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Light bends twice as much due to gravity as Newtonian theory predicts as discussed in this related question. The argument couched in general relativity, quote from here.

Since most objects move much slower than the speed of light, meaning that they travel much farther in time than in space, they feel mostly the time curvature. The Newtonian analysis is fine for those objects. Since light moves at the speed of light, it sees equal amounts of space and time curvature, so it bends twice as far as the Newtonian theory would predict.

But is it possible to show this without general relativity, by considering a uniformly accelerated rocket and looking at how much light bends? Under the equivalence principle this situation should be the same as light in a uniform gravitational field.

A simple argument for the rocket example is that light goes in a straight line while the rest of the rocket accelerates at 1g. Suppose there are is a series of rulers that measure the height of projectiles as they pass by (the rocket is way too slow to reach relativistic speeds in the time it takes to run these experiments). Let t be the elapsed time after any projectile hits the "top" of it's trajectory (highest "height" as read by the rulers). The rocket has moved ahead by (1/2)9.81 t^2 m. Regardless whether it's a thrown stone or a photon, the projectile is travels in a straight line and thus has fallen behind in our accelerated frame by the same amount at any given t. There seems to be a flaw in this reasoning because it not giving us the needed factor of 2. What is the "catch"?

Kevin Kostlan
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This is perhaps more a comment than an actual answer to the question, but I feel it is worth pointing out that in Nordstrøm's theory of gravitation, which is fully relativistic and internally consistent (but disagrees with experiment), gravity does not bend light at all (and, symmetrically, of course, light has no gravitational effect). This can be explained by the fact that, in Nordstrøm's theory, the source of the gravitational field is the trace of the matter tensor (which is zero for a photon gas), which is equated to the curvature scalar (trace of the Ricci tensor). Since Nordstrøm's theory is internally consistent, it is probably not possible to show using a simple thought experiment that a relativistic theory of gravity must bend light, let alone that it must do so twice more than in Newtonian theory.

There is probably something intelligent to be said about the fact that Norstrøm's theory makes gravity a spin-0 field whereas General Relativity makes it a spin-2 field, and that this somehow explains the fact that we get 0 times and 2 times the bending of light in the Newtonian theory, but I don't know how to make this precise.

Gro-Tsen
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But is it possible to show this without general relativity, by considering a uniformly accelerated rocket and looking at how much light bends?

No. The factor of 2 only comes out of the full formalism of General Relativity, in which there $g_{tt}$ term in the metric is not constant. Had there been a semi-Newtonian explanation of the factor of two, Eddington's confirmation of the GR prediction would not necessarily have been seen as a vindication of the whole GR theory.

Your seem to expect that the equivalence principle on its own should work, but that is not sufficient, because the bending of light involves the motion of a ray through a inhomogeneous gravitational field. The central field (Schwarzschild solution) is not spatially uniform, so its effects are not the same as the effects of an accelerating frame of reference.

Buzz
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If I understand right, the issue here is that you are experiencing time dilation. So inside the rocket, you see light falling at only the Newtonian amount of acceleration.

However a distant observer, not subject to time dilation, looking in through a window on the side of your rocket, would (I think) both

A: Estimate that your rocket is traveling faster than you think it is.

and

B: Perceive that the beam of light's path is bending by enough to keep up with that (higher) estimate of your speed/acceleration.

I could totally be wrong about that.

The short version is : I think it's because local and non-local observers disagree about how much the beam of light is curving.

jimbob
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