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We know that through GR, in a gravitational field light bends twice as much as predicted by strict Newtonian physics. In researching this I've found explanations like light bends on both the space and time components of spacetime on page 116. And in the PBS Spacetime at 10:50, he says that it is caused by the wave front of light refracting in the gravitational field.

In either case, if light is bending twice as far as predicted, is it falling twice as fast as say a bullet shot horizontally across the planet? So on the Earth, would light fall at 18.4 m/s^2?

Edit: This question on the effect of gravity on light and bagels has just popped up on my sidebar.

foolishmuse
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1 Answers1

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foolishmuse asked: "In either case, if light is bending twice as far as predicted, is it falling twice as fast as say a bullet shot horizontally across the planet?"

No, at the vertex (at $\rm y=r$ with the initial motion along the $\rm x$ direction) it would fall (vertically accelerate) by a factor of $\rm 2 \ v^2/c^2+1=3 \times$ faster:

acceleration of a photon

The $\rm d^2 y/dT^2$ at the direct flyby is $3 \times$ higher, but if you integrate that over a large $\rm \Delta x$ the resulting $\rm dy/dT$ (the area instead of the curve) is only $2 \times $ higher than under Newton since the radial difference is smaller than the transversal and farther away the radial component dominates.

The factor $2$ is also not exact, that is an approximation for almost straight paths with small deflections where $\rm \tan \phi \approx \phi$. In the strong field, for example if the photon grazes the photon sphere at $\rm r=1.5 \ r_s$, the deflection angle is even stronger than $2 \times$ Newton.

The $\rm x$ axis on the plot is actually the local shell time $\rm T$, but at a large distance like the initial $\rm y=1000 \ GM/c^2$ that is almost equal to the actual $\rm x$ direction if we don't count half pixels.

A very good explanation about how that fits with the equivalence principle is here:

Einstein Online wrote: "There is a fundamental restriction implicit in our use of the equivalence principle: Only in a small region of space, and over a brief time period, can we use the laws of special relativity and end up with a good approximation. Only in a small falling elevator can we assume that light propagates at constant speed, along a straight line. On the other hand, if we actually want to measure the deflection of light, we will have to look at the big picture. We must consider not just local, but global deflection"

If the earth was not round but infinite and flat the equivalence principle would even hold over long distances and times, and the vertical acceleration would be the same regardless of the initial velocity (in terms of the local time $\rm d^2 y/ d T^2$, while in terms of the proper time $\rm d^2 y/ d \tau^2$ would be higher by $\gamma^2$, as would be the $\rm g$ force if the particle travels on the ground in a straight line).

Yukterez
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