General Relativity and the precession of the perihelion of Mercury

[cianfa72]

TL;DR Summary

Test of GR: anomalous precession of perihelion of Mercury

Hi, as test of GR I'm aware of there is the "anomalous" precession of the perihelion of Mercury.

My question is: in which coordinate system are the previsions of GR verified concerning the above ? Thanks.

 

[Vanadium 50]

I am not sure this question is answerable. The perihelion advance is a physical thing. It's not coordinate dependent.

 

[Orodruin]

You don’t verify things in coordinate systems. You verify invariant predictions.

 

[cianfa72]

You don’t verify things in coordinate systems. You verify invariant predictions.

Can you be more specific in this statement ? Thanks.

 

[Orodruin]

Can you be more specific in this statement ? Thanks.

The statement is very specific already. what part of it confuses you?

 

[cianfa72]

Can you give an example of invariant prediction concerning the perihelion advance ?

 

[PeterDonis]

Can you give an example of invariant prediction concerning the perihelion advance ?

The position of Mercury on Earth's sky relative to other astronomical objects--the Sun, Moon, stars and other planets--as a function of time by standard Earth clocks.

 

[Orodruin]

The angle between the line connecting the Sun and Mercury and the line connecting Mercury to some selected fixed star at Mercury's perihelion on consecutive orbits.

Can you give an example of invariant prediction concerning the perihelion advance ?

 

[cianfa72]

How the derivation of such properties is carried out starting from EFE ?

 

[PeroK]

How the derivation of such properties is carried out starting from EFE ?

See, for example, section 9.3 in Gravity, by James Hartle.

 

[Sagittarius A-Star]

Can you give an example of invariant prediction concerning the perihelion advance ?

A measurement result is invariant. Urbain Le Verrier studied related astronomic measurements.

Urbain Le Verrier in 1859 on the perihelion precession of Mercury
...
While of the Sun we only have meridian observations subject to great objections, we have, over the space of a century and a half, a certain number of observations of Mercury possessing great precision - I want to speak of the internal contacts of Mercury's disc with the Sun's disc, when the planet comes to pass in front of this star. As long as the place where the observation was made is well known, and as long as the astronomer had a reasonable refracting telescope and his clock was accurate to within a few seconds, the knowledge of the instant when the internal contact happened must allow us to estimate the distance between the centre of the planet and the centre of the Sun without an error of more than one arcsecond. We possess, from 1697 until 1848, twenty-one observations of this type, for which we have to be able to satisfy in the most stringent manner whether the irregularities of the movements of the Earth and of Mercury were well calculated, and if the values attributed to the perturbative masses are correct.
...
But what is remarkable is that increasing the secular movement of the perihelion by 38 arcseconds was enough to show all the observations of the transits to within one second, and even the majority of those to within one half-second. This result, so clean, which immediately shows in all comparisons a greater precision than achieved until now in astronomical theories, clearly shows the increase in the movement of the Mercury's perihelion is indispensable, and that therefore the Tables of Mercury and of the Sun have all the desired precision.

Source:
https://pappubahry.livejournal.com/572713.html

 

[cianfa72]

The angle between the line connecting the Sun and Mercury and the line connecting Mercury to some selected fixed star at Mercury's perihelion on consecutive orbits.

The angle between these two lines is always evaluated at Mercury's perihelion on consecutive orbits. And this angle changes from a Mercury's orbit (given as completed at its perihelion) to the next, right ?

Edit: I found this interesting video

 

[Meir Achuz]

TL;DR Summary: Test of GR: anomalous precession of perihelion of Mercury

My question is: in which coordinate system are the previsions of GR verified concerning the above ?

I think this means, in which coordinate system are calculations made of the anomalous precession of the perihelion of Mercury made, which can then be compared to experiment.
This can be answered by putting "precession of mercury calculation" into Google.
The calculations given there generally use spherical coordinates, with a term 1/r^3 added to the differential equation.

 

[Orodruin]

The calculations given there generally use spherical coordinates, with a term 1/r^3 added to the differential equation.

It should be pointed out that it is only ”added” relative to the Newtonian case. It is an essential part of the GR equations of motion for the radial coordinate after integrating out the angular part.

 

[cianfa72]

It is an essential part of the GR equations of motion for the radial coordinate after integrating out the angular part.

From my understanding, the above GR equations of motion involve the Schwarzschild radius in Schwarzschild spacetime (in which the radius is centered in the Sun).

 

[PeterDonis]

From my understanding, the above GR equations of motion involve the Schwarzschild radius in Schwarzschild spacetime (in which the radius is centered in the Sun).

In the weak field approximation, which is sufficient to derive perihelion precession, it doesn't matter whether you use Schwarzschild coordinates or isotropic coordinates; the weak field terms are the same in both cases. (Or, to put it another way, the difference between areal radius and radial distance is small enough that it drops out of the weak field approximation.) AFAIK the solar system barycentric frame that is commonly used in astronomy uses isotropic coordinates.

 

[cianfa72]

AFAIK the solar system barycentric frame that is commonly used in astronomy uses isotropic coordinates.

