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

{{DISPLAYTITLE:Proper and improper conservation laws}}

''Plain-language counterpart: [[Noether's theorems/Explainers/A conservation law that is only wearing the costume|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 <math>I</math> admit a <math>\mathfrak{G}_{\infty\varrho}</math>, and let
<math>\mathfrak{G}_{\sigma}</math> 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

<math>\sum\psi_{i}a_{i}^{(\lambda)}=\mathrm{Div}\,\chi^{(\lambda)}\qquad\qquad\mathrm{(1)}</math>

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

<math>\mathrm{Div}\,B^{(\lambda)}=\mathrm{Div}\left(\sum\alpha\cdot\chi^{(\lambda)}\right)\qquad\qquad\mathrm{(2)}</math>

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

'''Divergence relations whose <math>B^{(\lambda)}</math> 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
<math>\mathfrak{G}_{\sigma}</math> follows from invariance under
<math>\mathfrak{G}_{\infty\varrho}</math>, and <math>\mathfrak{G}</math> becomes
a subgroup of <math>\mathfrak{G}_{\infty\varrho}</math>.

Hence: '''the divergence relations belonging to a finite group
<math>\mathfrak{G}_{\sigma}</math> are improper if and only if
<math>\mathfrak{G}_{\sigma}</math> is a subgroup of an infinite group with
respect to which <math>I</math> 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

<math>y_{i}=x_{i}+\varepsilon_{i};\qquad v_{i}(y)=u_{i}(x)\qquad\qquad\mathrm{(3)}</math>

so that

<math>\Delta x_{i}=\varepsilon_{i},\qquad\Delta u_{i}=0,\qquad\bar{\delta}u_{i}=-\sum_{\lambda}\frac{\partial u_{i}}{\partial x_{\lambda}}\varepsilon_{\lambda}\qquad\qquad\mathrm{(4)}</math>

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

<math>\sum\psi_{i}\frac{\partial u_{i}}{\partial x_{\lambda}}=\mathrm{Div}\,B^{(\lambda)}\qquad(\lambda=1,2,\dots n)\qquad\qquad\mathrm{(5)}</math>

Noether calls the '''energy relations''', since the conservation laws
<math>\mathrm{Div}\,B^{(\lambda)}=0</math> belonging to the variational problem
correspond to the energy theorems and the <math>B^{(\lambda)}</math> to the
energy components.

The criterion then reads: '''if <math>I</math> admits the translation group, the
energy relations are improper if and only if <math>I</math> 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:

<math>y=x+p(x);\qquad v_{i}=u_{i}+\lg\left(1+p'(x)\right)=u_{i}+\lg\frac{dy}{dx}\qquad\qquad\mathrm{(6)}</math>

<math>\Delta x=p(x);\qquad\Delta u_{i}=p'(x);\qquad\bar{\delta}u_{i}=p'(x)-u'_{i}p(x)\qquad\qquad\mathrm{(7)}</math>

The dependencies (N16) here become

<math>\sum_{i}\left(\psi_{i}u_{i}'+\frac{d\psi_{i}}{dx}\right)=0\qquad\qquad\mathrm{(8)}</math>

and the improper energy relations

<math>\sum\left(\psi_{i}u_{i}'+\frac{d\left(\psi_{i}+\mathrm{const.}\right)}{dx}\right)=0\qquad\qquad\mathrm{(9)}</math>

A simplest invariant integral of the group is

<math>I=\int\frac{e^{-2u_{1}}}{u'_{1}-u'_{2}}dx\qquad\qquad\mathrm{(10)}</math>

'''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 <math>n</math> 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
[[Noether's theorems/Applications/General relativity and the energy question|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
[[Noether's theorems/Sources|Sources]].''