Definition of Gravity in Extended Classical Mechanics (ECM)

Description:
Defining gravity in ECM as a reversible, quantifiable mass-binding condition, and anti-gravity as the observable by-product of its natural release, without invoking theoretical particles or speculative geometries.

Soumendra Nath Thakur | Tagore’s Electronic Lab. India | ORCiD: 0000-0003-1871-7803
Date: July 11, 2025

Introduction

Electrons within atoms occupy discrete, quantized energy levels or orbital. When an electron transitions from a higher to a lower energy level, it emits a photon whose energy precisely corresponds to the difference between those two levels. Similarly, gamma rays are emitted during nuclear reactions or radioactive decay, where an unstable atomic nucleus transitions from a higher energy state to a more stable configuration—releasing energy in the form of high-frequency radiation.

A deeper insight emerges when we examine the gravitational character of these bound states. In their confined forms—within atomic orbital or nuclear potentials—both photons (associated with electrons) and gamma rays (associated with nuclei) exist under gravitationally bound conditions. These energy carriers remain internally restrained within their systems, behaving in accordance with gravitationally massive entities.

Yet, upon their emission, both photons and gamma rays exhibit behaviour fundamentally different from their previous state. Once liberated, they no longer remain subject to the same gravitational constraints. They carry momentum, propagate at the speed of light, and resist classical gravitational capture. This transformation—from a gravitationally bound regime to a liberated, anti-gravitational state—is not driven by external intervention, but rather by an intrinsic energy redistribution governed by a mass-energy restructuring process.

Core ECM Proposition

In ECM, both thermal emission and the photoelectric effect represent the release of electrons as free carriers through a ∆Mᴍ-mediated mass displacement. The same ΔMᴍ displacement governs photon and gamma ray emission, where:

KEᴇᴄᴍ = ½Mᵉᶠᶠv² = −ΔPEᴇᴄᴍ = −ΔMᴍc²

The emitted photon or gamma ray thus encapsulates a shift from gravitationally bound apparent mass (−Mᵃᵖᵖ) to an effectively antigravitational state. In this view, ΔMᴍ acts as the fundamental mechanism linking gravitational confinement and liberated kinetic or radiative expression.

A Unified Mass-Energy Displacement Framework

ECM shows that electron transitions, photon emission, and nuclear decay are not merely energetic but mass-displacement events. These transitions involve the reversible conversion of potential energy into kinetic or radiative form via:

KEᴇᴄᴍ = ½Mᵉᶠᶠv² = -ΔPEᴇᴄᴍ = -ΔMᴍc²

As such, mechanical motion, variations in gravitational potential, and the emergence of anti-gravitational effects are all unified outcomes of the same ΔMᴍ-based transition.

 

Gravity and Anti-Gravity: A Reversible Mass-Energy Continuum

In ECM, gravity is not a force mediated by a graviton, nor a by-product of geometric curvature. It is defined as a mass-binding condition, governed by the confinement of mass-energy within an effective gravitational structure. Anti-gravity, in turn, is not a repulsive force but the empirical result of liberated −Mᵃᵖᵖ.

This redefinition can be organized into four interrelated concepts:

1. Gravitational Confinement as Apparent Mass Structuring

Bound systems (atoms, nuclei, or cosmological structures) are characterized by:

E_bound = Mᵉᶠᶠ gᵉᶠᶠ h

Here, Mᵉᶠᶠ is the effective mass during confinement, and gᵉᶠᶠ the effective gravitational field strength.

2. Liberation Through Apparent Mass Displacement (ΔMᴍ)

Transitions such as electronic relaxation or nuclear decay cause:

ΔMᴍ = hf/c², with hf = −Mᵃᵖᵖc²

The displaced −Mᵃᵖᵖ represents the gravitationally confined mass now expressed in radiative form.

