We know that light which travels perpendicularly to a gravitational field will bend, producing gravitational lensing. It won't speed up or slow down, but only its direction changes, right? But what if it travelled straight outward? We know it will redshift, but will it slow down, even a little bit, due to gravity, similar to that of a massive object, maybe in proportion to the redshift or something, or speed up as it falls inward? Consider it for these scenarios:
1) The Earth. Will light slow down as it leaves the Earth's surface due to Earth's gravity?
2) The sun, which is a few million times more massive than Earth. Will light slow down as it leaves the surface of the sun?
3) The redshift of the sun is about one part in a couple million, so what if the sun were a few million times more massive than it is? Would light slow due to its gravity?