So, where does this miracle live? Mathematics tells us that a dynamical system (a vector field of velocities or accelerations) in an 8-dimensional space with a neutral metric is the best candidate for this role [1]. Similar things have already been encountered, for example, in [2], but then dynamical systems in flat space-time were considered, which did not allow us to fully describe gravity and quantum-algebraic properties of particles. Now we hope to fill in these gaps. However, to do this, you need to go back to the ether, but not in 3-dimensional or 4-dimensional, but in 8-dimensional space.

1. Igor V. Bayak, Applications of the Linear Algebras of Vector Fields to the Modelling of Physical Phenomena, Journal of Geometry and Symmetries in Physics, 38 (2015)

2. Birkhoff G. D., Flat space--time and gravitation, 1944, Proc. Nat. Acad. Sci., v. 30 No. 10.

I will try to explain on my fingers how a pseudo-Riemannian manifold is obtained from the local algebra of vector fields. First, it must be understood that every algebra of linear vector fields is equivalent to the corresponding matrix algebra, and the Dirac matrix algebra is equivalent to the algebra of linear vector fields of an 8-dimensional space with a neutral metric (+4, -4). Secondly, it should be understood that the local algebra of linear vector fields is generated by curvilinear coordinates of an 8-dimensional space satisfying generalized Cauchy-Riemann conditions that provide local preservation of the Dirac algebra. In this case, the generators of the Dirac algebra are locally transformed, thereby generating a pseudo-Riemannian manifold. Note also that the curvilinear coordinates are generated by the current line of the vacuum vector field of matter.

A few words about the origin of an 8-dimensional space with a neutral metric. The fact is that we study dynamic flows on the surface of a 7-dimensional sphere, and therefore we need to use an 8-dimensional space to describe this dynamic system in evolution. At the same time, the globally minimal flow on a 7-dimensional sphere, which we associate with a vacuum flow, is a foliation on this sphere with a typical layer having the topology of a Clifford torus S^3 x S^3, and therefore in the algebraic theory of gravity we consider an 8-dimensional space with a neutral metric. At the same time, the Dirac algebra associated with this space is generated by the Lie algebra of linear vector fields tangent to the hyperspheres of an 8-dimensional space with a neutral metric. Note also that the hypersphere of this space has the topology R^4 x S^3 and therefore, in evolution, the observer has access to a 4-dimensional Minkowski space, and in statics, a 3-dimensional Euclidean space.

Igor Bayak "Since the generally accepted theory of gravity is GRT, one could try to derive it from some algebraic theory."

Before you try to derive it from something more basic, you should correct a principal fallacy. The problem is that GRT ignores the mass equivalent of the energy density contained in the gravitational field.

Considering that, the Poisson equation gets an additional term:

∇²ɸ=4πGρ=>∇²ɸ=4πG(ρ-(∇ɸ)²/(8πGc²)) or ∇²ɸ+(∇ɸ)²/(2c²)=4πGρ

ɸ is the gravitational potential, ρ is the mass density, G is the gravitational constant, c is the speed of light. The additional term is written with bold letters.

The extended Poisson equation then would be a good start to implement your idea.

Wolfgang Konle: The extended Poisson equation then would be a good start to implement your idea.

Actually, I wanted to expand the dimension of the Laplace equation and as a result get the Poisson equation as the limit of GRT.

Igor Bayak

The point is that natural laws, which are exactly fulfilled, can be derived from very basic relations. But if they are not fulfilled exactly, deriving them from something more basic is questionable.

If the Dirac algebra is considered as a dynamical system, then two phase spaces can be noticed there. One phase space (t*,x,y,z) is generated by the evolutionary parameter \tau (τ) and the Dirac matrices \gamma_0,\gamma_1,\gamma_3,\gamma_3 (γ0,γ1,γ2,γ3) and the other phase space (t,x*,y*,z*) is generated by the evolutionary parameter and the same Dirac matrices, but multiplied by an imaginary unit. In this case, the first phase space has a Minkowski space-time metric, and the second has an inverse metric. In my opinion, this is a good opportunity for a theoretical justification of the existence of an energy-momentum space. Indeed, the phase trajectory corresponding to the coordinate t* is unclosed, and the phase trajectory corresponding to the coordinate t is closed, and therefore the phase trajectory of the coordinate t + t* has a helical shape. Thus, the angular velocity of rotation in the phase space of our dynamical system can be compared with the rest energy of a material point. Similarly, the phase trajectory of the x coordinate is closed, and x* is unclosed, and therefore the phase trajectory of the x+x* coordinate has a helical shape, and therefore the angular velocity of rotation in this subspace of the dynamical system can be compared with the momentum of a material point.

