198 RELATIVITY, THE GENERAL THEORYequivalence principle only. Then, after the tensor equations of general relativity
have been derived and the equivalence principle has been understood to hold
strictly only in the small, the red shift is returned to and a proof is given that it is
sufficient for the derivation of the previous result to consider only the leading
deviations of g^ from its flat-space-time value. If the text is modern enough, one
is treated next to the niceties of second-order effects and to the extreme cases where
expansions break down. All this should be remembered in order to grasp better
Einstein's plight in 1911. He knows that special relativity is to be incorporated
into a more profound theory, but he does not know yet how to do that. With care
he manipulates his three coordinate systems in order to obtain Eqs. 11.1-11.4. He
knows very well that these equations are approximations, but he does not know
to what.
THE BENDING OF LIGHT
What and how can we measure? That prime question of science has a double
entendre. First of all it means, What is conceptually interesting and technically
feasible? Taken in that sense, Einstein's remarks on the red shift and the deflection
of light had given direction to the phenomenology of general relativity even before
that theory existed. The question has also a second meaning, What is a meaning-
ful measurement as a matter of principle? Also in that sense Einstein had con-
tributed by his re-analysis of simultaneity in 1905. In 1907 the study of the Max-
well equations in accelerated frames had taught him that the velocity of light is
no longer a universal constant in the presence of gravitational fields. When he
returned to this problem in 1911 he left aside, once again, these earlier dynamic
considerations. Instead, he turned to the interpretation of Eq. 11.4.
'Superficially seen, [this equation] seems to state something absurd. If light is
steadily transmitted from S 2 to S,, then how can a different number of periods per
second arrive at S, than were emitted in S 2? The answer is simple, however.' The
apparent trouble lay not with the number of periods but with the second: one must
examine with the greatest care what one means by the rate of clocks in an inho-
mogeneous gravitational field. This demands an understanding of the following
three facts of time.
The Clock Factory. One must first construct 'gleich beschaffene Uhren,' iden-
tically functioning clocks, to use Einstein's language. He does not state how this
is done. However, his subsequent arguments make sense only if the following
procedure is adopted. Construct a clock factory in a (sufficiently small) region of
space in which the gravitational field is constant. Synchronize the clocks by some
standard procedure. Transport these clocks, one of them (U,) to a position 1,
another one (U 2 ) to a position 2, etc.
Local Experiments. Observe the frequency of a spectral line generated at 1
with the clock Uj. Call this frequency v(l,l) (produced at 1, measured with U,).