# Theorem between Curvature and submanifolds

## Main Question or Discussion Point

I wonder if there are any theorems between changing curvature of some overall manifold and the equivalence of this to the creation of submanifolds.

It seems to me that this would be the missing link between the expanding universe of GR and the particles of String Theory. Perhaps this is also a type of symmetry breaking process.

Any comments or observations out there?

Thanks.

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Originally posted by Mike2
I wonder if there are any theorems between changing curvature of some overall manifold and the equivalence of this to the creation of submanifolds.
what do you mean by "changing"? remember that in GR, time is part of the manifold, so there is no way to say the curvature changes in time in a consistent way.

It seems to me that this would be the missing link between the expanding universe of GR and the particles of String Theory.
the link between string theory and GR is not missing, it is well known: String theory contains GR as a first order approximation.

Perhaps this is also a type of symmetry breaking process.
which symmetry did you have in mind to be broken, and what does this have to do with String theory, GR, submanifolds, or changing curvature?

Re: Re: Theorem between Curvature and submanifolds

Originally posted by lethe
what do you mean by "changing"? remember that in GR, time is part of the manifold, so there is no way to say the curvature changes in time in a consistent way.
I was refering to the "expansion" of the universe, how the volume of space is increasing with time. This is sometimes described as the "unwinding" of a curled up dimension. And this gives rise to the idea of a curvature of space dimensions changing with time as it uncurls.

the link between string theory and GR is not missing, it is well known: String theory contains GR as a first order approximations.
This sounds like a start. But is it fair to say which one include the other? And is both SR and GR "included"? And are higher order branes also included in this relationship?

Strings and the world-sheets thereof are submanifolds of the overall space-time. The more partilces/strings, the more mass density and the greater the curvature of spacetime. So this does sound like the start of a relationship between the curvature of something and the existence of submanifolds.

which symmetry did you have in mind to be broken
Don't know, really. But once you distinguish regions (as you do when you have submanifolds "somewhere" with respect to others) , then you can no longer say that every place is equivalent to any other.

Re: Re: Re: Theorem between Curvature and submanifolds

Originally posted by Mike2
I was refering to the "expansion" of the universe, how the volume of space is increasing with time. This is sometimes described as the "unwinding" of a curled up dimension. And this gives rise to the idea of a curvature of space dimensions changing with time as it uncurls.
the spatial curvature can be constant in an expanding universe. for example, it can be zero.

This sounds like a start. But is it fair to say which one include the other? And is both SR and GR "included"? And are higher order branes also included in this relationship?
it is certainly fair to say that string theory contains GR. like i said, string theory predicts einsteins field equation as a first order approximation.

on the other hand, GR does not contain string theory. in fact, up until the seventies, string theory was unknown, and yet GR had existed for many decades.

as far as special relativity is concerned, string theory contains that as well, but unlike general relativity, special relativity is put in by hand in string theory.

i don t think branes have anything to do with this question.

Strings and the world-sheets thereof are submanifolds of the overall space-time. The more partilces/strings, the more mass density and the greater the curvature of spacetime. So this does sound like the start of a relationship between the curvature of something and the existence of submanifolds.
strings and worldsheets are submanifolds of spacetime, i agree.

i don t see any straightforward relationship between number of submanifolds and curvature.

Don't know, really. But once you distinguish regions (as you do when you have submanifolds "somewhere" with respect to others) , then you can no longer say that every place is equivalent to any other.
R3 has infinitely many submanifolds. this does not imply that any region of R3 is somehow special.

Re: Re: Re: Re: Theorem between Curvature and submanifolds

Originally posted by lethe
i don t see any straightforward relationship between number of
submanifolds and curvature.
The reason I ask such questions is that it would seem that at the most
fundamental beginnings of the universe, we have both the expansion of the
universe (a manifold) and the creation of particles (submanifolds of the
univserse) that do not get any larger with time. This would seem to occur at
the very first instant at the starting point of very small size, possibly
even a single point. If this is not arbitrary, then it would seem that there
must be some connection between expansion and particle creation. The
question I have is if expansion gives particles, then was their ever an
expansion of space time without particles, the expansion of the one and only
"God" particle? It may be that multiple particles existed from the start. Or
perhaps due to some quantum mechanical effect, there may have been only one
particle to begin with, the "universal" particle, which latter fractured
into many.

R3 has infinitely many submanifolds. this does not imply that any region of
R3 is somehow special.
I suppose you'd have to have forces between particle (between these
submanifolds) before invariance with place is destroyed, hey?

i have pretty much no idea what you are talking about.

Isn't String Theory itself a description of how manifolds emerge from other manifolds and then join, since they describe the interaction of world-sheets which are manifolds? Isn't M-theory a study of how lower dimensional submanifold (strings) emerge from higher dimensional manifolds (membranes)? Perhaps M-theory itself is the theorem I'm looking for. Or perhaps the theorem I'm looking for would be the non-perturbative form of M-theory we would all like to see.

My earlier point was that if two different things are true in conjunction at the same place at the same time (as they would have to be at the singularity from which the big bang occured), then this implies that one proves the other so that there is an equality between them. So the question is whether the universe begins with both particle creation (emergence of submanifolds) and universal expansion simultaneously, or whether we could have expansion without particle creation.

Originally posted by Mike2
Isn't String Theory itself a description of how manifolds emerge from other manifolds and then join
no.

