# What is the relativistic cause of planetary orbital rotation.

• YummyFur
In summary, Newtonian physics correctly predicts that a planetary orbit will be an ellipse and general relativity correctly predicts that this ellipse will rotate, as was demonstrated with mercury.

#### YummyFur

Newtonian physics correctly predicts that a planetary orbit will be an ellipse and general relativity correctly predicts that this ellipse will rotate, as was demonstrated with mercury.

The only thing that I can think of is that because the orbit is an ellipse then the planet will be experiencing different degrees of gravity as it varies it's distance from the sun and will also be traveling at different speeds, also due to the orbit being an ellipse.

So there would be time dilation due to gravity and also time dilation due to orbital speed.

Is it this time dilation that is the cause for the orbit to rotate, or is the cause of the orbital rotation due to something entirely different.

The rotation of the axis of the ellipse is called "precession". The best explanation I've seen is right here on PF. Here's a paraphrase:

The space near the sun is curved according to GR.

Take a piece of paper. Lay it flat. Draw the sun on it, then draw a perfect ellipse for Mercury's orbit.

OK. Now...

Take scissors and cut the paper from the edge right to the point of the sun. This will allow you to pull the cut edges together so they overlap just a small bit - a half inch or so. This forms the flat paper into a very shallow funnel, with the sun at the point. Now it is a crude model of curved space centering on the sun.

But...

The cut across the paper slices right through the ellipse of Mercury's orbit. When in the funnel shape - the ellipse's two edges no longer match up. This means when Mercury follows its path around one year, it does not come back to where it first started. In fact, if you continue its path on this curved surface you will see it precess - just like a spirograph.

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OK that is a good explanation I will check it out.

However, a further question if I may. Is the precession caused solely by the curvature of space. Can the precession be predicted by the time dilation I have alluded to in my original post. In other words, can the curvature of the space that the ellipse resides in be ignored and instead be equally well describe din terms of the time dilation due to the varying effects of gravity. Hope I have been clear enough.

Or this this nothing at all to do with time dilation.

I'm speechless. That explanation has a nice sound to it, indeed. But is utterly false.

You cannot approximate the curvature of space in the Schwarzschild solution with a cone. What you described is known as an angular defect - the idea that if you go around in a circle you will have rotated by something less than 2π. The Schwarzschild solution does not have an angular defect. In fact each section r = const is a perfectly normal sphere, with surface area 4πr2. An angular defect represents a severe departure from the assumption in general relativity that spacetime is locally Minkowskian.

Please do not ever quote this explanation again!

The precession of the perihelion in planetary orbits is simply due to the fact that the gravitational potential is not precisely 1/r2, which is one of the few potentials that produces closed orbits.

Bill_K said:
Please do not ever quote this explanation again!

I think you're right in that it'd be wrong to describe that cone as representing 'space-time', but it does serve a useful purpose to demonstrate that 'local time' for the orbiting planet varies according to its distance from the Sun (viz. the orbit it is caused to take is longer than a 'planar' distance in the ecliptic).

...Actually, yeah, it's backwards anyway, isn't it? It should precess around in the same direction as the orbit itself (because it follows a longer path), whereas the cone cutting exercise shortens the orbit. Have I got that the right way around? Now I am getting confused!

Wait a minute can someone tell me if I'm going off the rails here. Like the twin paradox when one takes off and comes back into the future. Is what is happening something like that, as the planet makes it's way around to its starting point, because it has sped up on the way, so time has slowed down for it and when it got back to the starting point it was a little bit in the future as it were, or a little bit more precessed. Hope that came out right.

Time travel is as good an explanation as saying space is conical. However there's really nothing so magic about it. As I said, the Newtonian potential V ~ 1/r2 is almost unique in having the property that the orbits are closed. With ANY other potential the orbits will precess or spiral about the center. No elaborate explanation in terms of time dilation or spacetime curvature is required.

DaveC426913 said:
The rotation of the axis of the ellipse is called "precession". The best explanation I've seen is right here on PF. Here's a paraphrase:

The space near the sun is curved according to GR.

Take a piece of paper. Lay it flat. Draw the sun on it, then draw a perfect ellipse for Mercury's orbit.

OK. Now...

Take scissors and cut the paper from the edge right to the point of the sun. This will allow you to pull the cut edges together so they overlap just a small bit - a half inch or so. This forms the flat paper into a very shallow funnel, with the sun at the point. Now it is a crude model of curved space centering on the sun.

But...

The cut across the paper slices right through the ellipse of Mercury's orbit. When in the funnel shape - the ellipse's two edges no longer match up. This means when Mercury follows its path around one year, it does not come back to where it first started. In fact, if you continue its path on this curved surface you will see it precess - just like a spirograph.

I think you refer to this:

http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_space2.gif

The purely spatial curvature shown here is not the whole story though. Here is more:

http://www.fourmilab.ch/gravitation/orbits/
http://www.bun.kyoto-u.ac.jp/~suchii/eff.potent.html [Broken]

[PLAIN]http://www.bun.kyoto-u.ac.jp/~suchii/eff.potent.jpg [Broken]

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Bill_K said:
You cannot approximate the curvature of space in the Schwarzschild solution with a cone.
Why not? The spatial geometry is a Flamm's paraboloid:

http://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid

For a small range of r- coordinates this can be approximated with a cone surface. For a simple informal explanation you can replace most of it with a cone.

Bill_K said:
What you described is known as an angular defect - the idea that if you go around in a circle you will have rotated by something less than 2π. The Schwarzschild solution does not have an angular defect.
Of course it has an angular defect. A gyroscope orbiting a massive star will not point in the same direction after one circular orbit.

