Mike, the energy of a photon doesn't have to do with it's speed. They do gain energy when falling due to gravity because they gain momentum.

Momentum in classical mechanics is the product of mass and velocity.

zero x velocity = zero. No gain of momentum

For a particle with rest mass, momentum is simply p = mv, and force is the time rate of change of momentum, f = dp/dt = d/dt(mv) = m(dv/dt) = ma. However, at non-relativistic speeds, f = ma = m(dv/dt) ≠ ma. Increasing speed near the speed of light increases the mass; so dp = mdv + vdm; dp/dt = m(dv/dt) + v(dm/dt) ≠ ma. However, f = dp/dt remains valid for particles with rest mass at all speeds. When talking about speeds near c, you must forget about f = ma and use the

*correct* formula, f = dp/dt.

Einstein's general relativity formulas are written for Minkowski space-time. The fundamental difference between that and Euclidean space is that Minkowski redefined a straight line is as the path of light. In Euclidean space, gravity bends light, so light accelerates without gaining speed, because the velocity changes perpendicular to its direction of travel. In Minkowski space-time, gravity does not bend light; the velocity of light remains constant, so there is no acceleration.

I don't know why, but the concept of force is not part of general relativity; if it were, it would have to be f = dp/dt, and that would be valid for photons as well as for particles with rest mass. I also don't know why photons are considered to have zero mass in general relativity. The formula E = mc², yields m = E/c², which makes perfect sense to me.

Gravity changes the momentum of a photon, and the rate of change of momentum is force, so a photon does feel a gravitational force in a gravity field. That force is in the direction of the gradient of gravitational potential in Euclidean space, but in Minkowski space-time it is in the direction of the photon's travel. The velocity in general relativity is constant, so dv/dt = 0, and f = m(dv/dt) + v(dm/dt) = c(dm/dt). The force is the rate of change of mass of the photon times the speed of ligth. Work is force times the parallel component of the distance, so work is being done on the photon by gravity, and that is why the photon gains energy. That's not how it's usually explained in general relativity, but the result is the same.