<p>As far as I know Newtonian prediction is based on Soldner's work in 1801 and Einstein 1911 also predicted around $0.87$arcsec (he was also accused for plagiarizing Soldner's work the article:<a href="http://articles.adsabs.harvard.edu//full/1923PASP...35..185T/0000185.000.html" rel="noreferrer">Historical Note on the Problem of Light Deflection in the Sun's Gravitational Field</a> explains why the accusations were wrong).</p>

<p>The Newtonian approach accounted for the space curvature, as did special relativity which Einstein first used, but it missed the part that not only space is curved but space time is curved. So the factor of two was because of the spatial curvature, that's because by <a href="https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Flat_spacetime" rel="noreferrer">Minkowski metric</a> we consider that light travels flat so that $\rm{d}s^2=0$.The correct answer would be to use the <a href="https://en.wikipedia.org/wiki/Schwarzschild_metric" rel="noreferrer">Schwarzschild metric</a> and get the correct answer.</p>

<p>This paper gives more insight on the formulas and the correct approach. <a href="http://home.fnal.gov/~syphers/Education/Notes/lightbend.pdf" rel="noreferrer">Gravitational Bending of Light</a></p>

CC BY-SA 3.0
2017-05-04T19:47:02.800Z
If the Sun would have been twice as massive, would the outcome also be the double of the answer of Netwon? Or  under what conditions wouldn't the outcome be twice as big?
Marijn 
CC BY-SA 3.0
2017-05-04T20:19:22.197Z
The outcome should always be exactly double the Newtonian,  because the space-space part of the Schwartzschild metric tensor is of the same form as the time-time part, and the photon has a trajectory where both contribute equally, so you get a factor of 2.
kingW3
CC BY-SA 3.0
2017-05-04T19:08:04.820Z
2017-05-04T19:08:04.820Z
kingW3
<blockquote>
  <p>Einstein predicted a declination of 1.74 arc seconds, but according to Newton the declination should have been 0.86 arc seconds. So is it a coincidence that the prediction of Einstein is twice as large... ? </p>
</blockquote>

<p>No. Einstein's figure is twice Newton's because of the wave nature of matter. Check out Ned Wright's <a href="http://www.astro.ucla.edu/~wright/deflection-delay.html" rel="nofollow noreferrer">deflection and delay</a> article:</p>

<p><a href="https://i.sstatic.net/ndyzV.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ndyzV.gif" alt="enter image description here"></a>   </p>

<p>He says this: <em>"Newtonian mechanics predicts that a particle traveling at the speed of light which just grazes the edge of the Sun will be deflected by 0.875 seconds of arc"</em>. But that's a particle such as an electron, which we might have created from light in <a href="https://en.wikipedia.org/wiki/Pair_production" rel="nofollow noreferrer">pair production</a>. Pair production <em>"often refers specifically to a photon creating an electron-positron pair".</em> </p>

<p>Obviously an electron isn't the same thing as a photon. One essential difference is mass, another is charge, and another is the way it's deflected by gravity. If you threw an electron fast past the Sun, nearly as fast as light, its deflection would be half the deflection of a photon.  </p>

<p>You can understand why by thinking what would happen if you trapped the photon in a gedanken mirror-box such that it was going round and round a square path. Then if you put it in a gravitational field the light would be deflected like this:</p>

<p><a href="https://i.sstatic.net/OZdQ3.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OZdQ3.jpg" alt="enter image description here"></a></p>

<p>The horizontals curve down, and because of this the position changes. The photon in its gedanken mirror-box <em>falls down</em>. Then if you were to throw it fast past the Sun you'd say it was deflected - it's moving fast from left to right, but it falls down too. But note that the verticals are not affected. Only half the light path is curved by the Sun. The electron is affected in a similar fashion because of the <a href="https://www.google.co.uk/?gfe_rd=cr&amp;ei=EUM_VY2-Juiq8wfv_oCgDA&amp;gws_rd=ssl#q=wave+nature+of+matter&amp;spf=406" rel="nofollow noreferrer">wave nature of matter</a>. This is why the deflection of matter is half the deflection of light.    </p>

