How do you intuitively understand of the Einsteinian portion of the gravitational lensing equation?

The total gravitational lensing equation is an addition of the Newtonian Solution $$\left(\frac{2GM}{c^2r}\right)$$ and the Einsteinian Solution $$\left(\frac{2GM}{c^2r}\right)$$ thus the combined solution is $$\left(\frac{4GM}{c^2r}\right)$$.

Where the Newtonian Solution is derived from the following example. Consider the Sun with radius r and a photon of light skimming the surface and being diverted through an angle alpha. This angle is obtained by dividing the velocity of the photon in the diverted direction by its speed of light $$v/c$$.

Where velocity is obtained by multiplying the acceleration of the photon in the diverted direction $$\left(\frac{GM}{r^2}\right)$$ by the time it takes that photon to travel twice the radius of the Sun $$\left(\frac{2r}{c}\right)$$. Thus giving the velocity of the photon in the diverted direction $$\left(\frac{2GM}{rc}\right)$$. Then dividing this velocity by the speed of light $$\left(\frac{2GM}{rc}\right)$$ / $$c$$ to obtain the angle that the photon is diverted $$\left(\frac{2GM}{rc^2}\right)$$.

However, the Einsteinian portion of the gravitational lensing equation is not so intuitive, which is obtained by dividing the Schwarzschild radius of the Sun by the radius that the photon passes above the Sun Rs/r = $$\left(\frac{2GM}{c^2r}\right)$$ which contributes further to the photon's deflection and thus further increases the angle alpha. But it leaves out what is actually taking place with the photon to cause this further contribution to the angle alpha and thus harkens back to the days of "just shut up and calculate".

With the Newtonian Solution I can see and understand the mechanics that are involved but with the Einsteinian Solution the mechanics allude me. I know that it's because space is changing around the gravitating mass as a function of distance from that mass and thus the situation can be likened to a changing refractive index around that gravitating mass where $$n(r)$$ = $$\left(\frac{1}{1-\frac{2GM}{c^2r}}\right)$$ represents the changing refractive index (which I do not understand how to incorporate). Which in turn is changing the direction of the photon's wavefront and thus changing the photon's propagation angle.

But as to how the mechanism intuitively functions is beyond me and thus any help in understanding this matter would be greatly appreciated.

1 Answer

The second term (which, unlike the first, is independent of the speed) comes from the angular defect in the spatial geometry around the sun, the same thing that causes the anomalous precession of Mercury's orbit.

Here are some illustrations from "Relativity Visualized" by Lewis Epstein, which has a good treatment of this.

On the top left is a constant-Schwarzschild-time slice of the Schwarzschild geometry (exterior + interior, though the latter isn't relevant to this problem). On the top right is an approximation of it by a cone surrounded by flat space. On the bottom left, the cone has been cut along a radius so it can be flattened, and a spacelike geodesic drawn on it. The missing pie slice is the angular defect. On the bottom right is a top-down view of the embedding and the geodesic once the cone has been taped together again.

Note that this is not a gravity well. A slower-moving particle would not bend more. There is no speed in this diagram; the deflection angle is set by the geometry.

You can approximate the geodesic of the actual test particle by a zigzag path with (constant-Schwarzschild-time) spacelike and (constant-Schwarzschild-position) timelike parts. On the timelike parts, it experiences a quasi-Newtonian acceleration, and on the spacelike parts it follows something like the path shown in this image. If the deflection angle is small, the total length of the timelike parts depends on the speed but the total length of the spacelike parts doesn't.

(See also this answer.)