1.3.4 Gravitational redshift

The 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.
Figure 1.8, shows a cross section of the latticework across the mass at the centre, and a signal coming from outer space whose frequency is left constant throughout the gravitational field. As spacetime in the vicinity of large masses is stretched, than both units used to measure time or space are also stretched, hence, either the clock for frequency or the meter for the wavelength, would measure a higher frequency or a shorter wavelength respectively. Considering time for a start, and assuming that each small square in Figure 1.8 represents one second, from the figure it is possible to see that with the same second two different frequencies are measured depending how close the observer is to the mass.
 

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. 
In order to be absolutely sure that the model agrees completely with the formulas, a further check needs to be done by, this time, using space instead of time. To do this, it is simply necessary to measure the wavelength rather than the frequency, and prove that the redshift is also proportional to the wavelength, as:

(1.3)

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=0.25-0.2=0.05m and for the bottom signal, where l was 1 meter, Dl=1-0.8=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. 
As it has been shown, the three dimensional latticework model for gravity, could really be mathematically described by the same formulas used for the gravitational redshift with the only assumption that frequency is not affected by gravity as for the speed.

Figure 1.9 - One small square of Figure 1.8, is reported to show consistency with the gravitational redshift equation.