01 Sep 16
Originally posted by DeepThoughtThanks for the attempt at explaining it. I admit not having understood it all.
The Friedmann Robinson Walker (FRW) metric is obtained on very general principles. By assuming a spatially uniform matter density, meaning the matter sector is homogenous and isotropic you can show that there are three possible space-times - one where space like slices are 3-spheres, one where they are flat and one where they are hyperbolic. All these ...[text shortened]... ically symmetric, which means the metric has a particular form, the density doesn't affect that.
06 Sep 16
Originally posted by DeepThoughthow can a one dimensional anything have a curve? If it had a curve it would by definition be sticking itself into another dimension.
It's not obvious to me that a two dimensional flat Euclidean space (forget time for now) is capable to holding all possible one dimensional curves. To see this consider a curve on a plane embedded in the three dimensional space. Since this is Euclidean space we can choose axes so that the plane is the xy plane, in other words the set of points with z = ...[text shortened]... an infinite dimensional embedding space to fit all curved spaces of a given finite dimension in.
07 Sep 16
Originally posted by sonhouseThe one dimension refers to points on the curve ie the curve does not have a width. So a parabola on a plane is one dimensional. A sphere is a two dimensional surface. Points on the surface of the earth can all be specified with just latitude and longitude - two dimensions.
how can a one dimensional anything have a curve?
07 Sep 16
Originally posted by twhiteheadTechnically speaking it is still one dimensional but a realistic image of it would require 2 or even 3 dimensions since a physical realization would have it contaminating higher dimensions I would think. For instance, if it was a helix it would contaminate 3 dimensions in order to properly display it even though the curve itself is 1 dimensional.
The one dimension refers to points on the curve ie the curve does not have a width. So a parabola on a plane is one dimensional. A sphere is a two dimensional surface. Points on the surface of the earth can all be specified with just latitude and longitude - two dimensions.
07 Sep 16
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Originally posted by sonhouseThe thing is that all such shapes may be embedded in higher dimensions. The Earth is actually in at least four dimensions, since time is a dimension too. There may be more dimensions as well.
Technically speaking it is still one dimensional but a realistic image of it would require 2 or even 3 dimensions since a physical realization would have it contaminating higher dimensions I would think. For instance, if it was a helix it would contaminate 3 dimensions in order to properly display it even though the curve itself is 1 dimensional.
A rotating helix requires four dimensions to fully describe.
08 Sep 16
Originally posted by twhiteheadYeah, that makes sense, even if the helix is a one dimensional construct, I think, right?
The thing is that all such shapes may be embedded in higher dimensions. The Earth is actually in at least four dimensions, since time is a dimension too. There may be more dimensions as well.
A rotating helix requires four dimensions to fully describe.
17 Sep 16
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Originally posted by sonhouseIn mathematics, at least, a "curve" may be called a "topological 1-manifold," which means that locally the curve is homeomorphic to intervals of the real number line. In essence that means that sufficiently small pieces of the curve may be deformed, or bent, to become line segments. Thus, curves in a plane, or in space, or in n-space, are all considered "one-dimensional." From a functional point of view, this means that only one independent variable is needed to characterize the curve. The function
how can a one dimensional anything have a curve? If it had a curve it would by definition be sticking itself into another dimension.
r(t) = (cos t, sin t, t)
is a helix in three-dimensional space, and only the single variable t is needed to trace it. It is thus considered a one-dimensional object, though it does not fit in two-dimensional space.
The surface of a sphere is considered in Mathematics Land to be two-dimensional, though it can only "fit" into n-dimensional space for n at least 3.
18 Sep 16
Originally posted by SoothfastSo r(t) = (cos t, sin t) is just a circle? Minus the z part?
In mathematics, at least, a "curve" may be called a "topological 1-manifold," which means that locally the curve is homeomorphic to intervals of the real number line. In essence that means that sufficiently small pieces of the curve may be deformed, or bent, to become line segments. Thus, curves in a plane, or in space, or in n-space, are all considered ...[text shortened]... Land to be two-dimensional, though it can only "fit" into n-dimensional space for n at least 3.
18 Sep 16
Yes indeed. We have x=cos t and y=sin t, so:
x^2+y^2 = (cos t)^2 + (sin t)^2 = 1,
which is the equation of a circle of radius 1 with center at (0,0).
Another helix, only with elliptical coils:
r(t) = (2cos t, 3sin t, t).
Here x=2cos t and y=3sin t, so x/2=cos t and y/3=sin t, and thus
x^2/4 + y^2/9 = (cos t)^2 + (sin t)^2 = 1.
This is the equation of an ellipse with semiminor axis 2 on the x-axis and semimajor axis 3 on the y-axis. Center is still at (0,0).
18 Sep 16
Originally posted by SoothfastSo if the cos t and sin t are not equal, it would be some kind of spiral? like (cos t^3+ (sin)t^2 =1?
