Homogeneity is uniformity, isotropy is uniformity in all orientations. What's to misread?
Isotropy refers to geometry being independent of direction. Homogeneity does not imply isotropy; a good example of that is a cylinder, which wraps around in one direction but not in another, but is still homogeneous. Thus, by ignoring that condition, you're greatly misinterpreting the statement.
BTW, in more common geometric usage unqualified 'isotropy' is the same as 'global isotropy'; I was just a bit more explicit.
Yes, I meant ball in the mathematical sense, somebody else said a universe that is spatially a 3-sphere, and I took this to be a sphere in the common-usage sense. A non-mathematician thinks of a sphere as a ball rather than the surface of a ball.
It isn't, and that's a dangerous attitude to take in a context that's heavily mathematical. Particularly since it's responsible for some of your confusions, including one below.
Please explain. Use a universe that isn't expanding for simplicity, and imagine we shine a light beam in a straight line.
Let's take a particular flat 2-torus for simplicity. Imagine a square of side length 2, with Cartesian coordinates with origin at the center and axes aligned in the obvious way (parallel to sides). But unlike a square, the sides are identified: (x,-1)~(x,+1) and (-1,y)~(+1,y). The coordinates refer to the same point on the manifold. A more convenient definition is ℝ²/ℤ, or if we're talking about topology, a 3-torus is the product of three circles S¹×S¹×S¹, though both of these may be more cryptic for people unused to these areas of mathematics.
That means if you're at the origin and you shine a light beam along the x-axis, it will eventually hit you in the back: it will go from (0,0) to (1,0), which is the same as (-1,0), and proceeds still along the x-axis to (0,0). Similarly for many other directions, but not all: if the slope of the light beam direction is irrational in these coordinates, then it will never reach the origin again, although it may come arbitrarily close. Thus the flat torus fails to be isotropic, as not all directions behave in the same way.
A flat 3-torus is completely analogous. Note in particular there's no mention of any higher-dimensional space. Any embedding to a higher-dimensional manifold would be purely a convenience for some purposes and not at all necessary for any intrinsic properties.
OK I understand what you're saying now. The globally is the key word, and it makes the statement somewhat tautological. You can render it down to if the universe is flat and edgeless, it is infinite.
Not so. The flat torus is flat and edgeless, but not infinite. In a sense all valid mathematical theorems are tautologous, but your interpretation of it is still incorrect.
The error is in the word globally. It's loaded. Delete it.
Like I said, in the common geometric usage, unqualified 'isotropic' means 'globally isotropic' already.
See
wiki which says
Einstein's static universe is closed (i.e. has hyperspherical topology and positive spatial curvature). A hypersphere is a 4-sphere. Please explain why Einstein proposed a 3-sphere rather than a 4-sphere.
You're making essentially the same mistake as above. A 4-sphere is indeed a hypersphere, but a 3-sphere is
also a hypersphere. The usual sphere such as the surface of a common ball or (a somewhat idealized) Earth is a 2-sphere because it is 2-dimensional.
The topology of the Einstein static universe is ℝ×S³, one dimension of time and three dimensions of space, the latter of which form a 3-sphere.
I understand EM, Vorpal. Don't take F
μν as the fundamental object. See
this. But let's not get bogged down on electromagnetism. Let's talk cosmology.
Ok.
The GR stress-energy-momentum tensor features pressure, and dark energy was described as pressure by Phil Plait. If that "pressure" is counterbalanced at all locations because the universe is infinite, it cannot result in expansion, can it?
Of course it can. There's nothing wrong with the FLRW family of solutions.
Perhaps you're thinking of there being no movement through space whenever the pressure is uniform. But in the comoving frame of FRLW solutions where pressure is so 'balanced', the galaxies are stationary (on averfage) and space is expanding.
How can an infinite universe expand? Answers on a postcard please.
"On a poscard", huh? As an idealization, imagine an infinite line of equally spaced galaxies:
Code:
...-*-*-*-*-*-...
...-*--*--*--*--*-...
...-*---*---*---*---*-...
There's nothing whatsoever conceptually difficult about space between galaxies stretching.
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I'm afraid Farsight may be right about that. A torus is not a space, it's a 3-d surface imbedded in 3-space. Any point on the Torus surface can be defined with 3 euclidian coordinates. If you want to make a space out of a torus, you'll have to add a dimension.
That's like saying that a Euclidean plane is not a space but a surface embedded in 3-space. An embedding may be practically useful for certain things, but it isn't necessary. Besides, the flat 3-torus we're talking about
cannot be embedded in (Euclidean) 3-space. See above for one way to think of a flat 2-torus without any higher-dimensional space; a flat 3-torus is straightforwardly analogous and has been covered in
sol invictus's post.
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