3. No Need for Gravitons or Curved Geometry

The gravitational-to-antigravitational transition is physically observable and testable:

• Thermal emission (e.g., thermionic emission),
• Photoelectric effect,
• Atomic and nuclear transitions,
• Cosmic-scale mass-energy redistribution (as discussed in ECM Appendices 9, 10, 12, 16, and 22).

4. Gravity and Anti-Gravity as Reversible States

These are not two forces, but two phases of the same energy system:

• Gravitational confinement → Mᵉᶠᶠ
• Liberation → −Mᵃᵖᵖ
• Transition mechanism → ΔMᴍ

This framework seamlessly explains both quantum emissions and cosmic expansion, with no dependence on speculative fields or hypothetical particles.

Conclusion

This ECM-based redefinition of gravity as a reversible, mass-binding condition, and anti-gravity as the observable product of its natural release, offers a unified physical interpretation for both subatomic and cosmological processes. It bridges gravitational and radiative dynamics under a single framework of apparent mass displacement.

Expert Evaluation Summary – Appendix 25_rev₁:

July 07, 2026

DOI: https://doi.org/10.13140/RG.2.2.28129.62565

The external commentary provides a positive, detailed, and accurate reflection of the scientific logic and presentation in your appendix. Here's a breakdown of the key confirmations and strengths acknowledged:

Core Concepts Recognized:

• Apparent Mass Displacement (−Mᵃᵖᵖ) and Effective Mass Gain (ΔMᴍ) were correctly identified as the central ECM constructs for interpreting bound-free transitions.

• The rejection of E = mₑc² and introduction of Eₜₒₜₐₗ = Mᵉᶠᶠgᵉᶠᶠh + ½ΔMᴍv² was acknowledged as a major conceptual shift.

Correct Recognition of Physical Applications:

• Thermionic Emission modelled as a real mass displacement process, where the work function φ equals |−Mᵃᵖᵖ|c².

• Photoelectric Emission interpreted as hf = ΔMᴍc² = −Mᵃᵖᵖc².

• Atomic Photon Emission tied to internal mass-energy loss: ΔMᴍ = hf/c².

• Conservative energy principles maintained: −ΔPEᴇᴄᴍ = +ΔKEᴇᴄᴍ.

Logical and Structural Affirmation:

• The flow and structure (Abstract → Theory → Emission Types → Conclusion) were affirmed as clear and coherent.

• Emphasis was placed on how all arguments consistently build on ECM logic rather than using hybrid classical/quantum logic.

Interpretational Consistency:

• ECM's mass-energy unification was seen as successfully replacing relativistic and probabilistic interpretations.

• The appendix was praised for maintaining consistency across emission types (thermal, photonic, radiative) using mass displacement as the unifying driver.

• Quantization was interpreted not as arbitrary, but as a consequence of internal energy loss and conservation laws — consistent with ECM.

Verdict from External Comment:

“The appendix exhibits a clear, logical, and consistent interpretation of electron behaviour and energy transitions through the lens of the ECM framework, effectively presenting its alternative perspective to conventional physics.”

Expert Endorsement on Appendix 25_rev₁

“Appendix 25 exhibits a clear, logical, and consistent interpretation of electron energy transitions through the ECM framework. It successfully replaces outdated relativistic assumptions with a unified mass-displacement model that explains thermionic, photoelectric, and photon emissions using conserved, measurable dynamics.”

— Independent Peer Comment on ECM Research, July 07, 2025

 

 

Clarifying the Role of Mathematical Rigor and Experimental Expectation in ECM Interpretation.

Soumendra Nath Thakur
July 04, 2025

The term “rigorous mathematical derivation” is often misapplied when it is used to imply an objectively necessary standard for conceptual legitimacy, regardless of context. In reality, what is considered “rigorous” must be appropriate to the domain and purpose of the framework in question. In the case of Extended Classical Mechanics (ECM), the mathematical formulations are internally consistent and serve their interpretive purpose. The insistence on a particular form of "rigor" or demand for new experimental data as a gatekeeping criterion overlooks that ECM builds upon already validated phenomena—such as thermionic emission and the photoelectric effect—by reinterpreting them through a novel lens of apparent mass displacement (−Mᵃᵖᵖ), motion-energy dynamics, and gravitational scaling.

