Meaning of the distance between galactic objects in GR

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Discussion Overview

The discussion revolves around the interpretation of the distance to the Sagittarius A* black hole from Earth within the framework of General Relativity (GR). Participants explore the implications of using the Friedmann-Lemaître-Robertson-Walker (FLRW) model versus the weak field approximation at the galactic scale, as well as the nature of the galactic rest frame.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that the distance to Sgr A* can be understood as the proper distance at a given cosmological time using the FLRW model.
  • Another participant counters that the FLRW model is inadequate for galactic scales, proposing that the weak field approximation is more appropriate for describing spacetime in this context.
  • There is a question about whether the weak field approximation leads to a metric tensor on spacetime as a manifold.
  • Clarifications are made regarding the definition of the galactic rest frame, with some participants indicating it is the irrotational frame where the galaxy's center of mass is at rest.
  • Participants discuss the implications of the irrotational frame, questioning whether the galaxy's constituents rotate within this frame and how their motion is characterized.
  • It is noted that while the constituents of the galaxy move in approximately circular orbits, there are local variations to this motion.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the FLRW model versus the weak field approximation for galactic distances. The discussion remains unresolved regarding the precise implications of the irrotational frame and the nature of motion within it.

Contextual Notes

The discussion highlights limitations in the application of cosmological models at galactic scales, as well as the need for clarity on the definitions and implications of the galactic rest frame and irrotational motion.

cianfa72
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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 ?
 
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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?
 
Last edited:
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.
 
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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.
 
Last edited:
  • #10
Well, approximately circular. There are lots of local variations to that, but in general yes.
 

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