Eddington Exp vs Newton: Comparing Starlight Deflection

[Karl Coryat]

TL;DR Summary

Why did GR predict a deflection of starlight by the Sun that was double the Newtonian expectation?

The Eddington Experiment famously confirmed GR by showing, as Einstein had predicted, a deflection of starlight by the Sun that was double the deflection expected by Newtonian gravity. I don't understand where Einstein's 2x number came from. I make the following assumptions:
1. That a ray of light and an inertial test mass both take the same geodesic through spacetime
2. That the expected Newtonian deflection is calculated by determining how much the starlight "falls" toward the Sun enroute
3. That the acceleration of this falling is the same as the acceleration of a test mass, Galileo having established that everything takes the same acceleration regardless of mass.

One or more of those assumptions must be wrong.

To put this question another way, does a light beam parallel to the Earth's surface deflect downward at 19.6 m/s², i.e., 19.6 meters vertically for every 90 billion kilometers horizontally?

Thank you for your help!

 

[PeterDonis]

I don't understand where Einstein's 2x number came from.

The best way to understand that is to look at the actual GR math, which can be found in most GR textbooks.

1. That a ray of light and an inertial test mass both take the same geodesic through spacetime

This is not correct. Given a fixed starting event in spacetime and a fixed spatial direction, a light ray follows the unique null geodesic defined by those starting parameters, but an inertial test mass can follow any of a continuous infinity of timelike geodesics allowed by those starting parameters, depending on its initial velocity. None of those timelike geodesics are the same as the null geodesic the light ray follows (since null and timelike geodesics are fundamentally different).

2. That the expected Newtonian deflection is calculated by determining how much the starlight "falls" toward the Sun enroute

This is basically what the Newtonian calculation does, but note that in Newtonian physics the fact that light "falls" at all has to be assumed since it is not implied by the theory.

3. That the acceleration of this falling is the same as the acceleration of a test mass, Galileo having established that everything takes the same acceleration regardless of mass.

This is true locally at a point, but the light bending calculation is not a local calculation, it's a global calculation. One way of looking at where the factor of 2 comes from is that in the curved spacetime surrounding a massive body, the local frames all along the trajectory of a light ray passing near the body do not "fit together" the way they would in flat spacetime. Some sources will attribute this to "spatial curvature", which, if we adopt a particular frame of reference, can indeed be a valid description of the math.

To put this question another way, does a light beam parallel to the Earth's surface deflect downward at 19.6 m/s², i.e., 19.6 meters vertically for every 90 billion kilometers horizontally?

This is a meaningless question since the Earth's size is much, much less than 90 billion kilometers.

 

[PeroK]

TL;DR Summary: Why did GR predict a deflection of starlight by the Sun that was double the Newtonian expectation?

The GR calculation is non-trivial. It's in Hartle's GR book, culminating in the equation $$\delta \phi = \frac{4GM}{bc^2}$$ where $\delta \phi$ is the angular deflection and $b$ is the impact parameter.

The classical inverse square deflection can be found here, equation (4.20):

https://www.damtp.cam.ac.uk/user/tong/relativity/four.pdf

$$\delta \phi = 2 \tan^{-1}(\frac{GM}{bc^2})$$ If we take the case that $\frac{GM}{bc^2}$ is small (the Hartle equation was explicitly for this case) and use the Taylor expansion for $\tan^{-1}$, we get the :
$$\delta \phi = \frac{2GM}{bc^2}$$ which is half that calculated from GR.

Note also that these are approximations for the particular case of a small deflection. The maths, as it often does, gets messy.

 

[Karl Coryat]

This is a meaningless question since the Earth's size is much, much less than 90 billion kilometers.

I appreciate the explanations! Let me rephrase that last question as, Does a light beam parallel to the Earth's surface deflect downward at 1.96 x 10^–9 meters vertically for every 9 kilometers horizontally? (It's a conceptual question rather than a practical one.)

 

[PeterDonis]

I appreciate the explanations! Let me rephrase that last question as, Does a light beam parallel to the Earth's surface deflect downward at 1.96 x 10^–9 meters vertically for every 9 kilometers horizontally? (It's a conceptual question rather than a practical one.)

Measured locally, yes (assuming your numbers are correct, I haven't checked them). But if you do a global measurement of light bending by the Earth, you will get twice the value you would expect by just adding up all the local contributions and assuming that spacetime is flat (or, equivalently, that space in a frame in which the Earth is at rest is Euclidean).

 

[Karl Coryat]

Measured locally, yes (assuming your numbers are correct, I haven't checked them).

I think you mean no, because those numbers reflect twice the Newtonian prediction (19.6 m/s²). I think what you're telling me is, if we had instruments of arbitrary precision, in the limit of spacetime being locally flat, we would measure light to "fall" at the Newtonian rate; but if we zoomed out and considered the total deflection along the ray's route, we would find exactly twice that value. Is that right? (I apologize for any sloppy language.)

 

[PeterDonis]

I think you mean no, because those numbers reflect twice the Newtonian prediction (19.6 m/s²).

Well, I told you I didn't check the numbers. I assumed you were just expressing the standard "acceleration due to gravity" at the Earth's surface in different units.

I think what you're telling me is, if we had instruments of arbitrary precision, in the limit of spacetime being locally flat, we would measure light to "fall" at the Newtonian rate

Yes.

if we zoomed out and considered the total deflection along the ray's route, we would find exactly twice that value.

No. You can't compare the two things that way. You would find that the global light bending is twice the value you would get if we added up all the local "falling" on the assumption of flat spacetime (or Euclidean space in the Earth's rest frame). But there is no valid way to convert that global result into a local "rate of bending" that is twice the Newtonian value. If you try to do any such conversion, the only valid way to do it just undoes all the effects of curved spacetime (or non-Euclidean space in the Earth's rest frame) and gets you right back to the Newtonian local "rate of falling" that you measured. All of which is just another way of saying that GR predicts both things: that the local "rate of falling" is the Newtonian one, and that the global bending is what it is. There is no way to separate the two.