# Is it coincidence that Einstein predicted exactly twice the outcome of Newton for the gravitational deflection of light? [duplicate]

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?

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 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 we consider that light travels flat so that $\rm{d}s^2=0$.The correct answer would be to use the Schwarzschild metric and get the correct answer.

This paper gives more insight on the formulas and the correct approach. Gravitational Bending of Light

• 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 May 4 '17 at 19:47
• 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 May 4 '17 at 20:19

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 article:

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. 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:

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. This is why the deflection of matter is half the deflection of light.

• But how does the deflection of light vs matter relate to this question? – zephyr May 5 '17 at 14:20
• 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. – Rob Jeffries Apr 18 '19 at 16:44
• 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. – Rob Jeffries Apr 18 '19 at 16:47
• 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. – Rob Jeffries Apr 19 '19 at 10:28
• 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. arxiv.org/abs/gr-qc/0311010 – Rob Jeffries Apr 19 '19 at 10:34