Enquerencia said:
but if it does have a size, or ever did, but has always had its center everywhere at every moment, even as it's expanding, and if it doesn't have an edge, how can it be flat? I feel like in missing something really fundamental.
Understandable. We aren't talking "normal" geometry that most people are used to. In addition, there's the problem that General Relativity itself is a theory of variable geometry, something most people have never even heard of. I'll try my best to explain what I know, but be warned that I may be mistaken.
To start, you need to know just a bit about General Relativity. The really short explanation is that GR is a geometric theory (a theory of geometry) that describes space and time. Unlike what you're used to here on earth, where the exact geometry never noticeably changes, GR states that the geometry of spacetime changes based on the presence of mass. The end-effect of this is to cause gravitation, time dilation between observers, and several other effects that I won't go into. In addition, cosmology itself uses GR as the fundamental theory from which to model the universe as a whole. So all of this geometry stuff is extremely important to cosmology.
When we model the universe with GR, we run into the problem of having variable geometry. There are essentially three types of possible geometry: flat, open, and closed. Take, for example, the surface of a perfect sphere. The geometry on the surface of the sphere is
not "flat". Lines that are initially parallel will end up crossing if drawn on the surface of the sphere. The angles of triangles do not add up to 180 degrees like they do on a flat piece of paper, instead equaling more than 180. A straight line drawn on the surface will come back around to its starting point. The geometry is
closed.
The geometry of a flat piece of paper is, well, flat. This is the geometry you learned in school. Parallel lines never intersect or get further away from one another. Triangle angles add up to 180 degrees. A straight line never intersects itself.
An open geometry is like the surface of a saddle. Lines that are initially parallel will diverge. Angles of a triangle add up to less than 180 degrees.
Now, a key thing to realize is that all of these descriptions have so far talked about 2-D surfaces within 3-D space. We can obviously see the different geometries with no problems, as we live in a 3-D universe. There is an "up" and a "down" side of a ball on the ground, and of a saddle and a piece of paper. We can put the coordinates of any point on any of the lines on these surfaces in a 3-D coordinate system as well as a 2-D. Since we can describe all of this curvature by referencing how it behaves within a higher-dimensional space, we call this way of measuring curvature "
extrinsic".
But this poses problems to measuring and describing the curvature of 3-dimensional space itself. We live in 3 dimensions and don't have the option of referencing a higher-dimensional space to describe any possible curvature. Fortunately, there is another way. By measuring things like angles and seeing how things that move in straight lines behave over very large distances, we can come up with a way of describing and modeling the curvature of our universe without referencing a higher dimensional coordinate system. Using a mathematical tool known as a "manifold", we can describe this curvature purely in terms of our own three dimensions. We call this curvature "
intrinsic". This distinction between intrinsic and extrinsic is important because, at first, it seems like the universe must be embedded within 4-dimensional space in order to have a curvature. As far as we know, this is not true, and we have ways of measuring curvature that doesn't require us to reference 4-dimensional space. Whether there are other dimensions or not is not my point here, I merely want to explain that it is not required that there be other dimensions in order to have curvature.
But, what does curvature of 3-D space mean? Put simply, it means something similar to what it meant when talking about 2-dimensional surfaces. The behavior of lines, shapes, and other things embedded within that space. In real life one obviously can't draw a line on empty space, so it makes measuring the properties of space a little more difficult. We have to look at what happens to objects as they move around though space. In the context of the universe as a whole, we typically look at how light behaves. At the very largest scales the universe is homogeneous and isotropic, meaning that if you zoom out REALLY far, the universe looks pretty dull. You wouldn't be able to see all the little clumps of matter and dark matter. Everything would look the same no matter what way you looked. Because the universe is homogeneous and isotropic at the largest scales, we can look at the behavior of light that has been traveling very long distances and, based on the behavior of that light, determine the overall shape of the universe. Thus, one of the things astronomers look at when determining the shape of the universe is at the CMB.
Now, what do the different shapes mean for the universe? We still have our three types from earlier, flat, open, and closed. The different properties of each still apply. Parallel lines may converge, diverge, or remain parallel depending on whether the universe is closed, open, or flat respectively. The geometry of the universe does put constraints on its overall shape, but it does not determine whether the universe is finite or infinite except that a closed universe must be finite. A flat or open universe could still be either finite or infinite as far as I know.
Note that all of this ignores expansion. Expansion itself doesn't change the geometry of the universe, but it does complicate its measurements. A closed, finite universe can still expand.
Earlier I talked about variable geometry. This is important because by measuring the amount of matter and energy in the universe we can figure out whether our universe is closed, open, or flat. To quote wikipedia's article:
General relativity explains that mass and energy bend the curvature of spacetime and is used to determine what curvature the universe has by using a value called the density parameter, represented with Omega (Ω). The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat. Put another way
- If Ω = 1, the universe is flat
- If Ω > 1, there is positive curvature
- if Ω < 1 there is negative curvature
Put simply, if there's enough matter and energy, the curvature should be positive. If there's too little then it will be negative. If there's just the right amount then the universe will be flat.
Enquerencia said:
The one "known" I can understand is that the universe is certainly expanding, whatever size and shape it might be. It was explained to me that just because it used to be more dense than it is, and is getting less dense, does not imply that the size of the universe is changing or that it was ever smaller than it is, but I still can't really comprehend it. I get that there is no outside space, empty until the universe expanded into it, but I constantly see references to the big bang as being an infinitesimal point of infinite density suddenly bursting into existence, creating space and time,where before, neither existed. But in the very first instants of the big bang, as it the universe sprang into existence, how could it already be infinite and flat? Wouldn't it have to be closed, or spherical?
Ignore those references. Realize that in order to model the history of the universe we can only do so by basing our theories off of observations. Because the speed of light is finite, we can see further back into the past as we look further away. But the problem here is that we can only see so far, both in distance and in time, so we can't actually see what things were like at the very beginning. What we do know is that in the past the universe was more dense than it is now. If we construct a model and look at the density of the universe over time, we see that it approaches infinite density as we go further back in time. If we let it reach infinity we have a singularity, which is just what happens when our math stops working. I get a singularity when I try to divide by zero, so this isn't something that just popped up in cosmology. Now, if the entire universe, all of it according to the model, gets denser as we go back in time, then when the density is infinity and we get our "singularity" then this singularity is
everywhere. So any references you see that claim the big bang started at a single point in space is simply wrong.
As for the creation/beginning of the universe, we don't actually know anything about that. It's possible that the universe came into existence already infinite and flat. I mean, if we want to talk about things popping into existence from utterly nothing prior, then why would a finite universe popping into existence from a single point be any more plausible than an infinite universe? Either way you still have the building blocks of everything that will ever exist suddenly coming existence.