In our observable area of the universe, we find planets, stars, galaxies, clusters of galaxies, etc. Due to the uniformity of what we see, it is reasonable to infer, that this is what we would find beyond what we observe. What is commonly called our universe is what I call a space-time continuum. According to the standard model, the diameter of this continuum is thought to be 130 billion lightyears. Beyond this is the unknown and will never be known by those in our continuum. I see no reason to dought the existence of more. Therefore I consider the universe to be, the continuum we are in, and whatever else there may be.
Dragon, How do you suppose mainstream cosmologist arrived at the 130 billion LY figure for the diameter of the "universe". Do you believe that what is observable is all there is?
I don't know what lays beyond the universe, but people of such calibre that live in such realms are nothing but Gods.
BenTheMan: I am pretty sure that both your integrals represent infinite volumes, with one larger than the other. However, I do not remember enough of any pertinent math course to be certain on this issue. Billy T: The set of all real numbers between 0 and 1 has more members that the set of all integers. Cantor had a proof which was similar to the following. The digits associated with all the real number can be associated with an array: A11, A12, A13 . . . . A21, A22, A23 . . . . A31, A32, A33 . . . . . . . . . . . Each row of the above array corresponds to a real number: A1x is the first row, A2x, the second, and so forth. Now, if the number of memeber in the two sets are the same, then each row (a real numbers between 0 & 1) can be paired with an integerb]:[/b] 1 with Row A1x, 2 with A2x, 3 with A3x, et cetera. Now suppose we create a real number with digits B1, B2, B3 . . . . . . . as follows. Choose B1 different from A11 Choose B2 different from A22 Choose B3 different from A33 . . . . . . . I vaguely remember some requirement that the choices made for digits Bx, not allow all the digits beyond some point to be the same (or perhaps they cannot all be zeros or nines). The real number represent by B1B2B3 . . . . . . . differs from every real number in the above list, proving that the one-to-one pairing cannot be done. There will always be at least one real number unpaired.
No, but Einstein died about 45 years before there was any experimental evidence to the contrary. If he was alive today, he'd have to be a fool not to believe that the universe was flat. So, if you're comfortable talking out of your ass about physics that's irrelevant in light of new experimental results, have fun.
Why don't you look at the links I've posted. But if you are confined to the classroom, and have no way of ever leaving, or ever seeing outside of the classroom, and all of your theories tell you that there's another classroom, how can you say anything else? Ok... This is confusing. I had to do a bit of research, but the idea is this... The universe is 13.7 billion years old, but due to the periods of inflation, the actual distance is 78 billion light years. Which means diameter = 156 Gly. It's like adjusting the cost of something for inflation (in the economic sense). For example, a new car cost $4000 brand new in 1950, and $20,000 today.
MAX TEGMARK ON MULTIVERSES: Q: You state in your paper "the key question is not whether parallel universes exist (Level I is the uncontroversial cosmological concordance model), but how many levels there are." Are you suggesting that the existence of level I parallel universes is not a key question? In other words, do you argue that the number of universes is larger than n=1 (the lowest number we can obtain from observation)? Indeed. Although it's far from obvious that n=oo, I think the astronomical evidence is very compelling that n > 1. The curvature of space measured by the cosmic microwave background is so small that if space is a (finite) hypersphere, then it is large enough to contain at least n=1000 other Hubble volumes. If space is finite by connecting back on itself like a donut, the cosmic microwave background measurements again require the donut to be large enough to contain n > 1 Hubble volumes. Of course you can always postulate that space ends abruptly right outside the cosmic horizon with a big warning sign said "MIND THE GAP", but you'll have a hard time providing an elegant mathematical formulation of that theory, let alone convincing other people of its virtues. http://space.mit.edu/home/tegmark/multiverse.html
BenTheMan, My figure for the diameter of the space-time continuum may be outdated. My point was that we know much about what is beyond our observable area of the universe. I've read recently that some of the galaxies we are receiving light from are actually as far away as 47 billion lightyears. Don't ask me to explain how we know (I'm not a cosmologist). I understand it has to do with a couple of different types of redshift along with accelerated expansion. While we can not tell what these galaxies look like today, they have been observed. I consider this the limit of our observation. Beyond this we have to use reason and inference to deduce what the rest of the universe "looks" like.
No, your figure was close. You said 130, I said 156. The universe is 13.7 billion light years old, but the diameter of the observable universe is 156 billion light years.
