Isn't space expansion logically required?

[Finny]

The geometry of spacetime does not change; spacetime is a 4-dimensional geometric object that just is. It describes the entire history of the universe.

Does this apply to the inflationary period? pre inflationary??

 

[PeterDonis]

Hopefully that's correct.

Basically, yes. The only caveat is that null lines only appear as 45 degree lines on certain kinds of spacetime diagrams, the ones that are "conformally equivalent" to the usual SR spacetime diagram of Minkowski spacetime. For example, in the 3 diagrams of the universe that you linked to, the bottom one meets this criterion, but the others do not (as you can see from the fact that the "light cone" drawn on those diagrams is not marked out by 45 degree lines).

Is it then correct to say that in the early universe, the geometry of space was infinitesimal, but the topology of space was infinite?

The geometry is not "infinitesimal"; "infinitesimal" describes a small piece of spacetime, not its geometry as a whole.

The topology of space is not "infinite"; it's "unbounded" (or, more pedantically, a "manifold without boundary") and "non-compact" ("compact" is the precise term in topology for what we would think of as a "finite" region in ordinary language).

part of this discussion has centered on the semantics of whether or not a thing that must be 4d or 2d but with one dimension as time is a single thing that is not "split" or a thing that is split into distinct parts.

One dimension being timelike does not "split" anything; it's just a fact about the geometry of the spacetime.

the only reason I can do that is because the geometry of the planet supports that topology (if that is saying it right)

Not really. Topology is prior to geometry. The correct statement is simply that the topology of the manifold allows it.

the line representing the right and left side of the screen is just a visual artifact that the math doesn't need, but something seems suspicious about the idea that it is not located anywhere specifically (in which case the surface would not be smooth)

Why not?

 

[PeterDonis]

Does this apply to the inflationary period? pre inflationary??

It applies to the entire history of the universe except (possibly) for a period at the very beginning when the density was of the order of the Planck density and a spacetime description was not possible. I say "possibly" because we don't know for sure that there was such a period.

 

[Jimster41]

Basically, yes. The only caveat is that null lines only appear as 45 degree lines on certain kinds of spacetime diagrams, the ones that are "conformally equivalent" to the usual SR spacetime diagram of Minkowski spacetime. For example, in the 3 diagrams of the universe that you linked to, the bottom one meets this criterion, but the others do not (as you can see from the fact that the "light cone" drawn on those diagrams is not marked out by 45 degree lines).
The geometry is not "infinitesimal"; "infinitesimal" describes a small piece of spacetime, not its geometry as a whole.

The topology of space is not "infinite"; it's "unbounded" (or, more pedantically, a "manifold without boundary") and "non-compact" ("compact" is the precise term in topology for what we would think of as a "finite" region in ordinary language).
One dimension being timelike does not "split" anything; it's just a fact about the geometry of the spacetime.
Not really. Topology is prior to geometry. The correct statement is simply that the topology of the manifold allows it.
Why not?

So the metric was infinitesimal but the topology was "without boundary". Or is that still wrong?

Why not smooth? I was thinking that something different had to define that boundary exactly because it can't be arbitrarily close to the origin like any other line. But I can imagine there are mathematical/geometric proofs around this question, that show it can be different in that way wouldn't necessarily mean "not smooth", ways that rationalize it's location with respect to the origin some other way. In other words I can imagine that someone has proved that a topology can allow for that kind of connection (flying around flat-land one way on a plane). In fact I can imagine that someone could think up topological spaces of all kinds and prove formal characteristics for lots of interesting reasons. But then my question becomes (since there is always another question) is such a topology possible in the real geometry of spacetime?

Whether or not a geometric object like a triangle is "split" into two + one sides, does seem goofy, unless it's a right triangle. I would have used a tetrahedron as an example but I don't know what a "right tetrahedron" is called, or even if there is one - the point only being that there are geometric objects with internal asymmetry. Is spacetime not one of those? To be clear though, I don't have a need for it to be "split" I thought your post #46 sounded good. In fact I think I've been proposing all along that it's most interesting viewed as a single strangely flexible, but also constrained "geometric object".

I Learned a lot here Peter, by the way.

 

[PeterDonis]

So the metric was infinitesimal but the topology was "without boundary". Or is that still wrong?

The "metric is infinitesimal" part is. The metric doesn't have a size. Particular regions of spacetime have a size; a small enough region around a particular point could be considered "infinitesimal". When in doubt, try thinking of a simpler example like a Euclidean plane or the surface of a 2-sphere. You wouldn't describe the geometry or the metric of such surfaces as "infinitesimal"; you would use that word to describe the size of a very small piece of the surface.

I was thinking that something different had to define that boundary exactly because it can't be arbitrarily close to the origin like any other line.

The "boundary" of the square in the 2-dimensional "flat torus" analogy I gave (the Asteroids video game screen) is just an arbitrary boundary, like the prime meridian or the international date line on the surface of the Earth. It has no physical significance; it's just a convenience for human use in description. The origin is likewise an arbitrary choice.

What I think you may be trying to get at here is that those two arbitrary choices are not independent, just as the choices of the prime meridian and the international date line on the Earth (or more precisely the 180 degree meridian) are not independent. Once we designate some particular meridian on the Earth as "0 degrees", which meridian is "180 degrees" is automatically chosen (it's the one opposite the "0 degrees" meridian). Similarly, once we choose a particular point on the flat 2-torus as the "origin", which particular square is the "boundary of the video game screen" is automatically chosen: it's the set of points that are at the maximum possible distance from the origin.

there are geometric objects with internal asymmetry. Is spacetime not one of those?

No. The asymmetry of a right triangle, as opposed to an equilateral triangle, has to do with the group of transformations that leave the triangle invariant being smaller for the right triangle than for the equilateral triangle. The difference between timelike and spacelike in spacetime is a different kind of difference, so to speak; you can have a spacetime with the maximal possible set of transformations leaving its metric invariant (for example, flat Minkowski spacetime) that still has timelike, spacelike, and null vectors in it.