The bending of starlight is twice the Newtonian prediction

[PeterDonis]

you can add the equivalence principle to SR and say something about gravity.

Yes, you can. But "bending of light" is not the same as "gravity"; you don't even need to "add the equivalence principle to SR" to predict that light bending will be observed in an accelerating rocket in flat spacetime.

 

[Sagittarius A-Star]

Fair enough, you can add the equivalence principle to SR and say something about gravity.

A. Einstein did it in 1911 with SR and got the wrong result by a factor of 2 for the bending of light near the sun (see §4 in the link). Later, he did a new calculation, based on GR.
http://www.relativitycalculator.com...ation_on_the_Propagation_of_Light_English.pdf

 

[PeroK]

Yes, you can. But "bending of light" is not the same as "gravity"; you don't even need to "add the equivalence principle to SR" to predict that light bending will be observed in an accelerating rocket in flat spacetime.

We're not talking about accelerating rockets, we're talking about starlight being "bent" by the Sun. That's why we need a theory of gravity.

 

[PeterDonis]

We're not talking about accelerating rockets, we're talking about starlight being "bent" by the Sun. That's why we need a theory of gravity.

That still doesn't make it correct to say that, because there is no gravity in SR, there is no bending of light in SR. There is. And it occurs in accelerating rockets; you need the proper acceleration in order to derive the SR result. Which means you also need the proper acceleration to invoke the equivalence principle and make use of the SR result in the presence of gravity; locally a free-falling observer even in a curved spacetime will not see any bending of light.

 

[PeterDonis]

A. Einstein did it in 1911 with SR and got the wrong result by a factor of 2 for the bending of light near the sun (see §4 in the link).

The calculations in general in that paper are not really just doing it "with SR", because the concept of "uniform gravitational field" that Einstein uses is not SR. (It also turns out to be quite a bit more complicated than Einstein realized when he wrote that paper.) But even if we set that aside, section 4 extends the results to non-uniform gravitational fields like that of the Sun, which is certainly not SR.

 

[PeroK]

This calculation is not really just doing it "with SR", because the concept of "uniform gravitational field" that Einstein uses is not SR. (It also turns out to be quite a bit more complicated than Einstein realized when he wrote that paper.)

That, it seems to me, is precisely the point that @Sagittarius A-Star was making.

 

[Sagittarius A-Star]

That, it seems to me, is precisely the point that @Sagittarius A-Star was making.

Einstein applied in 1911 the equivalence principle globally. That was his error. Here is Einstein's GR calculation from 1916, see equation (74):
https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

 

[Sagittarius A-Star]

The calculations in general in that paper are not really just doing it "with SR", because the concept of "uniform gravitational field" that Einstein uses is not SR.

I thought, SR is the theory of flat spacetime. So Einstein should have assumed, a "uniform gravitational field" only locally to apply the equations of SR, but he did it globally.

 

[PeroK]

I was wrong about there being no bending of light. It can also happen in a prism:

 

[Sagittarius A-Star]

I found an explanation for the factor 2 between Einstein's calculations from 1911 and 1916:

Doubling the Deflection
...
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).
...

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

 

[Nugatory]

I found an explanation for the factor 2 between Einstein's calculations from 1911 and 1916:

And that is also the explanation provided by @Ibix in #2, the very first reply to the original post, four pages back.

 

[PeterDonis]

That, it seems to me, is precisely the point that @Sagittarius A-Star was making.

Saying "he did it with SR", which is what @Sagittarius A-Star said in the post I responded to, seems to me to be a strange way of saying "he didn't really do it with SR", which is what I said.

 

[PeterDonis]

I thought, SR is the theory of flat spacetime. So Einstein should have assumed, a "uniform gravitational field" only locally to apply the equations of SR, but he did it globally.

Yes, it is true that assuming there could be a "uniform gravitational field" globally was a problem with Einstein's 1911 calculation.

I think it's worth expanding some on why it is a problem. Take a step back and consider the flat spacetime of SR in a global inertial frame (which of course always exists in flat spacetime). It is obvious that in this global frame, there is no bending of light. In order to have bending of light in flat spacetime, you must be at rest in a uniformly accelerated frame (at least if we are talking about vacuum--with a material medium present you can, of course, have light bending in an inertial frame in flat spacetime, as @PeroK's example of a prism shows). But there is no global uniformly accelerated frame in flat spacetime. (Of course there isn't one in any curved spacetime either, but in 1911 Einstein hadn't quite gotten to curved spacetime conceptually.) We know that now because we know about Rindler coordinates (which were not discovered until decades later, IIRC), and we know that they only cover a portion of flat spacetime, not all of it.

