I have tried several time to write an OP that, in a suitably oblique fashion, states my confusion with the following comment, but I lack the mathematical sophistication to express my uncomprehension with the desired level of abstraction, so I guess I have to make request for clarification explicit:
As far as I understand the geometry of Riemannian manifolds and/or topological manifolds without any formal education in differential geometry, any (n-1)-dimensional manifold can serve as a bondary for an n-dimensional manifold; the "spatiality" of the dimension is largely, if not completely, irrellevant. However, I also realize that I must take care to acknowledge that the previous statement is about manifolds where the metric tensor in positive-definite and that, in the case of Riemannian spacetime manifolds, the metric tensor is in fact not positive-definite, which is the very distinction between space-like, light-like, and time-like directions on a Riemannian manifold.
My dispute with the above comment is that, when I refer to a manifold of lesser dimension's being the boundary of a manifold of of greater dimension, I do so without respect to the <space/light/time>-likeness of the direction. The dispute is therefore made in incomplete knowledge of whether an (m+n)-dimensional manifold is actually a boundary for and ((m+1)+n)-dimensional manifold are (m+(n+1))-dumensional manifold and in partial acknolwedgement therein.
tl;dr: I wonder how tight the analogy I make between "singly typed" dimensional manifolds and "doubly typed" dimensional manifold and their respective boundary manifolds actually is.
Quote:
Originally Posted by Brian-M (Post 9415759) Time is only the fourth dimension when you're dealing with three spacial dimensions. But if you're discussing something with more than three spacial dimensions, then the 4th dimension is not time. Let's pretend that the point of the balloon analogies is to get the idea across that the three-dimensional universe we observe is the surface of a four-dimensional n-sphere. (The fourth dimension here is not referring to time.) A 1-sphere (a circle) is a 2-dimensional "sphere" with a 1-dimensional surface. A 2-sphere (a ball) is a 3-dimensional sphere with a 2-dimensional surface. A 3-sphere is a 4-dimensional sphere with a 3-dimensional surface. What the argument is saying is that our 3-dimensional universe exists as the surface of an expanding 4-dimensional 3-sphere. (Remember, the 4th dimension here is not time.) The 3-sphere has a boundary (our universe exists at the boundary), but our universe doesn't have a boundary (because it loops around, like the surface of a ball). ETA: I'm not arguing that our universe actually is the 3D surface of a 4D sphere, this is just an example. |
As far as I understand the geometry of Riemannian manifolds and/or topological manifolds without any formal education in differential geometry, any (n-1)-dimensional manifold can serve as a bondary for an n-dimensional manifold; the "spatiality" of the dimension is largely, if not completely, irrellevant. However, I also realize that I must take care to acknowledge that the previous statement is about manifolds where the metric tensor in positive-definite and that, in the case of Riemannian spacetime manifolds, the metric tensor is in fact not positive-definite, which is the very distinction between space-like, light-like, and time-like directions on a Riemannian manifold.
My dispute with the above comment is that, when I refer to a manifold of lesser dimension's being the boundary of a manifold of of greater dimension, I do so without respect to the <space/light/time>-likeness of the direction. The dispute is therefore made in incomplete knowledge of whether an (m+n)-dimensional manifold is actually a boundary for and ((m+1)+n)-dimensional manifold are (m+(n+1))-dumensional manifold and in partial acknolwedgement therein.
tl;dr: I wonder how tight the analogy I make between "singly typed" dimensional manifolds and "doubly typed" dimensional manifold and their respective boundary manifolds actually is.
via JREF Forum http://forums.randi.org/showthread.php?t=263543&goto=newpost
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