Pentcho,
You asked that we look at things quantitatively, but you resort to quoting references and not a quantitative reply.
AUTHORITY. Talking of references, Einstein is the authority and not Brian Cox, Jeff Forshaw and David Morin. And Einstein as you have quoted and referenced before says light speed and frequency is reduced the nearer the path to a source of gravity. That is authority.
EXPERIMENT. For experiment, analysis of Pound and Rebka's experiment posted by me somewhere on this website shows that the frequency at the top of a tower or at a height, the frequency Fu will be higher than the frequency Fd at the bottom (subscripts u for Up, d for Down). Pound and Rebka's experiment showed in ALL their measurements that the frequency of the source is ALWAYS red and below that of the absorber in all cases, whether source is up or down (i.e. all minus sign). Why this should be so is left for other theories like 'tired light', 'extinction', etc. What is important here is that on the difference of averages, the frequency when the source is at the bottom, Fd is less than the frequency, Fu at the top of the tower by -5.13 x10^-15, which difference can therefore only be attributable to position in the gravitational field. However, despite reporting rightly that Fu is more than Fd is what was found experimentally, there is an erroneous (or mischievous) statement in the paper that light frequency increases as it falls, which is not corroborated by their own findings. This paper is now free to read and I have linked it several times, but I doubt if you want to spend time to refute what I have claimed or rather agree with quotes from Morin and others. The Gravity probe A gives results that also conform to the prediction from Authority, and I quote: "The 100 kg Gravity Probe A spacecraft housed the atomic hydrogen maser system that ran throughout the mission, and a microwave repeater to measure the Doppler shift of the maser signal. The satellite was launched nearly vertically upward to cause a large change in the local gravity seen by the maser, reaching a height of 10,000 km (6,200 mi). At this height, general relativity predicted a clock should run 4.5 parts in 10-10 faster than one on the Earth...Gravity Probe A confirmed the prediction that gravity slows the flow of time,...".
QUANTITATIVE 1. When an object is at a height, it has potential energy. The higher the altitude, the higher the gravitational potential energy, i.e. mgh or - GMm/h
This may be where there may be confusion in signage. There is no doubt that at a higher altitude, when h is higher, Potential energy is higher. Both mgh and -GMm/h indicate this. Now, from Newtonian mechanics, g = GM/h^2. But if substituted into mgh, without paying attention to the conventional change of sign, mgh may become GMm/h, which erroneously indicates that as h reduces in GMm/h, Potential energy is increasing and as h increases Potential energy is reducing. A minus sign resolves this, so the Potential energy of a mass in orbit or in a gravitational field is - GMm/h. Potential energy tends to infinity at infinite altitude and reduces with height in the gravitational field. Also may be better to use r instead of h.
Now, when an object drops or falls under gravity, it loses Potential energy, P.E. while its Kinetic energy, K.E. increases.
P.E. = - GMm/r, while
K.E. = GMm/2r = 1/2(mv^2)
Here, I quote from my old physics textbook by Nelkon and Parker:
"Hence, the kinetic energy of the satellite (...falling object) INCREASES when it falls to an orbit of smaller radius, that is the satellite is speeded up. This apparent anomaly is explained by the fact that the P.E. decreases by twice as much as the K.E. increases from the formulae. Thus on the whole there is a LOSS of energy, as we expect".
In summary, when objects fall, overall energy is lost and radiated away as heat. Can light fall likewise and lose energy? That is the question.
In the theory that light is particle, its energy, E is related to its frequency by E = hf, where h is Planck's constant. Therefore on this assumption that light is particle, when its energy falls like that of other falling particles, its frequency, f will fall as well. But there is a confusion.
QUANTITATIVE 2. When particles fall, their kinetic energy increases and therefore speed increases, so would light speed also increase based on this premise and support your position? Sounds reasonable. But does light have potential energy in a gravitational field that can be converted to kinetic energy? Can light be at rest in the field? Indeed, can something that is massless, i.e. m = 0, even have potential or kinetic energy in the real sense?
COMMON SENSE: Apart from speed, does a stone and other known particles falling to Earth have an associated frequency or wavelength to its fall? If not, then particles can fall under gravity without necessarily having a frequency attached to the event. If something is therefore suspected of being a particle, but it has an associated change in its frequency then perhaps it is not a particle, since other particles do not behave likewise by having an associated frequency. In Pound and Rebka's paper, they referred to the "apparent weight" of photons, but today Quantum Mechanics tells us photons have no mass. What is the implication then of this QM pronouncement of having no mass/weight on the Pound and Rebka assertion/ findings/ interpretation? It is for this and other reasons that I feel that emission/ particle theory will not endure eventually. Rather, the wave picture and a better understanding of it seems best.
EMPIRICAL EVIDENCE. If the wave picture prevails, then there must be a medium through which light propagates, this medium being a form of matter or empty space. If it is of matter, surely its density must be enhanced nearer the Earth's surface than at a height and this must influence light propagation. Going by the evidence from water, air, glass, etc, that the higher the matter density, the slower the speed of light, then light speed will reduce as it falls to Earth and conversely increase as it rises in the gravitational field. It is my proposal that such matter than can account for Einstein's prediction exists in the form of 'dark matter'.
Just in case, Steve is reading, if you look at the twists and turns made in the derivation of the deflection angle, 4GM/c2R you will see evidence of fraud and working towards a known answer. In what unit is 4GM/c^2R by dimensional analysis? When light enters a gravitational field, it is deflected. What of when it exits and leaves? Fermat's principle and the principle of reversibility of light's path imply light must bend again on exit, this time in the opposite direction to that on entry, was this taken into account? Observed deflection in Sobral, Brazil was about 1.98 arc seconds, more than two error bars above the predicted 1.75". How does this support GR? Most optical measurements (more than 12) till date are above 1.75" and average ~1.90", and C.M. Will in his review paper terms those high values anomalous, despite being preponderant (I can give references if needed). Radio measurements today use software to screen out 'anomalous' data and claim they have improved accuracy of the GR prediction to 0.00003% or something like that.
In summary, Steve, don't trust without checking is what science has become today unfortunately. Or as Reagan legacy quote: "Trust, but verify", which he is reported to always say at every meeting with Gorbachev during his tenure.
Regards,
Akinbo
