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Given a stress-free system as a perfect crystal with points or atoms ordered in a three dimensional lattice in the Euclidean reference space, any defect, external force or heterogeneous temperature change in the material connection that induces stress on a previously stress-free configuration changes the equilibrium configuration. A material has stress in a reference which does not agree with the intrinsic geometry of the material in the stress-free state. By stress we mean forces between parts when we separate one part from another (tailing the system), the stress collapses to zero for any part which assumes new configurations. Now the problem is that all the new configurations of the parts are incompatible with each other. This means that close loop in the earlier configuration now is not closed and that the two paths previously joining the same two points now join different points from the same initial point so the final point is path dependent. This phenomenon is formally described by the commutators of derivatives in the new connection of the stress-free parts of the system under the control of external currents. This means that we lose the integrability property of the system and the possibility to generate global coordinates. The incompatible system can be represented by many different local references or Cartan moving Euclidean reference, one for any part of the system that is stress-free. The material under stress when is free assumes an equilibrium configuration or manifold that describes the intrinsic “shape” or geometry of the natural stress—the free state of the material. Therefore, we outline a design system by geometric compensation as a prototypical constructive operation.

Given the stress-free system in the Euclidean reference space, any field of forces between particles due to gravity, electromagnetic, heterogeneous temperature, dissipation, or crystal defects will be called stress field. The defects or the physical fluxes change the material connection that induce stress on a previously stress-free configuration as in the holonomic system and as the equilibrium configuration or geometry change. Now the problem is that all the new configurations of the parts are incompatible with each other, with a geometry that differs from the intrinsic geometry of the system. This incompatibility creates defects in the reference. The coordinates of non-intrinsic geometry are not commutative and any loop cannot return to the initial value. This means that the integration operator is not unique and the system is not conservative. A simple example of incompatible geometry is given by rotation movement in the flat geometry. The geometry without curvature is not the intrinsic geometry of the rotation so stress forces appear as centripetal and centrifugal forces to compute the movement. When we use the intrinsic geometry for rotation as curvilinear coordinates, the reference is stress-free. The incompatible system for the defects (singularity) cannot be represented by a global reference but can be represented by many different references or Cartan moving references, one for any part of the system that is stress-free or locally compatible. The material under stress when is free assumes an equilibrium configuration or manifold that describes the intrinsic “shape” or geometry of the natural stress-free state of the material. The article underlines that the appearance of non-conservative facets in systems is a universal aspect which may be explained analyzing the structural links between quantum mechanics and Maxwell’s equations, and also between gravitation and Maxwell's equations, thus outlining a general theory of open and nonholonomic systems. All that generalizes “input” and “output” concepts in Systems Theory (every “law” is a systemic connection among a series of input/output(s), under specific boundary conditions) has already been overcome by Einstein geometry that radically changes the old Newtonian concept of input (force) and output (acceleration).

The main examples of the intrinsic geometry for gravity force as a stress are the Einstein general relativity with curvature (defects in rotations) without torsion and the example of Cartan moving reference in gravity is the “Teleparallel” with torsion (defects in translation) without curvature. In this paper, we use a Maxwellian-like generalized gauge approach to get the intrinsic geometry in different systems. Here, we follow Caianello’s idea [

With a moving local reference it is possible to detect the geometric nature of the system. Historically we remember the Galileo principle for which systems with constant velocity are all equal by local reference that moves with the system (inertial movement). Any local reference cannot be detected if the system moves and the velocity itself, too, cannot be detected. In this Galilean situation, any local reference has the same geometry of the global reference, the topology of the system is always the same (conservative system). In

The Cartesian reference has no defects and local geometry is the same of the global geometry. After the transformation, we have another reference that has the same properties of the original Cartesian coordinates (Definition: A system is compatible if the local geometry is the same as the global geometry).

