I'm by no means well-versed in the special theory of relativity, but I believe one of its results is that light seems to travel the same speed at you no matter what your own speed may be (since it's all relative anyways, there's no absolute frame of reference for you to really measure your own speed), as a consequence of the Lorentz transformations.

I think this was confirmed in an experiment where scientists measure the speed of light from the sun both during sunrise and sunset, since the relative speed between the sun and the measurement devices are different due to earth's rotation (sorry, don't have a link handy for all this, I probably got some details wrong but you get the idea).

I guess the theory of general relativity would extend this to gravity.