Chronos got me Thinking -- Finite or Infinite Universe

[Whack-a-Mole]

Are you sure Dr. Matrix? I’m no cosmologist/mathematician but I found the following (emphasis added). The part I highlighted below seems to suggest this particle/antiparticle creation can happen anywhere. The connection to Black Holes is there to show a potentially interesting outcome if this happens right at the edge of an event horizon (the black hole glows…if only very dimly).

http://math.ucr.edu/home/baez/physics/hawking.html

How does this work? Well, you’ll find Hawking radiation explained this way in a lot of “pop-science” treatments:

Virtual particle pairs are constantly being created near the horizon of the black hole, as they are everywhere. Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other. But near the horizon of a black hole, it’s possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation.

In fact this argument also does not correspond in any clear way to the actual computation. Or at least I’ve never seen how the standard computation can be transmuted into one involving virtual particles sneaking over the horizon, and in the last talk I was at on this it was emphasized that nobody has ever worked out a “local” description of Hawking radiation in terms of stuff like this happening at the horizon. I’d gladly be corrected by any experts out there… Note: I wouldn’t be surprised if this heuristic picture turned out to be accurate, but I don’t see how you get that picture from the usual computation.

The usual computation involves Bogoliubov transformations. The idea is that when you quantize (say) the electromagnetic field you take solutions of the classical equations (Maxwell’s equations) and write them as a linear combination of positive-frequency and negative-frequency parts. Roughly speaking, one gives you particles and the other gives you antiparticles. More subtly, this splitting is implicit in the very definition of the vacuum of the quantum version of the theory! In other words, if you do the splitting one way, and I do the splitting another way, our notion of which state is the vacuum may disagree!

[ZenBeam]

It was some weird shape that “tiled” negatively curved space.

If this is true, then the finiteness or infinitness of the volume of the universe isn’t tied to whether or not the universe eventually collapses.

I think what connects spatial curvature to the finiteness of the universe is the assumption of homogeneity. This is the assumption that on the largest distance scales, the universe looks the same no matter where you are. For a two-D analogue, a sphere is homgeneous, but the surface of an egg is not since the curvature varies.

If you find the Scientific Americanissue, pay close attention to whether they say if the curvature is constant. It may be that homogeneous negatively curved space must be infinite. Likewise, it may only be homogeneous positively curved space which must be finite. In 2-D, a paraboloid of revolution has positive curvature everywhere, but is infinite.

[Whack-a-Mole]

Source-- http://www.superstringtheory.com/blackh3.html

…quantum mechanics brings with it quantum uncertainty, and quantum vacuum fluctuations where particle-antiparticle pairs are always being created, then destroying one another, virtually, in the vacuum.

[John_Kentzel-Griffin]

Yeah, but the particles usually distroy each other before they can be detected. Also they are distroyed as often as created, so the net result is no new mass-energy is created.

[Chronos]

Jeff_42: OK, I thought you were talking about something else. Virtual particles can spring spontaneously into existence, but they go into debt in the process. Under ordinary conditions, there’s nothing else to pay off that debt, so they promptly go bankrupt, and disappear. No energy is actually produced in this process. Hoyle’s theory was that real particles could spontaneously appear, and last indefinitely. He used this to explain how the Universe could retain a constant density, and still be expanding. Nobody else thinks that the density is constant, but his pet theory (the Steady-State model) required it.

DrMatrix: Yeah, that sounds like the article in question. All of the examples in the article are 2-d, for simplicity, but there are indeed tiles which can tesselate a 3-d space of uniform negative curvature-- You’ll forgive me for not trying to describe them. The reason that you’ll find folks saying that constant constant negative curvature implies infinite space, is that it’s a fairly new idea that space might not be simply connected-- It hadn’t even occured to folks before.