Bending of light - Newton vs. Einstein

[pixel]

Using Newton's equation for gravity and assuming a corpuscular theory of light, one can calculate the angle that light would bend in a gravitational field. General relativity predicts a bend that is twice as large. In the Newtonian limit of GR (which includes weak gravity), does the GR prediction for bending of light approach the Newtonian corpuscular value, or is it still 2x (or some other factor) greater?

 

[PeterDonis]

In the Newtonian limit of GR...

...you can't make a prediction for the bending of light, because the Newtonian limit assumes low speeds (speeds much less than the speed of light) as well as weak fields.

In the weak field limit of GR, without restricting to low speeds, the prediction for light bending is the GR one (the "twice as large" one).

 

[Meir Achuz]

I think the 2X is related to Thomas precession, which occurs in GR or SR.

 

[1977ub]

Is it not simply the case that the 2x is due to space not being euclidian in the vicinity of a gravitating mass ?
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/

 

[A.T.]

Is it not simply the case that the 2x is due to space not being euclidian in the vicinity of a gravitating mass ?
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/

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

 

[pixel]

Is it not simply the case that the 2x is due to space not being euclidian in the vicinity of a gravitating mass ?

But does it become more Euclidean as you approach the Newtonian limit of GR?

 

[pixel]

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

This article actually addresses my motivation for asking my question - how the 2x deflection is reconciled with the equivalence principle and the accelerating elevator thought experiment.

 

[PeterDonis]

Is it not simply the case that the 2x is due to space not being euclidian in the vicinity of a gravitating mass ?

If the "2x" refers to the comparison done in the mathpages article linked to in post #5, yes, that's one way of looking at it. However, this way of looking at it has a couple of significant limitations.

First, "space" is frame-dependent. The "space" referred to in the article you link to and the mathpages article linked to in post #5 assumes standard Schwarzschild coordinates centered on the massive body. Other coordinates lead to different notions of "space", not all of which are non-Euclidean (for example, in Painleve coordinates in Schwarzschild spacetime, "space" is Euclidean). But the global light bending is invariant; it doesn't depend on which frame you choose.

Second, the issue that leads to the "2x" for global light bending, as compared to the "x" in a local "accelerating elevator" experiment, is really due to the way the local "elevator" frames fit together in spacetime, not space. Thinking of the path of the light as "bent in space" doesn't really capture that, because the local frames that have to "fit together" are not at different points in space; they are at different points in spacetime, the different points along the worldline of the light beam as it passes by the massive object. The "2x" arises because the spacetime is curved, so the local frames don't fit together the way they would in flat spacetime.

 

[1977ub]

If the "2x" refers to the comparison done in the mathpages article linked to in post #5, yes, that's one way of looking at it. However, this way of looking at it has a couple of significant limitations.

First, "space" is frame-dependent. The "space" referred to in the article you link to and the mathpages article linked to in post #5 assumes standard Schwarzschild coordinates centered on the massive body. Other coordinates lead to different notions of "space", not all of which are non-Euclidean (for example, in Painleve coordinates in Schwarzschild spacetime, "space" is Euclidean). But the global light bending is invariant; it doesn't depend on which frame you choose.

Second, the issue that leads to the "2x" for global light bending, as compared to the "x" in a local "accelerating elevator" experiment, is really due to the way the local "elevator" frames fit together in spacetime, not space. Thinking of the path of the light as "bent in space" doesn't really capture that, because the local frames that have to "fit together" are not at different points in space; they are at different points in spacetime, the different points along the worldline of the light beam as it passes by the massive object. The "2x" arises because the spacetime is curved, so the local frames don't fit together the way they would in flat spacetime.

for a nonmoving case, such as a rigid circle built around the sun with a rigid rod going through the middle of the sun to the other side, and the ratio not being pi, does the same / similar logic hold ?

 

[PeterDonis]

for a nonmoving case, such as a rigid circle built around the sun with a rigid rod going through the middle of the sun to the other side, and the ratio not being pi, does the same / similar logic hold ?

The "ratio" here is a different thing: you are comparing the arc lengths along two spacelike curves. Once you specify the curves, those two arc lengths and their ratio are invariant; and you can specify the curves in a way that doesn't depend on which coordinates you choose. But the curves are spacelike, whereas the worldline of the light ray in the light bending case is null.

 

[pervect]

Is it not simply the case that the 2x is due to space not being euclidian in the vicinity of a gravitating mass ?
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/

This is a bit confusingly worded. I think.

Are you trying to say that thee 2x is due to spatial curvature? If so, I'd agree.

 

[PeterDonis]

Are you trying to say that thee 2x is due to spatial curvature? If so, I'd agree.

With the caveat I gave in post #8.

 

[A.T.]

But does it become more Euclidean as you approach the Newtonian limit of GR?

Both contributions to deflection get weaker when gravity gets weaker. Their ratio remains 2.

 

[A.T.]

This article actually addresses my motivation for asking my question - how the 2x deflection is reconciled with the equivalence principle and the accelerating elevator thought experiment.

