Light deflection by the sun

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Dated: 2 January 2015

There was an RG discussion, in which the idea came up to account for the factor of 2 between the pure equivalence principle prediction for light bending (which is also the Newtonian prediction) and the full general relativistic one by calculating a contribution by refraction and one by acceleration. These would turn out to have the same size and adding them up would give a factor of 2! Unfortunately, these two contributions are not the same by accident, they are necessarily the same, because they correspond to two different views at the same physics. So there is no good reason to add them up. I give a complete calulation here, using Fermat's principle and obtain the full result, demonstrating that Fermat's principle can also be applied in curved spacetime. (Moreover, we do not have to write down the geodesic equations of motions of light explicitly to do the calculation.)

Note added (07.12.2023): It can be seen by inspection of the general formula that the second calculation must give a result that is by a factor of two larger than that of the first. This is simply a consequence of the particular form of the speed of light in the two calculations. The equivalance principle gives c(r) = c/(1+Φ(r)/c²), the isotropic Schwarzschild metric c(r) = c/(1+2Φ(r)/c²), and the final result is linear in the potential Φ.

Light deflection by the sun, 02.01.2015

Note added (07.12.2023): While my calculation gives the correct result, it is flawed, as I found out later. As long as I work with exact formulas, everything is correct (I believe), but when I solve the differential equation for the polar angle φ(r) perturbatively, I disregard that the "perturbation" containing a factor 1/(1-b²/r²)^1/2 actually becomes very large as r approaches the impact parameter b. I noticed that problem when trying to solve Eq. (22) numerically for my paper [1]. It can already be seen from the analytic representation Eq. (23): Because φ(r) varies between an angle close to 0 and an angle close to π, the cosine in the expression will take the argument π/2 somewhere on the trajectory. But then the argument of the sine becomes infinite, so we have infinitely fast oscillations of the ratio b/r. Nevertheless, even though the "approximate" solution given by Eq. (23) is not valid along the entire trajectory and becomes even qualitatively wrong through part of it, the pieces of the trajectory far from the sun seem still to be well approximated. At least the result for the angle is correct there. A way to obtain a good approximation to the whole trajectory is to work with r(φ) instead of φ(r). This is done in [1], essentially by starting from the second expression on the right-hand-side of Eq. (13) rather than from the third one, (Instead, one could simply rewrite Eq. (16) into a differential equation for r(φ).)

[1] K. Kassner, Classroom reconstruction of the Schwarzschild metric, Eur. J. Phys. 36, 065031 (20pp) (2015).

Next: Fermat's principle in general relativity   Up: Introduction Science Education Project    Previous: Particle in a (homogeneous) gravitational field

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