D O C . 1 7 G R AV I T Y A N D M AT T E R 1 3 9

Published in Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1919):

349–356. Submitted 10 April 1919, published 24 April 1919.

[1]In the plenary session (Gesamtsitzung) of the Prussian Academy of 15 May 1919, Einstein gave

a lecture on this paper. The following summary of the lecture appeared in Preußische Akademie der

Wissenschaften (Berlin), Sitzungsberichte (1919): 463: “Mr. Einstein talked about a visualization of

the relations in spherical space, and also about the field equations of the general theory of relativity

from the point of view of the cosmological problem and of the problem of the constitution of matter.

The lecture was essentially a report on . . . [Einstein 1919a]” (“Hr. Einstein sprach über eine Veran-

schaulichung der Verhältnisse im sphärischen Raum, ferner über die Feldgleichungen der allgemei-

nen Relativitätstheorie vom Standpunkte des kosmologischen Problems und des Problems der

Konstitution der Materie. Der Vortrag war im wesentlichen ein Referat über . . . [Einstein 1919a]”).

[2]For a discussion of the theory presented in this document and its relation to other attempts to

explain the stability of charge-carrying particles, see Pauli 1921, part V, especially sec. 66. Einstein

covered his new theory in a course on general relativity offered in the summer semester of 1919 (see

Doc. 19, [pp. 21–24]). In 1915, Einstein had already briefly adopted the hypothesis that gravitational

fields play a role in the structure of matter. Invoking this hypothesis, he could use the traceless energy-

momentum tensor for electromagnetic fields as the only source term in the field equations while

avoiding the implication that the energy-momentum tensor of matter would also have to be traceless

(see Einstein 1915g [Vol. 6, Doc. 22], p. 800).

[3]Mie 1912a, 1912b, 1913. For a discussion of Mie’s theory, see Pauli 1921, sec. 64. For a histor-

ical account, see Corry 1999b.

[4]Hilbert 1915, 1917. For historical accounts, see Corry 1999a, 1999b, Corry et al. 1997, and

Sauer 1999.

[5]Weyl 1918b. In Weyl 1919c, p. 122, the author showed how the terms added to the Maxwell-

Lorentz theory in Mie’s theory emerge naturally from the author’s own unified theory of gravity and

electromagnetism (first proposed in Weyl 1918a and criticized in Einstein 1918g [Doc. 8]). Einstein

and Weyl had discussed this aspect of Weyl’s theory the year before (see, e.g., Einstein to Hermann

Weyl, 16 December 1918 [Vol. 8, Doc. 673]).

[6]This is the first time that Einstein used this form of the field equations in his publications. Earlier,

instead of the term proportional to R on the left-hand side, he had written a term proportional to T on

the right-hand side (see, e.g., Einstein 1915i [Vol. 6, Doc. 25], eq. (6), Einstein 1918a [Doc. 1], eq.

(2), and Einstein 1918e [Doc. 4], eq. (1)).

[7]The tensor representing “gravitating mass” in the new theory proposed in this document does

depend on derivatives of the metric (see p. 353).

[8]In Doc. 19, [pp. 15–16], Einstein, following Weyl 1917, sec. 2, and Weyl 1918b, pp. 187–188,

derived the vanishing of this divergence from a variational principle. These relations can already be

found in earlier mathematical literature and are now known as the contracted Bianchi identities (see

Pais 1982, pp. 274–276, for historical discussion).

[9]

. See Einstein 1916e (Vol. 6, Doc. 30), pp. 798–799, for the definition of the

divergence of a mixed second-rank tensor. Eq. (2) is equivalent to .

[10]Einstein 1916b (Vol. 6, Doc. 27).

[11]In Poincaré 1906, so-called Poincaré stresses, conceived of as a pressure exerted on the electron

by the ether, were introduced as a stabilizing force in the purely electromagnetic model of the electron

given in Lorentz 1904a. The addition of the Poincaré stresses to the energy-momentum tensor for the

electron’s electromagnetic field guarantees that the divergence of the total energy-momentum tensor

vanishes. Similarly, in the theory presented in this paper, gravitational terms are added to the energy-

momentum tensor for the electromagnetic field to ensure that the divergence (as defined in eq. (2)) of

the total energy-momentum tensor vanishes. As in the case of the Lorentz-Poincaré electron, these

additional terms can be interpreted as pressure terms. For discussions of the Lorentz-Poincaré elec-

tron model, see Pauli 1921, sec. 63, Miller 1973, and Rohrlich 1973.

[12]Hilbert 1915. For a historical discussion of the relation between the theories of Mie and Hilbert,

see Corry 1999b.

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

[14]These field equations cannot be derived from a generally covariant Lagrangian that is simply

the sum of a term for the gravitational field and a term for matter as in Einstein 1916o (Vol. 6, Doc.

i

σ

–ggστTiτ ≡

–gTσi;σ 0 =