Flatlander analogy of spherical surface as observable universe

[Catastrophe]

TL;DR

Analogy of 'flatlander' inhabiting spherical surface suggests that 'universe' implies 'observable universe' and that an observer must be considered. Thus observable universe is a relative term.

I have addressed the analogy of a flatlander inhabiting the curved surface of a sphere. At first, I considered the 'radius' as representing time, but, latterly, I prefer the surface representing the entire (2 dimensions + time, i.e., flatlander observable universe).

 

[Nugatory]

TL;DR Summary: Analogy of 'flatlander' inhabiting spherical surface suggests that 'universe' implies 'observable universe' and that an observer must be considered. Thus observable universe is a relative term.

I have addressed the analogy of a flatlander inhabiting the curved surface of a sphere. At first, I considered the 'radius' as representing time, but, latterly, I prefer the surface representing the entire (2 dimensions + time, i.e., flatlander observable universe).

You are basically right about what is meant by "observable universe", and also that the word "observable" is often left out.
But I am not seeing how the analogy with a flatlander on the surface of a two-sphere is helping us visualize anything in modern cosmology. You might want to take a moment to consider how a flatlander could conclude that their two-dimensional universe is the surface of a sphere:
- Locally, if measured with sufficient accuracy the interior angles of a triangle will not add up to quite exactly 180 degrees; initially parallel straight lines will eventually intersect; the pythagorean theorem won't work; and the like.
- Globally, if the flatlander stays in the same place long enough they will see the back of their head somewhere out in the far distance.

None of this behavior corresponds well with the actual structure of the universe (as best we have been able to measure) so I would be very cautious about any intuition suggested by your analogy. To be useful, I think you would have to be more clear about the ways in which the analogy does match the real thing and which conclusions drawn from it can be trusted.

 

[Catastrophe]

You are basically right about what is meant by "observable universe", and also that the word "observable" is often left out.
But I am not seeing how the analogy with a flatlander on the surface of a two-sphere is helping us visualize anything in modern cosmology. You might want to take a moment to consider how a flatlander could conclude that their two-dimensional universe is the surface of a sphere:
- Locally, if measured with sufficient accuracy the interior angles of a triangle will not add up to quite exactly 180 degrees; initially parallel straight lines will eventually intersect; the pythagorean theorem won't work; and the like.
- Globally, if the flatlander stays in the same place long enough they will see the back of their head somewhere out in the far distance.

None of this behavior corresponds well with the actual structure of the universe (as best we have been able to measure) so I would be very cautious about any intuition suggested by your analogy. To be useful, I think you would have to be more clear about the ways in which the analogy does match the real thing and which conclusions drawn from it can be trusted.

QUOTE
But I am not seeing how the analogy with a flatlander on the surface of a two-sphere is helping us visualize anything in modern cosmology. You might want to take a moment to consider how a flatlander could conclude that their two-dimensional universe is the surface of a sphere
QUOTE

To me, the analogy suggests that an additional dimension, beyond that perceivable by the (ND) is required. That is (ND+1). That suggests that we may require an additional dimension (which we cannot access) to understand "expansion into what?".
I never suggested that a flatlander could conclude any such thing. In fact, such a conclusion would be impossible (on my definition).

 

[weirdoguy]

to understand "expansion into what?".

But we already understand the "expansion into what?" issue, i.e. there is no issue. Expansion by the very definition means that distance between two bodies that are not gravitationally bounded (e.g. two galaxy clusters) grows with (cosmic) time. There is no space (hehe) for "into what?" issues in the math. It's a non-existing problem.

I do understand that laypeople have issues, because they do not study the math of expansion, but some pop-sci sources that try to explain it by other means.

 

[Ibix]

To me, the analogy suggests that an additional dimension, beyond that perceivable by the (ND) is required.

It isn't. Curved spacetime is often visualised using a 2d sheet embedded in a 3d space, true, but the higher dimensional embedding space is entirely for the purposes of illustration. There's no need in the maths for the it (everything is defined with respect to measurements made in 4d spacetime) and no observational or experimental evidence for one.

In short, you can propose additional dimensions if you like, but unless you can find a measurable consequence of them it's just proposing undetectable unicorns.

 

[PeterDonis]

To me, the analogy suggests that an additional dimension, beyond that perceivable by the (ND) is required.

To me, this means you totally missed the point of the analogy. The analogy is all about investigating the properties of a surface without using any additional dimensions. In the case of something like the surface of the Earth, it's easy to be misled about this because we know there is an extra dimension present. But in the case of the universe, we have no reason to believe that's the case, and the Flatlander analogy lets us analyze the properties of the universe anyway.

 

[Catastrophe]

Not to worry. I am happy with my position on this. Thanks for your comments.

Cat :) :) :)