Dear Igor, maybe you will be interested in my article `The True Geometry of Nature (Hypothesis)`

https://vixra.org/abs/1804.0311

С пожеланиями успехов, Лев

Спасибо, интересно.

I think there is confusion between a relatively short distance space used in Special Relativity with large cosmological distances. Inertial frames are necesary for the first. Once you add gravitation this changes completly.

Juan Weisz Cosmological distances may well fit in a closed world and cosmological distances are not an obstacle to the local invariance of the shape of the isotropic cone. Another thing is that in a closed world, an isotropic cone is compactified into a product of a sphere and a bouquet of two circles.

I interprate a non inertial frame in the expansion of the universe...accelerating outwards.

@ Juan Weisz The expansion of the universe can be interpreted as an evolution of the shape of an isotropic cone, for example, as an evolutionary increase in the angle of opening of an isotropic cone. And the role of absolute time τ can be assigned to relative time t=log τ .

And why not discuss the issue of the relationship between geometry and the material world in the framework of this discussion. If the material world is understood as moving matter, then the gradient (covector field) of the quadratic metric interval of the pseudo-Riemannian manifold can be compared with the field of velocities or accelerations of particles of moving matter and thereby reconcile the etheric and geometric approaches to gravity.

Please Go To This Link :

https://en.wikipedia.org/wiki/Containerization

To illustrate the previous post about the relationship between vector fields and metrics, I will give an example with a deformation of a Lie algebra that is generated by a field y∂x+x∂x and a deformation of a pseudo-Euclidean square of the metric interval x²-y².

This Lie algebra admits deformation eφy∂x+e-φx∂y. At the same time, the corresponding deformation of a square of the metric interval e-φx²-eφy² means that a covector field formed as a gradient from this interval cancels corresponding generator of the deformed Lie algebra.

@ Abdelkader Mohamed Elsayed, container of the universe = receptacle of the universe

Let's try to generalize the question of deformation of the Lie algebra by extending it to Riemannian (pseudo-Riemannian) manifolds.

To do this, we first form linear vector fields that form Lie algebra, which zero the convolution with them of the differential 1-form obtained by taking the gradient from the corresponding quadratic metric interval

\begin{equation}

\nabla S^{2} = \frac{\partial g_{ij}x^{i}x^{j}}{\partial x^{i}}\,dx^{i}

\end{equation}

and then consider a local Lie algebra that locally cancels the differential 1-form obtained by taking the gradient from the differential quadratic metric interval

\begin{equation}

\nabla ds^{2} = \frac{\partial \left(g_{ij}(x')dx^{i}dx^{j}\right)}{\partial \left(dx^{i}\right)}\,dx^{i}

\end{equation}

where the independent variables are the coordinates of the tangent space of the Riemannian (pseudo-Riemannian) manifold $T_{x'}M^{n}$. Thus, the algebra of local linear vector fields of a manifold $M^{n}$ can in some cases induce a Riemannian (pseudo-Riemannian) metric on it, and then we talk about the Lie algebra $\mathcal{A}_{T}(T_{x'}M^{n})$ as about the Lie algebra of Killing vector fields.

See page 36

Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS

In connection with the above, the question arises - what metric does the Lie algebra of the Dirac matrix algebra induce? It turns out that this is a metric of an 8-dimensional space ℝ8 with a neutral metric, that is, a metric with a signature (+ 4, -4). The hyperspheres of this space, to which the vector fields of our Lie algebra are tangent, have the topology of the space S3×ℝ4 . And since the Lie subalgebra tangent to the spheres S3 generates the Lie group SO(3, ℝ), the integral lines of the vector fields of this subalgebra in ℝ8 are the coordinate lines of Euclidean space ℝ3.

