I'm standing on a black hole, is light approaching red or blue?

[Mordred]

There was a long recent thread on apparent event horizons and I both went into the discussion and came out of the discussion with the belief that for all practical purposes, apparent event horizons are mathematical mumbo jumbo which I feel free to equate with the tooth fairy.

I would be interested in that thread sounds intriguing. Particularly since I have numerous Unruh/Hawking, Parker, Schach-wolf effect etc articles on various forms of particle production.

How one defines a boundary and types of vacuum takes some digging into with regards to virtual particles. QM I'm still weak on but I'm studying it along with particle physics atm.

 

[PeterDonis]

That bolded part is an exaggeration. I don't think it suggests infinite extent so much as allows for the possibility. You could just as well say that it suggests that the universe is finite but bounded.

No, you can't; at least, you can't if you are talking about the actual models used in cosmology, which is what the quoted passage was talking about. Those models require that if the universe is flat (more precisely, if it is *spatially* flat), it is infinite in extent (more precisely, in *spatial* extent). If the universe is finite but unbounded [Edit: corrected from "bounded"], it must have positive spatial curvature.

Of course all this assumes that our current cosmological models are basically correct; but AFAIK nobody has proposed an alternative model that allows the universe to be both spatially flat and spatially finite but unbounded [Edit: corrected as above]. (People have speculated about the universe having a flat geometry but a different spatial topology, like a flat 3-torus; but AFAIK these speculations have not led to any actual testable models.)

 

[phinds]

No, you can't; at least, you can't if you are talking about the actual models used in cosmology, which is what the quoted passage was talking about. Those models require that if the universe is flat (more precisely, if it is *spatially* flat), it is infinite in extent (more precisely, in *spatial* extent). If the universe is finite but bounded, it must have positive spatial curvature.

Of course all this assumes that our current cosmological models are basically correct; but AFAIK nobody has proposed an alternative model that allows the universe to be both spatially flat and spatially finite but bounded. (People have speculated about the universe having a flat geometry but a different spatial topology, like a flat 3-torus; but AFAIK these speculations have not led to any actual testable models.)

I don't understand your talking about finite but bounded. It is my understanding that there is wide agreement that if the universe IS finite then it is UNbounded. Finite and bounded would imply an edge and that just doesn't make sense.

 

[PAllen]

Help! I'm standing on a black hole and I can't get out!

 

[WannabeNewton]

The terms "finite" and "unbounded" are so ambiguous that it boggles my mind why they are used in some contexts. In the FLRW cosmological model, the simplest 3-manifold corresponding to the $k = +1$ constant sectional curvature case is the 3-sphere $S^{3}$ which is a closed manifold meaning it is compact and has empty manifold boundary. This is what "finite" and "unbounded" commonly seem to refer to in whatever context they show up with regards to the aforementioned cosmological model. However it is of course true that $S^{3}\subseteq \mathbb{R}^{4}$ is bounded in the metric sense (a subset of a metric space is bounded if it can be contained in some open ball) and is also closed in the sense that $S^{3}\subseteq \mathbb{R}^{4}$ is closed in the euclidean topology on $\mathbb{R}^{4}$ (this in fact is one way to verify that $S^{3}$ is compact since every closed and bounded subset of $\mathbb{R}^{n}$ is compact by Heine-Borel). Like I said, the terminology is definitely ambiguous from a mathematical standpoint so it may very well be that Peter was referring to how $S^{3}$ is bounded in the second sense mentioned above.

 

[Mordred]

That ambiguity is one reason that I am glad none of my cosmology textbooks (12) of them nor any recent thesis papers I've read (lost count on the number). Utilize the bound/unbound terminology anymore. They simply state finite or infinite is unknown then discuss the 3 geometries and describe their effects on light paths.

 

[WannabeNewton]

That ambiguity is one reason that I am glad none of my cosmology textbooks (12) of them nor any recent thesis papers I've read (lost count on the number). Utilize the bound/unbound terminology anymore. They simply state finite or infinite is unknown then discuss the 3 geometries and describe their effects on light paths.

12 books wow! Quite the cosmology connoisseur I see ! And yes I agree that simply stating the 3 geometries should be enough since the local geometrical properties (but not global topological properties such as your torus example) follow suit from simply prescribing these 3 geometries. One book that I think discusses the terminology and "technicalities" in a nice way is the book George Jones recently recommended to me: "Gravitation: Foundations and Frontiers" - T.Padmanabhan

 

[PeterDonis]

I don't understand your talking about finite but bounded.

Oops, I meant to say "finite but unbounded". You're quite right, finite but bounded doesn't make sense. I have fixed my previous post.

 

[PeterDonis]

it may very well be that Peter was referring to how $S^{3}$ is bounded in the second sense mentioned above.

I was referring to $S^3$ being a manifold without boundary in the topological sense; but as phinds pointed out, I should have said "unbounded", not "bounded" (I have now corrected that in my previous post). I don't know if there's a more precise term that has that specific meaning without the ambiguities you mention. One more reason to be wary of expressing things in natural language instead of in math.

 

[phinds]

Oops, I meant to say "finite but unbounded". You're quite right, finite but bounded doesn't make sense. I have fixed my previous post.

I had to laugh at myself. After chiding you about this, I happened to notice that my own post #23 used bounded where I MEANT to say UNbounded (I've gone back now and edited it)

 

[Mordred]

Thanks for the reference WannabeNewton. I'll have to add that to my collection.