Einstein's Theory of Relativity: Why Planets Orbit Quicker Near the Sun

[Rorkster2]

Einstein predicted a slightly quicker orbit for planets very close to the sun that will deviate from Newton’s gravitational laws very slightly. Later observations, as we know, have proved correct and General Relativity holds up. But what is the perceived reason as to why this is? What is the explanation as to why planets have slightly quicker orbits when they are extremely close to a massive body?

 

[PeterDonis]

What is the explanation as to why planets have slightly quicker orbits when they are extremely close to a massive body?

By "slightly quicker orbit" I assume you are referring to the precession of the orbit's perihelion; it's worth noting that this is not quite the same thing as "the planet travels faster in its orbit".

As for "explanation", it depends on what you think counts as an explanation, but here's how I would describe it. If we idealize the gravitational field of the Sun (meaning, the particular curved spacetime caused by the Sun's mass) to be a weak, spherically symmetric field, then we can view the Sun's gravity, acting on the planets, as a "force"--but it is a force that does not "point" directly at the Sun, and that depends on the planet's velocity as well as its position. The extra effects due to these differences are what cause the perihelion precession.

 

[TheEtherWind]

that the speed of light isn't infinite

 

[bcrowell]

http://www.lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.2

See subsection 6.2.6, at "We can easily understand..."

 

[Bill_K]

From the reference...

Mercury spends more time near perihelion than it should nonrelativistically. During this time, it sweeps out a greater angle than nonrelativistically expected, so that when it flies back out and away from the sun, its orbit has rotated counterclockwise.

The advance of the perihelion per revolution is

ε = 24π³a²/c²T²(1-e²)

where a is the semimajor axis, e is the eccentricity and T is the orbital period. The effect persists even in the limit e → 0, so to ascribe it to what happens when the planet is near perihelion can't be right, since a circular orbit has no perihelion. The fact that Mercury's orbit is rather elliptical only serves to make the effect easier to observe.

The advance of perihelion occurs because the frequency of small radial oscillations is less than the orbital frequency. This is true not only at the perihelion but equally at the aphelion. In fact the advance occurs at a constant rate throughout the orbit.

 

[A.T.]

This topic comes up again and again. Is it even possible to formulate an intuitive yet correct "reason" for the precession? I posted this a while ago with no reply:

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:
https://www.physicsforums.com/showthread.php?t=224397

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

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

 

[Bill_K]

I have a deep abiding mistrust of "intuitive reasons" for things. Because 99 percent of the time they are total BS, and yet they acquire a life of their own and get repeated over and over again, just because they are so appealing.

However, in the present case...

As was pointed out in the thread you cited, one way of writing the effective potential is

V_eff = - GMm/r + (L²/2mr²)(1 - 2GM/c²r)

The terms in V_eff have an easy interpretation - the first is the Newtonian potential and the second is the centrifugal barrier. And we see that in GR the centrifugal barrier is weakened by a factor 1 - 2GM/c²r. This, right here, is what causes the frequency of small radial oscillations to be less.

So now if you trace back in the derivation and ask why the centrifugal barrier is weakened, it's ultimately because the dr² term in the Schwarzschild metric has a factor (1 - 2GM/c²r)^-1 in front of it, while the dφ² term does not. In other words, radial distances are stretched relative to angular distances by this factor, and so again this has the effect of making radial oscillations slower than the angular ones.

Well now, all you have to do is think of an intuitive reason why the dr² term in the Schwarzschild metric has a factor (1 - 2GM/c²r)^-1 in front of it while the dφ² term does not!

 

[A.T.]

Thanks, for your reply Bill.

So now if you trace back in the derivation and ask why the centrifugal barrier is weakened, it's ultimately because the dr² term in the Schwarzschild metric has a factor (1 - 2GM/c²r)^-1 in front of it, while the dφ² term does not. In other words, radial distances are stretched relative to angular distances by this factor,

This sounds like purely spatial curvature (dt doesn't appear here). Can we say, that the precession is completely explained by the non-Euclidean spatial geometry (Flamm's paraboloid)?

I remember some opposition to this approach and the intuitive simplified visualization shown here:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

I was told that gravitational time dilation also plays a role in the effective potential difference between Newton and Schwarzschild. But your comments seems to suggest that this difference can be explained solely by the distortion of spatial geometry.

 

[Bill_K]

Yes I agree, the orbital precession is due entirely to the spatial geometry, as represented by Flamm's paraboloid. I quickly have to add however that showing this picture brings with it a big risk of misinterpretation, as in "space is conical", as well as a marble rolling around on a rubber sheet. But it does give a correct explanation of the precession. Time dilation does not play a role, as it affects both the radial and angular motion equally.

 

[bcrowell]

From the reference...

The advance of the perihelion per revolution is

ε = 24π³a²/c²T²(1-e²)

where a is the semimajor axis, e is the eccentricity and T is the orbital period. The effect persists even in the limit e → 0, so to ascribe it to what happens when the planet is near perihelion can't be right, since a circular orbit has no perihelion.

Interesting point. But I disagree with your argument and would put it in a different form.

The fact that a circular orbit has no perihelion doesn't really tell us much. You can't define an advance of perihelion for a circular orbit, so if a particular explanation says that the advance is zero in that case, that's not a point against the explanation.

What's relevant is that in the limit as the eccentricity approaches zero, my explanation predicts that the advance approaches zero, whereas the actual result fro GR approaches a nonzero constant.

So anyway, I agree that this puts a bullet through the heart of my favorite explanation -- oh, well!

The pictures of the cones do seem to give results that match up better with the exact equation's dependence on its variables. This page uses the cone thing to derive $\epsilon=(\ldots)m/a$, where ... represents a unitless factor that they weren't hoping to get right. This seems about right, because the exact expression depends on $a^2/T^2$, which becomes 1/a if you substitute in Kepler's law of periods. It predicts that the result is independent of eccentricity, which is reasonably close to the actual dependence, which is weak (of order e²).

 

[bcrowell]

Well now, all you have to do is think of an intuitive reason why the dr² term in the Schwarzschild metric has a factor (1 - 2GM/c²r)^-1 in front of it while the dφ² term does not!

Isn't this simply an expression of spatial curvature? You have positive spatial curvature when the circumference of a circle is less than 2 pi times its radius.

 

[A.T.]

This page uses the cone thing...

That website tries to put math behind Epstein's book, which contains only conceptual pictures, like the ones I linked before:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

 

[A.T.]

Yes I agree, the orbital precession is due entirely to the spatial geometry, as represented by Flamm's paraboloid. I quickly have to add however that showing this picture brings with it a big risk of misinterpretation, as in "space is conical",

Well, the cone is an approximation of the paraboloid, with the advantage that you can build and visualize it more easily. But you have to point out that the actual spatial geometry is quite different, and has more dimensions.

as well as a marble rolling around on a rubber sheet.

That is ouf course bad. But note that Epstein:
- first explains the role of the time dimension, for the main effect of gravity. He has exactly one picture of the marble rolling around on a rubber sheet in his book, with the caption saying : "Common but completely wrong idea, better forget it now".
- when showing the purely spatial geometry, deliberately makes the cone pointing up (as a bump, dot a dent) to avoid the wrong "rolling into dents" idea. He makes clear that the direction doesn't matter.

But it does give a correct explanation of the precession. Time dilation does not play a role, as it affects both the radial and angular motion equally.

So if we combine Newtonian gravity, with the spatial part of the Schwarzschild metric, we get the correct shape of the orbits, as predicted by GR. Of course the time to traverse the orbits will be off, because we are ignoring gravitational time dilation, but the shape will be correct, right?