If the universe is infinite

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You are only partially correct. If the universe expanded at the speed of light, our universe would have long ago underwent a big rip. The CC is perfectly balanced at 10^-122 for stars and galaxies to form.

However, if we consider two receding galaxies moving away from one another, according to a certain point of reference of an observer, the receding velocity would indeed 'move' apart faster then light.

My copy of Gravitation by Misner, Thorne, and Wheeler is within twenty feet of me. Please provide the chapter and section numbers where it supports your claims. I'm particular interested in seeing where they said that a torus is flat.

I'm trying to picture parallel geodesics on a torus.

It comes to mind that you may be thinking of it as a manifold where at every point, a neighborhood may be mapped to a set in Euclidean space even though the object itself isn't Euclidean.

Misner's Gravitation is outdated. See: http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html

I'm not sure what it would mean to say "universe expanded at the speed of light". It seems rather nonsensical. Sure, two points sufficiently far apart from each other may be moving apart at the speed of light or faster, but that's not the same thing.

That is exactly what I meant, two galaxies moving apart from one another, can have a superluminal receding velocity from another in a given frame of reference.

That was something like ten or twenty seconds of discussion in the Special Topics in General Relativity I took back in the 1970s.

Regarding the links earlier to the interview with Joseph Silk and to the portion about how things have changed since MTW, much of that is similar.

What is strange is the discussion about the torus. In the one link, they show the familiar wrapping of a unit square in the plane into a cylinder. This does not change the geometry of the space because the curvature is still zero. That is, one does not need to stretch or constrict the space of the unit square to form the cylinder. In the plane, the curvature is always zero in every direction and thus the Gaussian curvature is trivially zero. In the cylinder, curvature along the axis remains zero and the maximum curvature, is the curvature of the circle perpendicular to the axis. Thus, the Gaussian curvature is zero everywhere on the cylinder.

But when you then fold that around to form a torus, you cannot do so without stretching or constricting the space. The curvature is no longer zero except along the two opposite circles along the sides of the torus. Along the outer portion of the torus, the curvature will be positive everywhere and along the inner portion of the torus, the curvature will be everywhere negative.

I see how they avoid this problem in the page with "Although this surface cannot exist within our three-dimensional space, a distorted version can be built by taping together top and bottom (see 2 above) and scrunching the resulting cylinder into a ring (see 3 above)." In other words, they acknowledge that it isn't really a torus, but a flat space that nonlinearly maps like a torus.

So when I think of a torus, I think of what they call the distorted version which is not flat while you are thinking of a flat piece of plane mathematically folded, but not warped, similar to a torus. It would be interesting to be able to visualize that other than as a unit square with the edges aligned in a certain way.

Note that my background was in the 1970s and is kind of stale. I was aware that there was some evidence that the cosmological constant may not be zero as we always assumed it was in the 1970s (another twenty or so seconds of Special Topics in General Relativity), but have never studied what happens if it is nonzero other than the most obvious implications.

And that is why it is possible for a universe with critical density to remain 'zero' and also be finite.



No. I think you are now mixing the global topology of the universe with the localized Einstein field equations.



They are referring to another finite yet flat topology, this is different from the torus manifold, although both allow photon feedback loops.



Well, this is the kind of stuff I do for a living.

that is therefore the reason that perpetual results equaling 12 (or 2 sixes) is not possible. ( you are talking in both a plural sense and a singular sense but i assume you mean 2 dice)

that is unfortunately incorrect.
if you are talking about a pattern, i assume you mean a relationship between the 2 die's results (which can only be their product), and considering that each die has a 16.6 % chance of showing any number, then when that is combined in a dual rolling situation and the question is asked "what may be a likely product of the 2 resultant die's values" then the highest probability must be 7.
i can make this simple and non algebraic so all can understand.

