Answer
As Ben says, it's nonsense. The most nonsensical claim is this: "both Special relativity and Galilean relativity work." Though what the author has in mind is more specific than arbitrary light orbits, and he is not completely wrong in seeing that the Newtonian calculation is hand-waving, but does not address the actual problems and certainly does not fix them. Usually, what is done in the Newtonian "toy model" exercise is to treat light as a (massive) corpuscle with velocity far from the Sun being c. Of course, that means that somewhere along the orbit, the light will have velocity higher than c, but there's no mechanism in Newtonian gravitation have a constant-speed unbound orbit, so that's a bit of a necessary evil. That's what is usually taken to be the Newtonian prediction for the deflection of light.
(Note: the author explicitly says that by "Newtonian theory", he means "Newtonian theory obeying Galilean relativity". That's exactly what standard Newtonian mechanics is, so I feel confident if there's any misunderstanding, it is due to the author.)
One of the author's mistakes is that he considers the path of the light ray trajectory to be parabolic just as it should be in Newtonian theory under a constant gravitational field. But for a parabolic orbit around a point, the speed far away from the gravitating body would be zero, which obviously cannot be the case for light. On the other hand, the orbit far from the gravitating body should be straight, so in the far limit are lines. Thus, the orbit must be hyperbolic.
At first glance, one might think: aha, what if one takes the velocity somewhere else along the orbit to be c, say at Earth or closest distance to the Sun, or whatever? But that can't affect the situation too much, because if the corpuscle has velocity v~c, then its kinetic energy is so utterly dwarfs the gravitational potential energy due to the Sun, that it will simply be irrelevant just where on the orbit the velocity is exactly c, because it will be extremely close to it everywhere on the orbit.
We can check this more explicitly. In the Newtonian assumption, the light corpuscle will have a hyperbolic orbit, and we can talk about the the angle θ between the asymptotes of that hyperbola. If the corpuscle is unaffected by gravity, then this angle be π radians, so a sensible measure of deflection is its deviation from π; for convenience let's call this 2ψ. The relevant relationship is: e=1+v2∞RpGM=1cosθ2=1sinψ≈1ψ+ψ6+…,
So we have v∞=c giving us some eccentricity and some ψ. Let's try velocity at periapsis vp=c instead, and compare the result. We know from conservation of mechanical energy that the velocity at r=Rp and velocity at infinity are related by 12v2p−GMRp=12v2∞, so plugging in vp=c gives: e′=1+(c2−2GMRp)RpGM=e−2.
With e so large (~105 or higher), the addition of 1 is similarly unimportant, thus we can also approximate the total deflection to be 2ψ≈e−1≈2GMc2Rp,
Even without the Newtonian toy models, there's a very straightforward and intuitive way that GTR's effect on light is twice that of Newton, and it can be seen clearly in the linearized approximation. Like for Newton, GTR's gravity couples to energy, which would be the relativistic mass (though that term is depreciated). But unlike it, GTR's gravity also couples to pressure, and for a light beam the pressure in the direction of motion is exactly equal to its energy, in units of c=1.
Addendum: A brief explanation of questions from the comments. In linearized gravity, one begins with the assumption that the metric is a perturbation of Minkowski spacetime, imposes a gauge condition, and proceeds in analogy to the retarded potential of electromagnetism. Using approximations appropriate to a nearly-Newtonian regime (energy dominates stress, negligible retardation), the result applicable here is: ds2=−(1+2Φc2)c2dt2+(1−2Φc2)(dx2+dy2+dz2)⏟dS2,
How it is associated with the "spatial curvature" can also been derived from the above. What we want is the Newtonian d2S/dt2=−∇Φ, and to do that, we can explicitly extremize the proper time of an orbiting particle, and again make use of Taylor series approximations, keeping terms only to O((v/c)2,Φ/c2): τ=∫dt√(1+2Φc2)−1c2(1−2Φc2)dS2dt2≈∫dt(1−1c2(12dS2dt2−Φ)),
The interesting part in the above is this: since we're dropping the higher-order terms, the deviation from Euclidean dS2 becomes completely irrelevant--it only contributes terms that are dropped this Newtonian limit. Newtonian behavior can just as well be described by the much simpler metric ds2=−(1+2Φc2)c2dt2+(dx2+dy2+dz2)⏟dS2,
Without even calculating the exact amount of angular deviation, the effective indices of refraction make the following pretty clear: the effect of the spatial curvature is to double the deviation of light it would be without it, and since Newtonian gravitation can be reproduced entirely by the purely temporal component of the metric, this is can be reasonably be interpreted as why GTR's effect is twice that of Newtonian gravity.
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