To know if a system has no defects or is compatible we take a local reference that we move to form a loop. If, after the loop, we return to the same point and to the same states, the system is compatible and conservative. We know that the Euclidean geometry in the Cartesian reference is a compatible geometry without defect for which any derivative commutes one with the other in this way

We can see the compatible property by this categorical diagram

Given the rotation system, we know that the tangent vector in any point in the Cartesian reference is given by the tangent vector

The directional derivative is given by the scalar product of the vector

With

For

We have that ^{2} + y^{2} = R^{2}

Intrinsic geometry of circles which derivative is

When we move on the intrinsic geometry circles (

The Cartesian coordinates and geometry are not the intrinsic geometry because include the (0,0) point that is a singular point or defect. In this situation, the commutator is different from zero. Because the direction derivative on the tangent vector to the circle is a derivative for polar coordinates that is the intrinsic geometry, we have no particular problem to introduce the singular point. The derivative is denoted as the Lie derivative. We remark that the Lie derivative can be obtained also by the differential form

In fact, we have
^{2} + y^{2} = R^{2}

In the Cartesian reference and geometry, the equation for inertial movement is

We can see that no force appears so the system is in the stress-free state. Given the transformation of the curvilinear coordinates q in the Cartesian coordinates x
^{i}^{i}^{μ}^{i}

We have the change of the velocity

In many cases this is true of only the local transformation of the derivative but, in general, it is impossible to write a global expression. So, it is true of only the transformation

The reference ^{i}_{μ}

The basis of the new reference is a function of the position q and time t in the Euclidean space.

We remark that the new moving reference in a Cartesian space loses the commutative property

In the new reference, the acceleration takes this form

Because the basis is orthonormal and complete, as we can see in

So, we have

By the inertial movement of the basis, we compute the movement in a stress-free intrinsic geometry whose geodesic equation computes the connection on the manifold where the basis moves, which value is given by the variables

The force ^{μ}

Spherical intrinsic geometry. Locally, the space is flat but globally we have a curvature for which the basis moving on the sphere is not commutative.

The dynamical equation of a Geodesic movement on a sphere is given by the previous equation, and this can be represented by

Geodesic triangle and geodesic trajectories. On the surface of the intrinsic geometry (spherical geometry), the geodesics are straight lines without curvature and so are stress-free.

Now, we have the problem to compute the derivative from the Cartesian coordinates to the general moving basis _{v}^{v}^{i}e_{i}

The derivative is

We remark that if

The connection terms

Now, in index notation, the covariant derivative of ^{i}^{μ}

If we have a point moving on a curve in time, we have
^{j} = x^{j} (t)

In the geodesic line we have

The derivative in the direction of the tangent vector is equal to zero. So, the geodesic is a line without stress.

Given the electrical circuit inertial equation (free from stress) for the voltages

When we change the reference from fixed and inertial movement for the voltage to the current moving reference we know that we have the relation
^{i} = R^{i}_{μ} di^{μ} = e^{i}_{μ} di^{μ}^{i} – R^{i}_{μ} di^{μ}^{i} – e^{i}_{μ} di^{μ}

The first equation is the phenomenological relation between currents and voltage by the resistor tensor ^{i}_{μ}^{i}_{μ}

For the compatibility between the phenomenological equation and intrinsic geometry, we have the identity

Given the connection term, we can design the resistor tensor in a way to have geodesic transformation and covariant derivative in the wanted space of the currents. For example, given the spherical geometry by the transformation

The tangent vector is

And the moving basis is

For the phenomenological identity, we have the resistor matrix

And the electrical circuits with current-controlled voltage sources (CCVS) and resistors (

Moving reference on the sphere.

We know that the connection terms of the intrinsic geometry are

Now, we represent the circuit with current-controlled voltage source (CCVS) ^{i}_{μ}

In ^{β}^{α}

Moving reference in the electrical circuit.

To complete possible electrical circuits, we have another three derivative transformations

Given a space where the general coordinates are ^{1}, ^{2},....., ^{n}

Angular displacement q = θ and mind control of initial and final positions.

