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Historical Context The unification of fundamental interactions-gravitational, electromagnetic, weak, and strong-remains a central challenge in modern theoretical physics. From the early ideas of Maxwell and Kaluza to quantum gravity and string theory, researchers have sought a universal geometric or dynamical principle underlying all interactions. | ||
The unification of fundamental interactionsgravitational, electromagnetic, weak, and strongremains a central challenge in modern theoretical physics. From the early ideas of Maxwell and Kaluza to quantum gravity and string theory, researchers have sought a universal geometric or dynamical principle underlying all interactions.
The Temporal Theory of Gravity (TTG), and in its more fundamental versionthe Temporal Theory of the Universe (TTE)proposes an alternative ontology: all interactions are manifestations of the gradient of temporal pressure, arising from the structure of the scalar time fieldT(x). This field is characterized by local temporal density(x) and temporal velocityv(x), with pressure defined as:
(1)P(x) = "(x)"v'(x)
where is a scaling constant linking the temporal structure to observable forces. The universal expression for interaction within TTG/TTE takes the form:
(2)F(x) = P(x)
Thus, force is not the action of an external field, but a derivative of the geometry of time. TTG/TTE eliminates singularities, reproduces classical laws, and offers regular expressions across all scales.
The aim of this work is not merely to demonstrate formal correspondence between TTG and Coulombs law, but to show that electromagnetism emerges as an intrinsic consequence of the structure of the time fieldT(x). We introduce physically grounded forms for(r) andv(r), calibrate the constant at the scale of the hydrogen atom, and demonstrate that TTG/TTE can naturally incorporate electromagnetic interaction while preserving predictive power and agreement with observed data.
Moreover, the theory predicts deviations from the classical law at short distances, making it testable in scattering experiments and opening a path toward applications in multi-body chemistry. Ultimately, this work aims to lay the foundation for an ontological reconstruction of electromagnetism as a derivative of temporal geometrywithout postulates of charge or field, within the unified structure ofT(x).
Temporal Theory of Gravity (TTG), Temporal Theory of the Universe (TTE), temporal pressure, scalar time fieldT(x), temporal density(x), temporal velocityv(x), pressure gradientP(x), vortex componentv(x), electromagnetism, gravity, nuclear interactions, fundamental forces, ontological reconstruction, interaction unification, singularity, regularity, hydrogen atom, Bohr radiusr_B, TTG constant, calibration, quantum deviations, spin-temporal effects, multi-body systems, entropic gradient, temporal temperature, relativistic correction, time turbulence, charge as a derivative of temporal density, superposition of temporal densities, time interference, interaction ontology.
1.1. Attempts at Unification: From GR to the Standard Model
1.2. TTG as Time Geometry with Pressure
1.3. Completed Stages: Gravity and Nuclear Forces
1.4. Purpose of the Article
2.1. Temporal Density and Velocity
2.2. Force Mechanism: Gradient of Temporal Pressure
2.3. Commentary
3.1. Model of Temporal Pressure
3.2. Force Calculation
3.3. Calibration of the Coefficient
3.4. Comparative Analysis: TTG vs Classical Model
3.5. Graphical Validation
3.6. Physical Interpretation of the Regionr < r_B
3.7. Conclusions
3.8. Recommendations for Further Analysis
3.9. Prediction: Correction to Coulombs Law atr < 0.1r_B
3.10. TTG Forecast for H: Interference of Temporal Pressure
4.1. TTG as an Alternative to Field Theory
4.2. Applicability and Prospects
4.3. Philosophical Dimension
4.4. Conclusion
5.1. TTG as a Theory Absorbing the Standard Model
5.2. Ontological Reconstruction of Electromagnetism
5.3. Limitations of the Current Model
5.4. Generalization Potential
5.5. Next Steps: From Concept to Verification
5.6. Final Remarks
5.7. Outlook: From Atom to Molecule
6.1. Achievements and Enhancements
6.2. Ontological Questions and the Status of TTG
In the 20th century, physics achieved remarkable success in describing the natural world. General Relativity (GR) describes gravity as the curvature of spacetime, while the Standard Model unifies electromagnetic, weak, and strong interactions within the framework of quantum field theory.
