Difference between revisions of "TFNR - Rotation - Curl"

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Revision as of 19:08, 11 March 2026

Rotation, or Rotational Action, is the instantiation through which Conservation manifests at the most elementary level of Physical Reality.

It is the third fundamental Mode of manifestation of the Elementary Action, its third Component. The third of the Modes of Physical Action.

From a causal perspective, we can represent it as the Rotation Operator, the Operational Agent responsible for the formation of vorticity, with its sub-components of chirality and axial orientations, in the elementary spatial fluctuations of the Elementary Field.

From a variational perspective, Rotation manifests as inhomogeneities (torsion / curl) in the distributions of chirality and axis orientation of circulation / vorticity of the minute pre-spatial fluctuations.

<thead> </thead> <tbody> </tbody>
Cause (expression of force) Sub-elementary dynamism Dynamic effect Observed geometric variation
Internal interaction between the Force and the Field Local perturbation of the density of τ-distributions of σ-fluctuations Gradient formation Variation of the infinitesimal event-point volume (local dilation or contraction)
Field reaction (counter-force) Gradient Translation / divergence Stretching of σ-fluctuations along the direction of divergence
Combined field reactions (composed counter-forces) Interaction of divergences Rotation / curl Vorticity of σ-fluctuations within the event-points


Rotation is the way in which the Elementary Field reacts to the interaction of Elementary Action in the Mode that I call here Translation, one of the ways in which the Field responds in order to restore homogeneity in the temporal distributions of Elementary Events.

It is the Field's response which, still driven by the tension to restore homogeneity, but due to micro‑interactions—occurring at the same event‑points—between Translations with different directions, produces the opposite effect: it preserves inhomogeneities, transforming them into dynamic micro‑torsions that tend to maintain anisotropies and more or less coherent structures over Time. Micro-torsions that in turn tend to produce regions of non‑zero Perturbation, which themselves become sources of Translation and centers of Rotation, and so on, in a progressive unfolding of dynamic complexity.

The causal expression and the variational manifestation—two sides of the same coin—of this Mode of Elementary Action are governed by the Principle of Conservation, which, at the Level of Elementary Action, takes the form of the Principle of Least Action.

Rotation is the root of vorticity, in its fundamental aspects of chirality and axis orientation.

Rotation:Chirality

Sub‑Mode of the Rotation Motion of the Elementary Action corresponding to the manifestation of inhomogeneities in the pre‑temporal distributions of the chirality of the vorticity that characterizes the sub‑elementary dynamism of the Field (pre‑spatial fluctuations, Elementary Events).

The Rotation:Chirality correlation within a region of the Field that, moment by moment, supports/sustains an InfoStructure is the root of the Charge of that Structure, whereas the Rotation:Chirality correlation independent of the Field (produced by other Structures with non‑zero Charge present in the Field itself) is the root of the electric potential (or electric potential energy).

Rotation:AxisOrientation

Sub‑Mode of the Rotation Motion of the Elementary Action corresponding to the manifestation of inhomogeneities in the pre‑temporal distributions of the orientation of the vorticity axis that characterizes the sub‑elementary dynamism of the Field (pre‑spatial fluctuations, Elementary Events).

The Rotation:Axis‑Orientation correlation within a region of the Field that, moment by moment, supports/sustains an InfoStructure is the root of the Spin of that Structure, whereas the Rotation:Axis‑Orientation correlation independent of the Field (produced by other Structures with non‑zero Charge present in the Field itself) is the root of the magnetic potential (or magnetic potential energy).

The causal perspective

Let us hold firm to what was stated regarding the causal perspective in the previous sections, dedicated to Perturbation and Translation, namely that:

  • the causal perspective focuses on the active aspect of the Source's Agency,
  • the Operators of Reality are functional / operational derivations of the Primary Agent, of the Fundamental Force acting upon the Elementary Field,
  • the Operators are Entities not in the classical sense, but in their function of producing Existence, a Phenomenon—something that occurs, something that exists in the very act of generating Existence, a product of the dynamics of the inner interaction of the two aspects of the Source of Reality.