With the basis vectors _{α,β}_{α}^{T}_{β}

Intrinsic geometry can have curvature and torsion that can be seen as an external element as we can see in the Euler Lagrange equation with torsion

Rotation and torsion geometry.

We provide that any deformation of the reference as a crystal totally ordered is given by the transformation
^{i}^{i}^{i}

The difference of the distance L between points (atoms) before and after the deformation is given by the expression
_{ij}

In the work by Ruggero and Tartaglia [

The deformed elements fit perfectly or they do not. In the latter case, we must apply a further deformation to re-compact the body. In the first case, we speak of a compatible deformation.

In the second case, we have an incompatible deformation. Let us imagine that during the deformation the coordinates are dragged with the medium. In the compatible deformation, the internal or intrinsic observer cannot see any difference as the Galileo internal observer for inertial system. In the incompatible deformation, the internal observer notices a change in the number of particles along a cycle in the medium as excess of holes or particles. The internal point of view is useful to find an incompatible deformation, due to the presence of defects. Mathematically, an incompatible deformation corresponds to the non-integrability of the differential form _{j}_{j}^{j}

In this situation, intrinsic geometry will no longer be Euclidean. The intrinsic view suggests that relations can be found between the geometric properties and the densities of defects that influence them. From ideal crystal or ideal reference as Cartesian reference, after the deformation we have Crystal incompatibility or disclination where there is deformation in the rotation and without torsion (

Change of reference or crystal medium by curved system where the center is a singularity or defect in the disclination.

With the scalar

Tensor

Torsion as defects in translation.

The torsion is a defect in translation (dislocation) as we can see in

Defects in translation or crystal dislocation.

The defect or singularity is given by defect in the reference due to translation transformation.

For the wave equation, we have

When we use the space time reference _{j} = (x_{1}, x_{2}, x_{3}, ict)

For ^{−ieφ}

We remark that

For the wave equation, the change of the wave function generates incompatible medium where there are defects as we can see in

From compatible medium on the left, there is incompatible medium on the screen.

The new derivative does not commute but the wave equation does not change its form and the wave sources are always the same. In fact, for

We have

In the Jessel book [

In fact, the transformation ^{−ieφ}

In the transformation, the derivatives are those in the compatible medium but we must change the source from S as the original sources of the wave to new artificial sources or secondary sources

That gives us the physical image of the incompatibility in the medium when we transform one field to another. By the new sources we can generate the new field with the same derivatives. M. Jessel uses the new sources to design a wanted field from the initial one. This is the beginning of a new computation where we design a new intrinsic geometry in the field by artificial sources. This is an example of field control by the active or secondary sources S* [

InVuksanovic and Nikolic [

Active noise control (ANC) by the algorithm or DSP for S*.

The active noise control (ANC) is the process of reducing an unwanted or incompatible sound by combining it with a sound of the same amplitude but of opposite phase. The proposed ANC system uses an active sound barrier of secondary sources S* to cancel the unwanted sound or incompatibility from the primary source at an array of error microphones. By cancelling the sound at the error microphones distributed across the controlled region, the secondary sources create a zone of reduced noise over this area as we can see in

In the work by Russer [

When we substitute the new variable, we have

For

When

The phase is constant in space and time and we have the compatible condition
_{μ}, D_{v}_{μ}∂_{v} − ∂_{v}∂_{μ}

We have no curvature and torsion and the medium has no defects. However, when

We have the incompatible condition

For a reference with torsion, we obtain the incompatible equation

When we solve this equation, we can provide a new geometric representation of the electromagnetic equation by torsion of the reference and defects in the medium. In crystal, there is a separation of the charges and the reference for the electromagnetic field is deformed by a torsion as in the dislocation of the crystal.