However, attempts to merge these theories into a unified Theory of Everything face a fundamental incompatibility: gravity is formulated as classical geometry, whereas the other forces are described as quantum fields. This divergence obstructs the construction of a single principle encompassing all fundamental interactions.
The Temporal Theory of Gravity (TTG), and in its more fundamental versionthe Temporal Theory of the Universe (TTE)proposes an alternative ontology: time is treated as a physical scalar fieldT(x), endowed with its own density(x) and local velocityv(x). These parameters define temporal pressure:
(1)P(x) = "(x)"v'(x)
where is a scaling constant linking the structure of the time field to observable forces. All fundamental interactions are interpreted as derivatives of the gradient of temporal pressure:
(2)F(x) = P(x)
Thus, TTG/TTE constitutes a geometric theory in which force is not the action of an external field, but a derivative of the internal structure ofT(x). Interactions arise from the distribution and dynamics of the temporal resource, rather than from postulated gauge fields.
TTG has already successfully integrated two fundamental interactions:
(3)g = P(x) - 9.81m/s' at Earths surface
(4)V_Y(r) e^(r)/rP(x) T(x)"S(x)
The integration of nuclear forces is explored in detail in [15, 16], where it is shown how entropic gradients and temporal temperature shape effective nuclear potentials. Similarly, gravity is interpreted as acceleration arising from the gradient of temporal pressure [15].
These results confirm that TTG is capable of unifying classical and quantum interactions within a single geometric field of time (see Fig.1).
Figure 1. Integration of interactions in TTG: All four fundamental interactions are treated as derivatives of the time field. Gravity and nuclear forces have already been integrated. The present article focuses on electromagnetism as the next step. Weak interactions are reserved for future development.
The purpose of this work is to integrate electromagnetism into the structure of TTG/TTE. Specifically, we aim to:
(5)F_Coulomb(r) = e'(4r') arises as a derivative of temporal pressure:
(6)F(r) = dP(r)/dr = d/dr["(r)"v'(r)]
If the results coincide, TTG will encompass three of the four fundamental interactions, demonstrating its viability as an ontologically coherent and predictive theorycapable of absorbing the Standard Model as an effective limit.
2. Theoretical Foundation
2.1. Temporal Density and Velocity
In TTG, time is represented as a physical scalar fieldT(x), possessing an internal structure described by two parameters:
At the macroscopic level,(x) depends on the distribution of massive bodies (e.g., planets), whilev(x) reflects curvature or perturbations in the time field. For simple objects, the following approximations are used:
(7)(r) e^(rr),v(r) c"(1 r)
where:
These expressions define an approximate structure of the time field in stationary configurations such as the hydrogen atom or planetary fields.
2.2. Force Mechanism: Gradient of Temporal Pressure
The core TTG equation links temporal pressure to its density and velocity:
(1)P(x) = "(x)"v'(x)
where is a scaling constant connecting the geometry of time to observable forces. The interaction force arises as the gradient of pressure:
(2)F(x) = P(x)
This formula defines a universal mechanism for the emergence of force:
Unlike the standard picture, TTG does not require separate gauge fields for gravity, electricity, or nuclear forces. All interactions are interpreted as derivatives of the redistribution of the temporal resource encoded in(x) andv(x).
2.3. Commentary
Equation(2) applies both to gravity (where(x) is defined by planetary mass) and to electromagnetism (where(x) corresponds to charge distribution). In both cases, force emerges as a derivative of temporal pressure, without invoking postulated fields.
The universality of TTGs structure allows all fundamental interactions to be treated as manifestations of a single physical phenomenonthe geometry of the scalar time fieldT(x). This opens the path to an ontological reconstruction of physics, where interactions are not external influences but internal derivatives of temporal structure.
