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

faces, this law is as fundamental as the validity of the theory of special relativity in

infinitesimal domains of space-time is for the theory of general relativity.

18. Influence of Gravitational Fields upon the Rate of Clocks.

Redshift of Spectral Lines

According to the theory of general relativity the equivalence of coordinate systems

is not limited to those of nonuniform parallel translation (rotation-free motion). If

the theory should eliminate the epistemological deficiency of classical mechanics

that was mentioned above, then it must be capable of treating every coordinate sys-

tem, whatever its state of motion relative to others may be, as “at rest,” i.e., the gen-

eral laws of nature must be expressed by identical equations relative to all systems,

whichever way they are moving.

Again, we start with a domain that has no

gravitational field relative to a coordinate system .

Therefore, is an “inertial system” in the sense of

classical mechanics. Next we introduce a second co-

ordinate system that uniformly

rotates[45]

relative

to ; we symbolize this systems as a circular disk

uniformly rotating relative to (see fig. 3). We now

imagine two identically constructed clocks where

one is located in the center of the disk, the

other on its periphery, such that it partakes in the rotational motion. Judging

the rate of these clocks from (i.e., from the point of view of the nonrotating sys-

tem), it is clear from a result of the theory of special relativity, which is valid in ref-

erence to , that runs more slowly than , because has a velocity relative

to (against the paper), whereas has not. That runs slower than should

also be noticed by an observer sitting on the disk (say next to ); therefore

runs more slowly than when judged from .

According to the theory of general relativity, we can also view the system as

“at rest.” However, then we have to perceive the field of centrifugal forces existing

relative to as a (real) gravitational field that acts upon all masses that are at rest

relative to in proportion to their masses. (For the sake of completeness, we have

to add that this gravitational field does not only consist of this 〈gravitational〉 cen-

trifugal field, but it also has other components that act upon moving

masses).[46]

When judged from the two clocks are situated at different points in a gravita-

tional field, and the latter is the cause that the two clocks run at different rates.

U2

U1

Fig. 3

K

K

K′

K

K

U1) (

U2) (

K

K U2

[p. 27]

U1 U2

K U1 U2 U1

U1 U2

U1 K′

K′

K′

K′

K′