l8o RELATIVITY, THE GENERAL THEORY
stein's starting question, 'which must occur to everyone who has followed the
applications of the relativity principle.' Then he gives the standard argument. A
reference frame E, is accelerated in the x direction with a constant acceleration
- A second frame E 2 is at rest in a homogeneous gravitational field which imparts
an acceleration —7 in the x direction to all objects. 'In the present state of expe-
rience, we have no reason to assume that... E, and E 2 are distinct in any respect,
and in what follows we shall therefore assume the complete [my italics] physical
equivalence of a gravitational field and the corresponding acceleration of the ref-
erence frame [E,]. This assumption extends the principle of relativity to the case
of uniformly accelerated motion of the reference frame.' Einstein noted that his
review was not the place for a thorough discussion of the questions which now
arose. Nevertheless, he made a beginning by applying his new postulate to the
Maxwell equations, always for uniform acceleration. He did not raise the question
of the further extension to nonuniform acceleration until 1912, the year he first
referred to his hypothesis as the 'equivalence principle' [E7],
- The Gravitational Red Shift. Many textbooks on relativity ascribe to Ein-
stein the method of calculating the red shift by means of the Doppler effect of light
falling from the top to the bottom of an upwardly accelerating elevator. That is
indeed the derivation he gave in 1911 (Chapter 11). However, he was already
aware of the red shift in 1907. The derivation he gave at that time is less general,
more tortured, and yet, oddly, more sophisticated. It deserves particular mention
because it contains the germ of two ideas that were to become cornerstones of his
final theory: the existence of local Lorentz frames and the constancy of the velocity
of light for infinitesimally small paths. The argument, restricted to small velocities,
small uniform accelerations, and small time intervals, runs as follows.
Consider two coordinate systems S (x,y,z,t) and E (£,i7,f,r) which at one time
are coincident and which both have velocity v = 0 (the symbols in parentheses
denote the respective space-time coordinates). At that one time, synchronize a
network of clocks in S with each other and with a similar network in E. The time
of coincidence of S and E is set at t — r = 0. System S remains at rest, while
E starts moving in the x direction with a constant acceleration 7. Introduce next
a third system S' (x',y',z',tf) which relative to S moves with uniform velocity v
in the x direction in such a way that, for a certain fixed time t, x' = £, y' = 77,
z' = f. Thus, v = yt. Imagine further that at the time of coincidence of S' and
E all clocks in S' are synchronized with those in E.
I. Consider a time interval o after the coincidence of S' and E. This interval is
so small that all effects O(5^2 ) are neglected. What is the rate of the clocks in S'
relative to those in E if 7 is so small that all effects 0(y^2 ) can also be neglected?
One easily sees that, given all the assumptions, the influence of relative displace-
ment, relative velocity, and acceleration on the relative rates of the clocks in E and
S' are all of second or higher order. Thus in the infinitesimal interval 5, we can
still use the times of the clocks in the local Lorentz frame S' to describe the rate
of the E clocks. Therefore, 'the principle of the constancy of the light velocity can
be ... used for the definition of simultaneity if one restricts oneself to small light
