a group she constructs precisely because it cannot be a coefficient transformation, the logarithm seeing to that:

$y=x+p(x);v_i=u_i+\lg (1+p^{\prime }(x))=u_i+\lg \frac{dy}{dx}\mathrm{(}\mathrm{6}\mathrm{)}$

$\Delta x=p(x);\Delta u_i=p^{\prime }(x);\overline{\delta }{u}_i=p^{\prime }(x)-u{\text{'}}_{i}p(x)\mathrm{(}\mathrm{7}\mathrm{)}$

The dependencies (N16) here become

$\sum _i({\psi }_i{u}_{i^{\prime }}+\frac{d{\psi }_i}{dx})=0\mathrm{(}\mathrm{8}\mathrm{)}$

and the improper energy relations

$\sum ({\psi }_i{u}_{i^{\prime }}+\frac{d({\psi }_i+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.})}{dx})=0\mathrm{(}\mathrm{9}\mathrm{)}$

A simplest invariant integral of the group is

$I=\int \frac{e^{-2u_1}}{u{\text{'}}_1-u{\text{'}}_2}dx\mathrm{(}\mathrm{10}\mathrm{)}$

Why she chose this example rather than gravity. It shows the phenomenon in a case with nothing physical attached, which is exactly her point about what the phenomenon depends on.

The remark that governs how the whole subject is read

Noether closes by observing that for Hilbert's assertion to hold literally, the designation "general relativity" must be taken more broadly than usual, extended to the groups depending on $n$ arbitrary functions treated above. And she endorses Klein's suggestion that the physicists' term Relativität would be better replaced by invariance with respect to a group.

The property is group-theoretic. General relativity is one theory that has it. It is not the reason. The consequences for gravitational energy are on their own page, which is downstream of this one and not the other way round.

Provenance: the definitions, the criterion, the translation specialisation, the footnote on classical mechanics, the worked example and the closing remark are all Noether's sixth section. Displays (1) to (10) are hers, set as running text in the original and numbered here. Reproduced in her notation from the public-domain German original; her prose is paraphrased. See Sources.