Just to add to this excellent answer: @john, when you're learning some theory, make sure that you keep in mind its validity domain. Newtonian mechanics isn't always valid, so if you push its formulae too far, you may get "strange" results that simply mean that you're stepping outside the validity domain. Zero mass, or high velocity, are two of those cases.

Light always moves with a speed of $c$ when measured locally. Photons can and do appear to be accelerated.

@ProfRob What does locally mean here ? And when does it accelerate?

@ProfRob it is only a change in the vector, the speed is always c, at least in mainstream models of gravitational red shift.

@Abbas see my comment to profrob

Right. And velocity isn't the same as speed and contrary to your answer, the velocity of light can change, it doesn't involve quantum mechanics and Special Relativity isn't relevant to a situation involving gravity.

@ProfRob I edited. But photons always need quantum mechanics and special relativity

Acceleration is not just change of velocity, it's also change of direction. The velocity of light is constant, but it can change direction. General relativity explains that this happens due to the curvature of spacetime.

@Barmar Slight correction (since this was highlighted in previous comments, too): acceleration is a change in velocity, but not necessarily in speed. Velocity is basically "speed and direction". If you change the direction, that is a change in velocity.

Gravity is a force that appears because one isn't using an inertial frame. Just like the centrifugal force: to wit, just like the centrifugal force appears because a (newtonian) body wants to continue on a straight line but we are looking at it in a spinning frame, the force of gravity appears because a (Einsteinian) body or photon wants to move along a geodesic and we aren't watching from a free-falling frame. Geodesics are the generalization of straight lines to curved spaces. So saying that a photon "obviously" accelerates is not really "obvious".

Now this gets a little bit more complicated because we cannot fall with the photon, but in a free-falling frame near the bending mass, the photon line will appear straight.

Well, we cannot see the photons trajectory, so I should have been a bit more careful in phrasing this. Assuming the universe is static, the emitting star and our eyes will appear to be in opposite directions to a free-falling observer who crosses the photon's trajectory in the vicinity of a gravitational lens i.e. something we perceive as to be bending the light (or "accelerating the photons" in this discussion). In this sense the photon's trajectory is straight.

@tobi_s this answer is discussing the difference between newtonian formalism and range and quantum + special relativity. GR is a different range of variables and beyond, imo, the question.

@annav thanks, i was responding to the other comments (as one shouldn't if one wants to maintain one's sanity),and I was pointing out that the Newtonian picture of "acceleration = curved trajectory" is not really helpful in GR where we have framework for discussing photons moving on (apparently) non-straight lines as was done by other commenters.

So @tobi_s when we see light bend and we are using Newtonian physics, how is that unaccelerated (in Newtonian physics)?

@ProfRob One cannot talk meaningfully about the propagation of light in Newtonian mechanics. Doubly so when it comes to gravitational lensing. Thus I won't attempt to do so. I already explained how what you consider acceleration disappears by choosing the proper inertial frame. What is gained by inventing some "Newtonian" "physics" that makes it reappear?

@tobi_s the question asks about interpreting the trajectory of light in terms of Newtonian physics. That can and has been done (and gets the wrong answer), but does not predict no acceleration.

@ProfRob the question clearly asks about photons, and photons are not light, they build up classical electromagnetism but have quantum mechanical properties.

@annav you continue to repeat that. Yet there were predictions of the acceleration (a change in velocity) of light, using Newtonian mechanics, prior to General Relativity and Quantum Mechanics. An example of that treatment is here arxiv.org/pdf/physics/0508030.pdf and it uses neither Special Relativity or Quantum mechanics. The answer is wrong by a factor of 2, but it is not zero.

@ProfRob the treatment in physics/0508030 is for a massive body that's fast enough for its velocity not to change significantly while traversing the gravity well. Note that it cancels $m$ just like the original question. In other words it's an exercise that teaches us nothing about nature. The problem of cancelling zero remains, and that's why (returning to the original question) this formalism is not suited to discussing the propagation of light in a gravitational field. I will heed my own advice regarding comment sections and see myself out. Have a nice day.

@tobi_s The title of the paper is "The Newtonian gravitational deflection of light revisited". It could hardly be more relevant to the original question. The velocity does of course change, whilst the speed remains constant.

@ProfRob I've read the title. I've also read the paper. I was commenting on its contents. I think the original poster can pride himself in identifying a mistake in their reasoning that someone who published their thoughts on the arXiv didn't. Anyway, I'll try to heed my own advice and see myself out. Nice talking to you, bye.