Bill_K said:
In fact each section r = const is a perfectly normal sphere, with surface area 4πr2.
That is trivially true because the radial Schwarzschild coordiante "r" is defined as circumference / 2*pi. But "r" is not the physical distance R from a point on that circumference to the center. Just like it is the case on a cone surface, you will find that: r < R. And the volume of the sphere at r = const more than 4/3 pi r3.

Bill_K said:
An angular defect represents a severe departure from the assumption in general relativity that spacetime is locally Minkowskian.
No it doesn't, because this not a local effect. The angular defect happens over a huge area of space.

Bill_K said:
The precession of the perihelion in planetary orbits is simply due to the fact that the gravitational potential is not precisely 1/r2, which is one of the few potentials that produces closed orbits.
That is most of the effect, but the spatial geometry also plays a role.

PS: Newtonian gravitational potential is ~ 1/r, not ~ 1/r2

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Bill_K said:
As I said, the Newtonian potential V ~ 1/r2 is almost unique in having the property that the orbits are closed. With ANY other potential the orbits will precess or spiral about the center.

Not quite. ~1/r is the only potential that gives closed orbits and decreases (in magnitude) with r. However, a potential V ~ r2 does also give closed orbits.

My objection to the "cone" explanation is that it is an oversimplification, and for that reason is deceptive. A newcomer to relativity might well assume that the perihelion advance and de Sitter precession can be explained in terms of this one concept.

All the "Flamm's paraboloid" represents is the spatial geometry: a slice at constant t. Thus ignoring time dilation, which plays as much a part. Also the effect of the angular velocity of the particle, which is normally taken to be orbital velocity. Quantitatively there is no agreement. The cone tangent to the spatial geometry predicts an angular defect of πm/r. But the perihelion advance is 6πm/r, and the deSitter precession is 3πm/r.

I'd say the cone explanation has about as much utility as the "old" Bohr quantization of the hydrogen atom. It leaves the student feeling he understands a problem, when in fact he does not.

@Bill_K, I will concur, especially after having recently finished David Lindley's Uncertainty; Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science and even more so with Manjit Kumar's Quantum; Einstein, Bohr, and the Great Debate about the Nature of Reality, which should be required listening (audiobook) for all high school students.

Bill_K said:
My objection to the "cone" explanation is that it is an oversimplification, and for that reason is deceptive. A newcomer to relativity might well assume that the perihelion advance and de Sitter precession can be explained in terms of this one concept.

Kip Thorne seems to like the cone explanation:
http://einstein.stanford.edu/SPACETIME/spacetime4.html#geodetic_effect

Bill_K said:
Quantitatively there is no agreement.
That is not the point of such explanations. The point is to explain the mechanism in an intuitive way, not to provide exact numbers.

Kip Thorne is explaining geodetic precession in the quoted example. In spite of the similarity of the names, it doesn't mean he is explaining the precession of Mercury's orbit via this mechanism.

I would think that a good explanation of "A causes B" would involve something that predicts both the correct direction and the correct magnitude of B.

So if you say that "A causes B", and a detailed calculation shows that A causes only 1% of the magnitude of the observed effect of B, then I don't think it's a good explanation.

I'm rather skeptical that the spatial curvature (which I agree does exist in a Schwarzschild time slice) is the actual explanation for the precession of Mercury's orbit in the demanding quantitative sense.

As far as I know, the spatial curvature is only important for high speed motions, such as the deflection of light. For instance, the PPN parameter gamma describes spatial curvature, and that PPN parameter is the only PPN parameter that's involved in light defiection.

I haven't done a detailed calculation, but I'm not really convinced at this point that spatial curvature causes the precession of Mercury's orbit in any meaningful sense, and by a meaningful sense I mean that one could actually calculate a nearly correct value of the precession.

To use the PPN formalism example, you'd want to show that the PPN parameter gamma caused precession, and that it was the dominant factor that caused precession.

pervect said:
I'm rather skeptical that the spatial curvature (which I agree does exist in a Schwarzschild time slice) is the actual explanation for the precession of Mercury's orbit in the demanding quantitative sense.
Yes, but once you understand how 2D spatial geometry can cause precession, you can extend this idea to more dimensions including the time dimension (which might be dominant quantitatively).

This geometrical explanation seems closer to the fundamental geometrical model of GR, compared to invoking the derived concept of effective potential. But I'm always in favor of explaining things in several different ways.

I looked up the PPN expression for perihelion shift:

(2-beta+2 gamma) / 3 * 6 pi M_sun / a(1-e^2)

So some of it "just appears", i.e. it's not related to any particular PPN parameter, at least not obviously, some of it can be ascribed to gamma, the space curvature, which was the original explanation, and beta subtracts, beta being "nonlinearity in the superposition of gravity".

So I'd say the space curvature explanation is simple but not really very accurate, for one thing you'd still have some perihellion shift even if it gamma was zero. But the good news it seems to be in the right ballpark, if you look at only the gamma term it gives 2/3 the right value, which isn't bad. But clearly there are other effects that also cause perihelion shift as well as space curvature, though it's not clear what they are from the PPN expression.

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I'm trying to come up with an intuitive, yet correct explanation based on the effective potential. The problem is that in the links I posted are not very clear about what "causes" the "extra dwell time" at the inner part.

I found a plot in the this old thread:

And sketched two "reasons" for the "extra dwell time" below it:

What is a better explanation? A, B, both or something else?

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