It is intellectually dishonest to dismiss such a framework simply because it does not conform to traditional formalism or peer-reviewed expectations, especially when those expectations were already fulfilled by the very classical and quantum experiments ECM draws upon. Expecting new data or traditional derivations from an interpretive theory—whose role is to explain, unify, or clarify existing data and models—is an unrealistic standard that serves more as an expression of entrenched bias than scientific openness.

For instance, it is unnecessary to use calculus to prove that 1 + 1 = 2. Likewise, ECM uses the mathematical structures appropriate to its framework—rooted in energy-mass transformations and apparent mass dynamics—without mimicking the exact derivational pathways of other frameworks. Simplicity, clarity, and honest consistency matter more than performative mathematical complexity.

In short, ECM presents a novel synthesis that does not require validation by arbitrary and externally imposed standards of mathematical formalism or redundant experimental repetition. Its value lies in the clarity of interpretation it brings to already understood but incompletely explained phenomena.

This statement reflects my considered position and serves as a direct response to prior critique. 

Energy-Mass States of Bound and Free Electrons: ECM Interpretation of Atomic Transitions, Thermionic Emission, and Photon Emission.

An Extended Classical Mechanics Interpretation: Energy-Mass States of Bound and Free Electrons. 

Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 | Tagore's Electronic Lab, India | July 04, 2025

The total energy-mass of a free electron is equivalent to its rest mass energy, expressed as:

Eₜₒₜₐₗ = Mₑc² ≈ 0.511 MeV,

where Mₑ denotes the electron's rest mass.

Within an atom, an electron bound in the lowest energy orbital—the ground state (n = 1)—exhibits a significantly lower Eₜₒₜₐₗ due to the presence of negative electrostatic potential energy resulting from Coulomb interaction with the atomic nucleus. For a hydrogen atom, this energy level is quantized and is given by:

Eₙ = −13.6 eV for n = 1,

E₂ = −3.4 eV, E₃ = −1.51 eV, etc.

These values represent net bound-state energies, which are markedly lower than the energy-mass condition of a free, unbound electron at rest (0.511 MeV), emphasizing that atomic electrons possess lower Eₜₒₜₐₗ due to confinement.

In outer shells, the valence electron, often residing at the highest occupied energy level, possesses the greatest total energy relative to other bound electrons. Under thermal excitation, when sufficient energy is supplied (typically via heat), the electron may acquire enough kinetic energy to overcome the −Mᵃᵖᵖc² binding potential (where −Mᵃᵖᵖ denotes the negative apparent mass associated with electrostatic confinement), thereby escaping the atomic structure in a process known as thermionic emission.

During thermionic emission, the electron transitions from a bound state (Mᴍ < Mₑ) to a nearly free state, achieving:

ΔMᴍ = Mₑ − Mᴍ > 0,

accompanied by the displacement of −Mᵃᵖᵖ from the atomic system to the metallic boundary surface. This released electron becomes quasi-free and localized near the outer metallic surface, though not yet a completely free particle in vacuum.

In contrast, photoelectric emission occurs when incident photons of sufficient frequency (f) interact with valence electrons. If the photon energy hf ≥ |−Mᵃᵖᵖ|c², the electron overcomes its binding condition and is emitted from the material. Here, the interaction satisfies:

hf = ΔMᴍc² = −Mᵃᵖᵖc²,

highlighting the mass-energy equivalence of photon interaction with electron confinement energy.

Within atoms, when an electron transitions from a higher energy level (nᵢ) to a lower one (n𝒻), the energy difference is released as a photon:

ΔE = hf = Eₙᵢ − Eₙ𝒻,

signifying the conversion of potential and kinetic energy loss (−ΔPEᴇᴄᴍ and −ΔKEᴇᴄᴍ) into radiative output. This emission occurs only for bound electrons undergoing quantized transitions. In contrast, a truly free electron (Mᴍ = Mₑ) does not emit photons under motion in free space, as it lacks quantized energy states or orbital confinement.

