Orbit precessions - General Relativity vs Newton

1. Feb 4, 2016

Cobalt101

What is the underlying feature of general relativity that, unlike newtonian mechanics, results in the correct calculation of orbits i.e. including precession (e.g. Mercury). I not asking for the mathematics (i.e. the additional term in the equation) but rather what underlying "physical" feature causes it to not just be a perfect ellipse ?

2. Feb 5, 2016

3. Feb 5, 2016

4. Feb 5, 2016

sandy stone

Those slides certainly help to visualize what is going on, but I have a question. The two slides treat curvature in the time dimension and in spacial dimensions separately. Is this an accurate description of the situation, or is it done to simplify the explanation? For instance, in the depiction of curved space surrounding the sun, is there also curvature in the time dimension as well, that is just too hard to illustrate?

5. Feb 5, 2016

A.T.

Yes. You can only show 2 of the 4 space-time dimensions in such diagrams. What you see in part 1 (space-time) happens locally everywhere along the orbit shown in part 2 (space-space).

6. Feb 5, 2016

pervect

Staff Emeritus
I don't think that precession can be attributed to a single underlying physical feature of general relativity. If onr consider pertubation of a circular orbit - i.e. one takes a particle in a circular orbit and gives it a small perturbing "nudge", the perturbed particle will appear to oscillate relative to the non-preturbed particle. Precession exists when the period of this oscillation is different from an orbital period.

7. Feb 5, 2016

Staff: Mentor

I would say it's the fact that in GR, gravity is not a simple inverse square central force the way it is in Newtonian physics. In general it isn't a force at all; but even in the weak field, slow motion limit, where it can be viewed as a force, it isn't a simple inverse square central force. There is also an extra piece of the interaction which is velocity-dependent (some sources refer to this as the "gravitomagnetic" interaction, and the inverse square Newtonian force as the "gravitoelectric" interaction). But only a pure inverse square force can produce closed elliptical orbits.

8. Feb 6, 2016

haushofer

Looking at that visualization, you can wonder how scalar theories of gravity account for precession (they do, in fact, but with the wrong amount). So I'm a bit sceptic about such a visualization.

9. Feb 6, 2016

m4r35n357

I would say that the precession really is caused by the additional term in the gravitational potential (derived from the Schwarzschild solution). There is simply "more gravity" very close to a black hole than to a sphere (or point) of the same mass in Newtonian gravity. See for example equation 7.48 in Carroll.

As I understand it, it is entirely possible (but masochistic) to use curved spacetime to model Newtonian gravity, so that can't be the explanation.

10. Feb 6, 2016

PAllen

Note, this formulation does not involve a spacetime metric at all, nor spacetime curvature. Only spatial components of the curvature tensor are non-vanishing, same for the connection.

11. Feb 6, 2016

m4r35n357

OK, thanks for the clarification!

12. Feb 6, 2016

haushofer

It isn't. In Newton-Cartan theory there is no precession. I agree with your statement that precession is caused by such an additional term, and I'm not sure if this visualization is really an explanation. As I understand, already for scalar theories in flat space(time) one can find that an orbit does not close after a full rotation, hence giving precession.

13. Feb 6, 2016

haushofer

Newton-Cartan can be casted in a metrical formulation, but this metric structure is degenerate because there is no spacetime interval being invariant under the local isometry group (rotations+boosts). But this is no problem; the Riemann tensor is then completely given by the degenerate metrics and an additional closed two-form.

14. Feb 6, 2016

Hornbein

Aha. So precession is greatest with Mercury because it has the greatest angular velocity amongst the planets?

15. Feb 6, 2016

Staff: Mentor

It's a combination of angular velocity and closeness to the Sun, because the Sun's "gravitomagnetic field" gets weaker further away, just like the "gravitoelectric" field does.

16. Feb 6, 2016

Hornbein

Right. But these quantities are directly related for an orbiting body.

17. Feb 6, 2016

Staff: Mentor

Note, however, that this "additional gravity" depends on the angular momentum of the test object, so it isn't that the ordinary Newtonian force gets larger; it's that there is a different kind of force present, which is velocity-dependent, and therefore works more like a "magnetic" force than an "electric" force.

18. Feb 6, 2016

PAllen

What I said is consistent with what you say. There is no spacetime metric. That is equivalent to saying there is no spacetime invariant interval. The artificial combination (which is wholly unnececssary to the formulation of the theory, anyway) is not a spacetime metric because it lacks essential properties. In my view, the logical formulation has connection (and curvature tensor constructed from it), but no metric. Connection is all you need for the geodesic equation and curvature. Trying to make the connection metric compatible adds nothing because the metric is not a true metric.

Last edited: Feb 6, 2016
19. Feb 6, 2016

Staff: Mentor

Yes, they are, but the relationship depends on the orbital energy and angular momentum, which means it depends on the mean distance from the Sun, and also on the orbital eccentricity. I actually should have mentioned eccentricity as another factor that increases Mercury's precession.

20. Feb 7, 2016

haushofer

If you think in terms of equations of motion. If you e.g. want to derive the geodesic equation from an action principle, you need the metric formulation. The metric formulation also emerges naturally if you consider Newton-Cartan theory as a gauge theory of the underlying Galilean algebra. So it depends on what you call "the formulation of the theory".