D O C . 2 0 R E L AT I V I T Y L E C T U R E N O T E S 1 8 9

[7]A factor 2 is omitted in the second term of this expression.

[8]In this equation as well as in the next one, “ds” should be “ .”

[9]The integral below is the first line on the verso of the document. The integral taken from the

value of at aphelion to the value of ρ at perihelion gives π plus the perihelion advance during

half a revolution. The result given here is twice that amount, giving the perihelion advance for a full

revolution. The quantities a, T, and e are the semimajor axis, the period, and the eccentricity of the

planetary orbit, respectively. The result was first given in Einstein 1915h (Vol. 6, Doc. 24).

[10]Hugo von Seeliger. In Seeliger 1895, the author argued that if Newton’s law of gravitation holds

exactly, only a finite region of the universe can have a nonvanishing matter density. Seeliger suggested

a modification of Newton’s law to escape this conclusion. Einstein had become acquainted with

Seeliger’s work only after he wrote Einstein 1917b (Vol. 6, Doc. 43) (see Einstein 1919b [Doc. 18],

p. 433, footnote).

[11]If the expressions for the components of the Ricci tensor for a static metric,

given at the beginning of the document, reduce to and . Einstein derived

this same contradiction in his lectures on general relativity in Berlin earlier in 1919 (see Doc. 19,

[p. 22]).

[12]If the cosmological term is added to the left-hand side of the field equations, the equa-

tions do allow a solution in which the matter distribution is static. This modification of the field equa-

tions and the solution describing a static universe with spherical spatial geometry were first

introduced in Einstein 1917b (Vol. 6, Doc. 43). In his course on general relativity in Berlin earlier in

1919, Einstein modified the field equations in a way that is equivalent to the introduction of the cos-

mological term following Einstein 1919a (Doc. 17) (see Doc. 19, [pp. 21–23]; for discussion, see

Doc. 19, note 72).

ds2

ρ 1 r ⁄ ≡

g44 f

2

1, = =

Rικ Ρικ = R44 –fΔf 0 = =

–λgik