On the Pound-Rebka results

Just a short blog post demonstrating the consistency between the proposed model of relativistic spatial dilation and the results found by Pound and Rebka.

Let's take a closer look at the results found by Pound and Rebka1 back in 1959. Recall that they conducted their experiment in a manner which allowed relative motion to precisely offset the gravitational Doppler shift over a height, , which happened to be around meters. By separating the emitter from the receiver by this distance approximately inline with the Earth's radial vector at that position and moving the apparatus along this same radial vector they were able to pit Einstein's gravitational Doppler shift against his theory of relativistic motion.

They found a result of

which follows with the equivalence principle preciously. If instead of accepting that light gains or loses energy depending upon it's path through a gravitational field, what if the light actually is unaffected, but it is space that is in motion?

In the context of relativistic spatial dilation

Let's now consider this result on the context of the relativistic model that inspired the development of ULLD. Recall that the equivalence principle in this model is attributed to the dilation of space, integrated over the radial vector . This radial vector however, would coincide with the Earth's surface at that position, which places the emitter at .¹

The integral to achieve the equivalence principle is simply the derivative of , reintegrated over . The advantage to this approach in other contexts was to permit the result to be used as a scalar. By taking the average of the integral of , we produce a quantity that can be used in the same manner that is used.

In this case however, we want to find the integral over . Because is so small in comparison to and , the difference will be negligible. We can find that as

This means that for every tiny shift in time, , the radius will lengthen by the formula given above. Because the Earth's surface in this model would be rising up to meet the photon as it moves towards the Earth's surface from the emitter at , the Doppler shift measured would follow an identical mechanism to the Newtonian Doppler shift, which is preciously what we see as:

This leaves the energry state of light unchanged by gravity, creating a Doppler shift and shift in due to the relative motion of the surface of Earth alone.