Suppose that a light beam is shone upwards from surface of a planet. So, due to gravitational redshift, the frequency of the light perceived by observer far from the surface will be lower than that perceived by observer on surface of the planet. So what would happen to the radiation pressure? Will the pressure be smaller for observer far from the surface? If this is the case, what happens to the momentum of the photon? Does the momentum decrease as the photon travels away from the surface? Does this exert a force on the planet?
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Yes, the radiation pressure would be less for the observer further from the surface. Just as the momentum of a ball thrown upward decreases as it moves up, so does the momentum of the photon. Just as the ball exerts a gravitational force on the Earth, so does the photon.
Another example where this comes into play is this: take an empty box, whose insides are perfectly reflecting, and hang it from a spring balance. Now fill the box with light and weigh it again. It will be heavier. One way to interpret this is using $E=mc^2$, and saying that we have increased the energy and thus the mass of the box. But the other way is to say that the photons exert radiation pressure on both the top mirror and the bottom mirror of the box, but slightly more on the bottom mirror/slightly less on the top mirror due to the gravitational redshift. This gives a net downward force which stretches the spring slightly more - we read this as an increase in weight.
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$\begingroup$ Wouldn't hurt to put in the equation (e.g. as on the wikipedia page) and calculate an approximate value for some planet. $\endgroup$ Commented Dec 28, 2014 at 13:26