Regardless of whether it was a singularity or not, if it’s expanding now it must have been smaller before. If it expands uniformly the expansion must be always centred relative to the smaller pre-expansion.
No. There is no center, and never was. I'm going to get a little bit mathematical, but it might actually help you understand what's going on if you can follow.
In geometry, a "metric" is a mathematical quantity which tells you how to calculate distances. The idea of a metric may seem a little pointless if you approach things from a Euclidean perspective, because in Euclidean space with cartesian coordinates, the metric is a rather dull and uninteresting quantity, and it doesn't change from point to point. But let's write it out anyways:
ds
2 = dx
2 + dy
2 + dz
2
where ds is the distance between the points, and dx, dy, and dz are the coordinate differences between the points. You might recognize this as a simple variation of the Pythagorean theorem. So what's the big deal? Well, if we go to non-Euclidean space, or if the space changes over time, then things can get quite a bit different, and quite a bit more interesting. And in general relativity, the metric is what describes everything there is to know about space.
So let's look at a simple metric that describes a Euclidean space (to keep things simple), but in which that space expands over time:
ds
2 = a*t
2*(dx
2 + dy
2 + dz
2)
where now "a" is some constant. Note, too, that x, y, and z are
coordinates, and
not distance. They describe locations within our space, but we cannot find the distance between two points by looking at the coordinates alone without consulting the metric. That's not actually any different than ordinary Euclidean space, though: we just tend to set up our coordinates to match distances. The only difference is that now that our space is expanding, we can't have distance and coordinates match except at one particular instant.
OK, so I said this space was expanding. How do I know? Well, look at the equation. Consider the distance between the same two coordinate points at t=1 and at t=2: they are twice the distance apart at t=2 than they are at t=1. What happens as t -> infinity? All points approach being infinitely far away from each other. What happens as t -> 0? All points approach being zero distance away from each other, ie, the space collapses towards a singularity.
But is there a center? Well, no. I didn't make this explicit before, but this is a metric for a Euclidean 3D space which is
infinite. Where's the center of an infinite space? It has none. Once could say that it's at the origin, but in an infinite space (meaning that x,y, and z are unbounded), the choice of origin is completely arbitrary. Picking a different origin changes nothing. In fact, the metric itself doesn't even NOTICE the origin, even if you specify it, because it's only interested in
differences. So one could say that there is no center, or that the center is everywhere, but what one
cannot say is that the center is here and not there.
I will take “uniform curvature “ as “curved in all directions simultaneously”. How does a thing travel along a uniform curvature without being torn apart by essentially travelling in opposite directions at the same time (whatever the scale)?
I think you misunderstand what I mean by "uniform curvature". It's not the curvature in all directions (which the universe should also be if it's isotropic) but the curvature at different points should be the same (homogeneous). A sphere has uniform curvature: every point is equivalent, and the curvature is the same at all points on the sphere. An oblate or prolate
spheroid does
not have uniform curvature. An oblate sphereoid, for example, has less curvature on the flattened ends than on the sharply rounded sides.
A problem I have with a finite, folded back on itself 3D universe is that there should be a location from which it’s not possible to move in a particular direction. Please don’t tell me about a 2D surface of a sphere that I can easily imagine but not accept as being valid.
First off, you're wrong about not being able to move. If the curvature is constant, that will make no difference. For example, consider a rigid triangle on the surface of a sphere (yes, I know you don't accept it as valid, but bear with me). Because it's on a curved surface, the angles of the triangle will want to add up to something greater than 180 degrees (the value will depend on the curvature of the sphere and the size of the triangle - let's say 185 for now). But we can slide our triangle anywhere on our sphere, and it will make no difference, the angles always add up the same way. There's no problem here.
But your intuition is telling you something real. Suppose it's not a perfect sphere, but is lumpy in places. Let's say we try to move onto a bump where the curvature is greater, and the angles of the triangle would need to add up to 190 degrees. Then we would indeed have a problem.
But there are two additional relevant facts. The first is that in the real world, relativity actually prohibits perfect rigidity. While our triangle might resist bending, if we push it hard enough, it MUST bend. So while varying curvature could make motion difficult, it can never make it impossible. The second fact is that even on stellar length scales, NOTHING is rigid
at all. Everything is fluid (that's why even solid planets are spheres). So it doesn't even matter in the real world: we don't get significant curvature differences at smaller length scales, and at larger length scales nothing will resist moving through different curvatures anyways. So while you weren't
completely wrong to be thinking along those lines, in reality it just never matters.
As for accepting the 2D sphere analogy, well,
why can you not accept it? The math works out
perfectly. Our everyday experience tells us to think in terms of Euclidean space, but that means nothing: the curvature of space itself is far too small for us to notice directly, just as an ant cannot tell that the earth is a sphere and not a lumpy plane. We should not rely on our everyday intuition to guide us in such matters. We should look to both math and experiments, and the experiments say that GR is correct, and the math of GR says the universe can (but might not) curve around on itself.