CC BY-SA 3.0
2017-05-05T14:20:55.893Z
But how does the deflection of light vs matter relate to this question?
zephyr
CC BY-SA 3.0
2017-05-06T17:19:43.500Z
@zephyr : the Newtonian deflection is the deflection of matter, and it's correct. But it's not correct to assume that the deflection of matter is the same as the deflection of light. The deflection of matter is only half the deflection of light, because of the wave nature of matter. Have a look at https://arxiv.org/abs/1508.06478 by van der Mark and (not the Nobel) 't Hooft for more about light in a gedanken mirror-box.
John Duffield
CC BY-SA 4.0
2019-04-18T16:44:27.657Z
The "particle" in Ned Wright's article could be a photon. He doesn't specify because it doesn't matter in Newtonian physics. There is no distinction in Newtonian mechanics or GR between a particle with mass, or one without mass, travelling very near or at the speed of light. Both theories give the same deflection for photon and particle, but the GR deflection is twice as big. Where they differ is that Newtonian gravity suggests the deflection of a non-relativistic particle is the same as that of light too whereas GR says that a non-relativistic particle has half the deflection.
ProfRob
CC BY-SA 4.0
2019-04-18T16:47:42.503Z
A mildly relativistic particle would have a deflection somewhere in between the Newtonian and GR values for light. I have given you the necessary references in a comment to another question.
ProfRob
CC BY-SA 4.0
2019-04-19T08:53:20.790Z
@Rob Jeffries : It's a reference to a rather obscure [Chinese paper](https://arxiv.org/abs/1512.04037) written in 2015. I was hoping for a reference from Einstein on this. I note your reference to MTW in your other answer, which many people would say is the next best thing. However I'm not a fan of MTW - I think it's contradicts Einstein in some important ways.
John Duffield
CC BY-SA 4.0
2019-04-19T10:28:59.893Z
What an extraordinarily racist comment. "Classical and Quantum Gravity" is an Instutute of Physics journal and is the leading publication on all aspects of gravitation. Maybe it is hard for you to accept that Chinese authors (along with everybody else with a passing acquaintance with GR) might know more than you. As for MTW - these are the three leading classical general relativists of their time, one a Nobel prize winner. Your casual dismissal of their text, widely acknowledged as *the* text on General Relativity, is simply ludicrous.
ProfRob
CC BY-SA 4.0
2019-04-19T10:34:21.317Z
And here's a reference (in the same leading international journal)  saying the same thing by authors from French and Irish institutions if that matters to you. https://arxiv.org/abs/gr-qc/0311010
ProfRob
CC BY-SA 4.0
2019-04-19T16:59:03.507Z
@Rob Jeffries: that wasn't a racist comment, it was factual. I said Chinese because it wasn't an Einstein reference. As for your further reference, I gritted my teeth at Kerr gravitational field (because of the CTCs and wormholes) and the cosmic strings, then gave up at _"We shall use units in which the velocity of light c = 1"._ I refer you again to [what Einstein said](https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22): _"the curvature of light rays occurs only in spaces where the speed of light is spatially variable"._  That matters to me.
John Duffield
CC BY-SA 3.0
2017-05-05T12:36:18.660Z
2017-05-05T12:36:18.660Z
John Duffield
<p>During a solar eclipse Sir Arthur Eddington measured the deflection of star light by the Sun. Einstein predicted a deflection of 1.74 arc seconds, but according to Newton the deflection should have been 0.86 arc seconds. So is it a coincidence that the prediction of Einstein is twice as large or is there a kind of law involved which makes Einstein's outcome of his GR twice as big as that according to Newton?</p>

<general-relativity>
Is it coincidence that Einstein predicted exactly twice the outcome of Newton for the gravitational deflection of light?
Marijn 
# Is it coincidence that Einstein predicted exactly twice the outcome of Newton for the gravitational deflection of light?

(Asked by: username_0 on 2017-05-04)

During a solar eclipse Sir Arthur Eddington measured the deflection of star light by the Sun. Einstein predicted a deflection of 1.74 arc seconds, but according to Newton the deflection should have been 0.86 arc seconds. So is it a coincidence that the prediction of Einstein is twice as large or is there a kind of law involved which makes Einstein's outcome of his GR twice as big as that according to Newton?

**Tags:** general-relativity

---

## 2 answers

### Answer by username_1 on 2017-05-04. Score: 5

As far as I know Newtonian prediction is based on Soldner's work in 1801 and Einstein 1911 also predicted around $0.87$arcsec (he was also accused for plagiarizing Soldner's work the article:[Historical Note on the Problem of Light Deflection in the Sun's Gravitational Field](http://articles.adsabs.harvard.edu//full/1923PASP...35..185T/0000185.000.html) explains why the accusations were wrong).

The Newtonian approach accounted for the space curvature, as did special relativity which Einstein first used, but it missed the part that not only space is curved but space time is curved. So the factor of two was because of the spatial curvature, that's because by [Minkowski metric](https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)#Flat_spacetime) we consider that light travels flat so that $\rm{d}s^2=0$.The correct answer would be to use the [Schwarzschild metric](https://en.wikipedia.org/wiki/Schwarzschild_metric) and get the correct answer.

This paper gives more insight on the formulas and the correct approach. [Gravitational Bending of Light](http://home.fnal.gov/~syphers/Education/Notes/lightbend.pdf)

> **Comments:**
> **username_0**: If the Sun would have been twice as massive, would the outcome also be the double of the answer of Netwon? Or  under what conditions wouldn't the outcome be twice as big?  
> **username_1**: The outcome should always be exactly double the Newtonian,  because the space-space part of the Schwartzschild metric tensor is of the same form as the time-time part, and the photon has a trajectory where both contribute equally, so you get a factor of 2.  

