D O C . 1 9 R E L AT I V I T Y L E C T U R E N O T E S 1 8 3

of being independent of the surface element chosen. That λ is independent of is then derived from

this assumption. In other words, as is stated explicitly in Reichenbach’s notes, isotropy implies homo-

geneity. In Einstein’s own notes, a derivation of this result can be found in the margin farther down

on this page (see notes 80 and 81).

[78]With this substitution, the left-hand side of the equation above becomes the contraction of two

expressions antisymmetric both in i and κ and in l and m.

[79]In contracting the equation above, Einstein specializes to the case of a three-dimensional space

relevant for his cosmological purposes.

[80]In the original, the next few lines—from “3 dim” to “ ”—appear in the left-hand

margin. This is Einstein’s proof that isotropy implies homogeneity. This theorem was first derived in

Schur 1886 (see Pauli 1921, sec. 23).

[81]This equation is obtained by combining the two preceding equations, the second of which is the

contraction of the first. Because of the contracted Bianchi identities, the divergence of the left-hand

side reduces to the divergence of the last term, which is given on the next line (see [p. 10] for Ein-

stein’s definition of the divergence of a symmetric tensor). On the following line, the second term of

this expression is rewritten (the factor ½ should be 1). It follows that λ must be independent of .

[82]The comparison is between the equation (with ) above and the

equation in the box at the foot of [p. 22], which, in the notation used on [p. 23], can be written as

.

[83]In the manuscript, the next six lines appear to the right of the preceding five lines, separated

from them by a vertical line. In these six lines, Einstein derives an expression for the spatial part of

the metric of his cosmological model by embedding the three-dimensional spherical space in a four-

dimensional Euclidean space (see Einstein 1917b [Vol. 6, Doc. 43], p. 150, eq. (12)).

[84]As in Einstein 1919a (Doc. 17), is set equal to the energy-momentum tensor for the elec-

tromagnetic field.

[85]The factor p on the right-hand side should be multiplied by .

[86]The following considerations on the role of gravitational fields in the stability of charge-carry-

ing particles can be found in Einstein 1919a (Doc. 17), p. 352.

[87]Substituting into the field equation with pressure term above, one recovers the

modified field equations introduced in the first box on [p. 22] for the special case that is the

energy-momentum tensor for the electromagnetic field.

[88]Below, the main steps in Weyl’s derivation of the Schwarzschild-Droste solution are given

(Weyl 1918b, pp. 199–202; see also Pauli 1921, sec. 58β). In Einstein 1922c (Doc. 71), Einstein also

followed Weyl’s derivation, which he called “particularly elegant” (“besonders elegant”; p. 60). Since

the line element is static, it has the form given at the foot of [p. 21]; since it is spherically symmetric,

its spatial part ( ) has the form given below (the minus sign in front of should be a plus sign),

and the functions f and l depend only on . To determine these functions (the latter

through the auxiliary function h), the Christoffel symbols for are computed (the minus sign in

front of should be a plus sign) and inserted into the Lagrangian for the metric field,

which is defined on [p. 15]. The functions f and h are found by imposing the condition that the vari-

ation of the integral of —or H, as it is called by Weyl—over a spherical shell vanishes. The

definition of the auxiliary quantity w that is introduced in this context should include the factor

coming from the volume element (proportional to ) of a thin slice of this shell.

[89]Inserting into this expression for m, one recovers the expression for m given at

the head of [p. 21].

[90]The material below, which in the manuscript appears in the right-hand margin, gives the main

steps in Weyl’s derivation of the formula for the perihelion advance of a planet (Weyl 1918b, pp. 202–

205). From conservation of energy (see the first line below) and conservation of angular momentum

(see the next two lines; “ ” should be “ ”), a differential equation is derived for the

motion of the planet in polar coordinates r (or, rather its inverse ) and ϕ. Integrating dϕ

between the extremal values and of ρ, one finds the angle between perihelion and aphelion.

Twice the difference between this angle and π gives the perihelion advance per revolution (see the

formula in the box below, which can also be found on [p. 21]). For Mercury, this formula gives an

advance of 43″ per century.

xi

λ konst. =

xi

Rim 2λgim + 0 = λ 1 a2 ⁄ =

Rim

–κρgim =

Tiκ

giκ

p 4κ ⁄ –R =

Tiκ

–γij δiκ

r x1 2 x2 2 x3 2 + + ≡

γij

2lrδiκ

∗

–g ⁄

∗

–g ⁄

r2

Δr2dr

κ 8πK c2 ⁄ =

dxi ds ⁄ d2xi ds2 ⁄

ρ 1 r ⁄ ≡

ρ1 ρ2