I Absolute meaning of spatial deviaton angle of light around the sun

[wnvl2]

TL;DR Summary

Clarification sought for the absolute meaning of the deviaton angle of light as calculated by Einstein around the sun

Einstein first calculated the bending of light rays that are touching the sun as 1.75 arc-sec. For the calculation I refer e.g. to https://www.mathpages.com/rr/s8-09/8-09.htm

I know that spatial angles in general relativity don’t have an intrinsic value (are not invariant). They are dependent on the choice of the coordinate system. Angles can be calculated using

$$\cos\theta = \frac{a^{i}b_{i}}{\sqrt{a^{i}a_{i}b^{i}b_{i}}}$$

with i indexing only the spatial components.

I would have expected that the outcome is determined by the chosen coordinate system with associated metrics and that his value has no physical meaning if you don't know which coordinate system + associated metric was chosen. But apparently that 1.75 arc sec seems to have some absolute meaning without having to specify the coordinate system. Is there any assumption I am missing?

In this context I would also like to refer to Einstein’s hole argument. By changing the coordinate system in a hole I would expect to obtain a different total deflection angle.

 

[PeroK]

In this context I would also like to refer to Einstein’s hole argument. By changing the coordinate system in a hole I would expect to obtain a different total deflection angle.

The calculation in the given coordinate system results in a prediction for a local observation on Earth. If you change the coordinate system, you must end up with the same local measurement. For example, if we could find a coordinate system where the light was not deflected, then the worldline of the observer with the telescope on Earth would be more complicated - such that the local measurement would be the same. I.e. the orientation of the telescope on Earth, relative to the Earth's surface, as measured locally, would be the same in order to detect the starlight.

The chosen coordinates merely give the simplest way of calculating the required orientation of the telescope. The deflection angle, although not invariant, is meaningful to observers on Earth using the rest frame of the Sun to analyse the solar system.

 

[wnvl2]

If in both situations (with and without sun) the coordinate system + metric locally at the Earth and the star are the same (Minkowski at Earth and star), then the total deflection of the light between star and Earth calculated by integrating all instantaneous deviations of the light (that is how Einstein does the calculation in the link in my first message) over the full traject between star and earthwill be the same independent of the coordinate system (+ corresponding metric acoording to Ein,stein equations) chosen in between Earth and star.

I mean that for some part of the traject (a hole) I can choose a very exotic coordinate system (+ corresponding metric determined by the Einstein equations) as long as it fits at the boundary, I will find mathematically exactly the same deviation.