I understand that, in general, homogeneity is the physical attribute of being uniform in composition (" of the same form at every point"), but I'm slightly confused when it is used in cosmology as there seems to be an overlap with the notion of isotropy, i.e. direction independence. I've read that in cosmological terms, homogeneous means that the universe "looks the same" at every point (at a given instant in time), implying that it has uniform density (on large scales) and is such that the same laws of physics apply universally. However, isotropy means that at a given point (at a given instant in time) the universe "looks the same" in every direction. This seems to overlap somewhat with the concept of homogeneity?!
Is the idea that homogeneity means that, if an observer observes the universe to look a particular way from their vantage point (i.e. that it has certain physical properties), then another observer at any other point in the universe will make the the same observations and conclude that it has the same physical properties. The point being that both of them could observe anisotropies and thus the universe could potentially be homogeneous, but not isotropic. The reverse does not hold however, as if both observe the universe to be isotropic, the it is necessarily homogeneous?!
Sorry for the long windedness of this question, it's just been bugging me for a while.
Answer
Homogeneity in cosmology means uniformity from point to point, not only in composition or content, but in geometry as well. An empty space with a singularity is still non-homogeneous. Isotropy at every point does imply homogeneity, but we are not in a position to observe the universe from every point. Mathematically, isotropy at any two distinct points already implies homogeneity, but in practice we need them to be sufficiently far apart, and we are not in a position to observe the universe even from two points that are sufficiently far apart (recall how telescopes resolved stellar parallax only in 1838, and its absence was used as an argument against moving Earth).
On the other hand, it is easily possible for the universe to be isotropic at some point, in particular at the point where we and the Earth are, without being homogeneous. Any non-constant spherically symmetric distribution of matter will do. To get homogeneity from isotropy-here one needs to invoke the so-called Copernican principle, which states that neither the Sun nor the Earth occupy a special position in the universe. Of course, the Copernican principle is effectively the contrapositive of homogeneity, all places are the same rephrased as no place is special. See discussion on Physics Forums.
So when it is said that the universe is "homogeneous and isotropic" there is no "overlap" because only isotropy-here is meant as an assumption. On the other hand, isotropy-here (the only thing we actually observe) and homogeneity imply isotropy everywhere as a conclusion. By rejecting the Copernican principle one can construct a curious giant void cosmology for example, which explains accelerated expansion of the universe without the dark energy, but "most scientists believe that it is not reasonable to adopt a cosmological model in which the universe is simply a joke played for the benefit of mankind".
It is also possible for a space to be homogeneous but anisotropic purely geometrically, without any matter content. A simple example to visualize is a two dimensional cylinder: every point looks the same, but vertical and horizontal directions look different (globally). A three dimensional example is Poincare dodecahedral space obtained by identifying certain points on the $3$-sphere. If our universe had this shape there would be observable patterns in the CMB radiation indicating it.
There is also a difference between local and global homogeneity, vicinities of all points, may look the same, but some points may still be special from the global point of view. Take open flat disk for example, metric tensor is everywhere constant, so small neighborhoods of all points 'look the same', but the center is special, there is no global isometry that maps it to any other point. There are less visual such examples that do not have a 'boundary'.
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