Bending of Light Along the Sun: Explaining Half w/ Equivalence Principle

[HansH]

TL;DR Summary

The bending of ligh around the sun according to general relativity is twice that of the value one can explain with the equivalence principle. question is where does the other half come from. and why is it exactly a factor of 2.

The equivalence principle tels us that we cannot distinguish between gravity and an accelleration. using that fact one can reconstruct the bending of light along the sun, but only half of it results from the equivalence principle together with the gravitational field around the sun. Is it possible to explain the other half of the bending of light in some similar easy way?

 

[PeterDonis]

When you use the equivalence principle to "reconstruct" the bending of light by the Sun, and get the answer that is only half the correct answer, you are implicitly assuming that all of the local frames in which you apply the equivalence principle "fit together" globally in a certain way. But they actually don't. When you correct your analysis to take into account how the local frames actually do "fit together" globally, you get the correct answer, twice as large as the equivalence principle answer.

How this shows up mathematically depends on how, specifically, you choose to do the global analysis. The most common way to do it is to choose coordinates in which "space" is not Euclidean, and the extra factor of 2 in the final answer comes from "space curvature". This is how you will often see it described in the literature. However, this analysis depends on projecting the actual path in spacetime of the light ray into a surface of constant time, which is not exactly what trying to fit local equivalence principle frames together globally would lead you to do.

A more coordinate-independent way of describing the effect would be that spacetime is curved around the Sun, and that spacetime curvature is what changes how the local frames "fit together" globally. This has the advantage that you can do the analysis using local frames along the actual path in spacetime of the light ray. However, I think the math in this case is more complicated.

 

[Sagittarius A-Star]

I found an explanation for the factor 2 between Einstein's calculations from 1911 and 1916:
https://www.mathpages.com/rr/s8-09/8-09.htm

 

[HansH]

I found an explanation for the factor 2 between Einstein's calculations from 1911 and 1916:
https://www.mathpages.com/rr/s8-09/8-09.htm

Thanks for both relies.

This mathpages link was also found by someone on the Dutch physics forum some years ago. There we got a a lot of discussion about the 2 'peaks' in the blue curve of 1915. That discussion did however not lead to a better understanding of what was going on. The remark was that based on that curve the bending was not simply a factor 2 along the every point of the path but we had a lot of discussion about that this was probably pure due to the mathematics used but we could not get that out of the equations.

So if someone here can give a better answer leading to a better understanding about these 2 peaks that would be very appreciated. At that time I simply would expect the factor 2 everywhere on the light path but I also could not exactly explain why. at least I assumed it was too much coincidence that the factor 2 appears as seen over the complete path but would vary over the position on the light path and still end up in exactly a factor 2.

 

[PeterDonis]

the 2 'peaks' in the blue curve of 1915

As the mathpages article notes, that is an artifact of using Schwarzschild coordinates, which are not isotropic; in isotropic coordinates, which is what most people will intuitively be thinking in, the "1915" deflection is just twice the "1911" deflection everywhere along the path.

 

[HansH]

When you correct your analysis to take into account how the local frames actually do "fit together" globally, you get the correct answer, twice as large as the equivalence principle answer.

is it possible to draw a simple picture of how this 'fit together' works in order to increase the understanding? I could draw a local frame such as an alevator and then accelerate that with a lightbeam passing but then how to connect that to the following frame? there are some nice illustrations of how light bends according to the equivalence principle, see for example https://www.quantumuniverse.nl/relativiteit-10-het-equivalentieprincipe
(in Dutch, sorry, but the pictures are luckily general language) but is it possible to bring the idea of this connecting together in in some way in such pictures?

 

[PeterDonis]

is it possible to draw a simple picture of how this 'fit together' works in order to increase the understanding?