Let us return to the question of the shape of the Universe. Earlier, we settled on an 8-dimensional space with a neutral metric. However, in fact, based on the requirements of the quantum world, the Universe has the form of a product S3×S3×T2 and therefore the hyperspheres of this space are homeomorphic to the product S3×S3×ℝ. And since in the process of evolution the hypersphere changes shape so that the diameter of one 3-dimensional sphere significantly exceeds the diameter of another 3-sphere, then only three spatial dimensions and time associated with the Minkowski metric are available to the observer.

Igor Bayak I appreciate your comment from six days ago, "And why not discuss the issue of the relationship between geometry and the material world in the framework of this discussion. " I think this is a significant issue. What are your thoughts on it, and are you able to contextualize them in terms of David Bohm's writings on wholeness and the relationship between implicate and explicate orders? In other words, if the explicate order demonstrates geometry, which we know it does, does that imply that the implicate order also has geometrical properties? Given that the implicate order is characterized by a higher degree of order as compared to the explicate order, how would we understand a higher degree of geometry in the implicate order as compared to the properties of geometry we observe in the explicate order?

The shape of any enclosure (ex. universe) underlies ”active structure“ (cf. SWT_paper_2) in the form of dedicated releases of structure waves deriving from a non-locally and enclosure-independently performing Consciousness. No enclosure can be understood from ”within“ (Enclosure Theorem). Inert algebra (ex. Group theory, quantum formalism...) while describing certain enclosure properties cannot make them obvious. Infinite ”modelizing“ can be replaced helpfully by the structural nature of Consciousness only.

Imagine a geometry in which to every point exactly one individual antipode point exists. According to this assumption a geodesic in any point in any direction is a circle with Radius R tangent to the direction vector and located on the plane, defined by the direction vector and the antipode direction. The antipode point is located on the opposite side of every geodetic circle perpendicular to the antipode direction. R is half the diameter of the universe. Because R is extremely giant, we locally cannot determine any difference to a flat space. But from differential geometry we know that we can calculate all properties of this closed space from the definition of the geodesics. We also see that the definition of the geodesics is homogeneous and isotropic in the complete space.

If we fill this virtual space homogeneously with mass, there is no tendency to shrink, because the attractive forces in all directions exactly cancel out.

If we consider a photon running along such a geodesic, we can calculate a centrifugal acceleration of about one centimetre per second per year with the current estimate of the size of the universe.

Jean Thill Does your idea of "active structure" have any similarities to the idea of "active information" articulated by Bohm and Hiley?

Hi Jensine, the definition of ”active structure“ is based on structural oscillators (structure waves: see paper 2) thus is not compatible with the mentioned author‘s concepts.

Wolfgang Konle, see similar thoughts in my previous topic

https://www.researchgate.net/post/Material_point_or_material_circle_Whats_better

The difference is, the distance between the antipodal points in my understanding is small.

Jensine Andresen , If I understand correctly, then you are interested in how I relate the local geometry of the microworld and the global geometry of the universe. Since the geometry is determined by the form of the flow of matter, then locally minimal flows determine the geometry of the microworld, and globally minimal flows determine the geometry of the universe. I do not think that this concept is somehow connected with the Bohm concept, especially since the minimal flows flow over the surface of the 7-dimensional sphere. As for my interpretation of quantum mechanics, please see the second section of the introduction to

Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS

Jean Thill , I will try to read your article.

Thanks, dear Igor!

Igor Bayak Jean Thill Thank you both for your considerate replies. I will try to read both articles.

Igor,

We can model the described space with unit quaternions Q. Cartesian space coordinates then correspond to the imaginary part of exp(Q) according to x

Wolfgang Konle ,The fact is that the quaternion algebra can be represented by an algebra of linear vector fields whose generators are tangent to the hyperspheres of 4-dimensional Euclidean space, that is, 3-dimensional spheres, therefore the streamlines of the generator algebra are closed in a circle. As for the radius of the 3-sphere and the local metric, this requires additional explanation, but a little higher I touched on this topic.

Igor Bayak,

this sounds reasonable. Does it lead to the same metric as the quaternion representation?