The 156 billion light-year diameter was due to a mistake made by a science writer for Space.com, an incorrect figure which has been widely quoted by other popular press outlets. Neil Carnish et all's 2004 paper calculated a minimum diameter of 78 a billion light-year diameter now, but most likely about 92-94 billion light-years in diameter for the observable universe. A pretty good explanation of how the diameter was calculated is available on a wiki page: http://en.wikipedia.org/wiki/Observable_Universe
This is from Physics Help and Math Help - Physics Forums. It gives the "best fit" for the size of the universe. Wikipedia isn't always the best source. George Smoot has sometimes taught Physics 139 at Berkeley, an advance undergrad course in Spesh-and-Gen Relativity with supplemental Cosmology. You can get his notes if you google "smoot notes geometry universe" The notes give a simple formula for the radius of curvature. It is incredibly simple but let's try it. At least this will give a first approximation. You just take the HUBBLE LENGTH which we usually say is 13.8 gly, and you divide it by a factor which is sqrt(Omega - 1). that is all. If we use the best fit Omega of 1.011, then Omega - 1 is 0.011 and the square root is 0.105 and if you divide 13.8 billion lightyears by 0.105 you get 130 BILLION LIGHT YEARS. So if we live in a threesphere, and you want to picture it contained in boyscout fourspace, and as having a radius, this is a possible guess about the radius. If I have made a mistake in reading Smoot's notes or in calculating please let me know Please Register or Log in to view the hidden image! The Ned Wright paper that I am having fun with is mentioned here: physicsforums.com/showthread.php?p=1220828#post1220828 New Wright's new paper just came out arxiv.org/abs/astro-ph/0701584 discussion section page 14: "Using all the data together gives the plot shown in Figure 5. The best fit model is slightly closed with Omega_tot = 1.011 and M = 0.315. " Wright is an WMAP princ. investigator. Here is Wright's paper Constraints on Dark Energy from Supernovae, Gamma Ray Bursts, Acoustic Oscillations, Nucleosynthesis and Large Scale Structure and the Hubble constant Edward L. Wright (UCLA) 16 pages, 8 figure "The luminosity distance vs. redshift law is now measured using supernovae and gamma ray bursts, and the angular size distance is measured at the surface of last scattering by the CMB and at z = 0.35 by baryon acoustic oscillations. In this paper this data is fit to models for the equation of state with w = -1, w = const, and w(z) = w_0+w_a(1-a). The last model is poorly constrained by the distance data, leading to unphysical solutions where the dark energy dominates at early times unless the large scale structure and acoustic scale constraints are modified to allow for early time dark energy effects. A flat LambdaCDM model is consistent with all the data." the initial announcement of WMAP3 "implications for cosmology" paper by Spergel et al already contained hint of this. warning: it doesnt prove anything. the INFINITE FLAT universe is still consistent, it just is not the best fit. the best fit is with nearly spatially flat, slight positive spatial curvature and therefore the best fit is spatially finite. flat would be Omega = 1.00 exactly, the best fit is Omega = 1.011 However as Ned Wright is careful to say: a flat model is "consistent" with the data
thanks - I see the proof, but if you give it again to someone, instead of your next paragraph, it would have been more clear to me if you said something like: "Each row of the "A matrix" corresponds to an interger, and conversely. I.e. there is no integer without a corrsponding row." Now here is how you can construct a real number that differs from all the rows in the A matrix .... I.e. there is no interger corresponding to it, the real so construced. I thought of much simpler proof that the reals are larger than the integers: For positive integers n > 2, let the 1-to-1 corresponding real be 1/n. For negative n < 2 let the 1-to-1 corresponding real be 1-1/n. For 2, 1, 0, -1, and -2 let the corresponding reals be 0.338, 0.337, 0.336, 0.335 and 0.334 respectively. Observer there is no interger corresponding to the infinite number of real between 0.339 and 0.666 May not be an especially pretty proof, but simple to understand and shows an infinte number of reals have no corresponding integer. This is more impressive than only showing one that does not have a 1-to-1 mate. To make this slightly thread related. - Volume of our universe is finite. There could be an infinite volume one so big that even an infinte number like ours could not fill it.
We will have to iron this all out. I'll ask the astronomers on monday when I go in to work. I'll report backPlease Register or Log in to view the hidden image!
Ok. I'll believe Smoot. He won the Nobel Prize. A couple of things to note... This is a first order approximation. Second, it is quite possible that the universe is flat, with omega ~ 1. A small change in the value that you use for omega means that there is a big change in the radius of curvature that you calculate. That is, the radius of curvature for omega = 1 is infinite, but the radius of curvature for omega = 1.011 is 130 billion light years.
BenTheMan, Of course I realize that the "size" depends on the model you use as well as the other variables that are not yet nailed down. That is why it's a best fit figure. Also did you notice that the 130 number was for the radius? That makes the diameter 260 billion LY. That is for the known universe (I prefer space-time continuum). For the observable area of the universe I would take the 47 billion LY figure (the furthest observed galaxies) as the radius. Making the diameter 94 billion LY.
My personal opinion is that the Universe has a finite volume, but there's no exterior to it. It's geometry is such that you cannot get out of it, it's akin to those computer games (Space Invaders comes to mind) in which you reach the edge of the screen only to appear at the other end.
Unfortunately showing an injection from integers to reals doesn't prove \(|Z| < |R|\), only \(|Z| \leq |R|\). For example, there is an injection from naturals to integers (identity function), but both sets have the same cardinality as there is a bijection: \(f: N \rightarrow Z, f(x) = \left\{\begin{array}{ccc} \frac{x}{2} & \mbox{for} & x \mbox{even}\\ \frac{1-x}{2} & \mbox{for} & x \mbox{odd}\end{array}\right.\) So what you need to show is that an injection from reals to naturals (or integers) is not possible, not that an injection from naturals to reals is.
