THE PRAGUE PAPERS 199Next determine v(2,2), the frequency of the same* spectral line produced at 2,
measured with U 2. One will find (Einstein asserts) that i»(l,l) = v(2,2), 'the fre-
quency is independent of where the light source and the [local] clock are placed.'
[Remark. This statement is not true in all rigor: even though we still cannot
calculate the displacement of spectral lines caused by local external gravitational
fields (we have no theory of quantum gravity!), we do know that such a displace-
ment must exist; it should be small within our neighborhood.]
Global Experiments. Determine v(2,l), the frequency of the same spectral
line produced at 2 but now measured at 1 with U]. As Eq. 11.4 implies, v(2,)
¥= c(l,l). Yet, Einstein insists, we should continue to accept the physical criterion
that the number of wave crests traveling between 2 and 1 shall be independent of
the absolute value of time. This is quite possible since 'nothing forces us to the
assumption that the ["gleich beschaffene"] clocks at different gravitational poten-
tials [i.e., at 1 and at 2] should run equally fast.' (Recall that the synchronization
was achieved in the factory.)
The conclusion is inevitable: the compatibility of Eq. 11.4 with the physical
criterion implies that the clock U 2 in 2 runs slower by a factor (1 + 0/c^2 ) than
Ut in 1. This is, of course, compatible with v(2,2) = v(l,) since the spectral
frequency in 2 also decreases by the same factor. After all, the spectral line is
nothing but a clock itself. In other words, as a result of the transport to places of
different gravitational field strength, clocks become 'verschieden beschaffen,' dif-
ferently functioning. This leads to a 'consequence of... fundamental significance':
where c, and c 2 are the local light velocities at 1 and 2 (the difference between c,
and c 2 is assumed to be small, so that the symbol c in Eq. 11.6 may stand for
either c, or c 2 ). Thus Einstein restored sanity, but at a price. 'In this theory the
principle of the constancy of light velocity does not apply in the same way as in
. .. the usual relativity theory.'
The final result of the paper is the application of Eq. 11.6 to the deflection of
a light ray coming from 'infinity' and moving in the field of a gravitational point
source (i.e., a \/r potential). From a simple application of Huyghens' principle,
Einstein finds that this ray when going to 'infinity' has suffered a deflection a
toward the source given (in radians) by
where G is the gravitational constant, M the mass of the source, A the distance
of closest approach, and c the (vacuum) light velocity. For a ray grazing the sun,
*I trust that the term the same will not cause confusion.