1.3.4 Gravitational redshiftThe previous paragraph, explained how the speed of
light stays constant whatever happens to the spacetime
medium. In this paragraph it is shown that if the
frequency of light is also assumed to be constant, then
some consistency between the gravitational redshift
formula (Equation 1.0) and spacetime arises. At the
end of the day, frequency is the rate at which the
electromagnetic field changes direction, hence, if the
speed at which the electromagnetic field travel through
space does not change, than why should the frequency. In
an antenna, the current is forced to alternate the
direction to produce changes in the electromagnetic
field, and therefore propagate through space. The
frequency at which the current alternates is determined
by the time constants within the oscillator device, which
in turn depend on spacetime. However, once the
electromagnetic wave leaves the antenna, that frequency
is held throughout the entire journey. Thus, supposing
that the frequency is constant is not a totally
unjustifiable assumption. To look it more closely, in Figure 1.9 it is reported
one small square only, and in it, there are also two
sinwave with different frequency. This has been done to
show that for a given gravitational field, the redshift
caused by it, is proportional to the frequency as
expected by Equation 1.0. Thus, looking at the top wave
first and measuring its frequency with the bottom non
stretched second, it results 4 Hz, whilst the bottom
signal, measured with the same second results 1 Hz. Now,
repeating the measurements using the top stretched
second, as the two signals approach the mass, it is
possible to see that the top signal is now 5 Hz, whilst
the bottom is 1.25 Hz. Reassuming, the top signal was 4
Hz and has become 5 Hz, hence, Df
is 1 Hz, at the same time, the bottom signal was 1 Hz and
has become 1.25 Hz hence Df is
¼ Hz. This clearly agrees shows that the higher the
frequency the higher the shift caused on the signal in
agreement with the previous statement about Equation
1.0,
and also, again as expected, the ratio Df/f is constant for both
signals.
which can be proved in the same way as the frequency, but substituting
the energy of the photon hf by hc/l. The fact that this time there is the final
wavelength (l2) rather than the initial (f1)
as in Equation 1.0 does not make any difference as it is simply a matter
of choosing which end is the transmitter. The positive sign means that
as the frequency increases, the wavelength decreases. So, still referring
to Figure 1.9, but considering the square to be one meter instead of one
second, the top signal wavelength, would measure 0.25 meters, and the
bottom signal 1 meter. Then, getting closer to the mass, the same meter
would stretch to become slightly longer as shown with the top double arrow
headed second. With that meter, the top signal measures 0.2 meters, and
the bottom signal measures 4/5 m or 0.8 meters. Again, for the top signal
where l was
0.25 meters, Dl=0.250.2=0.05m
and for the bottom signal, where l
was 1 meter, Dl=10.8=0.2m,
which prove that the longer the wavelength the greater the shift as expected
by the formula. Moreover, the ratio Dl/l2=0.25
is always constant and so is Dl/l1=0.2 as expected. 
Figure 1.9  One
small square of Figure 1.8, is reported to show
consistency with the gravitational redshift equation.
