General relativity vs. Newtonian gravity

[calinvass]

I understand that General Relativity can make a difference between a spinning and non spinning mass thus can make better prediction for planetary orbits for example. The effect is frame dragging.
However if we simulate a Newtonian gravitation and instead of representing a planet as a sphere with a mass, we use small spheres close to one another ( the diameter of the small spheres to choose should be a question of accuracy, that form a rotating sphere of the same diameter as the planet, can we get the same effect as in general relativity for low velocity objects ?

 

[Ibix]

You mean replace the planet with a set of spheres of varying sizes which (approximately) fill its volume? Then set the whole thing rotating and use $F=\sum_i GM_im/r_i^2$ where i indexes the spheres and m is the mass of sime free falling test particle?

My first thought is "no". If you take the limiting case of an infinite set of spheres exactly filling the volume then you're back to where you started. So any "frame dragging" effect that may or may not occur would be due to inaccuracies in your model.

 

[calinvass]

Yes, exactly. So, if a set of infinite set of spheres doesn't work then a finite number has no purpose since in principle it only reduces the accuracy.

 

[A.T.]

However if we simulate a Newtonian gravitation and instead of representing a planet as a sphere with a mass, we use small spheres close to one another

Using many point masses is actually the generally correct way to apply Newtonian gravitation. Using just one mass representing the whole planet is a special case, that happens to work for spheres.

( the diameter of the small spheres to choose should be a question of accuracy, that form a rotating sphere of the same diameter as the planet, can we get the same effect as in general relativity for low velocity objects ?

Why would you? Newtonian gravitation doesn't depend on the movement of the masses at all.

 

[calinvass]

Why would you? Newtonian gravitation doesn't depend on the movement of the masses at all.

I understand it is not dependent of the rotations of the masses but if a large mass object is moving it will influence the trajectory of a smaller one around.

For a homogeneous sphere to get a different trajectory for an orbiting small object, a different distribution of mass inside the sphere will definitely change the trajectory. From what you are saying I conclude that in the case of a infinite number of masses of the same value, the distribution will not change, whether they are revolving around the center or not. However since those masses are in motion I thought they might produce an effect since those masses and diameters cannot reach zero but an infinitesimal number.
In a computer simulation this will produce a different result but that can be due to processing capabilities because the computer will not be able to handle infinite numbers. A mathematical model should work better.

 

[jbriggs444]

For a homogeneous sphere to get a different trajectory for an orbiting small object, a different distribution of mass inside the sphere will definitely change the trajectory.

There is no such thing as a different distribution of mass inside a homogeneous sphere. Unless you change its total mass.

 

[pervect]

I understand that General Relativity can make a difference between a spinning and non spinning mass thus can make better prediction for planetary orbits for example. The effect is frame dragging.
However if we simulate a Newtonian gravitation and instead of representing a planet as a sphere with a mass, we use small spheres close to one another ( the diameter of the small spheres to choose should be a question of accuracy, that form a rotating sphere of the same diameter as the planet, can we get the same effect as in general relativity for low velocity objects ?

I would also say no. Consider a spherical object doing a relativistic flyby, and look at it's gravitational effects. The first thought that comes to mind is to consider the results from Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object", which analyzes the effects of such a flyby on a "background" set of test particles all at relative rest to each other. (The flyby has some relativistic velocity $\beta c$ with respect to this background). The results from GR computation are not the same as for the Newtonian case. The paper may be hard to get in its entirety, though the abstract which presents the main results is easy enough to tie down.

The second thought that comes to mind (which may be more closely related to the frame dragging effects you are interested in) is to look more closely at the magnetic-like effects of GR. One simple result is to look at the effects of parallel and anti-parallel relativistically moving objects. Taking the limit when the parallel and anti-parallel moving objects, move at c, we are led to consider the gravitational interation of parallel and anti-parallel light beams. It turns out that parallel propagating light beams do not attract, but anti-parallel light beams do attract (and more strongly than a quasi-Newtonian prediction would predict). So the effects of motion on "gravity" just do not fit in the Newtonian model. I believe there is a reference by Tollman on the parallel / anti-parallel light beams, but I don't have a link (and it's also very old, so while the results are correct the way the results are shown isn't in the spirit of modern GR.)

In your attempt to decompose the initial body into pieces, said pieces are moving as the body rotates. The Newtonian model of moving objects isn't right, and hence your model won't be right either.

 

[calinvass]

There is no such thing as a different distribution of mass inside a homogeneous sphere. Unless you change its total mass.

No, I'm sorry, you are wright, I wanted to say a homogeneous sphere will create a different effect than an unevenly distributed mass one of the same total mass.

 

[calinvass]

Thanks. I'll have a look at that book.

The second thought that comes to mind (which may be more closely related to the frame dragging effects you are interested in) is to look more closely at the magnetic-like effects of GR. One simple result is to look at the effects of parallel and anti-parallel relativistically moving objects. Taking the limit when the parallel and anti-parallel moving objects, move at c, we are led to consider the gravitational interation of parallel and anti-parallel light beams. It turns out that parallel propagating light beams do not attract, but anti-parallel light beams do attract (and more strongly than a quasi-Newtonian prediction would predict). So the effects of motion on "gravity" just do not fit in the Newtonian model. I believe there is a reference by Tollman on the parallel / anti-parallel light beams, but I don't have a link (and it's also very old, so while the results are correct the way the results are shown isn't in the spirit of modern GR.

This makes sense in a way because since photons don't seem to have rest mass, they when they travel in parallel they are at rest in respect to each other and when antiparallel they are not. The problem is we can't define frames with photons at rest. However when an observer will see them at the same speed vs. at a 2c difference. In Newtonian mechanics there is no way to create any gravitational field using light.

 

[PeterDonis]

when they travel in parallel they are at rest in respect to each other

No, they aren't; the concept of "at rest" doesn't apply to a photon. The fact that they are moving at the same speed in the same direction relative to a given observer does not change that.

 

[pervect]

No, they aren't; the concept of "at rest" doesn't apply to a photon. The fact that they are moving at the same speed in the same direction relative to a given observer does not change that.

Thanks. I'll have a look at that book.

This makes sense in a way because since photons don't seem to have rest mass, they when they travel in parallel they are at rest in respect to each other and when antiparallel they are not. The problem is we can't define frames with photons at rest. However when an observer will see them at the same speed vs. at a 2c difference. In Newtonian mechanics there is no way to create any gravitational field using light.

You have to be a bit careful, as photons don't have rest frames. It happily turns out that if you don't mind errors of 2:1 or 4:1 or so, you get close to the right answer by considering particles of constant total energy as they approaches "c". Because the total energy is constant , the rest mass goes towards zero. Understanding where the errors of 2:1 or 4:1 come frome requires more than putting a band-aid on Newtonian physics, though. At some point, to get the right answers, one has to learn GR.

As this is a daunting task, I'd suggest taking it in easier steps. First learn special relativity, and then learn how electromagnetic fields transform in SR. In particular, learn the fields of a relativisticially moving charge.

 

[calinvass]

The approximation is enough to describe the difference between parallel an antiparallel photons.

My original question was however for low speed objects where the relativistic effects are negligible.