From the causal point of view, Rotation, in both its sub-Modes, can be represented as the Operator Rotation, a functional derivation of the active aspect of the Source of Reality that organizes the interacting reactions of the Elementary Field represented by Translation into rotational patterns (vortical reorientation of the pre-spatial fluctuations), directed toward the conservation of the inhomogeneities within the Elementary Field in both the aspects of chirality and axis-orientation.

The variational perspective

Symmetrically, from the variational perspective, Rotation manifests as the reaction of the Field to Translation, to divergences (Micro-flows) in the pre-temporal distributions of the vortical aspects of the sub-elementary pre-spatial fluctuations, resonating around the Planck scale.

I resume here what was stated in the corresponding subsection (The variational perspective) of the previous section dedicated to Translation about Micro-gradients, Micro-flows, Micro-torsions (or Micro-vorticities), Micro-states, about the infinitesimal portions of the Field I call "event-points", and about the fact that inhomogeneities tend to cause the re-Action of the Field, toward the restoration of homogeneity. The Field's reaction to these inhomogeneities, which I here call Perturbation, is part of this spontaneous process aimed at restoring homogeneity. These reactions generate oriented flows—flows of orientation—of more or less intense reorientation of the vortical (axial) directions of the elementary vortical fluctuations, of the Elementary Events.

These torsions in the Elementary Events, precisely because they occur on extremely small scales and at the level of individual event points of the Field, can be called Micro-torsions or Micro-vorticities.

Rotation is therefore the Field's spontaneous way of reorganizing divergences (Translation) into curl patterns—localized rotational regions—that embody the Principle of Conservation: every propagation of variations tends to generate rotational persistence, every fluctuation tends to induce a torsional memory that affects its own existence and causal permanence.

In this context, Rotation is a form of Conservation produced by the Field in reaction to the Propagation of the most elementary Variations, to translation—a Variation that takes the form of rotational dynamics, of axial reorientation of the vortical aspects of sub-elementary fluctuations.

Rotation can be seen and understood as Propagation and Conservation in Time.

From the point of view of the Elementary Events and the Dynamics of Elementary Action

Rotation transmutes the basic forms of Action, Perturbation and mainly Translation, into an incessant Process of Conservation of Action, conservation of inhomogeneities in the Field, in its dynamics at the most elementary level of organization.

The key question is: what really varies? What varies is the pre-temporal distribution of the Elementary Events, the tiny sub-elementary sigma-fluctuations of the Field.

The essential question is: what conserves? What conserves are the inhomogeneities (in the forms of Perturbation and Translation) in the distributions of Elementary Events and the correlation schemes, the patterns of Relations among them.

The essential question is: what is conserved? What is conserved are the inhomogeneities—manifesting as Perturbation and Translation—in the distributions of Elementary Events, together with the correlation schemes, the patterns of Relations among them.



The essential question is: what propagates? What propagates are inhomogeneities of vorticity and axis‑orientation in the distributions of Elementary Events and in their correlation schemes. Any deviation from isotropy in the vortical patterns of the Field tends to propagate through Rotation, generating micro‑torsions and micro‑vorticities at every scale.

Let me resume here what was stated in the corresponding subsection (The variational perspective) of the previous section dedicated to Translation, adapting it to Rotation.

In the absence of Perturbation, the vortical orientation of elementary fluctuations is entirely random and stochastically independent among neighboring event‑points.

The presence of Perturbation introduces inhomogeneities not only in the temporal distribution of spatial amplitudes, but also in the temporal distribution of vortical amplitudes—the duration and intensity of the tiny vortical fluctuations that characterize each event‑point.

To each vortical amplitude corresponds a temporal amplitude: larger vortical fluctuations entail longer durations, and vice versa. We may imagine these amplitudes as normally distributed around a universal mean value, generating a corresponding temporal distribution that describes how the vortical dynamics of an event‑point evolves over Time.

With great simplification, this distribution may be expressed as a kind of quasi‑frequency of vortical pulsation. Although not truly periodic, it provides a statistical measure of how much an infinitesimal volume of the Field is, on average, more or less torsionally “excited” compared to a standard value.

If an event‑point exhibits, on average, more intense vortical fluctuations for longer durations, it manifests negative Perturbation in the vortical domain. By the General Equivalence Principle for Perturbation, this corresponds to:

  • a curvature of the Metric (slower temporal flow),
  • a non‑zero density of negative Mass,
  • a deviation of the limiting speed of propagation of Causality/Action/Information,
  • a deviation of the “temperature” of the Elementary Field.

