7 6 D O C . 9 E N E R G Y C O N S E RVAT I O N

Published in Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1918):

448– 459. Submitted 16 May 1918, published 30 May 1918.

[1]Schrödinger 1918a, Bauer 1918, Nordström 1918a. See note 6 for more on Gunnar Nord-

ström’s position.

[2]Einstein 1916o (Vol. 6, Doc. 41).

[3]This same example was given in Einstein to David Hilbert, 12 April 1918 (Vol. 8, Doc. 503).

[4]Appeals to the primacy of integral over differential conservation laws to justify using a non-

generally covariant quantity to represent gravitational energy-momentum can also be found in Ein-

stein 1916e (Vol. 6, Doc. 30), p. 806; Einstein 1918a (Doc. 1), sec. 6; and Einstein 1918b (Doc. 2).

[5]In Lorentz 1916 and Levi-Civita 1917b, it was suggested to define the left-hand side of the field

equations as the energy-momentum tensor for the gravitational field, thus avoiding the need for Ein-

stein’s pseudotensor. See Einstein 1918a (Doc. 1), sec. 6, for some objections to this proposal. The

same proposal was put forward by Friedrich Kottler (Kottler 1918; see also Friedrich Kottler to Ein-

stein, 30 March 1918 [Vol. 8, Doc. 495]), and by Felix Klein. In his paper on Hilbert 1915, Klein had

questioned the interpretation of Einstein’s pseudotensor as representing gravitational energy-momen-

tum (Klein, F. 1917, pp. 476–477). In a letter to Klein, Einstein defended his treatment of energy-

momentum conservation against Klein’s criticism, especially against the claim that eq. (1) is a math-

ematical identity (see Einstein to Felix Klein, 13 March 1918 [Vol. 8, Doc. 480]). In his response,

Klein, apparently unaware of Lorentz 1916 and Levi-Civita 1917b, wrote that in a paper presented to

the Göttingen Academy on 22 February 1918 he had suggested to define the left-hand side of the field

equations as the gravitational energy-momentum tensor (Felix Klein to Einstein, 20 March 1918 [Vol.

8, Doc. 487]). The paper was eventually retracted (Felix Klein to Einstein, 18 May 1918 [Vol. 8, Doc.

540]), but Klein continued to take a lively interest in the problem of finding a satisfactory formulation

of energy-momentum conservation in general relativity. He discussed the issue both in further corre-

spondence with Einstein and with other Göttingen mathematicians, especially Emmy Noether (1882–

1935) and Carl Runge (1856–1927), Professor of Applied Mathematics. With the latter he undertook

a systematic survey of the relevant literature. These endeavors resulted in two important papers on the

topic: Klein, F. 1918a, 1918b. Material documenting Klein’s work on energy-momentum conserva-

tion, including two drafts of the unpublished paper mentioned above (entitled “A Proposal Concern-

ing the Definition of the Energy Components in Einstein’s Theory of Gravitation” [“Ein Vorschlag

betreffend die Ansetzung der Energiekomponenten in der Einstein’schen Gravitationstheorie”]) and

notes of his meetings with Runge in March–April 1918, can be found in GyGöU, Cod. Ms. Klein 22b.

[6]In Nordström 1918b and Schrödinger 1918a it was shown that the pseudotensor can be made to

vanish for the field of a point mass, a result to which Einstein had already responded in Einstein 1918b

(Doc. 2). Contrary to what Einstein suggested in fn. 1, Nordström saw this as an important argument

against Einstein’s concept of gravitational energy-momentum and in favor of Lorentz’s concept (Nor-

dström 1918b, p. 1208, and Gunnar Nordström to Einstein, 22–28 September 1917 [Vol. 8, Doc.

382]). In Bauer 1918, it was shown that the components of the pseudotensor can be given nonvanish-

ing values for a Minkowski space-time.

[7]See Einstein 1918b (Doc. 2), note 3, for Einstein’s definition of a “Galilean space.” Einstein

argues that mass should be defined in terms of its asymptotic gravitational field, although his prescrip-

tion for matching between interior and exterior solutions is too simplistic by modern standards, since

the exterior space cannot be strictly flat or “Galilean” and since his approach outlined below would

not deal with mass loss due to a system radiating gravitational waves (see, e.g., Papapetrou 1974, pp.

128–133).

[8]In Felix Klein to Einstein, 1 June 1918 (Vol. 8, Doc. 554), Klein asked for clarification of this

proof. He found some gaps in the proof given in response to this query in Einstein to Felix Klein, 9

June 1918 (Vol. 8, Doc. 561), which he eventually filled himself in Felix Klein to Einstein, 5 July 1918

(Vol. 8, Doc. 581). In Klein, F. 1918b, he introduced the notion of a “free vector” (an integral of a

vector field over some hypersurface in space-time) that enabled him to give a much more elegant

proof of the vector character of (see Felix Klein to Einstein, 5 July 1918 [Vol. 8, Doc. 581], note

9, for more details).

[9]Einstein 1917b (Vol. 6, Doc. 43).

[10]The possibility of an elliptic geometry had been brought to Einstein’s attention shortly after the

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