Which are the isotropic coordinates ?

 

[PeterDonis]

Which are the isotropic coordinates ?

https://en.wikipedia.org/wiki/Isotropic_coordinates

 

[cianfa72]

From wikipedia link the spacelike congruence given by the vector field ##\partial_r## in the isotropic chart should be actually a geodesic congruence, i.e. $$\nabla_{\partial_r} \partial_r=0$$

 

[PeterDonis]

From wikipedia link the spacelike congruence given by the vector field ##\partial_r## in the isotropic chart should be actually a geodesic congruence, i.e. $$\nabla_{\partial_r} \partial_r=0$$

This is true in Schwarzschild coordinates as well, because the spacelike surfaces of constant time, which are the surfaces in which these curves lie, are the same in both cases.

 

[cianfa72]

So the invariant measurement of the precession of Mercury's perihelion can be calculated in the isotropic coordinate chart centered on the Sun.

 

[Orodruin]

… or any other coordinate chart. This is the point of being invariant.

(Then the difficulty of the computation may vary by chart, but that’s why picking a chart well suited for the problem is an art.)

 

[cianfa72]

I was looking at Carroll's lectures on GR. In chapter 7 he derives the equation for the motion of an object in Schwarzschild spacetime using the effective potential ##V(r)## that happens to have an "additional" ##-1/r^3## term compared to Newtonian case. He gets (7.47): $$\frac {1} {2} \left (\frac {dr} {d\lambda}\right )^2 + V(r) = \frac {1} {2} E^2$$ where ##E## is the energy of the massive object (for a massive object the affine parameter ##\lambda## can be chosen as its proper time ##\tau##). Based on this value one gets different behaviors for the solutions of 7.47 (e.g. the existence of "turning point" or not).

Is the "form" of the massive object's trajectory in space actually an invariant?

 

[PeterDonis]

Is the "form" of the massive object's trajectory in space actually an invariant?

What do you mean by "form"?

 

[cianfa72]

What do you mean by "form"?

The fact for example that the object's trajectory in space is closed or it is elliptical.

 

[PeterDonis]

The fact for example that the object's trajectory in space is closed or it is elliptical.

The bolded phrase should tell you the answer. Hint: is "trajectory in space" frame-dependent?

Also note that in GR, in coordinates centered on the Sun, the "trajectory in space" of Mercury is not closed or elliptical.

 

[Vanadium 50]

Elliptical orbits are closed. They form ellipses.

 

[cianfa72]

The bolded phrase should tell you the answer. Hint: is "trajectory in space" frame-dependent?

Also note that in GR, in coordinates centered on the Sun, the "trajectory in space" of Mercury is not closed or elliptical.

Ok, so "trajectory in space" is not frame-invariant.
And yes, in coordinates spatially centered on the Sun such a trajectory is not closed or elliptical.

Therefore, which are the Mercury's spacetime path invariants we are interested in ?

 

[Ibix]

The fact for example that the object's trajectory in space is closed or it is elliptical.

You can make the following coordinate independent (and theory independent) statements:

I see no Doppler shift in radar pulses I bounce off the Sun.

The last perihelion of Mercury (you may specify radar ranging for distance here too if you like) occurred directly between me and the Sun.

The same was/was not true of the perihelion before.

The first condition makes me a hovering observer. The second condition establishes a definition of "where" the perihelion happened. The third condition differentiates between a Newtonian closed orbit and a GR unclosed one in a coordinate independent way.

I suspect what people generally do is use Schwarzschild spatial planes and project orbital paths down the timelike KVF, which is probably the best analogue to the Newtonian "draw the path in space" idea.

 

[cianfa72]

I see no Doppler shift in radar pulses I bounce off the Sun.

The first condition makes me a hovering observer.

An hovering observer w.r.t. Schwarzschild spacetime, I believe (i.e. my worldline is described from fixed Schwarzschild coordinates ##r,\theta,\phi## and varying ##t##).

The last perihelion of Mercury (you may specify radar ranging for distance here too if you like) occurred directly between me and the Sun.

I.e. in "between" along the straight line connecting me with the Sun.

The same was/was not true of the perihelion before.
The third condition differentiates between a Newtonian closed orbit and a GR unclosed one in a coordinate independent way.

Therefore this is frame-invariant.

I suspect what people generally do is use Schwarzschild spatial planes and project orbital paths down the timelike KVF, which is probably the best analogue to the Newtonian "draw the path in space" idea.

Edit: do you mean the projection of Mercury's spacetime path down the spacelike hypersurfaces (constant Schwarzschild coordinate time ##t##) orthogonal to the timelike KVF (they have the same spacelike geometry since in Schwarzschild spacetime the timelike KVF is hypersurface orthogonal)?

 

[PeterDonis]

Ok, so "trajectory in space" is not frame-invariant.

Correct.

in coordinates spatially centered on the Sun such a trajectory is not closed or elliptical.

Yes.

which are the Mercury's spacetime path invariants we are interested in ?

The invariant that is actually measured is the angular position of Mercury in Earth's sky relative to other visible objects in the sky, like the other stars and planets, as a function of Earth's proper time.