Has anyone checked whether the speed of light $c$ varies according to gravitation? My physics is fairly basic, but I hope someone can answer without being too rude.
A transparent medium such as water or glass refracts light and also reduces its speed, so I was wondering whether since light can be bent by gravity as in gravitational lensing, its speed would also be affected.
 A: It is true that the speed of light varies in transparent substances, and this variation in the speed of light, combined with some conservation of momentum and wave optics, is what produces effects such as refraction, which is a bending of light.
It is also true that astronomers have observed the bending of light by various astronomical bodies, from the sun to distant galaxies, which often can make the distortions appear quite dramatic.  
What is not true, however, is that this effect can be explained by a "slowing" of light in the area around the gravitating body.  In particular, you don't see any dispersion effects (in most materials, the speed of light is different for different colors, which blurs multicolored images) when objects gravitationally lens, which is not observed for gravitational lensing.  
Now, you might say, what about a qualitatively different "slowing" effect?  This might be possible (I at least don't know of any direct measurements of the speed of light varying near massive bodies), but there is a compelling theoretical reason to think that this is NOT what is happening - General relativity is the currently accepted theory of gravity, and general relativity explains gravity by a curving of spacetime.  It therefore predicts a bending of light near massive bodies because the light is travelling through a curved spacetime.  The space is still empty, it's just been affected by the massive body.  
Relativity was invented to explain other anamolies in theory and experiment (an inconsistency between gravity and special relativity, along with a known flaw of Newtonian mechanics in explaining errors in the orbit of Mercury, amongst other things), and this bending is something that it actually PREDICTED -- Eddington's expedition to observe the bending of light by the sun, and his measuremnt agreeing with the predictions of general relativity was the point where the theory started to advance into mainstream acceptance.  Similarly, relativity was used to predict the expansion of the universe, in contradiction to most of the existing literature on the topic at the time.  
So, if you did away with the explanation of gravitational lensing in terms of general relativity, you'd have to find another way to predict all of the things that relativity gets right.  
A: It's actual speed remains constant as long as it is passing through a vacuum. Light slows down in various media because the photons are bouncing off of the atoms comprising the media. Therefore the denser the medium, the slower the light because these atoms and molecules are closer together. But the same does not apply in a vacuum. Whenever light is in a vacuum, its speed is C. It's direction can change due to gravity but its actual speed remains constant.