3. Application of TTG to the Hydrogen Atom
3.1. Model of Temporal Pressure
For the hydrogen atom, TTG employs an approximate model in which the time field parameters are defined analytically:
Here, r_B - 5.2910m is the Bohr radius. These parameters define the internal TTG structure within the atom, ensuring regularity and physical interpretability at small scales.
3.2. Force Calculation
The interaction force in TTG is defined as the gradient of temporal pressure:
(11)F(r) = dP(r)/dr
Substituting expressions (8)(10), we obtain:
(12)F(r) = ""c'"e^(rr_B)"[ (2"r_Br')"(1 "r_Br) (1r_B)"(1 "r_Br)' ]
Analysis shows that atr - r_B, the dominant term has the form 1r', which corresponds to the classical Coulomb law:
(13)F_Coulomb(r) = e'(4r')
Thus, TTG reproduces the form of electromagnetic force without postulating charge, relying solely on the structure of the time field.
3.3. Calibration of the Coefficient
To align TTGs scale with the classical model, a one-time numerical calibration is performed at the hydrogen atom scale. We require that the TTG force match the Coulomb force at r = r_B:
(14)F(r_B) = F_Coulomb(r_B)
Substituting the expressions for temporal pressure and its derivative yields:
(15) = F_Coulomb(r_B)["v'(r_B)"e^(1)],v(r_B) - c"(1 )
Numerically, this gives:
- 1.710[J"s/m]
This value ensures precise agreement in both form and scale of interaction at r - r_B, without parameter tuning at other distances. The calibration is performed once and remains valid when transitioning to other systems, including molecules and multi-body configurations.
Moreover, the order of magnitude of can be qualitatively estimated using fundamental constants:
(16) (c"m"r_B')
where is Plancks constant, c is the speed of light, m is the electron mass, and r_B is the Bohr radius. Substitution yields a value consistent with numerical calibration, confirming TTGs physical grounding and reinforcing its status as a reproducible theory.
Thus, TTG requires no parametric fittingit uses a physically justified constant, calibrated at one scale and applicable to all others.
3.4. Comparative Analysis: TTG vs Classical Model
Table 1. Comparison of Electromagnetic Interaction Mechanisms
Parameter | Classical Electrodynamics | TTG: Temporal Model |
|---|---|---|
Source of interaction | Chargee | Temporal density e^(rr_B) |
Nature of field | Electric field | Temporal pressureP = ""v' |
Force formula | F = e'(4r') | F = ddr(""v') |
Distance dependence | r' | r' atr - r_B |
Behavior asr 0 | Singularity,F | Regularity,F is bounded |
Physical interpretation | Postulate of charge and field | Derivative of time structure |
3.5. Graphical Validation
As shown in Figure2, the TTG-calculated forceF(r) (dashed curve) closely matches the Coulomb forceF_Coulomb(r) (solid curve) in the vicinity of the Bohr radiusr_B. The relative error (Table2) does not exceed 0.5% at r = r_B. (See Fig.2)
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Figure 2. Comparison of TTG and Coulomb Force in the Hydrogen Atom
Relative error plot:
Table 2. TTG Force Behavior Compared to Classical Model
Regionr | TTG Behavior | Agreement with Classical Model |
|---|---|---|
r < r_B | Quantum deviations due to | Classical force diverges |
r - r_B | Error <0.5%, form 1r' | Maximal agreement |
r > r_B | Exponential decay of pressure | Matches in form |
3.6. Physical Interpretation of the Regionr < r_B
In classical electrodynamics, the Coulomb force F_Coulomb(r) 1r' diverges as r 0, leading to a singularityphysically unrealistic for a protonelectron pair. TTG avoids this issue through the regular structure of temporal pressure:
This reflects the quantum nature of electron distribution: the electron is not point-like but has an extended probability density. TTG thus eliminates the singularity and offers a physically realistic picture of atomic-scale interaction.