You can picture the non-Euclidean space I referred to in post #2 in terms of the Flamm paraboloid:

https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm's_paraboloid

More precisely, this is a 2-dimensional "slice" through the equatorial plane of the 3-dimensional non-Euclidean space I referred to. Heuristically, the spatial path of a light ray passing near the object at the center will curve because of the curvature of the paraboloid; and for the same reason, "elevators" accelerating outward at different points along the path and seeing the light ray locally "bend" inside them will not "fit together" the way they would on a flat sheet of paper (i.e., a "slice" of a Euclidean 3-space).

I say "heuristically" above, though, because the illustration I just described won't necessarily give you the right answer if you try to analyze it mathematically using the paraboloid directly. It's just a heuristic picture that might help you to visualize things.

 

[Reggid]

... and still end up in exactly a factor 2.

But it is not exactly a factor of 2. It is a factor of 2 in the leading order expansion in the Schwarzschild radius of the star over its actual radius. This is easily precise enough for any realistic experiment, but in principle there are corrections of order r_s/R to that factor of 2.

 

[HansH]

You can picture the non-Euclidean space I referred to in post #2 in terms of the Flamm paraboloid:

https://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm's_paraboloid

I checked that but is already one or more steps in the level of knowledge too far for me, so would requires some study first. I am looking for an explanation as close as possible to the equivalence principle, but not sure if that way of explaining the effect is possible at all.

 

[Motore]

What'a wrong with:

Intuitively, Einstein’s 1911 prediction was only half of the correct value because he did not account for the cumulative effect of spatial curvature over a sequence of small regions of spacetime, within each of which the principle of equivalence applies. This can be understood from the figure below, which depicts a ray of light passing through a sequence of “Einsteinian elevators” near the Sun.

By evaluating the absolute spacelike intervals Δs along the top and bottom of each “elevator car” we find that the walls which are parallel in terms of the local coordinates of the car are not parallel in terms of the global coordinates (except for the central car, where the 1911 and 1915 calculations do predict the same rate of deflection). In general, the simple argument based on the equivalence principle gives the correct rate of deflection in terms of the local coordinates, i.e., it gives the difference between the angles that the ray of light makes with the opposite walls of a car, which are parallel in terms of the car’s local coordinates, but this doesn’t account for the fact that the walls are not parallel in terms of the global coordinates. As we’ve seen, when integrated over the entire path of a light ray, the total deflection from the standpoint of a global system of coordinates is twice the deflection that would be given by simply summing the deflections measured by occupants of a series of elevators along that path, each in terms of their respective local coordinates.

https://www.mathpages.com/rr/s8-09/8-09.htm

 

[HansH]

in addition to previous post: I could draw a series of elevators and calculate how they would react according to the equivalence principle. that would give the effect that the lightbeam passing each elevator would be part of a parabola and together that would give the factor 1 (I did that calculation some years ago) so then the question is how to connect each elevator to the previous one to get the other part 1 making 2. that should be a bending of the space then with the same angle that the light beam bends (then you get the factor 2). but why should I bend it in that way?

 

[HansH]

#11 and #10 crossed each other as I did not see it when 10 when I posted 11. for #10 my remaining question then is intervals Δs along the top and bottom of each “elevator car” being different. where does that come from? probably not the equivalence principle but something else underlying. or is it time flowing different at the bottom and the top of the elevator. if it is that, is there then also an underlying reason to make it understandable?

 

[PeterDonis]

the question is how to connect each elevator to the previous one to get the other part 1 making 2. that should be a bending of the space then with the same angle that the light beam bends (then you get the factor 2). but why should I bend it in that way?

One heuristic explanation of the factor of 2 that you will see in the literature is that the local "elevator" analysis implicitly takes into account the "time curvature" part of the metric (similar to your intuitive guess in post #12), but not the "space curvature" part. For weak fields, it turns out that those two parts are equal; more precisely, the correction to the "space" part of the metric is $2 \phi$, where $\phi$ is the Newtonian potential, and this is the same as the correction to the "time" part of the metric. So the full bending will be twice the "locally derived" bending.