, since they describe the interaction of world-sheets which are manifolds? Isn't M-theory a study of how lower dimensional submanifold (strings) emerge from higher dimensional manifolds (membranes)?
i don t know M=theory, but this description doesn t sound accurate to me.

Perhaps M-theory itself is the theorem I'm looking for.
M-theory is not a theorem

Or perhaps the theorem I'm looking for would be the non-perturbative form of M-theory we would all like to see.
yeah, perhaps.

My earlier point was that if two different things are true in conjunction at the same place at the same time
conjunction? what does that mean? this is starting to sound like astrology.

(as they would have to be at the singularity from which the big bang occured), then this implies that one proves the other so that there is an equality between them. So the question is whether the universe begins with both particle creation (emergence of submanifolds) and universal expansion simultaneously, or whether we could have expansion without particle creation.
as far as i know, particle creation has nothing to do with expansion, and even a classical universe with a constant number of particles will undergo expansion.

Originally posted by Mike2
Isn't String Theory itself a description of how manifolds emerge from other manifolds and then join

Originally posted by lethe
no.
Is that because the splitting and joining of world-sheets is itself all only one manifold with holes in it, and not creating separate manifolds?

conjunction? what does that mean? this is starting to sound like astrology.
I'm refering to logical conjunction of two propostions held to be true. Consider the following statements of propositional calculus:
(The symbol "$$\to$$" stands for material implication, and "$$\cdot$$" symbolized conjunction.)

$$(p \cdot s) \to (p \to s)$$

and by symmetry:

$$(p \cdot s) \to (s \to p)$$

Together they prove:

$$(s \to p) \cdot (p \to s) = (p=s)$$

as far as i know, particle creation has nothing to do with expansion, and even a classical universe with a constant number of particles will undergo expansion.
It seems obvious that particle creations imply expansion since we could not distinquish one particle from another unless they are separated in space which must expand in order for that distinction to exist.

And the expansion of any manifold from nothing implies at least the existence of one particle, the particle described by that initial manifold that expands.

So perhaps it's not such a great leap to conjecture that it is a general truth that the expansion of one manifold is equal to the creation of submanifolds.

Re: Re: Re: Re: Re: Theorem between Curvature and submanifolds

Originally posted by Mike2
The reason I ask such questions is that it would seem that at the most fundamental beginnings of the universe, we have both the expansion of the universe (a manifold) and the creation of particles (submanifolds of the univserse) that do not get any larger with time. This would seem to occur at the very first instant at the starting point of very small size, possibly even a single point. If this is not arbitrary, then it would seem that there must be some connection between expansion and particle creation.
For example, here's a thought I'm considering; maybe there is some study of this in the literature somewhere: We have for each instant of time the overall manifold of reality that changes from the previous moment. There is a hypersurface that marks the initial and final state from one time to the next. That hypesurface is one less dimensionality of the origional manifold. But if we consider the time interval as a whole, and make no distinction between one instance of time and another in that interval, then we might consider there to be an infinite number of alternative paths from initial to finial state each of which contributes to a final result. And this would give a Feynman path formulation for the propogation of the hypersurface. Thus the expansion of an overall manifold creates the propogation of submanifolds (interpreted as particles). This is a pretty rough sketch. Do you think it has any merit?

what you are describing sounds a little like the concept of a cobordism. perhaps you should look into that theory. it is formulated in the language of category theory.

you can read a little about them in Baez' seminar on Topological Quantum Field Theory here

Originally posted by lethe
what you are describing sounds a little like the concept of a cobordism. perhaps you should look into that theory. it is formulated in the language of category theory.

you can read a little about them in Baez' seminar on Topological Quantum Field Theory here
Thanks for that link. Yes it does sound like a cobordism.

Re: Re: Re: Re: Re: Re: Theorem between Curvature and submanifolds

Originally posted by Mike2
, then we might consider there to be an infinite number of alternative paths from initial to finial state each of which contributes to a final result. And this would give a Feynman path formulation for the propogation of the hypersurface. Thus the expansion of an overall manifold creates the propogation of submanifolds (interpreted as particles).
So what is this result that considers every possible "path"? Is this like saying that a space is equivalent to every possible path through it? Is this some way of considering the entire space?

Re: Re: Re: Re: Re: Theorem between Curvature and submanifolds

Originally posted by Mike2
The reason I ask such questions is that it would seem that at the most fundamental beginnings of the universe, we have both the expansion of the universe (a manifold) and the creation of particles (submanifolds of the overall manifold of the univserse). This would seem to occur at the very first instant at the starting point of very small size, possibly even a single point. If this is not arbitrary, then it would seem that there must be some connection between expansion and particle creation.
Now I wonder if there is not some relationship between particle (submanifold) creation and entropy. Or whether there is not some equivalence between a decrease in entropy and the creation of some submanifold. For it would seem that the creation of a submanifold where before it was not would have to represent some increase of order in the universe. The order, or decrease in entropy, being stored in the constant characteristics of the submanifold.

Entropy is not necessarily a purely physical phenominon and can be used in association with probabilities in general.

For example, if the information (Shannon information) of the entire universe as a whole cannot change, then as the universe expands, there becomes more possible states, and the entropy of the universe would increase unless some process also produces states of lower entropy to compensate. Just speculating, any thoughts?

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