---

### Answer by username_2 on 2017-05-05. Score: -3

> Einstein predicted a declination of 1.74 arc seconds, but according to Newton the declination should have been 0.86 arc seconds. So is it a coincidence that the prediction of Einstein is twice as large... ?

No. Einstein's figure is twice Newton's because of the wave nature of matter. Check out Ned Wright's [deflection and delay](http://www.astro.ucla.edu/~wright/deflection-delay.html) article:

[![enter image description here](https://i.sstatic.net/ndyzV.gif)](https://i.sstatic.net/ndyzV.gif)

He says this: *"Newtonian mechanics predicts that a particle traveling at the speed of light which just grazes the edge of the Sun will be deflected by 0.875 seconds of arc"*. But that's a particle such as an electron, which we might have created from light in [pair production](https://en.wikipedia.org/wiki/Pair_production). Pair production *"often refers specifically to a photon creating an electron-positron pair".*

Obviously an electron isn't the same thing as a photon. One essential difference is mass, another is charge, and another is the way it's deflected by gravity. If you threw an electron fast past the Sun, nearly as fast as light, its deflection would be half the deflection of a photon.

You can understand why by thinking what would happen if you trapped the photon in a gedanken mirror-box such that it was going round and round a square path. Then if you put it in a gravitational field the light would be deflected like this:

[![enter image description here](https://i.sstatic.net/OZdQ3.jpg)](https://i.sstatic.net/OZdQ3.jpg)

The horizontals curve down, and because of this the position changes. The photon in its gedanken mirror-box *falls down*. Then if you were to throw it fast past the Sun you'd say it was deflected - it's moving fast from left to right, but it falls down too. But note that the verticals are not affected. Only half the light path is curved by the Sun. The electron is affected in a similar fashion because of the [wave nature of matter](https://www.google.co.uk/?gfe_rd=cr&ei=EUM_VY2-Juiq8wfv_oCgDA&gws_rd=ssl#q=wave+nature+of+matter&spf=406). This is why the deflection of matter is half the deflection of light.

> **Comments:**
> **username_3**: But how does the deflection of light vs matter relate to this question?  
> **username_2**: @zephyr : the Newtonian deflection is the deflection of matter, and it's correct. But it's not correct to assume that the deflection of matter is the same as the deflection of light. The deflection of matter is only half the deflection of light, because of the wave nature of matter. Have a look at https://arxiv.org/abs/1508.06478 by van der Mark and (not the Nobel) 't Hooft for more about light in a gedanken mirror-box.  
> **username_4**: The "particle" in Ned Wright's article could be a photon. He doesn't specify because it doesn't matter in Newtonian physics. There is no distinction in Newtonian mechanics or GR between a particle with mass, or one without mass, travelling very near or at the speed of light. Both theories give the same deflection for photon and particle, but the GR deflection is twice as big. Where they differ is that Newtonian gravity suggests the deflection of a non-relativistic particle is the same as that of light too whereas GR says that a non-relativistic particle has half the deflection.  
> **username_4**: A mildly relativistic particle would have a deflection somewhere in between the Newtonian and GR values for light. I have given you the necessary references in a comment to another question.  
> **username_2**: @Rob Jeffries : It's a reference to a rather obscure [Chinese paper](https://arxiv.org/abs/1512.04037) written in 2015. I was hoping for a reference from Einstein on this. I note your reference to MTW in your other answer, which many people would say is the next best thing. However I'm not a fan of MTW - I think it's contradicts Einstein in some important ways.  
> **username_4**: What an extraordinarily racist comment. "Classical and Quantum Gravity" is an Instutute of Physics journal and is the leading publication on all aspects of gravitation. Maybe it is hard for you to accept that Chinese authors (along with everybody else with a passing acquaintance with GR) might know more than you. As for MTW - these are the three leading classical general relativists of their time, one a Nobel prize winner. Your casual dismissal of their text, widely acknowledged as *the* text on General Relativity, is simply ludicrous.  
> **username_4**: And here's a reference (in the same leading international journal)  saying the same thing by authors from French and Irish institutions if that matters to you. https://arxiv.org/abs/gr-qc/0311010  
> **username_2**: @Rob Jeffries: that wasn't a racist comment, it was factual. I said Chinese because it wasn't an Einstein reference. As for your further reference, I gritted my teeth at Kerr gravitational field (because of the CTCs and wormholes) and the cosmic strings, then gave up at _"We shall use units in which the velocity of light c = 1"._ I refer you again to [what Einstein said](https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22): _"the curvature of light rays occurs only in spaces where the speed of light is spatially variable"._  That matters to me.  

---

