Could gravity accelerate light? Gravity causes anything with energy to accelerate toward the source. Black holes, for example, have such strong gravity that they pull in light and don't let any escape. But can acceleration still apply to light? The speed of light is constant, of course, but why are photons affected by gravity yet aren't accelerated by it?
Edit: My main question is why photons aren't affected in the same way as most other particles. I'm perfectly aware that it cannot surpass lightspeed, but I want to know what makes it unaffected by acceleration while other particles are affected.
 A: Photons are blue-shifted when attracted by gravity (I mean - moving towards a mass, not moving at right angles to the gravitational field like in an orbit). They can't go faster, but their energy goes up.
A: You don't feel acceleration.  When onboard the ISS, you are accelerating towards the earth (down) due to gravity: if you didn't, you would just fly away from the planet.  Because you and the ISS are accelerating exactly the same way, you don't feel a thing.  You don't feel a force if it's accelerating you: you feel pressure caused by opposing forces.  Here on the ground, I feel the floor beneath my feet opposing my normal gravitational acceleration.  
If you run a thruster on the ISS, then the ISS starts accelerating differently than you do, and eventually, one of the walls is going to collide with you.  Then you'll feel that wall interfering with your own gravitation acceleration, and feel something like weight.
Light undergoes acceleration due to gravity: look up 'gravitational lensing' for that.  To understand how light can accelerate with a constant speed, you have to understand the difference between speed and velocity, and what acceleration really means.
Speed is a 'scalar', just a number with no direction.  If you're travelling 30 KPH, that's your speed.
Velocity is a 'vector', a number with a direction.  Driving 30 KPH north is much different than driving 30 KPH south: clearly, you'll end up in different locations regardless of your speed.
Acceleration is not a change in speed, it is a change in velocity.  Think of a car.  There are usually three ways to accelerate a car.  To increase your speed (scalar), step on the accelerator, and you'll feel your seat back pushing into you harder as it accelerates you with the car.  To decrease the speed (scalar), hit the brake and you'll feel your safety straps accelerating you with the car.
But what happens when you turn?  Your speed stays roughly the same (exactly the same if you're skilled enough), but you are changing your direction.  Your 30 KPH north is becoming 30 KPH west, and the change in direction is an acceleration.  Depending on whether your car is build to drive on the right or the left, you'll have a tendency to either push against your door or into your passenger's lap.  That's still acceleration.
If a photon is passing by something heavy, it will be accelerated towards that object, changing its course but not its speed.  If a photon is going towards or away from something heavy, it can't properly accelerate by changing speed.  I'm not a physicist, but I believe that it increases or decreases energy by changing its frequency.  In other words, things that you would expect to increase its speed will instead increase its frequency ('blue-shifting' it if it's visible light), and what you would expect to decrease its speed will instead decrease its frequency ('red-shifting' if it's visible light).
A: To talk about acceleration in space is a little bit dangerous without exact definition. One has to separate free fall and acceleration from an impulse.
Imagine, you are inside the ISS during an orbit correction. The impulse from the rocket engine you could feel, you get some weight, and and this is an acceleration. 
In all other time you are weightless and does not feel any acceleration. But without doubt there is something what held the spaceship in his orbit. To call it a force is not correct because you has to feel an acceleration (doe to Newtons formula F = m a). That is the reason why Einstein does not more talk about gravitational forces but about curvature of space.
What is about the curvature of space and time? Due any particle or body follow the same path, when starting from the same point in the same direction? This is not the case. The straightest reachable path is with photons, they have the maximum possible velocity, but still are influenced by the curvature of space. For all other body the path is more bended under the influence of gravitational masses.
It is possible to answer a question about does the light travel with different speed in space. 1) No, if answer is in a frame close to the particle. 2) Yes, if answer it from a point in deep space. Near a black hole the photon runs slower from the point of view of a far away from this hole observer.
A: According to Einstein and many others, light speed c0 in vacuum is universal and measures about 299,792,458 m/s. So it is never possible to change the light speed in vacuum, which is the absolute upper limit for everything.
Gravity does not affect a light ray, but the space and time through which the ray travels. 
In a strong gravitational field, the time passes slower, according to observers far away from the field. For observers affected by the field it is still c0 though. 
And the space in this field is expanded, stronger the closer we get to the center of the field. If we imagine space as 2D plane, we could say that it gets a deep dent where the mass is. The light ray that travels through this dent now travels perfectly straight in his view, but observers far away see a curved path because the space is curved.
The noticeable "gravitational redshift" (if light travels from an observer into a gravitational field) or "blueshift" (if light comes out of a gravitational field to an observer) for external observers has its cause because the photon gathers energy while entering the gravity well while it loses energy when it leaves it. This is potential energy, depending on the gravity potential the photon is located at. As masses create gravity wells with low potentials, the "absence" of a (strong) gravity field can be called high potential. This energy difference can't express itself in a speed difference (kinetic energy), because we already said that light speed is constant. Instead, the potential energy converts itself into light/electromagnetic (please correct me if the term is wrong) energy, as it can be described with E=hf. That means the photon's energy is proportional to its frequency. An increase of the electromagnetic energy of the photon, because it enters a lower gravity potential and converts its potential energy, therefore results in a higher frequency, which is visible as a blueshift.
A: I would suppose the short answer is no, but photons are affected by gravity. 
What is happening is naturally quite relativ to the observer. Suppose you are sitting on the photon, travelling past the gravity source with the speed of light. As has been argued above you would experience the force of the gravitational pull as an acceleration toward the source. Also, if you were to observe the passage of time during the journey, you would find that your clock is running slower than the clock of an observer travelling further away from the gravity source. To this "further away traveller" you would therefore appear to be travelling faster, even though your both travelling at the speed of light.     
Ultimately, what this means is, while the gravity-source affects the path of the photon, the speed isn't affected due to the relative change in time.
At least thats how I understand it anyway. :)    