The dynamics of the Elementary Field—its incessant vortical fluctuations and their temporal distributions—is a fundamental aspect for deriving the Fundamental Physical Quantities associated with Rotation:

  • from the point of view "of" the Field:
    • magnetic field and magnetic permeability,
    • torsional Motion and rotational inertia,
  • and from the point of view "in" the Field:
    • magnetic Spin and intrinsic magnetic moment,
    • electric and magnetic potentials.

Now that we have revisited the dynamics that lead to the emergence of Perturbation, we can see how from Perturbation arises Rotation, and how from gradients emerges curl.

If, due to some phenomenon, a region of the Field develops a vortical configuration—a micro‑vortex, a torsional imbalance—then in the surrounding region Rotation will manifest: a reorientation of the vortical axes of the elementary fluctuations around the center of that torsional configuration. Rotation can be represented as a non‑zero curl in the Field, centered on the volume exhibiting vortical Perturbation.

We may imagine an infinitesimal volume of the Field, comparable to the Planck scale, undergoing incessant torsional pulsation. The extension, shape, and orientation of this micro‑vortex can be described by a stochastic function whose mean and variance depend on the intensity of the Fundamental Force and the resistance of the Field to torsional deformation.

At scales just above the Planck scale, these micro‑torsions compensate, leaving Space flat; but at the level of Elementary Action they generate torsional inhomogeneities that the Field attempts to rebalance.

Rotation is the Field’s spontaneous way of reorganizing gradients of vortical Perturbation into curl patterns, localized torsional regions that embody the Principle of Conservation in Time. Every torsional variation tends to persist, generating rotational memory and influencing the causal permanence of the event‑points involved.

Rotation is therefore a form of Variation produced by the Field in reaction to the most elementary Variations, to Perturbation—a Variation that takes the form of torsional modulation, of axial reorientation of the elementary vortical fluctuations.

Rotation can be seen and understood as Propagation and Conservation in Time.

From the point of view of the Fundamental Principles of Evolutionary Dynamics

The Information / Energy perspective

Formulation in vector calculus

The phenomenological perspective

Charge and Electric potential

Electric Fields (E): Generated by electric charges; they exert forces on other charges.

Spin and Magnetic potential

Magnetic Fields (B): Generated by moving charges (currents); they exert forces on other moving charges.

Maxwell's Equations

The foundation of the theory, consisting of four equations that relate electric and magnetic fields to their sources (charge density and current density).

Electromagnetic Waves

Propagating waves of electric and magnetic fields that travel at the speed of light.




Rotation represent the specification of "Conservation" at the most elementary level of Physical Reality.

Rotation is a specification of Conservation. It is the third fundamental mode of expression of the Elementary Action.

In term of Elementary Events, Rotation represents any action that tends to conserve Perturbation in the Elementary Field.

Perturbations, inhomogeneities in the distributions of the intensities of the Elementary Events, the intensities of the deformations of the Elementary Field under the incessant pressure of the Fundamental Force, the products of the Elementary Events, "conserve", persist, showing rotation, chirality, vorticity in the Field.

In the Dynamics of the Elementary Action we call "Rotation ", or Rotational Action, the vorticity of the perturbations, produced by the action tending to conserve inhomogeneity, in the points of expression of the action itself.

Like Perturbation, we cannot say that Rotation is something that exists, an entity. It is a phenomenon, a product of the dynamics of the inner interaction of the two aspects of the Source of Reality.

Rotation is associated with the Operator of Reality that has the same name and is the expression of the Fundamental Principle that we call Principle of Conservation (the mutations described by the Principle of Variation, the incessant variations of the states of the Entities, the Events, always tend to conserve). The Operator Rotation organizes Perturbations, the fluctuations of the Acted / Fields, in vortices of Action directed to maintain homogeneity, isotropy of the Elementary Field. The Operator Translation is responsible for the formation of vortices / curl aimed at conserve gradients (inhomogeneities, anisotropies) in the Acted / Fields.