In the electromagnetic theory, we have that
_{μ}F_{vp} = J_{μ,vp}

Because we have
_{μ}_{v}_{ρ}_{v}_{ρ}_{μ}_{ρ}_{μ}_{v}

We have the invariant property for the currents
_{μ,vρ} + J_{v,ρμ} + J_{ρ,μv}

Given the Maxwell equations in the tensor form
^{v}_{x}_{y}_{z}^{μv} = ^{μ} ^{v} − ^{v} ^{μ} that can be connected with the commutators’ property and the incompatible condition that we have explained in the prrevious chapters.

The Maxwell-like scheme for an incompatible system is given by the set of equations

To introduce the new wave equation for gravity [_{μ}_{v}_{α}^{λ}_{αμv}V_{λ}

With the double commutator we have the dynamic equation
_{k}_{ν}

For the conservation of the current, after contractions, we have the equation

Most applications of differential geometry, including General Relativity, assume that the connection is “torsion free”: that is, vectors do not rotate during parallel transport. Because some extensions of GR do include torsion, it is useful to see how torsion appears in a modern geometrical language. The torsion corresponds intuitively to the condition that vectors must not be rotated by parallel transport. Such a condition is natural to impose, the theory of General Relativity itself includes this assumption. However, differential geometry is equally well.defined with torsion as well as without, and some extensions of general relativity include torsion terms. The first of these was “Einstein-Cartan theory”, as introduced by Cartan in 1922. We define the torsion tensor by the connection symbols in this way

We now show in an explicit way that it is possible to present the previous dynamical equation by a wave equation with a particular source where the variables include symmetric and anti-symmetric connection symbols as well as torsion in one geometric picture. In

Symmetry and its physical implications on conservation principles have a long history in physics. The same importance, if not higher, is shown by the concepts of symmetry breaking and local gauge as the constructive principle to characterize interactions as a “compensation mechanism”. In particular, this was made possible by a unified geometrical vision of fundamental interactions in Gauge Theories [

We stress the mathematical aspects which make this approach a “theory to build geometric-based unified theories”.

The conceptual core of the procedure can be expressed in a five-point nutshell:

The description of a suitable substratum and its global and local properties on invariance;

The field potentials are compensative fields defined by a gauge covariant derivative. They share the global invariance properties with the substratum;

The calculation of the commutators of the covariant derivatives in (b) provides the relations between the field strength and the field potentials;

The Jacobi identity applied to commutators provides the dynamic equations satisfied by the field strength and the field potentials;

The commutator between the covariant derivatives (b) and the commutator (c) (triple Jacobian commutator) fixes the relations between field strength and field currents.

We chose an example connected to the recent developments of the extended gravity theories in order to show the generality of the approach. Actually, the GR syntax seems to regenerate itself from

In conclusion, the geometrical approach here delineated has significant potential in relation to the classical themes of systemics (system/environment; contextuality; computation; logical openness) thanks to the strategy allowing, in a simple and general way, to recognise the gauging as cognitive compensation between known and unknown domains.

The authors thank the Editors for their critical reading, and suggestions.

The authors contributed in equal measure to the work.

The authors declare no conflict of interest.

Given the general covariant derivative

To know the connection terms we begin with the computation of the first commutator

So we have

In conclusion, we have

Now, we have that

So, we have that the field

Now, we have

The Lagrangian gravitational density is

The dynamic equation for Non Conservative Gravity can be obtained in this way

Now, we have

For _{?2} = _{??} we have

However, for the Lorentz-like gauge we have

When the currents are equal to zero we have that

The variable

Now we look at the currents

In conclusion, we show that the non-conservative gravitational field is similar to a wave for 64 variables _{ρ} Γ_{λ} , the Chern-Simons terms ( ∂_{ν} Γ_{ρ} ) Γ_{λ} and the Maxwell-like terms ( ∂_{ν} Γ_{ρ} ) ( ∂_{μ} Γ_{λ} ) are present. So, we have the mass terms, the topologic terms and the electromagnetic-like field terms. We can model the gravitational wave with torsion as a particle in a non-linear medium which gives the mass of the particle, in a way that can be compared to usual SSB processes of the standard model, for an orientation in extensive literature [