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 7 9

[17]Although the derivation of the geodesic equation is part of Einstein’s notes for the lectures of

the previous week (see [p. 5]), Reichenbach’s notes indicate that it was presented at this point in the

course.

[18]The overwrites in this expression show that Einstein originally wrote down the covariant deriv-

ative of and then changed it to the covariant derivative of the corresponding mixed tensor .

Below, expressions for the contraction of both these covariant derivatives are given, first for ,

then for . In the former expression, “ ” should be “ .” The latter expression

is rewritten for the special case that is symmetric. For a more detailed exposition of this topic,

see Einstein 1916e (Vol. 6, Doc. 30), pp. 798–799.

[19]There should be a factor of in the second term within the square brackets.

[20]See Einstein 1918h (Doc. 10), a review of Weyl 1918b, for Einstein’s comments on the impor-

tance of these contributions of Levi-Civita and Weyl. This is the first time that Einstein uses the con-

cept of parallel displacement in introducing the curvature tensor.

[21]In Reichenbach’s notes, the curvature tensor is introduced at a later point, just before the dis-

cussion of gravitational field equations (see [p. 14]). At that point, one finds the derivation that Ein-

stein began but did not finish at the foot of [p. 10], along with a similar derivation that can be found

on [p. 25]. On [p. 10], Einstein computes the components of a vector obtained through succes-

sive parallel transport of a vector along the infinitesimal line segments and . In the sec-

ond term in square brackets, “ ” should be “ ,” and in the second Christoffel symbol in the last

term, α and β should be interchanged. Subtracting from this expression the same expression with d

and δ interchanged, one finds the curvature tensor contracted with .

[22]In Einstein 1916e (Vol. 6, Doc. 30), p. 800, eq. (43), the expression to the left of the vertical line

is defined as the curvature tensor, and, in the notation used in this paper, would be called . The

notation introduced here is used for the negative of the same expression on [p. 25].

[23]The first line of this expression for should have an overall minus sign.

[24]The first line of the expression for to the left of the vertical line should be multiplied

by . On the second line, “ ” should be “ .”

[25]This definition of the Ricci tensor as agrees with Einstein 1916e (Vol. 6, Doc.

30), p. 801, where the Ricci tensor is defined as the negative of the expression to the left of the vertical

line (in the second term, “ν” should be “σ”).

[26]The introduction of general relativistic electrodynamics on [p. 12] follows Einstein 1916b (Vol.

6, Doc. 27). The same topic is covered in Einstein 1916e (Vol. 6, Doc. 30), sec. 30, pp. 812–815, but

only in coordinates for which In both papers, the notation for the covariant electromag-

netic field strength tensor is rather than as in these lecture notes.

[27]Here and in what follows, “ ” should be “ .”

[28]“ ” should be “ .”

[29]In Einstein 1916b (Vol. 6, Doc. 27), p. 187, eq. (8), the force density was defined with the

opposite sign. Here and in the following, “ ” should be “ .”

[30]

is the Lagrangian for the electromagnetic field.

[31]The law of energy-momentum conservation for a system of charges and their electromagnetic

field in a gravitational field is derived from the requirement that the action is invariant under

coordinate transformations , with on the boundary of the integration

domain. This type of derivation is discussed extensively in Klein, F. 1917, 1918a, whose name is men-

tioned in Reichenbach’s notes at this point. It can also be found in Weyl 1918b, as Einstein noted in a

letter to Klein (see Einstein to Felix Klein, 24 March 1918 [Vol. 8, Doc. 492]).

[32]The definition of the variation can be found in Weyl 1918b, p. 186. For an infinitesimal coor-

dinate transformation , . Rewriting this as

and evaluating the two terms in this expression to first order

in , one arrives at the equation for below (“ ” should be “ ”). The expression for

on the next line is obtained in a similar way.

Aμν Aμ ν

Aμν;ν

Aμ;ν ν Aαν αν –gAαν ≡

Aμν

–g

A″μ

Aμ dxμ δxμ

dxτ δxτ

Aσδxτdxλ

Bμνσ α

Rμ

νσ

α

Rμκ,νσ gακRμ

νσ

α ≡

Rμκ,νσ

1

2

-- -

κ σ

α

î þ

í ý

ì ü

κ σ

α

Rμσ –Rμ

ασ

α

g – 1. =

Fρσ ϕρσ

μν

μν

ν μ

μ

α

α

gμν, ϕμ, ν) (

dτ

ò

xμ x′μ xμ Δxμ + = → Δxμ 0 =

δ∗

x′μ xμ Δxμ + =

δ∗gμν

g′μν(xα) gμν(xα) – ≡ [g′μν(x′α)

g′μν(x′α) g′μν(xα)], – [ – –gμν(xα)]

Δxμ

δ∗gμν

Δx′μ Δxμ

δ∗ϕμ