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No matter how fast you go, you will aways perceive the speed of light as constant. Taking that as a fact, the special relativity theory was formulated. Now, for what I understand about general relativity, just by standing here on the surface of the planet under the effect of gravity, my frame of reference is a little different from someone in geostationary orbit, and time goes by a little slower for me.

So, considering me and someone in geostationary orbit, does both of us would perceive light at the same speed? If so, does that means that the light is slower down here than up there?

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Well, that other question doesn't involves gravity and general relativity... –  lvella Feb 13 at 2:38
    
But it does involve the constancy of the speed of light in different frames. –  Kyle Kanos Feb 13 at 2:41
    
But that is not enough for considering it duplicated, right? I mean, I take it for granted, while the other question ask how it can be... –  lvella Feb 13 at 2:45
    
Possible duplicates: physics.stackexchange.com/q/24319/2451 and links therein. –  Qmechanic Apr 1 at 20:56
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up vote 3 down vote accepted

General relativity reduces to special relativity locally. What this means is that given an error tolerance $\varepsilon$, you can find an extended region (perhaps just a small one) around any point in spacetime such that the laws of physics as tested only within that region match those of special relativity to within $\varepsilon$.

That means if you measure the speed of light, say with a ruler and stopwatch, doing the experiment right in front of you, then you will measure the speed of light to be the standard value. The same holds for anyone else anywhere else in the universe, as long as they also confine all their measurements to the same region.

What confuses things is if you try to use a stopwatch in one place and a ruler in another. Then the distance traveled by the photon divided by the time it takes to travel that distance can come out to be anything. Meters at point $A$ are compatible with seconds at point $A$, but not generally with seconds at point $B$.

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What confused me is that with special relativity, I could depict a space ship moving in straight line as stretched and with people moving slowly inside compared to me standing on Earth. Then I could depict how they would measure the same ray of light differently from me, and get to the same speed. With general relativity, I don't even know to what direction I would be stretched for a geostationary observer... –  lvella Feb 13 at 2:54
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Both observers would perceive the light at the same speed. Any observer in any frame of reference anywhere in the universe will see light travel at the same constant speed. In terms of an light under gravity - if the light is approaching the massive object it will have a Doppler blueshift, while if the light is going away from the massive object it will have a redshift. In both cases however, the light will have the same speed.

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My question was more about the geostationary man measuring with his reference frame (I don't know exactly how) the speed light here on the surface... –  lvella Feb 13 at 3:03
    
It seems I missed the point of your question. Nevertheless, just a little bit of secondary info in my answer. –  user40229 Feb 13 at 3:07
    
Good question - I voted +1 for you. –  user40229 Feb 13 at 3:09
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