Igor Bayak "the angular velocity of rotation in this subspace of the dynamical system can be compared with the momentum of a material point"

Better: "The "torque of inertia" multiplied with the angular velocity of rotation can be compared with the momentum of a material point."

But this is not especially astonishing because both quantities obey conservation laws.

Wolfgang Konle,

It means that a material point in space-time does not rotate and at the same time rotates in a doublet of phase spaces.

Finally, let us imagine the equation of the connection of a pseudo-Riemannian metric tensor with hyperbolic rotations φi of planes (xi,x*i) rolled into cylinders

2ηij=(miγi)(mjγj)+(mjγj)(miγi)

where

mi=exp(φi)

Then, with an equal hyperbolic rotation in all planes, we get the classical Newtonian approximation

η00=exp(2φ), η11=η22=η33=exp(-2φ)

where φ is the gravitational potential.

Igor Bayak

Your formulae look suspicious.

γi is not explained.

ηij=(miγi)(mjγj)+(mjγj)(miγi) According to the summation convention, this seems to be a scalar expression.

exp(φ) where φ is the gravitational potential is illegal. An argument of the exp function must be dimensionless.

Wolfgang Konle

An explanation has already been given in previous posts. Therefore, let me remind you that \gamma_i is a Dirac gamma matrix equivalent to the corresponding vector fields in an 8-dimensional space with a neutral metric. I will also explain that in the formula for the metric tensor, the sum of products (anticommutator) of arbitrary gamma matrices is always either 0 or a unit matrix, which in our formula is replaced by a numerical unit (1). As for the gravitational potential as an argument of the exponent, for linearization, decompose the exponent into a Taylor series and take the first two terms.

Igor Bayak "...vector fields in an 8-dimensional space with a neutral metric"

What is the physical meaning of the 8 dimensions in our 3+1 dimensional world?

"As for the gravitational potential as an argument of the exponent..."

Only a dimensionless variable can be an argument of a function. The argument then must be a potential divided by a norm potential. What is this norm potential?

Wolfgang Konle: "What is the physical meaning of the 8 dimensions in our 3+1 dimensional world?"

Since the phase trajectory of the particle is a helical line, then one closed coordinate should be added to each open coordinate of the 4D space. Due to the closeness of these coordinates, we do not see them.

@Wolfgang Konle: "Only a dimensionless variable can be an argument of a function."

We are building a model. Everything is possible here.

Igor Bayak "We are building a model. Everything is possible here."

Only in fairy tales everything is possible!

Wolfgang Konle,

Well, then explain why the arguments of the function must necessarily be dimensionless quantities. At the same time, I have already clarified that in our case (small argument values), the value of the function is equal to the value of the argument.

Igor Bayak "...explain why the arguments of the function must necessarily be dimensionless quantities"

The dimension of a potential is velocity squared. What do you think you get by exp(velocity²)? 2*exp(velocity) would be a first try.

About algebraic theory of gravity and "the Dirac algebra": the algebra of Dirac gamma matrices is matrix representation of the more general Clifford algebra - see very good review by Stefan Floerchinger "Real Clifford algebras and their spinors for relativistic fermions" Article Real Clifford Algebras and Their Spinors for Relativistic Fermions

Moreover, statement "the generally accepted theory of gravity is GRT" is not true.

1. Einstein's theory of general relativity is just the simplest version of modern theories of gravity. It is based on the Riemann metric.

2. The next class of more complex theories is based on the Riemann-Cartan metric, in which torsion is present. Weyl was the first to consider such theories.

3. The most general class of theories of gravity is based on the presence of a metric with torsion and affine connection.

- see, for example, review of Mikhail Katanaev "Geometrical methods in mathematical physics " https://arxiv.org/abs/1311.0733v4 (text in Russian)

There you can also read the basics of differential geometry, the theory of Lie groups, as well as the basics of the theory of spinors - the theory of Clifford algebras. This is the minimum set of mathematics required to discuss modern theories of gravity.

Vadim Monakhov

Without considering the mass equivalent of the energy density contained in gravitational fields all these higher sophisticated differential geometrical methods are applied in vain.

The next physics levels which are still missing is considering the influence of movements (gravitomagnetics) and the fact that an absolute negative energy density cannot exist.

All these mathematical gimmicks are only playthings without a completed physics basement.

Thank you Igor for recommending my newest paper Global and Local Duality in Grassmann Algebras, and for calling my attention to this very interesting discussion!

Wolfgang Konle: All these mathematical gimmicks are only playthings without a completed physics basement.

