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Just more of a conceptual question on the mutual inclusivity of the cosmological principle. That is to say, I was wondering if it were possible to have a Universe that were isotropic but NOT homogeneous OR a Universe that were homogeneous but NOT isotropic.

My spidey sense is telling me that it is possible that a Universe could be istropic but not homoegenous.

Surely, by default, if a Universe is homogenous then it follows that it has to be isotropic?

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    $\begingroup$ Does this help at all? Interesting look at isotropy vs homogeneity in materials $\endgroup$
    – costrom
    Jan 12, 2016 at 20:47
  • $\begingroup$ I'm thinking that we can have istropy in an inhomogeneous material (varying layers of density or concentric density rings, for example), but surely once we have homogeneity in a medium, then it is also istropic by default? $\endgroup$
    – MichaelJRoberts
    Jan 12, 2016 at 20:50
  • $\begingroup$ @costrom P.S. It helped, a little. $\endgroup$
    – MichaelJRoberts
    Jan 12, 2016 at 20:52

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Neither of the two cases are completely inconceivable:

A homogeneous, anisotropic universe

A universe with galaxies spread evenly all over, but all spinning in the same direction. This universe would look the same no matter where you lived, but have a net angular momentum, so looking in one direction you'd see all galaxies spinning along your line of sight, and in another direction, you'd see them spinning perpendicular to this direction.

Another example is a universe that had been permeated by density waves in one direction. In this direction, you'd see the density of galaxies alternating between high and low, and perpendicular hereto you'd see a constant density.

homo-noniso

Yesterday's papers on arXiv included a paper (Schucker 2016) that discusses the the possibility that we might live in another type of homogeneous, anisotropic universe, namely one in which the observed expansion rate depends upon the direction in which you look. This is called a "Bianchi I universe", and isn't just a hypothetical curiosity (although the results of this paper is statisically non-significant). See also @JonesTheAstronomer's answer.

An inhomogeneous, isotropic universe

As John Rennie has taught us, Big Bang didn't happen at a point. However, if it did, and we happened to live in the central region, we could observe the same in all directions, but see a gradually thinnening universe, or maybe increasing to some point and then decreasing, depending on exactly how this exsplosion came about. This scenario would however imply that we inhabit a special place in the universe, which would make Kopernikus sad. If a universe is isotropic from more than one location, is must also be homogeneous.

inhomo-iso

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  • $\begingroup$ Nice pictures, pela, +1. But I have to say we don't actually see galaxies like the above. As for JR's answer, "the universe doesn't have a centre" is an unsupported claim which is arguably at odds with "there is no space outside the universe". $\endgroup$
    – John Duffield
    Jan 13, 2016 at 18:26
  • $\begingroup$ @JohnDuffield: No no, I didn't mean that we see galaxies like the above. My point is that such universes in principle are physically conceivable — we simply find observationally that we happen not to live in such a universe. In contrast, it's harder to imagine a universe where, say, galaxies lie in narrow streams pointing out from a common center, such as Ned Wright's left figure. $\endgroup$
    – pela
    Jan 13, 2016 at 19:33
  • $\begingroup$ As for JR's answer, there is plenty of observational evidence (no proofs, of course, as is always the case in physics), and I don't understand what you mean that it's at odds with "there is no space outside the universe" $\endgroup$
    – pela
    Jan 13, 2016 at 19:33
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    $\begingroup$ @JohnDuffield: Yes, the "grapefruit" had a center, just like the observable Universe has a center, namely us. It wasn't all of space, just as the observable Universe isn't all of space. If it were, our Universe would be closed with a strong positive curvature, which definitely is ruled out by observations. I agree that space could be finite, but if so, it must be much, much larger than the observable Universe, lest we'd observe a $\Omega_\mathrm{tot}\gg1$. $\endgroup$
    – pela
    Jan 14, 2016 at 14:44
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    $\begingroup$ Wrt. a center or not, I'll give you that there isn't evidence that the whole Universe doesn't have a center, other than if so, the cosmological principle which gives many meaningful predictions is wrong, and everybody will be sad. There isn't evidence for a center either, though. $\endgroup$
    – pela
    Jan 14, 2016 at 14:46
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I was wondering if it were possible to have a Universe that were isotropic but NOT homogeneous OR a Universe that were homogeneous but NOT isotropic.

Most people will be happy with the UC Berkeley definition that says homogeneous means "looks the same at every location" and isotropic means "looks the same in every direction". And some will know that as per Ned Wright's article, these attributes aren't quite the same:

enter image description here

He says "the figure above shows a homogeneous but not isotropic pattern on the left and an isotropic but not homogeneous pattern on the right". However as far as I know pictures like this just don't apply to our universe scattered with galaxies.

My spidey sense is telling me that it is possible that a Universe could be isotropic but not homogeneous.

My spidey sense is telling me some guy 46 billion light years away might say the universe is neither isotropic nor homogeneous. Because when he looks up, half the night sky is black or something.

Surely, by default, if a Universe is homogenous then it follows that it has to be isotropic?

I agree with the gist of that. IMHO if an observer sees a homogeneous universe, he sees an isotropic universe too. Yes, one can find hypothetical scenarios wherein the universe is homogeneous but not isotropic. But they're only hypothetical. And let's not forget that it's only an assumption. If you lived in a forest would you assume that the world was covered in trees? Which look the same at every location and in every direction? It isn't a particularly scientific assumption. For all you know some guy lives near the edge of the forest. I think you're better off saying we just don't know.

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    $\begingroup$ Just to make a point regarding the LHS diagram of a homogenous but not isotopic Universe, surely it's not the same in every location? Because you could either be a brick or part of the grout? Red and white, respectively. $\endgroup$
    – MichaelJRoberts
    Jan 12, 2016 at 21:27
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    $\begingroup$ In similar vein you could either be in a galaxy or I the intervening space, or even in a star or not. What we're talking about is large-scale homogeneity. The universe isn't homogeneous or isotropic on the smaller scale. If it was, there wouldn't even be any gravity. Read this where Einstein described a gravitational field as space that was "neither homogeneous nor isotropic". $\endgroup$
    – John Duffield
    Jan 12, 2016 at 21:34
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    $\begingroup$ Still can't accept that those two diagrams represent the two distinct descriptions, I know they aren't yours. Just don't see it. $\endgroup$
    – MichaelJRoberts
    Jan 12, 2016 at 21:49
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    $\begingroup$ Don't worry about it Mike, because like I said, as far as I know pictures like that just don't apply to our universe scattered with galaxies. $\endgroup$
    – John Duffield
    Jan 12, 2016 at 22:16
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    $\begingroup$ Of course. I do often wonder, however, that with the fluctuations in the CMB and the measured dipole isotropy surely we can no longer underpin our "world" (Universe) view with the cosmological principle? $\endgroup$
    – MichaelJRoberts
    Jan 12, 2016 at 22:18
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Within the framework of General Relativity there are important solutions of the Einstein equations that are (a) homogeneous but anisotropic and (b) inhomogeneous yet isotropic (about a single point).

Class (a) are the Bianchi Cosmologies which are most simply described as homogeneous fluids that have different expansion rates in different directions, or some form of rotation. There don't appear to be any simple descriptions of these but at a technical level it's hard to beat George Ellis' Cargese lectures: http://arxiv.org/pdf/gr-qc/9812046.pdf

Class(b) solutions are the Lemaitre-Tolman-Bondi (LTB) solutions which have the same non-uniform density distribution in all directions about one point. See https://en.wikipedia.org/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric

Our present universe is on average both homogeneous and isotropic, but both types of solution (a) and (b) nonetheless play an important role in cosmology.

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