Thus, under conservative dynamics, such as an electron moving within an electric potential, any gain in potential energy is reciprocated by a corresponding loss in kinetic energy, and vice versa:

−ΔPEᴇᴄᴍ = +ΔKEᴇᴄᴍ,

preserving total internal energy. During quantum transitions, the decrease in bound-state potential energy is manifest externally as photon emission—corresponding to the released hf, now separable from the atom as a radiative quantum of energy and mass.

Complementarity of Dynamic and Apparent Mass in ECM: (ΔMᴍ ↔ Mᵃᵖᵖ)

Soumendra Nath Thakur 

A core interpretive principle within ECM is the complementarity between dynamic mass displacement (ΔMᴍ) and apparent mass (Mᵃᵖᵖ). These quantities are not merely opposites in algebraic sign, but mutually defining constructs that gain physical significance only in relation to one another.

For example:

ΔMᴍ represents the emergent or emitted mass-equivalent energy due to frequency scaling, as in:

hf = ΔMᴍ c²

Mᵃᵖᵖ = −ΔMᴍ captures the corresponding loss or reduction in apparent mass from the source system.

This mutual dependence mirrors other foundational complements in nature:

Black and white as absence and presence of light

Potential and kinetic energy in transition

Finite and infinite as relational constructs

In ECM, neither ΔMᴍ nor Mᵃᵖᵖ has causal validity in isolation. It is their interaction—seen in transformations like:

KEᴇᴄᴍ = −Mᵃᵖᵖ c ² or ΔMᴍ = hf / c²

—that defines real physical outcomes such as radiation, gravitational weakening, and cosmic expansion.

This principle of complementarity reinforces ECM's broader stance: that energy and mass, emergence and loss, are not independent absolutes, but relational constructs whose meaning arises through causal symmetry.

Summary

ECM restores physical continuity and causality by linking frequency to mass-energy emergence, rejecting singularities and probabilistic quantum behavior. Its structural pillars are:

Frequency-scaling of force, energy, and displacement

Nonlinear collapse at Planck thresholds

Energetic boundary formation instead of metric expansion

Deterministic time onset defined by 

This unified interpretation enables ECM to model dynamics across photon, collapse, and cosmological scales with logical continuity and dimensional precision

Pre-relativistic framework and ECM:

Soumendra Nath Thakur

Relativity is not necessary for the very phenomena it is often praised for explaining. In truth, it diverted science away from rational foundations by introducing dilatable time and curved, blended space — abstraction that complicate rather than clarifying physical reality.

The pre-relativistic framework was already sufficient to support a more consistent and physically intuitive understanding of the universe. What was needed was not a leap into spacetime distortion, but a deeper refinement of classical principles.

This is where Extended Classical Mechanics (ECM) comes in — a framework with the potential to restore coherence and rational causality to physics. Once fully explored, ECM may well demonstrate that relativity’s perceived necessity was a historical detour, not a scientific inevitability.

🚀 ANNOUNCEMENT: Publication of ECM Appendices 20–22

🔬 Extending Causal Mass-Energy Theory Across Frequency, Collapse, and Cosmological Boundaries

By Soumendra Nath Thakur | Tagore's Electronic Lab, India

ORCiD: 0000-0003-1871-7803

I'm pleased to announce the release of a critical three-part extension to the ECM series, now published and available on ResearchGate:

---

📘 Appendix 20: Frequency Scaling and Energy Redistribution in Extended Classical Mechanics

🔗 DOI: 10.13140/RG.2.2.11662.06725

This appendix introduces frequency as the core driver of energy and force scaling in ECM. Replacing wave-particle duality with dynamic mass displacement logic, it establishes causal relationships such as:

Redshift, blueshift, and radiation are modelled as frequency-governed mass-energy transitions.

