Undergrad Meaning of the distance between galactic objects in GR

[cianfa72]

TL;DR

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 ?

 

[Orodruin]

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.

 

[cianfa72]

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?

 

[Orodruin]

Does the weak field approximation give rise to a metric tensor on spacetime as a manifold ?

Yes.

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.

 

[cianfa72]

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.

 

[Orodruin]

What does it mean it is an irrotational frame/coordinate chart ?

Does not rotate with the galaxy.

 

[cianfa72]

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) ?

 

[Orodruin]

Do the galaxy's constituents rotate in that irrotational chart

Yes, they move around in approximately circular orbits.

 

[cianfa72]

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.

 

[Orodruin]

Well, approximately circular. There are lots of local variations to that, but in general yes.