# [Physics] Does ‘special relativity + newtonian gravity’ predict gravitational bending of light

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It seems to me that special relativity (SR) already predicts that gravity will bend light rather than this effect being a proof of general relativity (GR). Photons have energy proportional to their frequency and according to $E = mc^2$ they also have a tiny, but non-zero relativistic mass equivalence. I have read the semantic argument that gravity deals with invariant or rest mass, but this should apply to a hypothetical photon at rest, not to real photons at velocity $c$.

I have considered the possibility that the SR effect is much smaller than the GR effect. GR explains the equivalence of inertial mass and gravitational mass as inherent rather than being a puzzling coincidence, but it is true in Newton's gravity, SR, and GR, so the quantitative difference between GR and SR does not seem right.

Yes, it does, or to be more precise, Newtonian gravitation predicts photons will be deflected if you assume photons have some mass. However the amount of this deflection is just half of what GR predicts. And the observed amount of deflection is what GR predicts (within experimental error).

In particular the way this is studied is by using something called the Parameterized Post-Newtonian (PPN) framework. This is discussed on this Wikipedia page and also in the paper by Will I mention below. PPN is essentially Newtonian gravitation with a bunch of first-order corrections from GR added to it, controlled by various parameters, so it's useful for experimental tests of GR, and comparisons between GR and other metric theories of gravity, where the field is weak: it would not be useful for instance, for tests involving black hole collisions where the field is very much not weak!

I believe the first, simplified, PPN framework was derived by Eddington, specifically for the purpose of understanding how the deflection of light by the Sun differed between GR and Newtonian gravitation.

PPN has a significant number of parameters, but for the case of the deflection of light by a spherically-symmetric field only one matters, which is known as $\gamma$. The angle of deflection is then given (remember this is a first-order approximation which is valid for a weak field) by

$$\delta\theta = \frac{1+\gamma}{2}\frac{4 M_\odot}{d}\frac{1 + \cos\Phi}{2}$$

where $d$ is the distance of closest approach to the Sun, $M_\odot$ is the mass of the Sun, and $\Phi$ is the angle between Earth-Sun line and the incoming-photon line.

In this expression, Newtonian gravitation would say that $\gamma = 0$, and GR would say $\gamma = 1$. So you can see that GR predicts exactly twice the deflection that Newtonian gravitation predicts. And this is what Eddington et al measured on the 29th of May, 1919, and discovered that $\gamma = 1$ (to within a fairly large uncertainty at the time, but it was clear that $\gamma = 0$ was ruled out): this made Einstein famous.

There is a discussion of this in the Wikipedia article on tests of GR, and the paper by Clifford M. Will has more detail I think (disclaimer: I have not checked the latter in detail: it looks like section 3.4 & specifically 3.4.1 may be what you want). My expression for $\delta\theta$ above is lifted from the paper by Will.