Would an infinite universe has a finite diameter?

[newjerseyrunner]

I know that space is expanding, so the further away you go from my location, the faster space is expanding, asymptomatically approaching the speed of light. I also know that as relative velocities approach the speed of light, the length of space contracts. From this I come up with a limit for the diameter of the universe, relative only to the rate of expansion and the universe's age. Is this a correct way to look at the universe on the grandest scale or am I missing something important?

 

[Orodruin]

asymptomatically approaching the speed of light. I also know that as relative velocities approach the speed of light

This is nonsense, the expansion of the Universe has a rate (measured in units of 1/time) and not a speed (measured in units of length/time). You can relate the rate to how fast distances are growing, but there is no inherent movement associated with this expansion.

 

[newjerseyrunner]

Yes, I know that expansion is a rate, but doesn't that translate to motion at a distance? If I'm standing on a disc that's uniformly doubling in size every second, at 1 unit from me, everything is moving relative to me at a rate of 1 unit per second. At 2 units from me, everything is moving 2 units per second. At a calculable distance away, the edge is moving away from me at nearly the speed of light? Why wouldn't it contract?

 

[phinds]

Yes, I know that expansion is a rate, but doesn't that translate to motion at a distance? If I'm standing on a disc that's uniformly doubling in size every second, at 1 unit from me, everything is moving relative to me at a rate of 1 unit per second. At 2 units from me, everything is moving 2 units per second. At a calculable distance away, the edge is moving away from me at nearly the speed of light? Why wouldn't it contract?

Even if you just limit yourself to the observable universe, recession rates are already WAY about the speed of light. About 3c actually.

In answer to your original question, that topology does not make sense. Any infinite universe does not wrap around. A universe that wraps around is finite but unbounded. (Well, it COULD be bounded, but no one I know of believes we live in a finite and bounded universe.

 

[newjerseyrunner]

I'n not following, I assumed an infinite universe. I wrote out my math and hopefully you can explain what I did wrong. I'm pretty sure that the math is correct, I'm sure it's my understanding of relativity.

I imagine a universe where I can describe things around my as a sphere with radius r. No matter how far away I go, I can always go further. The difference between distance 1 and distance 2 is the delta. Here is my math: step by step.

1) The diagram shows a circle with radius r and a second circle with radius r₀ + delta r₀. The velocity of objects moving away from us at that location is defined by r₀ times some scalar factor (E), it's exact value makes no difference as long as it's positive.

2) r₀ is the REST distance to the edge

3) Δr (relativistic) is Δr₀ times γ.

4) Replace the v variable of γ with the formula for finding v₀ from r₀.

5) Take the limit of the function as r approaches infinite.

6) I got a Δr of 0, meaning at the sphere defined with radius of infinity has a limit to it's size.

Wouldn't this imply that two galaxies really really far away from us (billions of light years) but one is millions of light years further than the other, from our perspective are essentially flatten paper thin and basically on top of each other?

 

[pervect]

Good news / bad news.

Good news - there's a paper that talks specifically about some of the issues of the "size" of the expanding universem, and how it's compatible with SR. Specifically, "Expanding Confusion", http://arxiv.org/abs/astro-ph/0310808. Among other things, the paper introduced four different concepts that could be related to the "size" of the universe, amongst them the event horizon, the particle horizon, the Hubble radius, and "the observable universe". The paper comes complete with several graphs of these concepts for the current "best fit" cosmology, called Lambda-CDM.

Bad news: the four concepts described above are all different, and none of them is probably quite the OP's notion of "diameter".

Bad news - at least on of the authors of the above paper has reportedly changed their mind on the explanation of how GR and SR fit together in cosmology.

Good news: GR and SR are definitely compatible.

Good news: there's another paper that suggests a possible explanation of the difficulty, namely Baez's "The Meaning of Einstein's equation"

Bad news, the explanation is rather abstract, and not particularly novice-friendly. It basically suggests that the concept of velocity itself is ambiguous unless two particles occupy the same position in space-time. To quote the relevant section:

Before stating Einstein's equation, we need a little preparation. We assume the reader is somewhat familiar with special relativity -- otherwise general relativity will be too hard. But there are some big differences between special and general relativity, which can cause immense confusion if neglected.

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

 

[PeterDonis]

3) Δr (relativistic) is Δr0 times γ.

4) Replace the v variable of γ with the formula for finding v0 from r0.

This is wrong; you can't apply special relativistic formulas to the universe as a whole. Those formulas only work in flat spacetime, and the spacetime of the universe as a whole is not flat. The "recession velocities" that cosmologists talk about are not special relativistic "relative velocities", and their kinematics don't work the way SR velocity kinematics work.

 

[newjerseyrunner]

Thanks, I'm reading the paper now and I understand it well enough to understand why my calculations were wrong.