1 4 2 D O C . 3 1 I D E A S A N D M E T H O D S

Quantitatively, one can conclude from this example that the rate of one and the

same clock is proportional to , where is the gravitational potential at the

location of the clock.

From this follows that a clock on the surface of a celestial body runs more slowly

than the same clock when it floats (at rest) somewhere out in space or when it sits

on the surface of a smaller celestial body. Every system is to be considered as a

“clock” which by virtue of internal laws and periodically occurring processes is en-

dowed with a specific frequency, that is, e.g., an atom that can emit or absorb a cer-

tain spectral line. When compared to the spectral lines generated by an element on

earth the spectral lines generated or absorbed by the same element at the surface of

the sun then must show a shift toward the red that is equivalent to a Doppler effect

of about 0.6 km/sec. It still is doubtful if this necessary consequence of the theory

is realized in nature; but according to recent investigations by two physicists at

Bonn, Grebe and

Bachem,6[47]

the reliable verification should be expected soon.

It should be noted that the law just mentioned can also be derived from the spe-

cial case discussed above, that is in a uniformly accelerated translatory motion

instead of being in rotation.

That identically constructed clocks, at rest relative to each other in a gravitation-

al field, run at different rates at different locations is, in principle, of great principal

significance to us because from this fact it follows that in the theory of general rel-

ativity, time cannot simply be measured by identically constructed, suitably adjust-

ed clocks that are at rest relative to each other, as is the case in the theory of special

relativity. The direct physical meaning of time is thereby lost. In the following, we

shall show that a corresponding situation exists with respect to spatial coordinates

such that again we are confronted by the necessity to revise the physical interpre-

tation of space and time.

19. Invalidity of Euclidean Geometry in the Theory of General Relativity

If an observer measures the circumference and the diameter of a circular disk

(at rest) with a measuring rod practically infinitesimal compared to , then the

ratio from the two measurements is .This result can be taken as safe-

6

Einstein left space here for a footnote, presumably to fill in the location of this refer-

ence later, but never did so.

1

Φ

c2

---- - +

Φ

K′

[p. 28]

U D

D

U

D

--- - π 3.14…) (