BFP Rauch: 1 revision imported

2026-09-10T13:29:56Z

1 revision imported

← Older revision	Revision as of 13:29, 10 September 2026
(No difference)

NT>NoetherVolumeBuild: Noether theorems volume, initial import (H-02)

2026-09-10T00:00:00Z

Noether theorems volume, initial import (H-02)

New page

{{DISPLAYTITLE: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
[[Noether's theorems/Sources|Sources]].''

Noether's theorems/History/Hilbert and Klein - Revision history

BFP Rauch: 1 revision imported

NT>NoetherVolumeBuild: Noether theorems volume, initial import (H-02)