Open topology and conservation laws

[RCP/CRT/RRT]

If it is assumed, hypothetically, that the Universe has an open topology, ignoring prominent evidence that it's flat (ya-ya), can it still be possible to be an isolated system? There by retaining conservation of energy laws (TD2) of a closed system?

 

What if you have an infinite potential of space (unbound) but finite metric of spacetime?

[RCP/CRT/RRT]

I wonder why I have not received any response to this?

Did I pose a question too obscure, too obvious, or just too confusing to offer dialog?

[modest]

If it is assumed, hypothetically, that the Universe has an open topology, ignoring prominent evidence that it's flat (ya-ya), can it still be possible to be an isolated system? There by retaining conservation of energy laws (TD2) of a closed system?

 

That's an interesting question, and I think difficult to answer.

 

I don't see how the first or second law of thermodynamics could reasonably be applied to infinite and unbound space except on some finite area. The energy content would be infinite so how could a change in internal energy be considered for an infinite system? I don't know. Likewise with the second law, the amount of information that an infinite system could hold is infinite so the entropy would be infinite... would entropy increase from infinite? It doesn't seem meaningful to consider things in those terms.

 

For any finite region of infinite space I'd say conservation certainly applies. I also recall Hawking did work showing that his black hole entropy equation could be applied to cosmological horizons...

 

http://cosmos.asu.edu/publications/papers/HowFarCantheGeneralizedSecondLaw%2078.pdf

 

That might be of interest.

 

~modest

[Turtle]

... The energy content would be infinite so how could a change in internal energy be considered for an infinite system? I don't know. ...

 

~modest

 

i don't know even more than you but i immediately thought of a koch snowflake, and by extension all fractals, when i read your comment. so, maybe flat finite areas of infinite bounds could give a way to consider this. :shrug: then again, maybe that would be bass-ackwards? :(

 

...The Koch curve is continuous everywhere but differentiable nowhere.

 

The area enclosed by the Koch snowflake is , where s is the length of one side of the original triangle, and so an infinite perimeter encloses a finite area.[1] ...

Koch snowflake - Wikipedia, the free encyclopedia

 

i have even less idea if this piece applies to what you guys are after, but it was interesting to read and takes a biological view rather than cosmological to conservation of energy in natural systems. so conservation...plant a tree seed today. :doh: :hyper:

Boundary Value Problems for Infinite Metric Graphs: A Study Motivated by Biological Networks

 

dialog offered. :cup:

[lemit]

I wonder why I have not received any response to this?

Did I pose a question too obscure, too obvious, or just too confusing to offer dialog?

 

Duh, part A and part C?

 

Do I get a prize or anything?

 

--lemit

[Qfwfq]

It doesn't seem meaningful to consider things in those terms.

Density. Continuity equations. For a finite region, the usual Stokes and Gauss theorems, a relation between the internal integral and flux through the frontier.

 

Flat would of course be the Robertson-Walker metric in the k = 0 case, the boundary between elliptic and hyperbolic geometry universes.

[modest]

Density. Continuity equations. For a finite region, the usual Stokes and Gauss theorems, a relation between the internal integral and flux through the frontier.

 

Ok... I'm lost with that. That would be like the entropy of a black hole (or an observable universe) being proportional to the surface area of the hole's horizon (or the cosmic horizon)?

 

Flat would of course be the Robertson-Walker metric in the k = 0 case, the boundary between elliptic and hyperbolic geometry universes.

 

Yes, and FLRW has no change in entropy being adiabatic expansion, but that would not be a good approximation as it ignores gravitational clustering and other perturbations, yes?

 

~modest

[Qfwfq]

Ok... I'm lost with that.

The conservation of a quantity such as energy can be expressed in differential form (continuity equation): divergence of flux gives the time derivative of density. This can be cast into integral form for a given region of finite volume: ingoing flux through enclosing surface gives time derivative of total inside.

 

That would be like the entropy of a black hole (or an observable universe) being proportional to the surface area of the hole's horizon (or the cosmic horizon)?

Actually that's a quite different thing.

 

As for the continuity equation, entropy not being a conserved quantity but one that can increase in total, the relation becomes a disequality. The time derivative must be at least the ingoing flux.

 

Yes, and FLRW has no change in entropy being adiabatic expansion, but that would not be a good approximation as it ignores gravitational clustering and other perturbations, yes?

These are ignored in all cosmological models which assume homogeneity at the cosmic scale. Galaxies are specks of dust, clusters and superclusters are grains of sand or tiny pebbles. This large scale homogeneity is by no means confirmed but it makes the model simpler, all too simple perhaps.