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I am taking an astrophysics course, and my astro professor said that "we need to introduce dark matter because Einstein's general relativity doesn't quite work without it".

I wanted to clarify/confirm with you experts 2 points based on my previous study of GR:

  1. It's not that GR doesn't work, it is that dark matter is needed when using a $k = 0$ FLRW model of the universe. So, it's the FLRW assumption that is problematic.

  2. Other, inhomogeneous models like Lemaitre-Tolman-Bondi or Swiss-cheese models are not well tested on cosmological scales.

Would love to hear thoughts on the above 2 points.

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    $\begingroup$ A simple analogy is with the discovered of Neptune in Newtonian gravity. It's presence is 'needed for Newtonian gravity to work' in the same sense. $\endgroup$
    – Eletie
    Dec 10, 2023 at 10:44
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    $\begingroup$ The amount of matter in the universe and its distribution is merely a model parameter, and initial conditions, not part of the theory itself. Dark matter is no more than a "calibration", like setting the overall constant to match Newton's gravitation. There is only any controversy because stupid people enjoy believing that "Einstein was wrong". $\endgroup$
    – m4r35n357
    Dec 10, 2023 at 11:17
  • $\begingroup$ The Sun is also needed to cause the exactly right amount of perihelion precession. And keep us in orbit. $\endgroup$
    – my2cts
    Dec 10, 2023 at 16:02
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    $\begingroup$ Very closely related: physics.stackexchange.com/q/174732/226902 physics.stackexchange.com/q/222275/226902 physics.stackexchange.com/q/278387/226902 and links therein. GR works very well and therefore we add dark matter! (if you do think that GR doesn't quite work then you look for alternatives to GR, while dark matter does not change GR but the matter sector). See: physics.stackexchange.com/a/222153/226902 $\endgroup$
    – Quillo
    Dec 11, 2023 at 9:04

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You're right about 1, sort of. GR works if you assume dark matter exists. Phrased alternatively, if you assume GR is correct, then you need dark matter. However it's worth pointing out that evidence for dark matter comes from more angles than just cosmology - so even if the FLRW universe is problematic, it does not mean there's no dark matter.

About 2, the key question as with all science is whether the models match observations. Since you're asking about inhomogeneous models, the question becomes, is the observable universe homogeneous & isotropic (which are the key assumptions that go into FLRW)? The answer is yes to a high degree. The canonical example for isotropy is the CMB, which is very nearly completely isotropic; see also this paper. Homogeneity can also be tested, and the universe appears to be highly homogeneous (example). This is why we believe in FLRW.

However, it's also worth pointing out that the current Standard Model of Cosmology is showing cracks, especially the Hubble tension but also a few other observations (like S8 tension). There are a myriad possibilities to resolve these problems, and one of them is to give up on the homogeneous & isotropic FLRW universe. Certainly people are thinking about it (example). The ending of that story is yet to be written.

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    $\begingroup$ Thanks @Allure for that. I am still a bit confused on the statement: " GR works if you assume dark matter exists." ... By GR, don't we mean the EFE? If you take a vacuum solution, I don't specify dark matter anywhere. GR still "works" doesn't it? $\endgroup$ Dec 10, 2023 at 9:08
  • $\begingroup$ @ThomasMoore modifying the EFE is effectively modifying GR. For a theory as a whole to "work", it has to work over a range of scales & regimes. $\endgroup$
    – Eletie
    Dec 10, 2023 at 10:45
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    $\begingroup$ But @Eletie , We're not talking about modifying the EFE. :) You have different solutions to the EFE. One only tests different solutions, correct? $\endgroup$ Dec 10, 2023 at 14:20
  • $\begingroup$ @ThomasMoore depends on what you mean by "works". If you mean it matches observations, then GR does not work (for a variety of astrophysical observations) unless dark matter exists. It's very similar to Eletie's comment about Neptune being needed for Newtonian gravity to work. If Neptune didn't exist and Newtonian gravity needed to be modified slightly at solar-system scales, you could still say it "works" on Earth-length scales. $\endgroup$
    – Allure
    Dec 10, 2023 at 23:44
  • $\begingroup$ @Allure , GR is not tested directly with respect to observations: solutions of the EFE are. Where would you "put dark matter" in the EFE? What is the form of the energy-momentum tensor? I believe most of the cosmological models have modeled dark matter as a perfect fluid. Then they test that assumption, and it all seems to fall apart starting with FLRW! As another example, when you say GR doesn't work unless dark matter exists, should we not use Kerr metrics for rotating black holes? $\endgroup$ Dec 11, 2023 at 1:54
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I very much agree with your first point.

Many people don’t understand general relativity, and this is very prominent unfortunately in the astrophysics community, that mostly always sees things from a Newtonian perspective.

When people talk about GR working/not working, they mean solutions of the field equations that are then tested against observations. Einstein’s equations themselves simply relate curvature of spacetime to the energy-momentum tensor.

The current standard model of cosmology relies on spatial homogeneity and isotropy, which yields solutions of the EFE that have 6 Killing vectors/a six-dimensional isometry group. In addition, they also consider spatially flat models. To make “these” models fit with observations, they then introduce dark matter/energy.

I don’t believe a high degree of isotropy indicates isotropy. Something is either isotopic or it isn’t. Further, even from an astrophysical perspective, it is clear that the existence of black holes in the universe implies that the universe is inhomogeneous. There was a lot of work done on the Swiss Cheese models that were shown to challenge the dark energy assumptions. This also leads to the important matching / boundary conditions of how to join a S-metric universe with FLRW.

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  • $\begingroup$ Even if universe is inhomogeneous and anisotropic on local scale we can ask till what extent it looks isotropic and homogeneous on cosmic scale (on average). Ig its more of a technical limitation with simulating non-linearities that we resort to post newtonian pov even though it is inaccurate. So one shouldn't interpret the failure of simplified (and probably wrong) approximation as an indication for failure of GR. $\endgroup$
    – KP99
    Dec 11, 2023 at 11:00
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I think the point that your professor is getting at is not at all concerning the validity of the general theory but touches instead on the fact that neglecting to include "dark matter" into one's standard application of GR to the problem at hand produces less satisfactory results.

Essentially, researchers found that when they included greater amounts of matter into the stress-energy tensor, the results of theory better matched the observations. The point to take away from the professor's comment is that dark matter is needed within the paradigm of standard general relativity.

P.S. It is worth noting that the idea of a flat universe is not very consistent with a universe that contains any kind of matter, dark or otherwise, as matter and energy content are responsible for inducing curvature into the theory. Technically, a flat cosmological model is an absolute vacuum.

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    $\begingroup$ you can certainly have Bianchi Type I models and k=0 FLRW models which have all sorts of matter in them, perfect fluids for the latter, and anisotropic shear stresses in the former.Both of these models have flat spatial sections. $\endgroup$ Dec 11, 2023 at 0:12
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Dark matter is more about the galactic rotation curves (so more a local issue). Whereas at a cosmological level its "dark energy", aka the cosmological constant, that is also the one that enters as a free parameter in GR models derived from Einstein equations under simplifying assumptions. Please correct me if I m wrong.

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    $\begingroup$ As I understand it dark matter is far from a local issue, in cosmology it relates mostly to a "global" network of filaments that attracts matter to clump into galaxies. $\endgroup$
    – m4r35n357
    Dec 10, 2023 at 20:24

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