'THE HAPPIEST THOUGHT OF MY LIFE' l8l
paths.' The trick of using three coordinate systems is ingenious. On the one hand,
S and S' are inertial frames and so one can use special relativity. On the other
hand, during a small time interval the measurements in S' can be identified with
those in E up to higher-order effects.
II. How do clocks in two distinct space points of E run relative to each other?
At t = T = 0, the two E clocks were synchronous with each other and with
clocks in S. The two points in E move in the same way relative to S. Therefore
the two E clocks remain synchronous relative to S. But then (by special relativity)
they are not synchronous relative to S' and thus, by (I), not synchronous relative
to each other. We can now define the time T of E by singling out one clock in
E—say, the one at the origin—and for that clock setting r = t. Next, with the
help of (I) we can define simultaneity in £ by using S': the simultaneity condition
of events 1 and 2 in E is
where, again, v = yt = yr. Let 1 correspond to the origin of E and 2 to a space
point (£,0,0) where the clock reading is called a. Introduce one last approximation:
the time T of S' — E coincidence is also taken small so that O(r^2 ) effects are
negligible. Then x 2 — xl = x' 2 — x( = |, £, =r, t 2 = ff, so that Eq. 9.2 becomes
a formula that is found—albeit derived differently—in modern textbooks.
The application of the equivalence principle to this equation is also familiar. It
says that for a resting frame in a homogeneous gravitational field in the | direction:
where $ is the gravitational potential energy difference between (£,0,0) and the
origin. [Note. Here and in what follows gravitational energy always refers to unit
mass so that $ has the dimension (velocity)^2 .]
Einstein at once turned to the physics of Eq. 9.4: 'There are "clocks" which are
available at locations with distinct gravitational potential and whose rates can be
controlled very accurately; these are the generators of spectral lines. It follows
from the preceding that light coming from the solar surface... has a longer wave-
length than the light generated terrestrially from the same material on earth.' To
this well-known conclusion, he appended a footnote: 'Here one assumes that [Eq.
9.4] also holds for an inhomogeneous gravitational field' [my italics]. This assump-
tion was of cardinal importance for Einstein's further thinking. He would explore
its further consequences in 1911.
- Maxwell's Equations; Bending of Light; Gravitational Energy = me^2.
Indomitably Einstein goes on. He tackles the Maxwell equations next. His tools
are the same as those just described for the red shift. Again he compares the