there is only one combination that would yield a product of 2 (1+1)
there are 2 combinations that could result in a product of 3 (1+2 and 2+1)
there are 3 combinations that could result in a product of 4 (1+3 and 2+2 and 3+1)
there are 4 combinations that could result in a product of 5 (1+4 and 2+3 and 3+2 and 4+1)
there are 5 combinations that could result in a product of 6 (1+5 and 2+4 and 3+3 and 4+2 and 5+1)
there are 6 combinations that could result in a product of 7 (1+6 and 2+5 and 3+4 and 4+3 and 5+2 and 6+1)
there are 5 combinations that could result in a product of 8 (2+6 and 3+5 and 4+4 and 5+3 and 6+2)
there are 4 combinations that could result in a product of 9 (3+6 and 4+5 and 5+4 and 6+3)
there are 3 combinations that could result in a product of 10 (4+6 and 5+5 and 6+4)
there are 2 combinations that could result in a product of 11 (5+6 and 6+5)
and only one combination that could result in a product of 12 (6+6)

so you see, the product of 7 is the most likely product to occur in the rolling of 2 dies over any number of instances.
that is what i was asserting. that only combinations of likelihoods gives rise to predicatability.

He said a "pair of dice". The roll of one honest dice is random. But the products of a pair of random honest dice thrown together does have a pattern (certain sums are more likely than others.). A pair of six sided dice are more likely to come up seven (1/6) than anything else, but only 1/36 as either 2, or as 12.

err i am not sure whether you are agreeing with me or trying to correct my interpretation by restating what he said.
what you said was in accordance (as far as 7 is concerned) with what i said, but i still have no idea. i know you have a high iq (probably more than 130), but what you just posted was not exceptionally distinct.

so i say, that in the multitudes of possibilities of combinations of theoretical possibilities, only one combination is the one that resulted in manifestual reality, and if one can work out how the "conspiracy" of probabilities intersect with each other, and isolate impossibilities and reject them from accounting, then maybe one can sit at home and view the universe in a relaxed manner with no need even to move a muscle.

but when i die i will be no more and no one will be the benefactor that carries my beliefs on to eternity because the future is so big and long that anything i am will dissipate into nothingness eventually as it unfolds.

is it really necessary to try to chart infinity?
it can never be done,and your life is not long enough to try no matter how smart you are.
you've just got to die like everyone else and either "sleep" forever, or suddenly realize every answer to every question you ever had and go from there.

Agreeing with you.

A couple days ago I was gonna jump in and post that. Then I got caught up in making grids on paper to see what kind of odds you get with different kinds of dice(like if you make six on regular six sided dice equal zero instead of six- how does that affect it, and like that), and never got around to posting. Then I remembered "oh yeah, I was gonna post that reply to Eric76". I know my statement is now kinda redundant because of your reply to him, but what the heck. I posted it anyway .

yeah well stuff happens.
it doesn't matter.

b9 wrote "if you are talking about a pattern ..."

Think of the two dice as being dice a and dice b. Each roll you record what each dice did separately. You could label them as a tuple (x,y) where x was the value on dice a and y the value on dice b.

Thus (3,4) is not the same as (4,3). And two rolls, (1,6) (3,4) would be distinct from (3,4) (1,6) or from (3,6) (1,4) or from (6,1) (4,3) and so on.

The specific pattern I mentioned was all double sixes: (6,6) (6,6) (6,6) ...

It could have just as easily been (1,2) (2,3) (3,4) (4,5) (5,6) (6,1) (1,2) (2,3) ... repeating indefinitely

In other words, consider all possible sequences of a billion rolls.

(By the way, think of one dice as being colored red and the other as being colored white if that helps you keep them separate.)

With one pair of dice, each roll could have 36 outcomes. Out of a billion rolls, you would have 36^1000000000 outcomes. One of those outcomes is to have (6,6) for every one of the billion rolls. Another is (3,4) a billion times in a row which would be different from (4,3) a billion times in a row and those would both be different from (3,4) followed by (4,3) and repeated half a billion times.

I readily admit that it would have been simpler to have just rolled one dice two billion times.