That's right, without a fundamental physical principle, all this mathematics does not work. In other words, the principle of least action should be raised to a metaphysical level. In this regard, it makes sense to talk about minimal flows of a seven-dimensional sphere, minimizing the length of the trajectory of each particle of matter moving along its surface.

Igor Bayak "...it makes sense to talk about minimal flows of a seven-dimensional sphere"

Exploiting any aspect of our three+1 dimensional world must be the first priority, before we turn to higher spheres.

The aspects we neglected up to now, are the energy density of force fields and the fact that an absolute negative energy density cannot exist.

It should be easy to understand that if we cannot successfully handle our few dimensions, what then gives us the confidence that a higher dimensional consideration could be promising? Only one thing is sure. More dimensions increase the complexity.

I would like to highlight the special role of the time coordinate of the algebraic theory of gravity. The fact is that in an 8-dimensional space with a neutral metric, a minimum flow is given, the current lines of which coincide with the time coordinates of the algebraic theory of gravity. For example, the time coordinates of the globally minimum vacuum flow are formed by the differential 1-form of the gradient of the function \log(\frac{(x_1)^2+(x_2)^2+(x_3)^2+(x_4)^2}{(x_5)^2+(x_6)^2+(x_7)^2+(x_8)^2})

Igor Bayak "function \log(\frac{(x_1)^2+(x_2)^2+(x_3)^2+(x_4)^2}{(x_5)^2+(x_6)^2+(x_7)^2+(x_8)^2})"

How is function "frac" defined? What is the physical meaning of the eight coordinates?

How can we see that this is not only a pipe dream?

Wolfgang Konle

Actually, this is a function equal to the natural logarithm of the ratio (\log\frac{}{}) of the square of the radius vector in a subspace with a positive metric to the square of the radius vector in a subspace with a negative metric. Its gradient is dual to the vector field, which is equivalent to the zero Dirac gamma matrix. As for the absence of physical manifestations of the 8th dimension, this is due to the fact that only non-compact (unclosed) basic vector fields are observed, and there are just 4 of them

An attentive reader of this discussion has already noticed that the algebraic theory of gravity is smoothly transitioning into the ethereal theory of gravity. I have already tried to describe gravity using vector fields in a compactified Minkowski space.

Article Gravity Model for Topological Features on a Cylindrical Manifold

Igor Bayak "Actually, this is a function equal to the natural logarithm of the ratio (\log\frac{}{}) of the square of the radius vector in a subspace with a positive metric to the square of the radius vector in a subspace with a negative metric. Its gradient is dual to the vector field, which is equivalent to the zero Dirac gamma matrix."

Your answer to the question "What is the definition of function frac?" only raises new questions:

What is a negative metric? What means "being dual"? How can function "frac" appear in the definition of function frac? Is it a recursive definition?

The negative metric is determined by the negative square of the vector. The concept of duality of 1-forms and vectors is a basic concept of linear algebra. The function of the logarithm of the ratio of squares is chosen so that its gradient coincides with the differential form corresponding to the zero Dirac gamma matrix, but in general any harmonic function in a neutral metric will do, since it satisfies the condition of minimality of the flow.

Igor Bayak "The negative metric is determined by the negative square of the vector."

How do you get a negative square of the vector? Does this vector have imaginary component values?

In an 8-dimensional space with a neutral metric, the square of the vector x is

(x_1)^2+...+(x_4)^2-(x_5)^2-...-(x_8)^2

Igor Bayak

Iteresting, what physical entities are represented by the components 1..4 and the components 5..8?

A pseudosphere in an 8-dimensional space with a neutral metric has the topology R^4xS^3, where the non-compact part of the product R^4 is associated with the Minkowski space, and the compact 3-sphere is responsible for gauge symmetries.

If we take into account that we study minimal flows on a seven-dimensional sphere, then we must keep in mind that the globally minimal flow has the form of a torus S^3xS^3 where one 3-sphere lies in the space x_1,...,x_4 and the other in the space x_5,...,x_8

Note also that the compactification of the hypersphere of an 8-dimensional space with a neutral metric onto the 7-sphere of an 8-dimensional Euclidean space necessarily leads to quantum mechanics, since the Minkowski space is thus compactified into a cylindrical manifold.

Igor Bayak

Sorry, but concerning your various explanations about higher dimensions only one question bothers me. Why and for what is that relevant?

Wolfgang Konle

This is important for those who will follow. String theories have not coped with their task, so we need to look for new approaches. In my opinion, vector fields on the 7-sphere are the most relevant thing.