Jheriko said:
the other point that people try to make is that with the balloon analogy it has volume, and the volume has a centre. comparing this to volume in the universe completely missing the abstraction the balloon analogy makes to simplify things - specifically using 2D as a substitute for 3D because people struggle with visualising and understanding 4D representations. in that sense, if the universe was an unbounded 3-sphere then we have a 'hypervolume' inside of it which also very much has a centre in that 'fictional' 4-space - note, that this 4-space is not a minkowski style space-time but a 'fictional' euclidean 4-space, in the same sense that the 3-space the balloon is embedded is not a 2,1 space-time, but euclidean 3-space - or alternatively an extension of the sphere's local coordinate system that replaces intrinsic curvature with extrinsic curvature by embedding the system in a space with higher dimensionality.
The point here is that you don't actually need that hypervolume to have a fully defined geometry of space.
That is, if we're talking in terms of the 2D analogy, you don't have to postulate the existence of the 3rd dimension to describe the geometry of the 2D surface as spherical, toroidal, etc., and without the 3rd dimension there can be no meaningful notion of a centre in the 3rd dimension, or of a volume.
An easy to relate example of how this can be true is the way some video games are made - in particular Asteroids, or similar ones in which a player's avatar can move off a computer screen and reappear from the opposite edge. This is an example of a toroidal geometry of space, and yet the screen remains flat. It is easy to calculate the hypothetical dimensions of the actual 3D torus that would exhibit the same geometry, and talk about its centre(s) of curvature, but those would remain 'virtual', or as you say 'fictional' dimensions, and would not indicate that there exists some actual 3rd dimension to the space displayed on the computer screen.
Similarly, the 3D geometry of our universe may turn out to be that of a hypersphere (but it is
not supported by observations so far!), but it doesn't necessitate the existence of a higher, fourth spatial dimension for it to be embedded in. One can talk about radius of the curvature of the universe, but it's a virtual notion, not pointing to an actual centre of a 4D hypersphere.
That's why readers introduced to the balloon analogy are always asked to focus on the 'flatland' of the surface and disregard completely the 3rd dimension. It's not necessary for the analogy to work, and can make the reader leave with an erroneous notion that from the expansion of the universe follows that there must be a fourth spatial dimension.As a side note, and I keep hammering this in but nobody ever seems to care, 'unbounded' has a precise mathematical meaning, and in the context of finiteness of spaces means the same as infinite. So surface of every sphere is always bounded (but has no boundary). There isn't such a thing as an unbounded sphere.
Seeking said:
In a 4d space-time, why is the center not the limit as time approaches 0? In this view the center does not exist in the present (not on the surface of the balloon).
zylon said:
Then the center of the universe is "under" every point in space (since the big bang), the same way that the center of the Earth is "under" every country on earth.
The fourth dimension of space visualized here may not be physically real, but it should clear up the "center of the universe" confusion. Do you agree?
Of course, this would cause us to ask what is outside the aforementioned hypersphere, which would get into the subject of things existing before they are created.
These posts propose essentially the same idea - to place the centre of the universe in the fourth spatial dimension.
If we do that, we are postulating that the fourth spatial dimension actually exists, which is unnecessary to describe the geometry and expansion of the universe. Could there exist a fourth dimension in which the universe is embedded? Sure. But when presented with two models, you always need to choose the one with the least assumptions, unless those assumptions are necessary to explain observations. The fourth dimension is unnecessary.
Additionally, and
perhaps more importantly, this looking for the centre of the universe in the centre of a hypersphere assumes a priori that hypersphere is the shape of the universe, for which there is no indication. The universe looks flat, and keeps looking flat despite ever improving measurements.
That is, coming back to the balloon analogy, it would be more appropriate to use an infinite rubber sheet as a model that would better represent the shape of the universe as it is known today, but then the analogy would not work so well for its intended purpose, which is not to present the geometry, but how expansion makes everything look like receding away from everything else - no matter the vantage point.
So, again, the analogy always specifically asks to focus on the 2D surface only, and not try to forcefully extract from it any conclusions that it was not meant to convey. Trying to give the centre of the balloon any meaning is one such conclusion.
(Blame 
)