Genady

By definition, it means the following: for every real number d>0 there exists such natural number N that |(sum from 1 to m of 1/2^n) - 1| < d for any m>N.

Could you please point to the page in the book for this? (I mean, the page number)

Following this definition (Manifold - Wikipedia): @Markus Hanke: Since homeomorphism is a part of the definition, it does not require anything more to be in place for that to work. Right? If the boundary of n-dim region in Euclidean space is (n-1)-dim region, the homeomorphism caries this property into the manifold. @studiot: Dimensions are defined, through the homeomorphism, the same way as dimensions in Euclidian space, i.e., the mapping of neighboring points has to be continuous. Here is the procedure described by MTW:

These questions cannot be answered with the currently known laws of physics. Even the phrase 'before the big bang occurred' does not have clear meaning, because time is defined in the universe which we observe, and which exists of course only after the big bang occurred. There are many theoretical suggestions to extend the known laws into these unknown territories, but for the lack of evidence, no one is better than others.

Yes, a simpler procedure is described in MTW on page 10. While they just describe it, clearly and 'obviously', I posted this question hoping that we can arrive to it or other(s) and clarify on the way what the minimal requirements and assumptions are. The starting point is that the 'space' is manifold. Since each manifold has a definite dimensionality, the question is well defined in this case. I am glad to have you guys participating.

Much obliged

It is not valid because GR does not allow it. Homogeneous isotropic space cannot be stable. That's why the prediction of expanding space happened a few years before Hubble's observations.

It could be an intrinsic property which adds to the expansion, but the large-scale expansion would happen without it anyway. The large-scale expansion happens without cosmological constant either.

It's a Mathematics forum

Right. Let's see. If we can mark points with a marker, and we have a piece of a line (a ruler with no marks), will it be enough?

I don't understand why it is needed. In GR, a universe homogeneously filled with inert dust, expands. My question is, in a universe with local inhomogeneities of such dust and thus not necessarily expanding locally, is the large scales expansion an emergent property?

Imagine that you are given a smooth space of unknown geometry. What kind of constructions would you use to figure out the number of dimensions of this space?

I don't know how to apply Gauss in this case. When gravity is described by a non-flat spacetime metric, what is the 'flux'?

I don't see why it would. (We cannot use Newton's law of gravity here. Expansion appears only in the Einstein field equation.)

Somebody just went to an old thread and marked -1 four of my posts in a row: starting with this one: I am very curious, what makes these posts deserving negs. And why extra image appears in my post when viewed in the activity lists? Here, the "IMG Lifestyle" image:

I understand 'emergence' in the same way. I think of this emergence as a result of averaging uncountable local changes in curvature, expansions and contractions caused by dynamics of local mass-energy inhomogeneities. On the large scales where these inhomogeneities become 'invisible', the net effect of all these local effects becomes uniform expansion of space.

These need to be considered separately. Vacuum energy is not a model of expansion. GR alone models expansion. Vacuum energy models acceleration of expansion. And in fact, effect of vacuum energy on the cosmological scales would be acceleration of expansion while on lab scales it is something else, an extra repulsion unrelated to any expansion.

The answers are here: Simple harmonic motion - Wikipedia

On page 4 of Gravitation by MTW I read, Here seems to be a mistake. Although motion, or more generally, causality is local, there exists other physics, which is non-local. This 'jump' from motion to physics seems wrong. Was such a 'jump' Einstein's mistake?

I have trouble doing this. Dimensions (mathsisfun.com)

It had nothing to do with faith.

I think that the biological nature of our existence requires us to be alive. The other three natures, i.e., the physical, the chemical, and the social, do not.

Would it be correct to say that cosmological expansion is an emergent property? I think so because it appears only on scales of hundreds of megaparsecs where the space becomes homogeneous and isotropic on average.

I've it added to my favorites!