Noether 1918

Emmy Noether's Invariante Variationsprobleme was presented to the Göttingen Scientific Society on 26 July 1918 and published in that year's Nachrichten, pages 235 to 257. It contains two theorems relating the symmetries of a variational problem to the structure of its equations, and it settled a question about energy conservation in general relativity that had occupied Göttingen for two years.

Provenance note

The paper is ESTABLISHED on Sources: the full text was obtained with its mathematics intact. But the text obtained is Tavel's 1971 English translation, which is under copyright, while the 1918 German original is public domain.

So this page follows the volume's translation rule. Noether's equations are reproduced elsewhere in her own notation; her prose is paraphrased here and never transcribed. Nothing on this page is a quotation.

The question

General relativity, published in 1915, made trouble for energy conservation. In the older physics, energy conservation is a dynamical law with a definite local content. In the new theory the quantity that plays the role of gravitational energy turned out to be coordinate-dependent, and no satisfactory local expression could be found.

Hilbert asserted that this failure was characteristic of general relativity rather than a defect of anyone's construction. The claim was debated in Göttingen through 1917 and 1918, with Klein and Einstein both involved. It was a claim without a proof.

Noether had been brought to Göttingen by Hilbert and Klein in 1915, in the face of faculty opposition to a woman lecturer, and was working on invariant theory. She was well placed to answer a question about what invariance implies.

What she proved

The paper contains two theorems, stated together in §1 and proved in the sections following.

Theorem I treats invariance under a group with finitely many parameters. If the action integral is invariant under a group with $\rho$ parameters, then $\rho$ independent combinations of the Lagrange expressions are divergences. Imposing the field equations turns those divergences into conservation laws. This is the result universally known as Noether's theorem, and it is developed on Noether's first theorem.

Theorem II treats invariance under a group depending on arbitrary functions. The conclusion is different in kind: instead of conservation laws, such invariance produces identities among the field equations, holding before any equation of motion is imposed. This is developed on Noether's second theorem.

Both theorems carry converses, which Noether proves and which are usually omitted when the first theorem is quoted.

The answer to Hilbert

§6 is the section the paper was written for, and it is the least quoted.

Noether classifies divergence relations as proper or improper, according to whether the conserved quantity is genuinely dynamical or is built out of the field equations and their derivatives — an identity in the costume of a conservation law. She then proves that the energy relations of a theory are improper exactly when the theory is invariant under an infinite group containing the displacement group as a subgroup.

General relativity is such a theory. Its energy conservation is therefore improper, and no local gravitational stress-energy tensor of the classical kind exists. Hilbert's assertion is correct.

But Noether's result is stronger than his claim, and in a direction that diminishes the role of relativity in it. The property belongs to any theory invariant under a group of arbitrary functions. She remarks that for the assertion to hold literally, the term general relativity would have to be read more broadly than usual, extending to every such group. The content is group-theoretic; gravity is one instance.

She closes by endorsing an observation of Klein's, that the word relativity as physicists use it is replaceable by invariance with respect to a group.

Why the second theorem was forgotten

Theorem I gives physicists a tool of enormous generality and is quoted constantly. Theorem II tells them that a class of conservation laws they might have wanted does not exist in the expected form, and is quoted rarely.

The imbalance misrepresents the paper. The two theorems are stated together, proved by the same method, and the second is the one that answers the question Noether was asked. A reader who knows only "symmetry implies conservation" has the corollary and not the argument.

The transmission was slow in any case. Tavel's English translation appeared in 1971, fifty-three years after publication, and the results reached physics gradually through intermediaries rather than through the paper itself.

Relation to the rest of this volume

Noether's theorems explain results this volume derived piecemeal before reaching them.

earlier result	which theorem explains it
conserved momentum for a cyclic coordinate, at the Euler-Lagrange equation	Theorem I, translation group
the Beltrami identity, at the Beltrami identity	Theorem I, time translation
Hamilton's seven classical integrals, at Hamilton 1834 and 1835	Theorem I, translations and rotations
the string's energy density and flux, at Continuum systems: the vibrating string	Theorem I, in one space dimension
charge conservation, at Classical field theory	Theorem I, internal phase group
the singular Hessian of the electromagnetic Lagrangian, at the electromagnetic Lagrangian	Theorem II, gauge group
the contracted Bianchi identities, at the Einstein-Hilbert action	Theorem II, coordinate group
the reparametrisation identity for arc length, at Geodesics	Theorem II, her own Weierstrass example

Eight results, one paper. That concentration is the reason this volume treats the 1918 paper as a structural pivot rather than as a historical episode.

Biographical note

Noether was not permitted a salaried position at Göttingen at the time she wrote this paper, and lectured under Hilbert's name. She was dismissed from the university in 1933 and emigrated to the United States, where she taught at Bryn Mawr until her death in 1935.

These facts are recorded because they bear on the paper's transmission: work published from an unsalaried position, by an author outside the ordinary academic channels, in a journal read mainly by mathematicians, took decades to reach the physicists it concerned.

Provenance: nothing on this page is quoted. Noether's theorems and her §6 result are paraphrased from the ESTABLISHED translation under the rule stated on Sources. The Göttingen debate, the dates, and the biographical facts are editorial summary; Klein's and Hilbert's 1918 notes are not sources of this volume and are not cited.