Yes indeed. We have x=cos t and y=sin t, so:
x^2+y^2 = (cos t)^2 + (sin t)^2 = 1,
which is the equation of a circle of radius 1 with center at (0,0).
Another helix, only with elliptical coils:
r(t) = (2cos t, 3sin t, t).
Here x=2cos t and y=3sin t, so x/2=cos t and y/3=sin t, and thus
x^2/4 + y^2/9 = (cos t)^2 + (sin t)^2 = 1.
This is ...[text shortened]... th semiminor axis 2 on the x-axis and semimajor axis 3 on the y-axis. Center is still at (0,0).
18 Sep 16
Originally posted by sonhouseFor any given t ∈ ℝ cos(t) and sin(t) are guaranteed to be different. Do you mean if the arguments are different, so in general:
So if the cos t and sin t are not equal, it would be some kind of spiral? like (cos t^3+ (sin)t^2 =1?
r(t) = (cos(f(t)), sin(g(t))
where f(t) and g(t) are functions. In that case yes, one will get a curve that is not a circle. Depending on the properties of f and g it may not even be closed.
To get a spiral we need x^2 + y^2 = a^2, where a is decreasing. This would be:
r(t) = (a(t)cos(t), a(t)sin(t))
so that x^2 + y^2 = a(t)^2 cos^2(t) + a(t)^2 sin^2(t) = a(t)^2 [cos^2(t) + sin^2(t)] = a(t)^2
To get a spiral that increases in radius linearly we can use a(t) = t.
21 Sep 16
Originally posted by twhiteheadThe surface of the Earth is spherical and yet has no boundary. Taking the surface of the Earth as the "universe" would be an example of a two-dimensional spatial realm that is both bounded and edgeless.
I disagree. To talk of a spherical shape at all implies edges which simply don't exist - at least I have never seen any scientific theory that suggests such a thing is possible. Now you may mean spherical in higher dimensions, but if so, then say so.
The universe is either flat and infinite or curves back on itself, but not in the three dimensions but a higher dimension. In the three dimensions it has no edge.
The actual universe, as I understand it, may be thought of as a 3-sphere (a three-dimensional sphere). A circle is a 1-sphere, a sphere as it's understood on the street is a 2-sphere. Continue the analogy to the next level to get a 3-sphere, though I doubt you'll be able to visualize it since a 3-sphere can only be embedded in R^4 (4-space). This brings up another important point: if the universe is a 3-sphere, it must be thought of as a 3-sphere "in isolation" -- that is, not embedded in a "larger" space such as R^4. This can be done mathematically (i.e. you can model and study the 3-sphere without referencing a space "containing" it), and I suspect it is the case physically.
Sonhouse's original hypothesis tacitly embeds our universe in a larger space, which I don't think is the prevailing view in cosmology today.
21 Sep 16
Originally posted by SoothfastAt this point things get technical...
The surface of the Earth is spherical and yet has no boundary. Taking the surface of the Earth as the "universe" would be an example of a two-dimensional spatial realm that is both bounded and edgeless.
The actual universe, as I understand it, may be thought of as a 3-sphere (a three-dimensional sphere). A circle is a 1-sphere, a sphere as it's unders ...[text shortened]... s our universe in a larger space, which I don't think is the prevailing view in cosmology today.
The geometry of the universe depends on the energy density. If the energy density is greater than the critical density the universe will, as you said, be the direct product of a three sphere and the real line for time (I'll call this a 4-cylinder - I don't know if that is standard terminology). However there are two other possibilities: if the energy density is exactly the critical density then it will be infinite and be identical (up to local bumpiness) with Minkowski space. If the energy density is less than the critical density the space-like part will be hyperbolic. In both the last two cases the universe has infinite extent - at least in the simplest models. One could imagine a situation where the universe has regions which are hyperbolic and regions of extreme positive curvature (black holes etc) which allow it to remain topologically equivalent to a 4-cylinder.
The standard model of Cosmology does not include higher dimensions. It is based on what we have evidence for. However, if brane-world scenarios are true then the four dimensional universe we see would be embedded in a higher dimensional space. So there are speculative theories (not that you'd know they are speculative if you heard an advocate talking about them) in which the universe we see is a brane embedded in a higher dimensional space.
22 Sep 16
Originally posted by DeepThoughtVery interesting. But, would it be fair to say that the temporal cross-section of the universe that constitutes "now" is a 3-sphere -- or at least is homeomorphic to one, modulo local anomalies such as black hole singularities?
At this point things get technical...
The geometry of the universe depends on the energy density. If the energy density is greater than the critical density the universe will, as you said, be the direct product of a three sphere and the real line for time (I'll call this a 4-cylinder - I don't know if that is standard terminology). However there are ...[text shortened]... king about them) in which the universe we see is a brane embedded in a higher dimensional space.