Chromotron t1_j6mjqvg wrote
Reply to comment by The_A4_Paper in ELI5: If the universe is flat, and therefore infinite in size, how could it have been condensed to a single point in the past? by joeyo1423
> If the universe as a whole(including outside of the observable universe) then it wasn't a single point, infinite was always infinite and will always be infinite.
This misconception is weirdly common and contradicts basic topology:
If every bubble of, say, 10^10 ly, was once a single point, then the entire universe was once a single point.
Proof: assume x,y are any two points. Connect them by a path of finite (but potentially extremely large, even by universe standards) length. Overlay that path with finitely many of those 10^10 ly bubbles, such that they overlap, forming a "chain". Let A, B be two neighboring overlapping bubbles. Then once all points of A were the same point a, and all of B were once the same point b. But now look at any point p in their intersection: p was once both a and b, thus a=b! Doing this iteratively with the chain of bubbles, we arrive at the conclusion that any two points were once the same! [ ]
And indeed, the infinite(!) 3D (or 4D) space is contractible, it can be contracted into a single point in finite time. Even at locally finite & bounded speed.
Anyway, there are quite a few models of the Big Bang where the universe was always infinite, just with also infinite density at the beginning. The Big Bang needs not necessarily be a single point in the usual sense.
RhynoD t1_j6nia1p wrote
Conversely, I have heard arguments that the misconception was that any part of the universe, infinite or otherwise, was a single point. Rather, all points were infinitesimally close, but not occupying the same space and not truly one point. In that case, the universe would continue to be infinitely large and infinitesimally small.
But I don't know enough about math or physics to know or argue one way or another.
Chromotron t1_j6nqjc7 wrote
There are ways to work with infinitesimal numbers just as with real numbers, but that does not really do the trick, as physics is not based on that. To my understanding (not an expert in cosmology or that deep into physics) the energies/fields of back then can cause infinite densities.
UntangledQubit t1_j6nnydr wrote
>Even at locally finite & bounded speed. >
How would that work? It seems like in the actual transition from a point into a space, speed wouldn't even be definable. Is there an explicit construction of such a contraction?
Chromotron t1_j6ntihj wrote
If inside the same space:
Let t be a real number running from 0 to a(ge of universe). We contract space to a point inside itself by sending each vector v at time t to (a-t)·v. So at t=0, v is wherever it should be, and at t=a we get 0 (italic to denote it is the vector 0, not the number 0) regardless of v. And in-between, it moves towards that inevitable 0.
This does not "jump" (it is continuous), but from an external view, v does move with speed |v|/a (with |v| the distance of v from 0) all the time. So points far away move arbitrarily fast, similar to how some parts of the universe move away from us faster than the speed of light. But "locally", so if every point only observes those close to it, points have almost the same speed and direction. So within a close bubble, the rest of space moves only slowly.
Now we have the issue that the real universe is not contracting/expanding "within itself". This requires some slight fixes and makes the calculations a bit more ugly (hence why I did the above first):
One should think about the universe at each time t as its separate thing: imagine the Universe as a planar flat thing; now draw a time-scale in another dimension (so we need 4D if we do the real thing); lastly, fix a random point B ("Big Bang") at distance a from U and draw all the "rays" starting in B towards each point of the universe.
If U were a perfect circle, this gives you an actual cone, and this is indeed called the cone construction. Whatever U was though, one can now contract towards B in this cone(ish) thing we made as before.
Now that is still pretty far from what General relativity tells us, but I hope it explains how one can model such a thing.
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