Perturbation is the way in which the Elementary Field react to the Elementary Action of the Fundamental Force, the way in which the Elementary Field receive the expression of the Fundamental Force, its the product of the expression of Causality at the most elementary level of Reality. Variation is the root of the emergence of the Dimensionalities of Reality that we call Spatiality and Temporality, the fundaments of Space and Time.

Rotation is the way in which the Elementary Field reacts to the appearing of Perturbations produced by the Action of the Fundamental Force, the way in which the Elementary Field receives the expression of the Fundamental Force, it is the product of the expression of Causality. Rotation is the root of the emergence of the Dimensionality of Reality that we call Temporality, the fundament of Time.

The best question with respect to Translation undoubtedly is: what conserve? What conserve is the inhomogeneity in the distributions of the Elementary Events and the patterns of correlations among them. Any departure from an homogeneous distribution of Events as any change in that distribution (and in their stochastic independence / correlation) represents a variation that tend to conserve to maintain inhomogeneity. At any level of complexity, from the most elementary to the Universe as a whole, Reality can be described as a net of more or less independent / correlated stochastic processes that conserve in the Fields.

The Modes of Action are intimately connected with the Dimensionalities of Reality: Physical Causality and Physical Variationality, that produce the Dimensions of Physical Reality: Cause, Space and Time. They represent three fundamental operators in the Formation of Physical Reality. The Operator Perturbation can be seen as the result of expression of Physical Causality, the active side of Physical Agency, while Translation and Rotation can be seen as expressions of the two aspects of its passive side, Physical Variationality: Physical Spatiality and Physical Temporality.

The Operator Rotation is responsible for the formation of curl in the Elementary Field, conservation of inhomogeneities, anisotropies, in the Field and their dynamics.

In vector calculus, Rotation is embodied by the vector operator called "Curl" (CURL or ROT), which represents a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

Rotation is:


Rotation Sub-modes / Sub-components

Rotation:Chirality

Rotation:Chirality is the first of the two sub-modes of Rotation. It refers to chirality, the sense of circulation of Elementary Action, the sense of circulation of the Elementary Events in the Elementary Field. It is the root of Charge and electric phenomena.

Rotation:AxisOrientation

Rotation:AxisOrientation is the second of the two sub-modes of Rotation. It refers to the spatial direction of orientation of the axis of vorticity in circulation of Elementary Action, the direction of the orientation of the axis of vorticity in circulation of the Elementary Events in the Elementary Field. It is the root of Spin and magnetic phenomena.


Charge

Rotation:Chirality is the causal origination of Charge, while Translation is the variational effect produced by the presence of different sources of Rotation in the Field, which produces attraction / repulsion of charges and the electric domains.


Spin

Rotation:AxisOrientation is the causal origination of Spin, while Translation is the variational effect produced by the presence of different orientations of the axis of Rotation in the Field, which produces attraction / repulsion of spins and the magnetic domains.



Rotation:Chirality

Parameter that expresses "the chirality of the spatial fluctuations (spatial distances) of the Elementary Field" (spatial chirality of the variational effects on the Elementary Field produced by the causal Action of the Fundamental Force, chirality of the Elementary Action, the most elementary form of Existence, primary manifestation of the Source of Reality).

The correlation between the Rotation:Chirality values ​​in a given portion of the Elemental Field (Relationship between Elementary Action of a contiguous set of event points) defines the Elemental Energy of that portion in the Component Rotation:Chirality, which we call Charge, the root of electric phenomena.


Rotation:AxisOrientation

Parameter that expresses "the orientation of the axis of rotation/chirality (spin axis) of the spatial fluctuations (spatial distances) of the Elementary Field" (orientation of the axis of rotation/chirality - spin axis - of the variational effects on the Elementary Field produced by the causal Action of the Fundamental Force, orientation of the axis of rotation/chirality - spin axis - of the Elementary Action, the most elementary form of Existence, primary manifestation of the Source of Reality).

The correlation between the Rotation:AxisOrientation values ​​in a given portion of the Elementary Field (Relationship between Elementary Action of a contiguous set of event points) defines the Elementary Energy of that portion in the Rotation:AxisOrientation Component, which we call Spin, the root of magnetic phenomena.


Rotation represent the specification of "Conservation" at the most elementary level of Physical Reality.

Rotation is a specification of Conservation. It is the third fundamental mode of expression of the Elementary Action.