For comparison we can consider another property:

Given two points x1,x2. The geodesic through both is the circle through x1,x2,x1’, and x2’, which has radius R. It is easy to proof that (x1’,x2’) is parallel to (x1,x2) and has the same length.

Summary

A proposal for a container C of the material world or for a closed topology of the universe. The coordinate system used in this proposal is cartesian, derived from unit quaternions Q multiplied by R, which represents the size of the universe. The container has the following properties:

(1) Each coordinate (x, y, z) in C corresponds to the exponential of a unit quaternion Q in the following way: q=R*exp(Q), x=i-part of q, y

Everything is not as simple as it may seem. The three-dimensional sphere is just a part (albeit substantial) of the shape of the vacuum vector field of the matter flow. In the framework of the proposed concept, matter moves along the surface of a 7-dimensional sphere, and the global flow geometry, which is determined by the geometry of the vacuum vector field, is defined by the Clifford evolutionary torus S3 ×S3. In the process of evolution, the radius of one sphere increases and the radius of another decreases, and therefore, at the moment of evolution, only three dimensions are available to the observer. The vacuum covector field that changes during evolution is expressed through the gradient from the quadratic metric interval of the 8-dimensional space with a neutral metric

S2=eτx12+eτx22+eτx32+eτx42-e-τx52-e-τx62-e-τx72-e-τx82

where τ - evolutionary parameter.

Indeed, an eight-dimensional evolving space is not simple. Can you separate the purely conceptual part of your proposal from the part, which has been derived from something existing?

Why do we need eight dimensions? Three dimensions is all what we can imagine for a container. Three dimensional curved and closed space even seems to overcharge our imagination.

Another approach are the requirements. What properties does the space need to be an appropriate container for our world?

Naturally, in choosing the appropriate space, we rely on physical reality. And physical reality tells us that space, together with the matter moving in it, should provide us with the classical principle of least action, and the quantum principle of least action, and gravity and so on. Therefore, if we want to correspond to physical reality in terms of the spatiotemporal symmetries of a quantum particle, then we will have to work with an 8-dimensional space with a neutral metric in which the Dirac matrix algebra is realized.

In this I think questions of scale are most important for this issue.

For example quantum theory does not modify the normal concept of space.

Almost all QM takes place at rather short range, chemical if you like, so normal unbent space is fine.

If you go to special relativity, you assume also inertial space similar to Newton,

but not with absolute time and space. Still forces and gravity are absent.

Only good for short range.

If you go to Cosmology, you are not in inertial system at all. It is expansion.(well, according to current ideas) You use methods closer to GR. There we can bend space.

So the space we talk about depends on what physical events we wish to discuss.

Juan, we definitively talk about long range space.

Hi Igor. I'll have a read of your Notes, because I think we may be attacking many of the same problems from slightly different angles. This is probably partly because we have developed expertise in different parts of the subject. Consequently, I think we could have some interesting exchanges on some of the topics above.

For example, you have clearly thought more deeply than I have about the hyperbolic geometry of spacetime. My first question, then, to kick this off, is: how are boosts represented on the isotropic cone?

A "container" for all matter without exception is, by definition, immaterial in the absolute sense of this word. Hence, such a "container" is the Absolute or, in other words, God. And His form, by definition, is indefinable.

Why is uncertain, on the contrary - there should be a well-defined form of absolute space. But the form of matter moving in this space is indeed not well defined.

We can assume a wonderful isotropic and homogeneous container:

Consider a three-dimensional space, which to every point P in the space has an adjoint point P’ on the opposite side of the space. The cartesian distance between P and P’ is 2R. Then consider any two points P1, P2 in that space. We now assume that a circle with radius R through the four points P1, P2, P1’, and P2’ is a geodesic. If the four points are exactly on a straight line, the orientation of the circle in the plane perpendicular to (P1, P2) is undefined.

Mathematically the cartesian coordinate system (x,y,z) in that space is homologue to the imaginary part of the exponential of unit quaternions Q multiplied with R.

