Bingo the Einsteiniano: As light falls in a gravitational field, its energy increases by exactly the same fraction that it increases for any other thing we could imagine dropping but the analogy stops here - the speed of light does not increase at all. No it doesn't. Why? Simply because Divine Albert said the speed of light is constant, yes we all believe in relativity, relativity, relativity:
http://www.amazon.com/Why-Does-mc2-Should-Care/dp/0306817586 Why Does E=mc2?: (And Why Should We Care?), Brian Cox, Jeff Forshaw, p. 236: "If the light falls in strict accord with the principle of equivalence, then, as it falls, its energy should increase by exactly the same fraction that it increases for any other thing we could imagine dropping. We need to know what happens to the light as it gains energy. In other words, what can Pound and Rebka expect to see at the bottom of their laboratory when the dropped light arrives? There is only one way for the light to increase its energy. We know that it cannot speed up, because it is already traveling at the universal speed limit, but it can increase its frequency."
Bingo the Einsteiniano: Divine Albert predicted that, as light falls in a gravitational field, its speed increases twice as fast as predicted by Newton. Why? Simply because Divine Albert is twice as great as Newton, yes we all believe in relativity, relativity, relativity:
http://www.speed-light.info/speed_of_light_variable.htm "In the presence of gravity the speed of light becomes relative. To see the steps how Einstein theorized that the measured speed of light in a gravitational field is actually not a constant but rather a variable depending upon the reference frame of the observer: 'On the Influence of Gravitation on the Propagation of Light', Annalen der Physik, 35, 1911. Einstein wrote this paper in 1911 in German. It predated the full formal development of general relativity by about four years. You can find an English translation of this paper in the Dover book 'The Principle of Relativity' beginning on page 99; you will find in section 3 of that paper Einstein's derivation of the variable speed of light in a gravitational potential, eqn (3). The result is: c'=c0(1+phi/c^2) where phi is the gravitational potential relative to the point where the speed of light co is measured. Simply put: Light appears to travel slower in stronger gravitational fields (near bigger mass). You can find a more sophisticated derivation later by Einstein (1955) from the full theory of general relativity in the weak field approximation: (...) For the 1955 results but not in coordinates see page 93, eqn (6.28): c(r)=[1+2phi(r)/c^2]c. Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911."