From the point of view of mathematics, there seems to be no obstacle for a noncompact manifold to be embedded in a closed manifold (for example, the winding of a torus everywhere densely filling a torus). However, this is not enough, since such an investment must be "revived", that is, populated with material objects and made to move them. It seems that this area is open for research, but I will be happy if you indicate to me the work corresponding to this topic. In turn, I can offer you a drop, which has the form of a 7-dimensional sphere, which is populated by a vector field of velocities of particles of a continuous medium. In this model, the topological singularities of a vector field should serve as material objects (particles), since linear vector fields form the Dirac matrix algebra. However, interesting consequences of this model from the viewpoint of physics also arise in the two-dimensional case. Thus, in the study of a random walk along broken lines of a winding of a classical sphere, one can obtain a generalized one-dimensional Schrödinger equation.
Mathematical notes on the nature of things (in Russian)
Preprint Математические заметки о природе вещей
Dear Igor,
I think that you are thinking in the theorem of Banach-Tarski, which goes against our common sense and in fact it corresponds to cut with parts having without measure.
https://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf
Daniel, I did not think about theorem of Banach-Tarski .
In addition, there was an incident of machine translation: instead of "However, this is not enough, since such an investment must be "revived", that is, populated with material objects and made to move them " should be replaced by "However, this is not enough, since such an embedding must be "revived", that is, populated with material objects and made to move them ". Have in mind that the manifold R3xS3 embeded in S7
Igor,
Are you thinking in a Hopf fiber bundle
S7 -> S4 and fiber S3 ?
That is to say, the one associated to the quaternionic numbers. Do you think in the topological solutions? What do you think?
No, I think in terms of distributions. In the space R8 the vector fields, tangent to the spheres in 8-dimensional Euclidean space and hyperspheres 8-dimensional space with a neutral metric form a lie algebra of a matrix algebra of Dirac. On the other hand, the product of S3xS3 spheres lies at the intersection of the sphere and the hypersphere, and therefore it is assumed that the distribution of vacuum vector fields is tangential to the space R3xS3.
@ Igor Bayak,
One can think in two ways:
(1) If the drop belongs to the universe, how can we put the universe in that drop? Is it a paradox?
(2) Contractions in topological spaces provide a direct answer: Yes.
The homotopy spheres, differential spheres, embeddings and differential structures for motions are technical issues.
In general: If X is a manifold ( closed or open),( with or without boundary) of dimension n.
Elementary isomorphisms(or homeomorphisms) guarantee the existence of a manifold Y such that
X and Y are isomorphic ( or homeomorphic), and you can design the metric of the atlas to ensure that X ⊆ Y. As a particular case, X = Universe and Y= drop. All undergraduate students can see that (a,b) ≡ R ( isomorphic). In other words, the whole real line R can be transformed into an open interval with a little length. The same construction works for any dimension.
Regards
@ Issam Kaddoura,
> (1) If the drop belongs to the universe, how can we put the universe in that drop? Is it a paradox?
A drop is just a beautiful image of the universe, but in fact the universe is a flow (vector field of particle velocities) on the sphere S7.
> (2) Contractions in topological spaces provide a direct answer: Yes.
It is assumed that R3\times S3 fills all S7. Thus, we believe that the dimension of the universe is smaller than the dimension of the drop.
@ Igor Bayak,
Our belief is not strict. It is variable with the discoveries.
The scientists consider the space as un Euclidean one, and the sphere is not the usual one, it is the exotic sphere ( homotopy sphere ).
Kervaire and Milnor ( Groups of homotopy spheres, Ann.Of Math 77 (1963)
504 - 537) show that in dimension n = 7 we have 28 homotopy spheres, one of them is S7 . The topic is not obvious.
Now, which structure we should choose that appropriate the model of our universe.
In general for any dimension n > 6, we have several structures to describe the drop.
As well as the universe itself.
Best regards
Mathematics, of course, allows arbitrariness in the choice of the model of the universe, but physics will leave among this set of models the only one that corresponds to nature. We can only choose the right one.
By the way, if we take the flow of a continuous medium on the 7-sphere as the model of the universe, then taking into account the evolutionary component of the flow (that is, its dependence on time), it is necessary to consider vector fields in the 8-dimensional space. And just in R8 there exists an algebra of linear vector fields that is isomorphic to the Dirac matrix algebra.
@ Igor Bayak
(*) [We can only choose the right one. ]
Which model is the right one to choose? And who decides?
No unique answers are available. If we answer one question hundred of questions arise immediately.
(*)[ And just in R8 there exists an algebra of linear vector fields that is isomorphic to the Dirac matrix algebra. ]
I think you didn't get my previous answer about n dimensional homotopy spheres. There are 28 spheres with dimension 7
∑i7 ⊂ R⁸ for i = 1, 2, 3,.....,28. all are in the 8th dimensional Euclidean space. These discoveries came after the famous Einstein SRT and GRT.
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