Big masses like planets and stars have gravitational fields. Gravitational fields curve space around them. So in theory a particle having mass moving at fixed speed, from its own perspective, will accelerate when moving closer to such a planet. It'll be moving at fixed speed in a curved space. This is similar to having a car move "straight" on a curved road (although I'm talking about moving toward and not beside). Light is also affected by such a mass. We know this because light curves away (changes direction) when passing close to such a mass. If light is affected, then it should also be affected when moving directly towards the mass. So it should be that although from the single photon perspective it's moving in constant speed (light speed), from an observer's perspective it's actually moving faster as it nears the planet, hence light changes its speed. Is this true? If not, what's the explanation?