1 7 8 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

[3]These gravitational redshift calculations closely follow Einstein 1911h (Vol. 3, Doc. 23), sec. 3,

pp. 491–494.

[4]The angle α is the angle of aberration for light propagating horizontally and registered by an

observer in the vertically accelerated reference frame.

[5]These bending-of-light calculations closely follow Einstein 1911h (Vol. 3, Doc. 23), sec. 4, pp.

496–498. In Reichenbach’s notes, it is mentioned at this point that the formula for α derived below

gives a value of 0.85″ for light grazing the surface of the sun, that a more exact calculation gives 1.7″,

and that the result will be tested by Eddington on 31 May 1919.

[6]As can be inferred from Reichenbach’s notes, this “confirmation” (“Bestätigung”) refers to the

following: in accordance with special relativity, the rate ν of a clock on the circumference of a disk

of radius r rotating at a small angular velocity ω is lower than the rate of a clock at rest in the

center by—to first-order approximation—a factor . If, according to the equivalence prin-

ciple, the centrifugal force is interpreted as a gravitational force, then the potential difference Φ

between the center and the circumference in this gravitational field is . It then follows that

the rate of the rotating clock is lower than the rate of the clock at rest by approximately the factor

, in accordance with the result found above for uniform accelerations. In an appendix written

in 1920 for Einstein 1917a (Vol. 6, Doc. 42), Einstein used this argument from rotation rather than

the argument from uniform acceleration to give a simple derivation of the redshift formula (see p. 89).

Reichenbach’s notes contain a more extensive discussion of the rotating disk, showing that in his 1919

lectures Einstein also drew attention to some problems with the above argument: the potential for the

centrifugal force does not satisfy the Poisson equation, and the Coriolis force is not taken into

account. The conclusion drawn from these observations is that, if the equivalence principle is to hold

for rotation, the gravitational field cannot be represented by a scalar potential such as in Newtonian

theory. As in Einstein 1916e (Vol. 6, Doc. 30), p. 775, the example of the rotating disk is used further-

more to argue for a generalization of special relativity. For a historical discussion of the importance

of such considerations in the genesis of general relativity, see Stachel 1980.

[7]For the published version of this so-called point-coincidence argument, see Einstein 1916e (Vol.

6, Doc. 30), p. 776.

[8]For a more elaborate statement about the heuristic importance of general covariance, see Ein-

stein’s response to Kretschmann 1917 in Einstein 1918e (Doc. 4), p. 242.

[9]The fourth term should read “ .”

[10]The interior brackets around the +Φ are in the manuscript. For practical purposes they should

be ignored, so that the equation reads as the integral of .

[11]This special case of a variable speed of light corresponds to the theory for static gravitational

fields in Einstein 1912c (Vol. 4, Doc. 3) and Einstein 1912d (Vol. 4, Doc. 4). In a note added in proof

to the second paper (p. 162), Einstein first gave the derivation given here. This was an important step

toward a more general theory of gravitation based on the metric tensor. (For further discussion, see

Vol. 4, the editorial note, “Einstein’s Research Notes on a Generalized Theory of Relativity,” sec. II,

pp. 193–194.)

[12]The indices written under and over other indices indicate where Einstein is changing the names

of summation variables to obtain a common factor of .

[13]The indices within square brackets stand for Christoffel symbols of the first kind.

[14]The indices within curly brackets stand for Christoffel symbols of the second kind. For this der-

ivation of the geodesic equation, see Einstein 1916e (Vol. 6, Doc. 30), sec. 9, pp. 790–791. The quan-

tity w is defined as .

[15]In the manuscript, these considerations for the special case of orthogonal transformations

appear in the left margin.

[16]“ ” at the end of the line should be “ .”

ν0

1

1ω2r2

2

------------- -

c2

- –

1

2

- –--ω2r2

1

Φ

c2

---- - +

dX4

2

L Φ +

δxν

ds dσ ⁄

aν Aν