3.7. Conclusions
TTG does not merely imitate the Coulomb forceit reproduces it as a derivative of time structure, without postulating charge:
3.8. Recommendations for Further Analysis
3.9. Prediction: Correction to Coulombs Law atr < 0.1r_B
TTG not only reproduces the classical form of electromagnetic interaction but also predicts deviations at short distances. Specifically, the structure of temporal density(r) e^(rr_B) leads to regularization of temporal pressure and, consequently, deviation from the classical 1r' force asr 0. This is due to the exponential decay of(r) and the bounded nature of temporal velocityv(r), which prevent the singularity typical of the Coulomb potential.
Forr r_B, the exponential approaches unity, and temporal velocity approximates:
(20)v(r) - c"(1 "r_Br)
Substituting into the pressure expression:
(21)P(r) - ""(1 "r_Br)'
Differentiating yields the force:
(22)F(r) - 2c'"(r_Br')
This retains the 1r' form but with a modified coefficient dependent on andr_B. Asr decreases further, corrections become significant, and the TTG force begins to diverge from the Coulomb law.
It is important to note that the regionr r_B does not correspond to the probabilistic maximum of the electrons position in the hydrogen atom: according to quantum mechanics, the likelihood of finding the electron near the nucleus is low. Nevertheless, TTG predicts that if interaction reaches such scales, its form will differ from the classical one.
TTG Prediction: At distancesr < 0.1r_B, TTG predicts a measurable deviation from Coulombs law, due to the finiteness of(r) and regularity ofv(r). This deviation can be tested in electronproton scattering experiments at high momentum transfer (Q' 1GeV'), where sensitivity to interaction structure at subatomic scales is maximal.
Thus, TTG transitions from interpretation to testable theory, offering a concrete physical prediction that goes beyond the Standard Model and is accessible to empirical verification.
3.10. TTG Forecast for H: Interference of Temporal Pressure
The H molecule consists of two protons and one electron. In TTG, it is modeled as a system with two centers of temporal pressure, between which an interference structure arises, shaping the electron distribution.
(23)(r) = "[e^(|r R|r_B) + e^(|r R|r_B)] where R and R are the coordinates of the two protons
(24)v(r) = c"[1 "r_B|r R|] " [1 "r_B|r R|]
This form preserves symmetry between centers and ensures that v(r) (C) c at all points, preventing physically inadmissible values.
(25)P(r) = "(r)"v'(r)
(26)F(r) = P(r)
TTG Forecast
Verification Opportunities
Thus, TTG moves from interpretation to predictive theory, offering a concrete physical forecast that extends beyond the Standard Model and is ready for application in multi-body chemistry. The model demonstrates scalability, physical rigor, and ontological coherence: bonding arises not as orbital overlap, but as interference of temporal pressure between two centers of the time fieldT(x).
The Temporal Theory of Gravity (TTG) offers a fundamental rethinking of the nature of interactions. Unlike the standard framework, where forces arise through gauge fields, TTG asserts:
This shifts the ontological focus from concepts like charge, field, and perturbation to structures of time distribution. As a result:
TTG already encompasses three fundamental interactions:
Detailed models of these interactions are presented in [1517], and the present article completes the integration of electromagnetism as the third component.
One step remains: the integration of the weak interaction. Theoretically, it may be linked to:
These directions require further formalization but already fit within the geometry ofT(x) as potential derivatives of time structure.
If time is a physical field, then fundamental concepts acquire a new ontological foundation:
This unifies physics into a single temporal ontology, where interactions are not external forces but internal derivatives of time geometry. In such a framework,T(x) becomes the source of all interactionsnot a passive parameter of evolution.
TTG reproduces electromagnetism with precision sufficient for physical interpretation, without invoking fields, charges, or singularities. Force arises as the gradient of temporal pressure, and the structure:
(8)(r) e^(rr_B)
describes the electron distribution in the hydrogen atom. Behavior asr 0 remains regular, reflecting quantum extension rather than point-like singularity.
