Both Newton's emission theory of light and Einstein's general relativity predict that the speed of light is variable in a gravitational field. However the emission theory gives the right and general relativity the wrong prediction. Here I am going to explain that in detail.
Newton's emission theory of light predicts that, if the top of a tower of height h emits light downwards, the light will fall with the acceleration of ordinary falling matter:
http://www.youtube.com/watch?v=FJ2SVPahBzg "The light is perceived to be falling in a gravitational field just like a mechanical object would. (...) 07:56 : (c+dc)/c = 1+(g/c^2)dh [as predicted by Newton's emission theory of light]"
http://bouteloup.pierre.free.fr/vulg/relge.pdf "Considérons une fusée posée sur le sol terrestre, donc immobile dans un champ de gravitation. Déja, à cause du principe d'équivalence, la lumière tombe vers le bas avec la même accélération qu'un caillou, vue par un observateur immobile dans la fusée."
That is, an observer on the ground will measure the speed of the light to be:
c' = c(1 + gh/c^2)
This means that the frequency measured by the observer on the ground will be:
f' = c'/L = f(1 + gh/c^2)
where f=c/L is the initial frequency (measured by an observer at the top of the tower) and L is the wavelength.
The frequency shift predicted by Newton's emission theory of light, f'=f(1+gh/c^2), is exactly the frequency shift that Pound and Rebka measured, that is, their experiment confirmed the emission theory in a straightforward way. Even Einsteinians admit that:
http://courses.physics.illinois.edu/phys419/sp2013/Lectures/l13.pdf University of Illinois at Urbana-Champaign: "Consider a falling object. ITS SPEED INCREASES AS IT IS FALLING. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, WE SHOULD OBSERVE THE SAME EFFECT FOR LIGHT. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction. Consider a light beam that is travelling away from a gravitational field. Its frequency should shift to lower values. This is known as the gravitational red shift of light."
http://www.einstein-online.info/spotlights/redshift_white_dwarfs Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."
Einstein's general relativity predicts that, if the top of a tower of height h emits light downwards, the light will fall with twice the acceleration of ordinary falling matter. That is, an observer on the ground will measure the speed of the light to be:
c' = c(1 + 2gh/c^2)
This shift in the speed of light predicted by general relativity is incompatible with the frequency shift f'=f(1+gh/c^2) measured by Pound and Rebka, given the formula:
(frequency) = (speed of light)/(wavelength)
That is, the Pound-Rebka experiment, while confurming Newton's emission theory of light, has in effect REFUTED GENERAL RELATIVITY.
References showing that, according to Einstein's general relativity, the speed of light varies in accordance with the equation c'=c(1+2gh/c^2):
http://arxiv.org/pdf/gr-qc/9909014v1.pdf Steve Carlip: "It is well known that the deflection of light is twice that predicted by Newtonian theory; in this sense, at least, light falls with twice the acceleration of ordinary "slow" matter."
http://www.speed-light.info/speed_of_light_variable.htm "Einstein wrote this paper in 1911 in German. (...) ...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. (...) You can find a more sophisticated derivation later by Einstein (1955) from the full theory of general relativity in the weak field approximation. (...) Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911."
http://www.ita.uni-heidelberg.de/research/bartelmann/Publications/Proceedings/JeruLect.pdf LECTURES ON GRAVITATIONAL LENSING, RAMESH NARAYAN AND MATTHIAS BARTELMANN, p. 3: " The effect of spacetime curvature on the light paths can then be expressed in terms of an effective index of refraction n, which is given by (e.g. Schneider et al. 1992): n = 1-(2/c^2)phi = 1+(2/c^2)|phi| Note that the Newtonian potential is negative if it is defined such that it approaches zero at infinity. As in normal geometrical optics, a refractive index n>1 implies that light travels slower than in free vacuum. Thus, the effective speed of a ray of light in a gravitational field is: v = c/n ~ c-(2/c)|phi| "
http://www.mathpages.com/rr/s6-01/6-01.htm "Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential phi would be c(1+phi/c^2), where c is the nominal speed of light in the absence of gravity. In geometrical units we define c=1, so Einstein's 1911 formula can be written simply as c'=1+phi. However, this formula for the speed of light (not to mention this whole approach to gravity) turned out to be incorrect, as Einstein realized during the years leading up to 1915 and the completion of the general theory. (...) ...we have c_r =1+2phi, which corresponds to Einstein's 1911 equation, except that we have a factor of 2 instead of 1 on the potential term."