Hilbert and Klein

Two men at Göttingen worked on the energy question before Noether did. Their papers are the immediate context of hers, and one of them names her in its closing paragraph.

Hilbert's two communications

Hilbert presented Die Grundlagen der Physik to the Göttingen Scientific Society in two parts: the first on 20 November 1915, published in the Nachrichten for 1915 at pages 395 to 407, and the second on 23 December 1916, published in the Nachrichten for 1917 at pages 53 to 76.

The first communication derives the gravitational field equations from a variational principle. Hilbert writes down an invariant, integrates it over spacetime, and requires the integral to be stationary. This is a different route to the same equations Einstein had reached, and it makes a structural fact visible that the other route hides.

The fact is a count. Hilbert states, as his Theorem I, that four of his fundamental equations follow from the remaining ones. They are not independent. He states this for the particular invariant he is working with, and he states the identities that produce it only for the case where the invariant depends on the metric and its derivatives.

The second communication returns to it and says plainly what it costs. Under the conditions that hold in general relativity, knowledge of the physical quantities in the present and past does not determine their values in the future — not because the theory is indeterministic, but because some of the freedom in the solution is freedom in the choice of coordinates and carries no physical content.

Hilbert also asserted, in the exchange Klein later published, that the failure of proper energy theorems is a characteristic feature of general relativity. He asserted it. He did not prove it in general.

Klein's three notes

Klein published three notes bearing on this.

The first, Zu Hilberts erster Note über die Grundlagen der Physik, appeared in the Nachrichten for 1917 at pages 469 to 482 and was presented at the session of 25 January 1918. It consists largely of an exchange of letters with Hilbert, and it derives Hilbert's identities in general form for the first time, by the method of infinitesimal transformations.

The second, Über die Differentialgesetze für die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie, appeared in the Nachrichten for 1918 at pages 171 to 189, presented on 19 July 1918. This is the substantial one. Klein takes the several expressions for gravitational energy proposed by Einstein, Hilbert, Lorentz and Weyl, and shows that all of them follow from one identity, differing only in which additional divergence term has been absorbed. He remarks at the outset that he no longer has to calculate, only to apply the most elementary formulas of the classical calculus of variations. That is the tone of the whole paper: it is a paper about bookkeeping, and it demonstrates that the disagreement in the literature had been bookkeeping all along.

The third, on the integral form of the conservation laws, appeared in the same volume at pages 394 to 423.

What Klein's second note establishes, and what it does not

Klein's method is to vary the action under an infinitesimal change of coordinates without requiring anything of that change at the boundary of the region. Earlier treatments, including his own first note, had required the change and its derivatives to vanish there. Dropping that requirement produces boundary terms, and Klein's paper is the examination of those terms.

He obtains two families of identities, and sorts out which are the sufficient conditions for invariance and which are not. The count runs to 140 equations, and Klein does not investigate what dependencies hold among them.

What he has, at the end, is a complete account for one invariant, obtained for one group — the coordinate transformations — by a method that works. What he does not have is a theorem. Nothing in his paper says what happens for an arbitrary invariant under an arbitrary group.

He says as much in his closing section, where he thanks Noether for her participation in the work and states that the mathematical ideas he had adapted to the physical question have been worked out in general, and will appear shortly in the same journal.

The direction of influence

It is often written that Hilbert and Klein asked Noether to solve their problem and she obliged.

The primary texts do not say that. Klein's closing section credits her participation and announces her forthcoming paper. Noether's own opening remarks say that Klein's second note and her own work influenced each other, and refers the reader to his concluding remarks. Neither text describes a commission. What both describe is two papers written alongside one another, by people in the same building, each shaping the other.

Hilbert settled the credit himself in 1924, when he reprinted both communications in the Mathematische Annalen. In the notes to that reprint he attributes the general proof of the theorem in question to Noether, and the first general derivation of the identities to Klein.

Provenance: this page contains no verbatim quotation. Hilbert's papers were located but the only electronic copy reached is OCR-damaged at word level, and Klein's papers were read in a modern English translation whose terms of reuse have not been established. Both are therefore summarised, not quoted. See Sources.