 What metric do you use to model a Galaxy

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I'm not an expert in GR, I'd like to if there is a standard practice in using a particular metric to model an isolated rotating Galaxy. I thought about Kerr-Newmann metric but I guess is not the standard practice. I guess is common practice to use Newton + a correction term, but in that case I'd like to know what term is it and where is comes from. Can someone help me with some references?

Thank you

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This depends on what do you need it for. You have to answer question such as

1. Do you want to describe motion at non-relativistic velocities near the galaxy?
2. Does the motion occur inside the galaxy or outside?
3. What do you assume about the morphology of the galaxy? (What type of Galaxy is it?)

If the answer to 1. is yes, you can typically use the metric in the Newtonian limit, i.e.

$$ds^2 = -(1 + 2 \Phi) dt^2 + (1 - 2 \Phi)(dx^2 + dy^2 + dz^2)$$

where $\Phi$ is the Newtonian potential sourced by the galactic matter. Now you "just" need to find the right $\Phi$. The answer depends a lot on questions 2. and 3. For instance, an almost spherically symmetric galaxy can be modeled by a spherically symmetric potential that is given simply by $\Phi = -GM(r)/r$, where $M(r)$ is the mass enclosed in the radius $r$. On the other hand, the outside potential of a flattened, almost axially symmetric galaxy can be modeled by potentials such as the Vinti potential or generally a multipolar expansion.

answered Mar 17 by (1,635 points)

Thank you Void. I think I got the idea and I would upvote your answer if I could... Unfortunately it seems I'm not yet able to vote :(

I don't know if any of the below URLs are useful?

1. The Cosmological Principle
https://ned.ipac.caltech.edu/level5/Peebles1/frames.html

2. Galaxy rotation curves with log-normal density distribution
https://arxiv.org/ftp/arxiv/papers/1502/1502.02949.pdf

3. Astronomy: a Rotating System of Satellite Galaxies Raises Questions
https://www.unibas.ch/en/News-Events/News/Uni-Research/A-Rotating-System-of-Satellite-Galaxies-Raises-Questions.html

@JevanPipitone interesting links but not related to the question

I am trying to find relevant material however I am posting a comment not an answer so I am not claiming to be providing an answer. I found this:

https://en.wikipedia.org/wiki/Galaxy_rotation_curve
under section "Halo density profiles"
"In order to accommodate a flat rotation curve, a density profile for a galaxy and its environs must be different than one that is centrally concentrated. Newton's version of Kepler's Third Law implies that the spherically symmetric, radial density profile $\rho(r)$ is:"

https://dept.astro.lsa.umich.edu/resources/ugactivities/Labs/tully_fisher/tf_intro.html
Rotation of Spiral Galaxies

https://physics.stackexchange.com/questions/26625/formula-for-rotation-curves-of-galaxies
Formula for Rotation curves of Galaxies

https://www.astro.umd.edu/~richard/ASTRO620/QM_chap5.pdf
Rotation Curves

http://astronomy.swin.edu.au/cosmos/R/Rotation+Curve
Example rotation curves for 1) a solid body, 2) the Solar System and 3) a spiral galaxy.

Void's post to the wikipedia article about Galaxy Types and morphology has some interesting and easy to understand information and I can see from the link there are types of galaxies such as Ellipticals, Shell, Spirals, Barred spiral, Super-luminous, Peculiar, Ring, Lenticular, Irregular, Ultra Diffuse, Dwarfs, Interacting, Starburst, Active, Blazars, Radio, Liners, Seyfert, Quasar, Luminous infrared. However it does not give a formula describing the characteristics of each of these galaxy types. For example this could be a formula for the visible shape such as a spiral, or it could be the distribution of ultraviolet and mid-infrared for a Super-luminous spiral, which would presumably require more than one formula.

I'm not an expert on this either like the original poster Dac0 wrote, I'm just interested in discussing physics, which is why I am writing comments rather than answers.

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