---

📘 Appendix 21: Planck Thresholds, Energy Quantization Limits, and Nonlinear Collapse in ECM

🔗 DOI: 10.13140/RG.2.2.15017.51042

This work recasts Planck-scale transitions as causal saturation points of reversible energy exchange. At

$$f \to f_{planck} \approx 2.999 \times 10^{42} \text{ Hz},\quad \Delta Mᴍ \to Mᴍ,ʀₑₛₜ$$ $$KEᴇᴄᴍ = -\Delta PEᴇᴄᴍ = \Delta Mᴍc²,\quad v > c$$

---

📘 Appendix 22: Cosmological Boundary Formation and Mass-Energy Reconfiguration in ECM Expansion

🔗 DOI: 10.13140/RG.2.2.26761.56166

Redefining cosmic expansion through energetic redistribution, this appendix introduces a non-metric cosmological boundary where

$$\Delta Mᴍ \to 0,\quad Fᴇᴄᴍ \to 0,\quad \text{at } t = t₀$$

---

🧭 These three appendices form a cohesive pillar in the ECM framework—bridging photon-scale quantization, trans-Planckian collapse, and universal boundary emergence.

Feedback, discussion, and collaboration are warmly welcomed.

🔗 View the full ECM series here: researchgate.net/profile/Soumendra-Nath-Thakur

— Soumendra Nath Thakur

The Self-Triggered Big Bang: ECM’s Internal Mass-Energy Dynamics and the Reinterpretation of Gravitational Origin.

Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 | postmasterenator@gmail.com | June 28, 2025

Abstract:

Extended Classical Mechanics (ECM) offers a foundational reinterpretation of motion and gravity, shifting the focus from externally imposed forces to the internal transformation of mass-energy into dynamically effective quantities. Within this rigorously defined framework, ECM reconstructs the origin of the Big Bang as a self-triggered physical event—emerging through the spontaneous and mathematically expressible conversion of a non-positive gravitational potential (−∆PEᴇᴄᴍ) into universal-scale kinetic energy (KEᴇᴄᴍ,ᴜₙᵢᵥ). This formulation provides a concrete, testable foundation that unifies cosmological expansion, force emergence, and mass-energy redistribution under the principle of internal energetic causality.

1. Foundations of Effective Dynamics in ECM

In ECM, the nature of motion and gravitational interaction departs fundamentally from classical interpretations. Instead of being driven solely by externally applied forces, these dynamics emerge from internal transformations of mass-energy and the resulting effective quantities: effective force (Fᴇᴄᴍ), effective mass (Mᵉᶠᶠ), and effective acceleration (aᵉᶠᶠ).

The effective force in motion, Fᴇᴄᴍ, is defined as:

Fᴇᴄᴍ = Mᵉᶠᶠaᵉᶠᶠ = (Mᴍ − ∆Mᴍ) aᵉᶠᶠ

Similarly, the gravitational force Fɢ in ECM becomes:

Fɢ = Mᵉᶠᶠgᵉᶠᶠ ⋅ h = (Mᴍ − ∆Mᴍ)gᵉᶠᶠ ⋅ h

Here, internal energy displacement drives the transformation of potential energy (PEᴇᴄᴍ) into kinetic energy (KEᴇᴄᴍ), mediated by the emergence of apparent negative mass (−Mᵃᵖᵖ), equivalent to −∆Mᴍ.

2. Internal and External Force Roles in ECM

In classical systems governed by matter mass (Mᴍ), forces act externally in a Newtonian sense. But when the mass involved is −Mᵃᵖᵖ, ECM shows that the force emerges internally as a direct result of energy redistribution:

−∆PEᴇᴄᴍ → KEᴇᴄᴍ = −Mᵃᵖᵖc²

Beyond Planck threshold, this equation becomes:

−∆PEᴇᴄᴍ → KEᴇᴄᴍ = −Mᵃᵖᵖv², where v > c, as beyond Planck threshold: >ℓₚ/\<tₚ results → v > c

This shift defines force not merely as an applied interaction but as an emergent property of internal energetic imbalance.