 

[PeterDonis]

the correction to the "space" part of the metric is $2 \phi$, where $\phi$ is the Newtonian potential, and this is the same as the correction to the "time" part of the metric. So the full bending will be twice the "locally derived" bending.

Note, though, that there is a further wrinkle here: IIRC, if you actually look at the details of the math, you will find that the "space curvature" correction comes in with a factor $v / c$, where $v$ is, heuristically, the speed of the object passing by the Sun (or other massive body) at closest approach. For light, of course, $v / c = 1$ and you get the full "space curvature" correction. But for, say, a comet coming from outside the solar system and making a close approach to the Sun before flying back out again, the "space curvature" correction to the bending will be much smaller. In the low speed limit, you just get the Newtonian answer since the "space curvature" correction goes to zero.

 

[haushofer]

Indeed, that can be confusing in the Newtonian limit: space is curved, but the coupling of the spatial velocity to the spatial curvature in the geodesic eq. is of higher order. Particles don't sense spatial curvature...unless they travel ultrarelativistically.

Which brings you outside of the Newtonian limit of course :P

 

[Thelamon]

I remember this. @HansH only actually, coz I know him a bit for a while.

I see that you have also grabbed the two peaks of PP again, sigh .

But that's also just a diagram, not a representation of an actual geodesic. Those peaks have no physical meaning whatsoever, as explained by Prof. Emeritus Dale Gray at the time.

Why this factor of 2 for light with respect to a Newtonian calculation or with regard to the equivalence principle alone has been explained as well (but I got to admit it was a chaotic mess on that forum with countless topics about it in the end .. rediculous, sometimes literally ROFL), but a short repetition:

In Einstein's 1911 calculation, he only used the time component (basically what we now call gravitational time dilation) and that's all the equivalence principle includes. It concerns an acceleration, so an artificial uniform gravitational field is "created", just like with rotation (centrifuge), but no actual gravitational field as in "curved space-time" (see that video once again, where those guys about "time dilation causes gravity or gravitational attraction", which is complete nonsense, are corrected a little bit).

In 1915 he realized (just in time) that there also had to be a spatial component (later elaborated into several components that make up the metric, so no solution like Schwarzschild did exist yet) again using Huygens principle.

The crux is that for light this component contributes as much to the "deflection" as the time component. Because light in vacuum gives a "null-like separation". So in a Minkowski diagram it always goes with 45°. Locally curved spacetime is always just Euclidean so you can always use that Minkowski diagram locally.

And a very easy analogue is that when with slow movements (in the Newtonian limit) the world lines in such a diagram run almost vertically along the time axis light always with 45°, if you roll this up then the world line of a slow movement can be bent almost exclusively in the time direction (gravitational time dilation), while that of light (in vacuum) is bent just as much in the time direction as in the spatial direction.

Of course this is an analogue. I could give you the full explanation again, but to be honest, I think a textbook GR will be to much of a struggle. But I remember giving you "Relativity Visualised" and with the free (mathematical) elaboration "Epstein explains Einstein" by Eckstein (Dreistein) you'll at least have some solid understanding of basic relativity. Instead of, I'm sorry to say specially since it's only logical when you been around that WF too long, but instead of sort of fantasizing uhm trying to visualise the impossible in some pre-relativity way. Not sure how to say it, but I wanted to react in that other topic and I got to agree that you're trying to build a skyscraper starting at the 100th floor or learning how to swim by immediately jumping in the middle of the ocean. And trying to solve a 1000 pieces puzzle while only having 10 random pieces in your possession. .. Your thoughts about lots of physics, mostly relativity in cosmology, go in all directions but your lost it seems. Imho you need good guidance so not on that WF. (That's very obvious to me at least.)

(Written in Dutch and used Google translate, but I feel like there are not too many language errors.)

Hope this helps a little.