Can you be more specific in this statement ? Thanks.Orodruin said:You don’t verify things in coordinate systems. You verify invariant predictions.
The statement is very specific already. what part of it confuses you?cianfa72 said:Can you be more specific in this statement ? Thanks.
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.cianfa72 said:Can you give an example of invariant prediction concerning the perihelion advance ?
cianfa72 said:Can you give an example of invariant prediction concerning the perihelion advance ?
See, for example, section 9.3 in Gravity, by James Hartle.cianfa72 said:How the derivation of such properties is carried out starting from EFE ?
A measurement result is invariant. Urbain Le Verrier studied related astronomic measurements.cianfa72 said:Can you give an example of invariant prediction concerning the perihelion advance ?
Source:Urbain Le Verrier in 1859 on the perihelion precession of Mercury
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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.
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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.
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 ?Orodruin said: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.
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.cianfa72 said: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 ?
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.Meir Achuz said:The calculations given there generally use spherical coordinates, with a term 1/r^3 added to the differential equation.
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).Orodruin said:It is an essential part of the GR equations of motion for the radial coordinate after integrating out the angular part.
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 said: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).
Which are the isotropic coordinates ?PeterDonis said:AFAIK the solar system barycentric frame that is commonly used in astronomy uses isotropic coordinates.
https://en.wikipedia.org/wiki/Isotropic_coordinatescianfa72 said:Which are the isotropic coordinates ?
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 said: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. $$\nabla_{\partial_r} \partial_r=0$$
What do you mean by "form"?cianfa72 said:Is the "form" of the massive object's trajectory in space actually an invariant?
The fact for example that the object's trajectory in space is closed or it is elliptical.PeterDonis said:What do you mean by "form"?
The bolded phrase should tell you the answer. Hint: is "trajectory in space" frame-dependent?cianfa72 said:The fact for example that the object's trajectory in space is closed or it is elliptical.
Ok, so "trajectory in space" is not frame-invariant.PeterDonis said: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.
You can make the following coordinate independent (and theory independent) statements:cianfa72 said:The fact for example that the object's trajectory in space is closed or it is elliptical.
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##).Ibix said:I see no Doppler shift in radar pulses I bounce off the Sun.
The first condition makes me a hovering observer.
I.e. in "between" along the straight line connecting me with the Sun.Ibix said:The last perihelion of Mercury (you may specify radar ranging for distance here too if you like) occurred directly between me and the Sun.
Therefore this is frame-invariant.Ibix said: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.
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)?Ibix said: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.
Correct.cianfa72 said:Ok, so "trajectory in space" is not frame-invariant.
Yes.cianfa72 said:in coordinates spatially centered on the Sun such a trajectory is not closed or elliptical.
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.cianfa72 said:which are the Mercury's spacetime path invariants we are interested in ?