The equivalence principle accounts for the local bending. But the global spatial geometry is such that even a locally straight path (spatial geodesic) would be deflected globally, which accounts for the additional deflection.

 

[PeterDonis]

the global spatial geometry is such that even a locally straight path (spatial geodesic) would be deflected globally

This is a bit misleading as you state it. The worldline of the light ray (which is a geodesic, so it is "locally straight") is null, not spacelike. The reason the light ray's path looks "locally bent" in an accelerating elevator is that the elevator is accelerating--the elevator's path is the one that is not a geodesic.

 

[1977ub]

The "ratio" here is a different thing: you are comparing the arc lengths along two spacelike curves. Once you specify the curves, those two arc lengths and their ratio are invariant; and you can specify the curves in a way that doesn't depend on which coordinates you choose. But the curves are spacelike, whereas the worldline of the light ray in the light bending case is null.

The fact that the ratio of a circle about the Sun to a diameter through the Sun's center is not pi - this suggest non-euclidian space, no ?

 

[A.T.]

This is a bit misleading as you state it. The worldline of the light ray (which is a geodesic, so it is "locally straight") is null, not spacelike. The reason the light ray's path looks "locally bent" in an accelerating elevator is that the elevator is accelerating--the elevator's path is the one that is not a geodesic.

I'm explicitly talking about spatial geodesics (locally straight spatial paths) there, not about space-time geodesics (locally straight worldlines). A spatial geodesic through Flamm's paraboloid would be globally deflected. That is one way to understand the additional global deflection.

 

[PeterDonis]

A spatial geodesic through Flamm's paraboloid would be globally deflected. That is one way to understand the additional global deflection.

Are you saying that such a spatial geodesic is somehow a projection of the worldline of a light ray?

 

[A.T.]

Are you saying that such a spatial geodesic is somehow a projection of the worldline of a light ray?

No. The light ray's spatial path deviates from a spatial geodesic (local deflection). But a spatial geodesic itself is deflected globally.

 

[pervect]

With the caveat I gave in post #8.

I'd generalize this from saying "space is curved in Scwarzschild coordinates" to to "space is curved in any static reference frame near a single massive body", but I agree with the cautions you express in #8. More could be said about what I mean when I say "static reference frame", but it'd jump the thread to A-level.

 

[Wes Tausend]

As a matter of trivia, Einstein apparently initially miscalculated the bending of a starlight ray. Having little or no equipment himself, he had started encouraging astronomers to look for this particular aspect of proving his GR. His error was that he had initially calculated the bend as only half of what it should be.

Although it first frustrated Einstein that vindication was taking so long, in the end he was fortunate that WWI intervened or he would have botched his original proof. After the war, astronomers again began to freely travel the world in search of a suitable eclipse and met with success. Fortunately they had the correct calculations by then. I'm not sure it is correct, but thought a crude explanation was that Einstein had modified Lorentz distance but failed to equally modify time.

No one should be too embarrassed about their own published papers as it sometimes takes more than one shot. I believe Einstein actually had promoted several slightly modified versions of GR over the latter years between SR and a final GR, at least in lectures. He made a lot of minor mistakes but it seems he eventually got his ideas right. Most importantly, he never gave up.

I'll be an amateur Relativity fan for forty years this spring and I cannot remember all the sources I've first come across. But I did find a partial reference here. There is a photo of a letter Einstein sent, that when enlarged, reveals the miscalculation.

Wes

 

[PAllen]

As a matter of trivia, Einstein apparently initially miscalculated the bending of a starlight ray. Having little or no equipment himself, he had started encouraging astronomers to look for this particular aspect of proving his GR. His error was that he had initially calculated the bend as only half of what it should be.

Although it first frustrated Einstein that vindication was taking so long, in the end he was fortunate that WWI intervened or he would have botched his original proof. After the war, astronomers again began to freely travel the world in search of a suitable eclipse and met with success. Fortunately they had the correct calculations by then. I'm not sure it is correct, but thought a crude explanation was that Einstein had modified Lorentz distance but failed to equally modify time.

No one should be too embarrassed about their own published papers as it sometimes takes more than one shot. I believe Einstein actually had promoted several slightly modified versions of GR over the latter years between SR and a final GR, at least in lectures. He made a lot of minor mistakes but it seems he eventually got his ideas right. Most importantly, he never gave up.

I'll be an amateur Relativity fan for forty years this spring and I cannot remember all the sources I've first come across. But I did find a partial reference here. There is a photo of a letter Einstein sent, that when enlarged, reveals the miscalculation.

Wes

The earlier results were based on precursors of GR, not the final equations (nothing before 1915 is equivalent to the 1915 theory). Whether the mistake would have carried over to full GR is a separate question (to which I don't know the answer).

 

[PeterDonis]

Whether the mistake would have carried over to full GR is a separate question (to which I don't know the answer).

According to Wikipedia, it didn't. Once Einstein had come up with the final (November 1915) field equation, he re-calculated the bending of light by the Sun and got the "2x" answer.

https://en.wikipedia.org/wiki/Tests_of_general_relativity#Deflection_of_light_by_the_Sun