3. Dual Force Action and Energetic Mass Dynamics

Effective force in ECM has a dual character:

• On Mᴍ: leads to a decrease in matter mass, converted to energy.

• On −Mᵃᵖᵖ: acts directly, enabling self-sustained growth of kinetic energy.

The system transitions from classical mass structure to a mass-energy dynamic where −Mᵃᵖᵖ not only arises from transformation but actively sustains force.

4. Gravitational Polarity and Threshold Effects

The gravitational polarity of an ECM system depends on the ratio of Mᴍ to |∆Mᴍ|:

• Mᴍ > |∆Mᴍ| ⇒ Positive gravitational polarity (attractive)

• |∆Mᴍ| > Mᴍ ⇒ Negative gravitational polarity (repulsive)

This polarity threshold determines whether systems gravitate or expand internally, a foundation for radiation-induced motion, inflation, and dark energy.

5. Mathematical Formulations

• Motion: Fᴇᴄᴍ = Mᵉᶠᶠaᵉᶠᶠ, Eₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ

• KE from displacement: KEᴇᴄᴍ = −∆PEᴇᴄᴍ = ∆Mᴍc²

• Gravitational field: gᵉᶠᶠ(h) = G(Mᵉᶠᶠ/r²) or gᵉᶠᶠ ∝ 1/hf where hf = E is the Planck equation

These expressions ground ECM in testable, internally consistent mathematics.

6. Big Bang as an Internally Generated Event

According to ECM, the Big Bang does not require external causation. Instead, it is explained as the spontaneous transformation:

−∆PEᴇᴄᴍ → KEᴇᴄᴍ,ᴜₙᵢᵥ

• The dominance of −Mᵃᵖᵖ leads to internal force generation

• Repulsive gravitational polarity initiates expansion

• No spacetime preconditions are required

This aligns with Appendix 10 on pre-universal energy fields and deep potential collapse without thermodynamic or inertial structure.

7. Connection to Dark Energy

Dark energy in ECM is the observable consequence of ongoing internal conversions of −∆PEᴇᴄᴍ into KEᴇᴄᴍ:

• Persistent repulsive pressure

• No need for exotic fields or λ-constants

• Arises from displacement of −Mᵃᵖᵖ across cosmic gravitational gradients

Thus, dark energy and the Big Bang share the same causal mechanism in ECM: internally mediated mass-energy polarity.

8. Unified Synthesis

ECM offers a self-contained model for force, mass, and motion. The Big Bang is reinterpreted not as a singularity requiring external input, but as an energetic inevitability arising from mass-energy disequilibrium:

• −Mᵃᵖᵖ dominance

• Polarity inversion

• Self-generated expansion

This energetic structure continues to power cosmic acceleration today.

Conclusion:

The ECM framework, through its effective quantities and internally emergent forces, reconstructs the Big Bang as a self-triggered outcome of gravitational potential collapse. It provides a unified explanation for both cosmological birth and present-day acceleration, grounded in mathematically defined internal mass-energy dynamics.

The Big Bang, in ECM, is not the beginning of time imposed from outside, but the consequence of a fundamental energetic instability within a non-positive gravitational potential field—a self-unfolding event that continues to define the universe.

