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I don't really understand what we are talking about when we say space is homogeneous. What we are measuring? My notion is: it should depend on the entity and with respect to that entity one can decide space is homogeneous or not! May be I'm asking stupid question, actually the fact is I don't understand the phrase "space is homogeneous" or "space-time is isotropic"

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Suppose we measure the density of matter as a function of position. Then we will come up with some function for the density $\rho(r, \theta, \phi)$ where $r$, $\theta$ and $\phi$ are polar coordinates with ourselves at the origin.

  • The density is isotropic if it is the same in all directions i.e. it is independent of $\theta$ and $\phi$ and just a function of distance, $\rho(r)$.

  • The density is homogeneous if we can move our origin, i.e. move to any other point in the universe and our density function $\rho(r)$ is unchanged.

The simplest such function is if $\rho$ is just a constant since it then automatically the same in all directions and the same at every point.

I've used density as an example, but this applies to any property of the universe that is a function of position.

Our universe is obviously not isotropic and homogeneous because in same places there is a vacuum while in others there are stars. However we know it started out as very close to isotropic and homogeneous from measurements of the cosmic microwave background radiation. And if we take average density on a large enough scale then we believe it is still on average isotropic and homogeneous (though this is a somewhat controversial issue).

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  • $\begingroup$ I don't think the OP is asking about the distribution of matter in the universe. $\endgroup$ – sammy gerbil Aug 25 '17 at 18:33
  • $\begingroup$ Great answer @John Rennie. +1. But one item I think it overemphasizes an issue which is more detail than the basic idea. The controversy on homogeneous/isotropic in the large scale has to do with small deviations. The microwave background settled the big issue, the smaller variation we now can see and model for the universe evolution is not a controversy, it is precision cosmology doing better. $\endgroup$ – Bob Bee Aug 26 '17 at 3:49
  • $\begingroup$ @John Rennie: that means one always decides homogeneous/isotropic w.r.t physical entity, like in your example you used density. I find no meaning of only saying "space is homogeneous", rather it sounds meaningful to me of saying"space is homogeneous with respect to ..." $\endgroup$ – sid Aug 29 '17 at 13:50
  • $\begingroup$ @sid: space, or rather spacetime, isn't a thing. It's a mathematical construction so it can't be homogeneous. It's the stuff that fills spacetime, whatever that stuff may be, that is homogeneous and isotropic. So assuming I understand you correctly you are quite correct that it is meaningless to say space is homogeneous. $\endgroup$ – John Rennie Aug 29 '17 at 14:00
  • $\begingroup$ Thanks a lot Sir! that is the answer I was expecting. It has been disturbing me since past couple of weeks. Thanks by Heart! :) $\endgroup$ – sid Aug 29 '17 at 17:40
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You are correct in the sense that there are physical quantities, observables, like scalar and vector fields that are, in general, varying in space.

However, that's not what the phrase "spacetime is isotropic" is meant to convey. What this means is that, in the absence of anything to break the symmetry, each point in spacetime is no different than any other point in spacetime.

Without reference to anything else, spacetime is considered as homogenous everywhere.

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