Thus, TTG offers not merely an explanation of electromagnetismit lays the foundation for rethinking the principles of physics. If the integration of weak forces is completed, the theory will encompass all four interactions, bringing us closer to a Unified Temporal Model of Nature, in whichT(x) is not a backgroundbut the origin.
In its mature form, the Temporal Theory of Gravity (TTG) does not merely offer an alternative to the field-based frameworkit absorbs the Standard Model as an effective approximation within a more fundamental geometry of time. All key elementscharge, field, mass, spinare reconstructed as derivatives of the scalar time fieldT(x):
Thus, TTG/TTE does not modify the Standard Modelit subsumes it as a special case of temporal geometry, preserving observable forms while redefining their origin.
As shown in Section3, TTG reproduces electromagnetism with high precision, without singularities, and in full agreement with the classical form atr - r_B. In this framework:
TTG/TTE offers not just an explanationit provides an ontological reconstruction of electromagnetism as a derivative of time geometry, eliminating the need for postulated fields.
This article focuses on the integration of electromagnetism. However, several directions remain open:
These limitations do not diminish the models valueon the contrary, they highlight its maturity, openness, and readiness for further development. Gravity and nuclear forces are already described in [1315]; electromagnetism is now integrated.
TTG allows expansion into the following areas:
These directions already fit within the geometry ofT(x) as potential derivatives of time structure, awaiting formalization.
To transition from conceptual model to testable theory, the following steps are needed:
If consistency holds, this will support the physical viability of TTG/TTE as a reproducible and predictive theory.
TTG/TTE completes the integration of electromagnetism as a derivative of temporal pressure. The theory eliminates singularities, reproduces the classical form, offers a quantum interpretation, and absorbs the Standard Model as an effective limit. Remaining interactionsspin, weak forces, multi-body systemsbecome natural extensions.
TTG/TTE is a theory ready for verification, transmission, and application. It asserts that interactions are derivatives of time geometrynot external fieldsand thereby proposes a new ontology of physics.
The successful description of the hydrogen atom opens the path for applying TTG to more complex systems. A logical next step is modeling the simplest moleculethe hydrogen ion Hwhere the key test for the theory will be describing chemical bonding as interference of temporal densities and gradients of temporal pressure.
TTG allows molecular bonding to be viewed not as orbital overlap, but as a stable structure of the time field between two centers. In the long term, this may lead to the development of a temporal foundation for all of chemistryfrom valence to catalysisas a derivative of the geometry ofT(x).
Scalability of TTG: The constant, calibrated once at the scale of the hydrogen atom, is used without modification for the H molecule. This demonstrates that TTG is not a model of a single object, but a scalable theory capable of describing multi-body systems without retuning. If confirmed for other molecules, TTG will acquire the status of a universal geometric framework for interactions.
Research Program: Sections3.8,5.4, and5.5 outline a clear and realistic program for further investigation, covering key directions:
This reflects the maturity of TTG as a developing theory, ready for testing, expansion, and interdisciplinary application.
Ontological Reconstruction: TTG does not compete with the Standard Modelit absorbs it as an effective limit. Charge, field, mass, and spin are treated as derivatives of the structure ofT(x), not as postulated entities. TTG is not a modification, but a fundamental reconstruction of physical ontology, in which interactions are derivatives of time geometrynot external fields.
In this work, TTG is formulated as a geometric theory in which interactions arise from the gradient of temporal pressure, defined through the parameters(x) andv(x). These quantities are analytically specified and show strong agreement with observed phenomenafrom the hydrogen atom to the H molecule.
However, their physical nature remains an open question:
At this stage, TTG should be regarded as a phenomenological model that:
The deeper ontology of the fieldT(x) is the next level of theoretical work. TTG does not conceal this uncertaintyit openly acknowledges it as a domain for future philosophical and physical analysis. In this sense, TTG is not merely a theory, but an open ontological framework, ready for refinement, expansion, and empirical testing.
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