On Thu, Jan 30, 2014 at 7:53 PM, Edgar L. Owen <[email protected]> wrote:
> Jesse, > > Your first paragraph is correct. My theory, or at least this part of the > theory, makes the prediction that the universe is a 4-dimensional > hypersphere with p-time its radial dimension, i.e. that Omega is very > slightly >1. See my previous post of today in response to Ghibssa for > considerably more detail and why p-time implies this. > > Re your second paragraph, if Omega is actually the curvature of space, it > should be possible to calculate the RADIUS of that positively curved space > for every >1 value of Omega should it not? If so, then what is the formula? > > Make the assumption that the geometry of the universe is 4-dimensional > hypersphere with the time from the present back to the big bang as it's > radial dimension. Is there any way to use Omega to calculate what the > radius of that time dimension would be making that assumption? > > Just make the assumption even if you don't agree with it and tell me if > that calculation is possible please.... > > As I said, in an FLRW metric it's possible to slice the 4D spacetime into as series of curved 3D hypersurfaces in such a way that the density of the "perfect fluid" filling space is totally homogenous in each slice--what are known as "surfaces of homogeneity". For positive Omega, each of these curved hypersurfaces is equivalent to the curved 3D surface of a 4D hypersphere in a 4D Euclidean space. There's a parameter R(t) that appears in some of the FLRW equations, and in the case of a closed FLRW universe, I believe it's equivalent to the radius of the hyperspherical universe at any given time, if you imagine "embedding" the curved 3D space in a Euclidean 4D space, like the curved 2D surface of a sphere sitting in ordinary 3D space (though such an embedding space is not actually necessary to describe a curved surface mathematically, you can describe curvature purely in terms of the lengths of paths within the surface itself, see https://en.wikipedia.org/wiki/Differential_geometry#Intrinsic_versus_extrinsic). You can indeed write an equation that relates R(t) to Omega(t), see the last two equations that appear prior to the section "Evolution of Energy Density" on p. 3 of the paper at http://www.astronomy.ohio-state.edu/~dhw/A5682/notes4.pdf . The equation involves a(t) and R0, and they mention earlier that a(t) is the scale factor, so that the radius at any time t is given by a(t)*R0 where R0 is the radius when a(t)=1. The equation also involves the energy density which is another function of time, but the "Evolution of Energy Density" section discusses how energy density can be defined in terms of a(t) so it's not really an independent parameter. But if your idea is to take the series of hyperspheres representing space at different times and nest them like an onion, treating time as the radius, there's a problem with this: the radius would *not* be directly proportional to the cosmological time t, where t is defined in terms of the proper time (clock time) of an observer who has remained at rest with respect to the local cosmic fluid around her since the Big Bang. In a closed FLRW universe the radius expands more quickly at earlier times than later times, until eventually the universe reaches a maximum radius and then it starts to contract again. Also, you didn't really address my question about deviations from the perfectly homogenous "perfect fluid" assumed in the FLRW model. In terms of slicing the 4D spacetime into a series of 3D slices, I think this would mean that instead of perfect hyperspheres you'd at best get approximate hyperspheres with "dimples" of various sorts, and I can't think of any straightforward rule for deciding *how* to slice the 4D spacetime into 3D slices (how to "foliate" the spacetime) akin to the simple choice of picking slices where the fluid has a perfectly uniform density in the FLRW model. So if you are presented with multiple choices of how to foliate the spacetime, each of which has the property that each slice is approximately a hypersphere at large scales, but with different foliations differing in the fine-grained details of which pairs of events in the neighborhood of gravity wells are assigned to the same slice, how would you decide which slicing represented "true" simultaneity, if any? Or are you just throwing out general relativity's claim about concentrations of matter causing local curvature in spacetime, and saying the curvature of space will be a perfect hypersphere regardless of how non-uniform the matter distribution is? Jesse > > On Thursday, January 30, 2014 6:07:59 PM UTC-5, jessem wrote: >> >> Edgar, if Omega=1 the universe wouldn't have the geometry of a >> hypersphere, 3D space would be "flat"--it would be more like a >> "hyperplane". Only if Omega is greater than 1 would it have the positive >> curvature of a hypersphere (and if Omega is less than 1 space would have a >> hyperbolic geometry with negative curvature, whose 2D analogue looks >> something like a saddle--see the 3 basic types of geometry shown at >> http://www.astro.ucla.edu/~wright/cosmo_03.htm ) >> >> Also, the Friedmann-Lemaître-Robertson-Walker metric which Omega appears >> in makes an important simplifying assumption: instead of a bunch of >> localized stars and galaxies, it treats all of space as being filled with a >> "perfect fluid", and it assumes there is a way to define simultaneity in a >> way that results in a bunch of 3D slices where the density of the fluid is >> perfectly uniform in each slice (though it decreases from earlier slices to >> later slices as the universe expands). So even if you decide to define your >> "present" in terms of such a slicing, it's obvious that in the real >> universe there are variations in density of matter, so this doesn't give us >> a guide as to what the correct definition of simultaneity would be in the >> neighborhood of a given localized clump of matter. In particular, there are >> many different coordinate systems with different definitions of >> simultaneity that are used by physicists to describe the neighborhood of a >> black hole--Schwarzschild coordinates, Eddington-Finkelstein coordinates, >> and Kruskal-Szekeres coordinates being some of the most commonly-used >> ones--so what experiment would you propose for deciding which definition of >> simultaneity is the "correct" one? >> >> Jesse >> >> >> On Thu, Jan 30, 2014 at 5:51 PM, Edgar L. Owen <[email protected]> wrote: >> >>> Liz, >>> >>> Good question. Give me the formula to get the radius of a 4-dimensional >>> hypersphere from the curvature and I'll tell you. I asked for this already >>> and Brent gave me a formula that seems to make some extraneous assumptions. >>> The problem is that Omega doesn't simply seem to be the curvature in the >>> ordinary sense of hyperspherical geometry so some sort of initial >>> conversion of Omega needs to be made first and I don't know what that must >>> be... >>> >>> Edgar >>> >>> >>> >>> On Thursday, January 30, 2014 3:37:20 PM UTC-5, Liz R wrote: >>>> >>>> On 31 January 2014 04:03, Edgar L. Owen <[email protected]> wrote: >>>> >>>>> Richard, >>>>> >>>>> I've already answered this same questions on multiple occasions. >>>>> >>>> >>>> :-) >>>> >>>>> >>>>> There isn't any direct mathematical relationship so far as I can see >>>>> though we should be able to compute p-time from Omega, the curvature of >>>>> the >>>>> universe. >>>>> >>>> >>>> Omega = 1 (http://en.wikipedia.org/wiki/Curvature_of_the_universe) to >>>> a very good approximation, according to the latest measurements - so what's >>>> p-time? >>>> >>>> -- >>> You received this message because you are subscribed to the Google >>> Groups "Everything List" group. >>> To unsubscribe from this group and stop receiving emails from it, send >>> an email to [email protected]. >>> To post to this group, send email to [email protected]. >>> Visit this group at http://groups.google.com/group/everything-list. >>> For more options, visit https://groups.google.com/groups/opt_out. >>> >> >> -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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