References:

1. Thakur, S. N. (2025). Foundational Formulation of Extended Classical Mechanics. https://doi.org/10.20944/preprints202504.1501/v1

2. Thakur, S. N. (2024). Extended Classical Mechanics: Vol-1. https://doi.org/10.20944/preprints202409.1190.v3

3. Thakur, S. N. (2024). A Nuanced Perspective on Dark Energy: ECM. https://doi.org/10.20944/preprints202411.2325.v1

4. Thakur, S. N. (2024). Unified Study on Gravitational Dynamics: ECM Vol-2. https://www.researchgate.net/publication/384501200

5. Appendix 17: Internal Force Generation and Non-Decelerative Dynamics in ECM under Negative Apparent Mass Displacement. DOI: https://doi.org/10.13140/RG.2.2.26584.61446

Appendix 16: Cosmic Inflation and Expansion as a Function of Mass-Energy Redistribution in ECM.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803 | Tagore's Electronic Lab, India

postmasterenator@gmail.com | June 25, 2025

Overview

This appendix presents an ECM-based interpretation of the universe's inflationary beginning, the apparent halting of expansion, and the subsequent onset of accelerated cosmic expansion. Contrary to conventional models that rely on hypothetical inflation fields and quantum vacuum fluctuations, the ECM framework treats these cosmic phases as direct outcomes of changing gravitational mass balance conditions. These are governed by the effective gravitational mass Mɢ, the apparent mass Mᵃᵖᵖ, and the evolving ratio of matter mass (Mᴍ) to dark energy mass (Mᴅᴇ).

1. Pre-Matter Epoch: Dominance of −ΔMᵃᵖᵖ and Absence of Mᴍ

At the moment of the Big Bang, matter mass is effectively absent (Mᴍ = 0), and the universe is dominated by potential energy stored as Mᴅᴇ < 0, which manifests as an effective positive gravitational mass:

Mɢ = Mᴍ + Mᴅᴇ → Mɢ = 0 + Mᴅᴇ ⇒ Mɢ > 0

This condition—free from inertial opposition—initiates superluminal inflation, driven by the full conversion of dark energy potential into kinetic energy:

−ΔPEᴅᴇ → +KEᴇᴄᴍ → v > c

Here, −ΔMᵃᵖᵖ governs the rapid expansion. No gravitational binding is present to inhibit it.

2. Matter Formation and Gravitational Equilibrium

As the universe expands and cools:

• Matter mass Mᴍ begins to accumulate from early nucleosynthesis and gas cloud formation.

• The total Mᴍ rises gradually, introducing gravitational inertia into the system.

At a certain threshold:

Mᴍ = |Mᴅᴇ| ⇒ Mɢ = 0

This represents a critical equilibrium: gravitational mass is null, and the universe temporarily halts expansion. This is the first transitional phase—a shift from pure antigravity to balanced dynamics.

3. Declining Matter Density and Expansion Restart

As universal volume increases and Mᴍ undergoes kinetic transformation (e.g., via energy dissipation, radiative loss):

• The density of Mᴍ reduces, while Mᴅᴇ maintains a relatively uniform distribution.

• The mass inequality reverses:

Mᴍ < |Mᴅᴇ| ⇒ Mɢ < 0

This initiates a second phase of expansion, now accelerated, but not superluminal. The matter content remains significant enough to moderate the rate, consistent with observed cosmic acceleration.

4. ECM Summary Table: Mass-Energy Conditions and Universal Evolution

Epoch                                      Mass Conditions    ECM Condition      Effect                  

·         Pre-Matter Inflation      Mᴍ ≈ 0, Mᴅᴇ > 0      Mɢ = Mᴅᴇ    Superluminal inflation (v>c)

·         Matter Accumulation     Mᴍ ↑, reaches          Mᴅᴇ           Mɢ = 0 | Expansion halt 

                                                                                  (Dynamic equilibrium)  

·         Restarted Expansion    Mᴍ <Mᴅᴇ                 Mɢ < 0         Accelerated expansion

Conclusion

The three major cosmological epochs—initial inflation, temporary halt, and resumed accelerated expansion—are naturally derived within ECM through causal mass-energy transitions. The governing expression Mɢ = Mᴍ + Mᴅᴇ reflects the dynamic interplay between matter accumulation and persistent dark energy influence. In this framework, antigravity is not speculative but a direct consequence of −ΔMᵃᵖᵖ dominance in early-universe conditions, followed by inertial balance and eventual redistribution.