In term of Elementary Events, Rotation represents any action that tends to conserve Perturbation in the Elementary Field.

Perturbations, inhomogeneities in the distributions of the intensities of the Elementary Events, the intensities of the deformations of the Elementary Field under the incessant pressure of the Fundamental Force, the products of the Elementary Events, "conserve", persist, showing rotation, chirality, vorticity in the Field.

In the Dynamics of the Elementary Action we call "Rotation ", or Rotational Action, the vorticity of the perturbations, produced by the action tending to conserve inhomogeneity, in the points of expression of the action itself.

Like Perturbation, we cannot say that Rotation is something that exists, an entity. It is a phenomenon, a product of the dynamics of the inner interaction of the two aspects of the Source of Reality.

Rotation is associated with the Operator of Reality that has the same name and is the expression of the Fundamental Principle that we call Principle of Conservation (the mutations described by the Principle of Variation, the incessant variations of the states of the Entities, the Events, always tend to conserve). The Operator Rotation organizes Perturbations, the fluctuations of the Acted / Fields, in vortices of Action directed to maintain homogeneity, isotropy of the Elementary Field. The Operator Translation is responsible for the formation of vortices / curl aimed at conserve gradients (inhomogeneities, anisotropies) in the Acted / Fields.

Perturbation is the way in which the Elementary Field react to the Elementary Action of the Fundamental Force, the way in which the Elementary Field receive the expression of the Fundamental Force, its the product of the expression of Causality at the most elementary level of Reality. Variation is the root of the emergence of the Dimensionalities of Reality that we call Spatiality and Temporality, the fundaments of Space and Time.

Rotation is the way in which the Elementary Field reacts to the appearing of Perturbations produced by the Action of the Fundamental Force, the way in which the Elementary Field receives the expression of the Fundamental Force, it is the product of the expression of Causality. Rotation is the root of the emergence of the Dimensionality of Reality that we call Temporality, the fundament of Time.

The best question with respect to Translation undoubtedly is: what conserve? What conserve is the inhomogeneity in the distributions of the Elementary Events and the patterns of correlations among them. Any departure from an homogeneous distribution of Events as any change in that distribution (and in their stochastic independence / correlation) represents a variation that tend to conserve to maintain inhomogeneity. At any level of complexity, from the most elementary to the Universe as a whole, Reality can be described as a net of more or less independent / correlated stochastic processes that conserve in the Fields.

The Modes of Action are intimately connected with the Dimensionalities of Reality: Physical Causality and Physical Variationality, that produce the Dimensions of Physical Reality: Cause, Space and Time. They represent three fundamental operators in the Formation of Physical Reality. The Operator Perturbation can be seen as the result of expression of Physical Causality, the active side of Physical Agency, while Translation and Rotation can be seen as expressions of the two aspects of its passive side, Physical Variationality: Physical Spatiality and Physical Temporality.

The Operator Rotation is responsible for the formation of curl in the Elementary Field, conservation of inhomogeneities, anisotropies, in the Field and their dynamics.

In vector calculus, Rotation is embodied by the vector operator called "Curl" (CURL or ROT), which represents a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

Rotation is:


Rotation Sub-modes / Sub-components

Rotation:Chirality

Rotation:Chirality is the first of the two sub-modes of Rotation. It refers to chirality, the sense of circulation of Elementary Action, the sense of circulation of the Elementary Events in the Elementary Field. It is the root of Charge and electric phenomena.

Rotation:AxisOrientation

Rotation:AxisOrientation is the second of the two sub-modes of Rotation. It refers to the spatial direction of orientation of the axis of vorticity in circulation of Elementary Action, the direction of the orientation of the axis of vorticity in circulation of the Elementary Events in the Elementary Field. It is the root of Spin and magnetic phenomena.


Charge

Rotation:Chirality is the causal origination of Charge, while Translation is the variational effect produced by the presence of different sources of Rotation in the Field, which produces attraction / repulsion of charges and the electric domains.


Spin

Rotation:AxisOrientation is the causal origination of Spin, while Translation is the variational effect produced by the presence of different orientations of the axis of Rotation in the Field, which produces attraction / repulsion of spins and the magnetic domains.

Links to the tables of contents of TFNR Paper


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