Undergrad Newtonian path of light in a gravitational field

[BiGyElLoWhAt]

I think this is the appropriate subforum.

I'm curious as to what approaches have been taken. I know this prediction isn't correct. I can think of at least a couple ways that I could go about this. They may or may not give the same prediction.

One approach would be to simply use kinematics, and force a rescale after every interval dt. I haven't tried it. This might be a computational thing.
What I mean is start with an initial velocity and position, use $a = GM/r^2$ and integrate. However, we need to maintain that the velocity is always c, so we would need to include a factor of something like $\frac{c}{|c + \int adt|}$.
I think I'm going to end up with something like:
$\frac{c}{|c + \int adt|} (\int (adt) + v_0)$ for the velocity function, which I could then integrate again to get a position function. I could write $v_0$ as $c<cos(\theta),sin(\theta)>$ and now I have a parameter to vary to get different paths.

Another approach I found on stack exchange is to use a test mass and take the limit as m-> 0.

I'm interested in thoughts on these as well as other possible ways you might try to tackle this.

Edit:
I think my rescale equation isn't quite right. That should work computationally, not analytically (assuming there is an analytical solution). The issue is that this isn't a rescale every dt, it's only a rescale at the end. I'm currently thinking about how I should put it in so that it also works analytically. Will post back when I think of it.

 

[Ibix]

In Newtonian gravity you can simply note that if $F=ma$ and $F=GMm/r^2$ the $m$s cancel out and the acceleration is independent of the mass. Then you can just feed $v=c$ into the standard orbital equations.

Alternatively you can take an effective potential approach, in which light turns out to be unaffected by Newtonian gravity.

Fundamentally, the problem is that Newtonian gravity is inaccurate for things travelling near $c$. It's really a case of what inaccuracy you want to accept.

 

[Orodruin]

However, we need to maintain that the velocity is always c

This is impossible in Newtonian mechanics as it rests upon the Galilean transformations. If the speed is c in one frame, it will not be in another. Hence there is no motivation for normalising speed to c.

 

[BiGyElLoWhAt]

For context, I am joining a project that aims to measure lensing during the solar eclipse in about 6 weeks. What they are currently attempting to do currently is come up with a non-relativistic prediction. I've read that allegedly it's off by a factor of 2. So far there are 4 possible ways to make predictions that satisfy this criteria in this thread.
As far as the normalization goes, would that not be appropriate for an initial wrong prediction?

Also, since we've all gathered here, for a more accurate prediction, my assumption is that I can just use the schwartzschild metric, calculate the geodesics using an arbitrary parameter (not tau) an use $\frac{dx^{\mu}}{d\lambda} = <0,\vec{c}>$.
Is that the general process for obtaining light like paths?

 

[Orodruin]

The deflection angle is very small since the Solar radius is pretty far away from the Sun's Schwarzschild radius. It is perfectly fine to just use first order perturbation theory in the weak field limit. This makes the computation very easy indeed.

 

[Ibix]

I think we discussed this before, and the predictions Eddington compared were GR and some form of proto-GR that neglected the curvature of the spatial planes. The latter gives predictions for the deflection of light that are the same as Newtonian gravity with a ballistic model of light as a particle that has speed $c$ at infinity.

I don't have a reference for that, unfortunately, but some searching of PF may turn it up.

 

[haushofer]

This is impossible in Newtonian mechanics as it rests upon the Galilean transformations. If the speed is c in one frame, it will not be in another.

Except if c is infinitely large

 

[A.T.]

I think we discussed this before, and the predictions Eddington compared were GR and some form of proto-GR that neglected the curvature of the spatial planes. The latter gives predictions for the deflection of light that are the same as Newtonian gravity with a ballistic model of light as a particle that has speed $c$ at infinity.

I don't have a reference for that, unfortunately, but some searching of PF may turn it up.

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

 

[Sagittarius A-Star]

Johann Georg von Soldner did a calculation in 1801, based on Newton's theory. He got a deflection angle, which is 1/2 of the result from GR.

On the deflection of a light ray from its rectilinear motion, by the attraction of a celestial body at which it nearly passes by.
...
If we substitute into the formula for $\text{tang} { \ \omega }$ the acceleration of gravity on the surface of the sun, and assume the radius of this body as unity, then we find ${\omega } = 0^{"}.84$. If it were possible to observe the fixed stars very nearly at the sun, then we would have to take this into consideration.

Hopefully no one finds it problematic, that I treat a light ray almost as a ponderable body. That light rays possess all absolute properties of matter, can be seen at the phenomenon of aberration, which is only possible when light rays are really material. — And furthermore, we cannot think of things that exist and act on our senses, without having the properties of matter.

Source:
https://en.wikisource.org/wiki/Tran...on_of_a_Light_Ray_from_its_Rectilinear_Motion

See also in the mathpages link, that @A.T. provided in posting#8:

Now, Soldner’s computation was based entirely on Newtonian physics for ballistic light particles, which unambiguously gives half of the relativistic value, and indeed this is the numerical value that Soldner gave (i.e., 0.84 seconds of arc for a ray grazing the sun). The “extra” factor of 2 appearing in most of his formulas has been attributed to a mere difference in notation, since it was common in the German literature of that time to define the symbol for “acceleration of gravity” as half of the modern definition (e.g., the distance traversed by a dropped object in time t was written as gt2 instead of (1/2)gt2.) The fact that this extra factor was missing from some of the formulas was evidently just due to a printing error.