1 4 0 D O C . 3 1 I D E A S A N D M E T H O D S

lation shows that a light ray passing closely by a celestial body of the size of the

sun must suffer a deflection in the order of magnitude of one second of arc. It is,

after all, this result the two English expeditions confirmed last year. However, it

should be pointed out here that this simple consideration provides numerically only

half of the actual value of

deflection.[43]

This is connected to the fact that the grav-

itational field in the theory of general relativity is not represented just by a vector

field but rather by a formally more complicated structure, such that the transition

from the parallel field—that obtains in reference —to the spherically symmetric

field of a celestial body is not as simple as it might appear at first glance. This will

be clarified later.

Second, this consequence shows that the law of the constancy of the speed of

light no longer holds, according to the general theory of relativity, in spaces that

have gravitational fields. As a simple geometric consideration shows, the curvature

of light rays occurs only in spaces where the speed of light is spatially variable.

From this it follows that the entire conceptual system of the theory of special rela-

tivity can claim rigorous validity only for those space-time domains where gravita-

tional fields (under appropriately chosen coordinate systems) are absent. The

theory of special relativity, therefore, applies only to a limiting case that is nowhere

precisely realized in the real world. Nevertheless, this limiting case ⟨also⟩ is of fun-

damental significance for the theory of general relativity; because the fact from

which we started out, namely that no gravitational field exists in the immediate vi-

cinity of a free-falling observer, this very fact shows that in the vicinity of every

world point the results of the theory of special relativity are valid (in the infinites-

imal) for a suitably chosen local coordinate system.

This connection can be illustrated with a geometrical comparison from the the-

ory of surfaces that turned out to be of decisive significance for the ⟨finding and⟩

implementation of the

theory.[44]

Metric relations in the plane are described by two-

dimensional Euclidean geometry; i.e., the constructions of Euclidean geometry can

be executed by means of a compass and ruler such that a compass (set to a certain

way) plays the role of a fixed distance and the ruler that of a straight line. If one

wants to do geometry on a curved surface, e.g., on a sphere or an ellipsoid, instead

of in the plane, then there are certain laws of construction with the compass (and a

ruler osculating to the surface), but these laws are no longer expressed by those of

the Euclidean geometry of two dimensions. In its geometrical properties, a small

piece of the surface better approximates those of the plane the smaller it is ⟨an in-

finitesimally small piece of the curved surface approximates in its properties, with-

out limit, those of an infinitesimal piece of the plane⟩. The Euclidean ⟨plane⟩

geometry of two dimensions applies to constructions with a compass and ruler on

curved surfaces in infinitesimal domains. For Gaussian geometry of curved sur-

K′

[p. 26]