So in extending his 1911 calculation globally, Einstein was in fact doing invalid math; he just didn't realize it.

 

[vanhees71]

The point is that when using the linearized Einstein field equations, which is for sure legitimate for our Sun, that it's not sufficient to take into account the perturbation of $g_{00}=1+h_{00}$ as it is for "slow" massive "test particles" (planets), because the naive "photon" is massless, and thus you have to take into account also the spatial corrections. For the radially symmetric vacuum solution in the usual spherical coordinates what you get is the corresponding approximation of the Schwarzschild solution,
$$\mathrm{d} s^2 \simeq (1-r_S/r)^2 \mathrm{d} t^2 - (1+r_S/r)^2 \mathrm{d} r^2 - r^2 (\mathrm{d} \vartheta^2 + \sin^2 \vartheta \mathrm{d} \varphi^2),$$
valid for trajectories with $r\ll r_S$.

 

[Sagittarius A-Star]

Yes, it is true that assuming there could be a "uniform gravitational field" globally was a problem with Einstein's 1911 calculation

I must correct by above posting #78. Einstein did not assume globally a "uniform gravitational field", but along the path of the light a gravitational field, of which the magnitude of gravitational acceleration depends on the distance to the sun. So he applied the equivalence principle locally, but assumed a wrong global model for the gravitational field, because in his model, it's direction is every the same.

 

[PeterDonis]

Einstein did not assume globally a "uniform gravitational field", but along the path of the light a gravitational field, of which the magnitude of gravitational acceleration depends on the distance to the sun.

Can you point out the specific place in the 1911 paper where Einstein makes this assumption (the one I've bolded in the above quote?

 

[Sagittarius A-Star]

Can you point out the specific place in the 1911 paper where Einstein makes this assumption (the one I've bolded in the above quote?

The distance to the sun is $r$ in the integral under equation (4):
http://www.relativitycalculator.com...ation_on_the_Propagation_of_Light_English.pdf

This integral is explained from the equivalence principle after equation (1) in

Now, we can read directly off the figure that the direction of the wave front changes by an amount equal to –∂c/∂y per unit of distance along the direction of the wave. Note that the total deflection is extremely small, so to the first order of approximation we can consider just the x component of its motion as depicted below.

The partial of c with respect to y is

To compute the total deflection of the wave front along the entire path, Einstein simply integrated this over the path.
...
(See below for a comment on how Einstein actually performed this integration.)

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

 

[PeterDonis]

The distance to the sun is $r$ in the integral under equation (4)

Ah, got it.

 

[Sagittarius A-Star]

Ah, got it.

I must also correct my posting #85. He makes no wrong assumption about the direction of the global gravitation field at a certain location. The only error is the missing spatial curvature.

 

[H_A_Landman]

I don't think so, but I'd need to double check the maths.

Is it the case that as the line of sight of the star moves further and further away from the star, the bending approaches the Newtonian prediction? If so, why are we told that the GR prediction is always precisely twice the Newtonian one?

It is not the case that in the weak-field limit, bending gets Newtonian. In that limit (of say the Schwarzschild metric) there is still a spatial term that ranges from 0% to 100% as large as the temporal term, depending on speed.

To get Newtonian gravity, you have to take the weak-field AND low-speed limits. The spatial term goes to zero at v=0, but equals the temporal term at v=c. So light cannot ever bend by just the Newtonian amount; because it travels at the speed of light, the spatial term is always at max, and the bending is exactly twice what Newton predicts.

At intermediate speeds, the spatial term is always smaller than the temporal term. Thus it is NEVER correct to simplify "gravity is due to the curvature of space-time" to "gravity is due to the curvature of space", but it is often correct to simplify it to "gravity is due to the curvature of time". For example, the gravity holding your butt in your chair is at least 99.9999% due to the curvature of time.

 

[haushofer]

That mathpages treatment ("doubling the deflection") is really insightful. It shows that the total deflection is doubled w.r.t. Newton, but not necessarily at every point of the photon's path; that depends on your coordinates.