I Meaning of the distance between galactic objects in GR

AI Thread Summary
The discussion centers on the distance of Sagittarius A* from Earth, which is approximately 26,996 light-years, and how this is interpreted within General Relativity (GR). It is noted that while the Friedmann-Lemaître-Robertson-Walker (FLRW) model applies to cosmological scales, it is not suitable for galactic scales due to the gravitational binding of galaxies. Instead, the weak field approximation is used to describe spacetime at the galactic level, providing a proper distance in the galactic rest frame. The concept of the galactic rest frame is clarified as the irrotational frame where the galaxy's center of mass is at rest, with constituents moving in approximately circular orbits. Overall, the discussion emphasizes the complexities of measuring distances and understanding motion within the framework of GR at different scales.
cianfa72
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About the meaning of the distance between galactic objects in the Universe in the context of GR
Hi, I was thinking about the claim that for instance Sagittarius A* (Sgr A*) black hole is a at 26996±29 light years from the Earth from a GR point of view.

Assuming a FLRW model for the Universe, maybe the above meaning is that at a given cosmological time ##t## (the "present" time) the proper distance between the Earth and Sgr A* evaluated on the spacelike hypersurface of constant cosmological time ##t## is actually 26996±29 light year.

Does it make sense ?
 
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The FLRW model is not a good model at the galactic scale as the galaxy is a gravitationally bound system. Instead, the spacetime on galactic level is pretty well described (apart from locally near extreme objects like black holes) by the weak field approximation. Sgr A* being some 27000 ly away is the corresponding distance in the galactic rest frame.
 
Orodruin said:
Instead, the spacetime on galactic level is pretty well described (apart from locally near extreme objects like black holes) by the weak field approximation. Sgr A* being some 27000 ly away is the corresponding distance in the galactic rest frame.
Does the weak field approximation give rise to a metric tensor on spacetime as a manifold ?

Edit: is the galactic rest frame the spacetime coordinate chart in which on average the galaxy's constituents are "at rest" -- i.e. have timelike worldlines with constant spatial coordinates?
 
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cianfa72 said:
Does the weak field approximation give rise to a metric tensor on spacetime as a manifold ?

Yes.


cianfa72 said:
Edit: is the galactic rest frame the spacetime coordinate chart in which on average the galaxy's constituents are "at rest" -- i.e. have timelike worldlines with constant spatial coordinates?
No. It is the irrotational frame where the galaxy com is at rest.
 
Orodruin said:
No. It is the irrotational frame where the galaxy com is at rest.
Could you be more specific ? As far as I can understand, it is the coordinate chart in which the galaxy's COM (Center of Mass) is "at rest" -- i.e. its timelike worldline has constant spatial coordinates in that chart (only varying timelike coordinate time ##t##).

What does it mean it is an irrotational frame/coordinate chart ? Thanks.
 
cianfa72 said:
What does it mean it is an irrotational frame/coordinate chart ?
Does not rotate with the galaxy.
 
Orodruin said:
Does not rotate with the galaxy.
Sorry, I don't grasp the point. Do the galaxy's constituents rotate in that irrotational chart (i.e. their timelike worldlines parametrized by the coordinate time ##t## rotate in that chart) ?
 
cianfa72 said:
Do the galaxy's constituents rotate in that irrotational chart
Yes, they move around in approximately circular orbits.
 
Orodruin said:
Yes, they move around in approximately circular orbits.
Ah ok, that means galaxy's constituents paths (parametrized by the coordinate time ##t##) as described in the irrotational chart move like circular orbits around the galaxy's COM.
 
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Well, approximately circular. There are lots of local variations to that, but in general yes.
 
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