Well, since the streets are safe to walk again…
And since warmgun hasn’t posted a reply…
I’ll venture a guess: PROOFof the fact than you can build a three-dimensional model representing our current universe without using 4 dimensions. That was after all the intent of the OP.
Regardless, this has been an exciting and enlightning trip.
I would like to here Jeff_42’s take on warmgun’s last post and some final thoughts from Chronos.
Seriously, thank you all for your insites.
Sorry, I meant last thoughts from Phobos.
…“hear from Jeff_42”
Don’t I feel like the dumb-ass?!
Comtemplating an accurate model of curved 3D space is where I’ll need to hit the books for more insight (beginning with Einstein).
When I first started contemplating cosmology & spacial topography, I immediately was drawn to the idea of a 4th, higher dimension of space in order to explain the curvature of 3D space. Seemed like a good idea…from the balloon analogy, we need to keep in mind that there is another 3rd dimension that is unseen to the 2D residents of the balloon’s surface…so perhaps the same is for us.
But as I studied more, I continue to get the impression that it is not an unseen dimension of space, but the known dimension of time that curves space. Step outside the universe for a moment (arg! foul! foul!) and picture one of those light cone/world line diagrams. As time progresses forward, the diagram expands and changes. No time, no change in space. No change in space, no gradient to invoke gravity. (side thought…or, with Quantum Mechanical eyes, no movement of gravitons.)
Anyway, that’s my impression of a very complex subject. I’m still learning about Relativity so this is by no means my final conclusion on the matter.
I will keep an eye out for relevant web links & post them here.
Phobos:
I’m no Einstein, but I’m pretty sure that it’s not ‘time’ that curves ‘space’, but that ‘space-time’ is curved. We know that the cause of curved space-time is mass, but when it comes to explaining the mechanism, even Einstein was silent. M-theory does offer a possible answer…
Yeah, but I’m wondering if saying “space is curved” is meaningless…you need the time factor (“spacetime is curved”).
Yeah, I need to read Greene’s book one of these days.
It’s finally dawned on me… What we’re arguing over is a problem of terminology. In everyday language, what we mean when we say “curvature” is not the same as what we mean in Riemann geometry. The common sense of the word “curvature” requires higher dimensions to curve into; the Riemannian sense does not. Einstein’s heated table, provided that it affects all measuring devices in the same manner,is curved in a Riemannian sense, but not in the common sense. On the other hand, the surface of a cylinder is commonly considered curved, but not by Riemann. There is some overlap: What Riemann geometry calls “spherical curvature” in n-space can, indeed, correspond to the n-dimensional surface of an (n+1)-sphere embedded in a Euclidian (n+1)-space, which would also be considered curved in the common definition. However, it does not need to. All “curvature” means, in a Riemann sense, is a relationship between the distances between points in a space. Since only n coordinates are needed to describe each of those points, the space is n-dimensional. I cannot create a model of a curved 3-space in Euclidian 3-space, but I can create such a model in curved 3-space. To reiterate another point, a homogeneous, isotropic negatively curved n-space cannot be considered to be the surface of some (n+1)-dimensional figure embedded in a Euclidian (n+1)-dimensional space. To all of the folks challenging me to produce a Euclidian 3-dimensional model of curved space, I challenge you to produce one using four dimensions. Except in special cases, it can’t be done.
day-yumn…
Point well taken, Chronos. I believe you are right about the terminology issue. I see you are on to something that I overlooked. Still it does eliminate the possibility.
And if you meant me when refering to “all the folks”, I certainly didn’t mean to single you out, it just that your reputation precedes you. And justly so.
But I still feel the fact it can’t (hasn’t?) be done is the hoofprints-in-the-sand proof of a fourth dimension.
No math, just a hunch.
But this weekend I’m off to the library!
Like ‘Ahnald’ - I’ll be back.
Grrrr…
What I meant of course was …“still it does not eliminate the possibility”.
Sorry.