P(x,y,z)=R*Im(exp(Q(r,i,j,k))); Im means x->i,y->j,z->k. |Q|=1. Q+Q’=2(+0i+0j+0k)

Because every geodesic is a circle, the space is closed. But there is no outer rim. Every point in the space is a centre of gravity in respect to the whole space. Homogeneously filled with mass, this space/container does not tend to collapse. Filled with a hot gas it also has no tendency to expand.

Filled with a gravitational field, which causes the space curvature, the radius R of the curvature depends on the intensity of the field. Higher intensity leads to a smaller R.

Wolfgang Konle ,

Let me make a comment to expand your understanding of space. The fact that you associate Euclidean space with a three-dimensional sphere is already good, but this is not enough. It is also necessary to compare space-time and the product of a three-dimensional sphere and a circle S3xS1. Moreover, physics still has a space of global gauge symmetries, so we need one more extension of the absolute space. It turns out that if we take a doublet of Minkowski spaces and wind it around the product of the Clifford torus S3xS3 and the time torus S1xS1, then unitary symmetries can also be described in this absolute space.

Igor Bayak, you are right, considering space and time is an expansion of what I had posted. However, in this thread a container has been mentioned. In my understanding a container is spatial. Accordingly focusing on a closed spatial topology, which is not a sphere and even has a higher symmetry, is adequate.

But why do you claim that I had described a sphere?

Wolfgang Konle : " But why do you claim that I had described a sphere? "

Because the unit of quaternions can be compared to a three-dimensional sphere.

Wolfgang Konle > However, in this thread a container has been mentioned. In my understanding a container is spatial.

From the point of view of the author of the topic, here we are talking about the container of matter, which has a spatial essence in a deeper sense of the word. The author's interpretation is as follows. Matter moves along the surface of a seven-dimensional sphere. At the same time, the vacuum form of matter is a Clifford torus S3xS3 with three-dimensional spheres moving in the process of evolution in circles so that the radius of one sphere increases and the radius of the corresponding circle decreases, and vice versa, the radius of the second sphere decreases, and the radius of the corresponding circle increases.

Igor Bayak, "The authors interpretation is as follows…"

This statement is a censorship, which tries to restrict valid answers to the scope of the authors interpretation. If you expect a confirmation of your interpretation, instead of an open discussion, you obviously have selected the wrong medium.

Igor Bayak "Because the unit of quaternions can be compared to a three-dimensional sphere."

It seems that you have completely misunderstood my contribution. First, quaternions are dimensionless like real numbers. Second, I am talking about unit quaternions. These are quaternions with the absolute value of one. I am not talking about the units of quaternions because these do not exist. The relation of the unit quaternions to spatial dimensions is achieved by the multiplication with R, which has the unit/dimension of length.

Number stability yields the frame of the “material” world container, while number form instability overcomes it: cf. Number form theory.

Wolfgang Konle

It seems that I misunderstood you, but if you will allow me, then I will still state here my understanding of the algebra of quaternions. Firstly, it is the Clifford algebra, into which the three-dimensional Euclidean space is embedded, and secondly, this algebra has a representation in the form of an algebra of linear vector fields, the generators of which are tangent to three-dimensional spheres.

Igor Bayak , the unit quaternions topologically describe the surface of a four dimensional sphere.

However, the unit norm quaternion belongs to the three-dimensional unit sphere in the four-dimensional real space of quaternion coordinates.

Igor Bayak, the unit quaternions belong to the surface of a four dimensional sphere. Their imaginary part is three dimensional and can be mapped to a three dimensional coordinate system. If the exponential function is used in that mapping, we can be sure that the addition, then equivalent to a multiplication, remains in the space of the unit quaternions. (The product of two unit quaternions also is a unit quaternion.)

How would you define the mapping of a three dimensional sphere (x²+y²+z²)

Wolfgang Konle > How would you define the mapping of a three dimensional sphere (x²+y²+z²)