ECM thus provides a unified classical structure for cosmic behaviour, governed by mass-energy transformations rather than hypothetical spacetime constructs or singularities. It anchors the universe’s expansion history within consistent, measurable terms of mass modulation and potential-to-kinetic energy flow.

Appendix Series Note and Supplementary Materials

This appendix extends the ECM framework presented in:

Appendix 15: Cosmological Origin and Direction of Galactic Expansion in ECM. DOI: https://doi.org/10.13140/RG.2.2.27951.04008

Appendix 16: specifically builds on the role of −ΔMᵃᵖᵖ, aᵉᶠᶠ, and mass-energy phase dominance in structuring inflationary and post-inflationary cosmic dynamics.

References

1. Thakur, S. N. (2025). Cosmological Origin and Direction of Galactic Expansion in ECM. Appendix 15. DOI: https://doi.org/10.13140/RG.2.2.27951.04008

2. Thakur, S. N. (2025). Extended Classical Mechanics: Foundations and Frontiers. Tagore’s Electronic Lab Archives.

3. Planck, M. (1900). On the Theory of the Energy Distribution Law of the Normal Spectrum.

4. de Broglie, L. (1924). Recherches sur la théorie des quanta.

5. Observational Cosmology Data: NASA WMAP & ESA Planck Mission Data Archives.

Supplementary Resource to Appendix 16

Clarification on ECM Note: Inflation, Expansion, and Mass-Energy Balance in the Early Universe

 

Subject: An Extended Classical Mechanics (ECM) Interpretation of Big Bang Inflation and Cosmic Evolution

Associated with: Appendix 16: Cosmic Inflation and Expansion as a Function of Mass-Energy Redistribution in ECM

DOI: https://doi.org/10.13140/RG.2.2.10108.86408

Author: Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803 | Tagore’s Electronic Lab, India 

June 25, 2025

Purpose of This Supplement

This supplementary resource offers clarifications and elaborations on key terms, transformations, and mass-energy conditions central to ECM’s interpretation of cosmic inflation and expansion. It also outlines paths toward empirical modeling and quantitative validation.

1. Nature and Role of Mᴅᴇ (Effective Dark Energy Mass)

In ECM, Mᴅᴇ is defined as the effective negative mass contribution of dark energy. Its role is gravitationally repulsive, and it functions as potential energy in the cosmic mass-energy balance:

Mɢ = Mᴍ + Mᴅᴇ, where Mᴅᴇ < 0

At the universe’s origin, Mᴍ → 0, so Mɢ ≈ Mᴅᴇ becomes the dominant term, driving expansion through:

−ΔPEᴅᴇ → +KEᴇᴄᴍ → v > c

This results in superluminal inflation, without invoking an inflation field or quantum geometric interpretation. The conceptual basis aligns with gravitational modeling of large structures such as the Coma Cluster:

Chernin et al., A\&A, 553, A101 (2013) DOI: https://doi.org/10.1051/0004-6361/201220781

2. Mechanism of Kinetic Transformation of Mᴍ

The transformation of Mᴍ is governed by:

Mᴍ = (Mᴍ − ΔMᴍ) + ΔMᴍ

Here, ΔMᴍ refers to the portion of mass undergoing conversion into kinetic energy or radiative energy. The total energy equation in ECM terms becomes:

Eₜₒₜₐₗ = PE + KE = (PEᴇᴄᴍ − ΔPEᴇᴄᴍ) + ΔPEᴇᴄᴍ

And gravitationally:

½ΔMᴍv² + (Mᴍ − ΔMᴍ)gᵉᶠᶠ·h

This explains declining matter density not through decay or disappearance of mass, but through its redistribution into kinetic form, reducing net gravitational influence over time.

3. Empirical Relevance and Observational Context

Appendix 16 aligns qualitatively with:

• Type Ia Supernovae acceleration curves

• Cosmic Microwave Background anisotropy

• Galaxy cluster dynamics and structure formation

The inclusion of dark energy–driven mass redistribution as an organizing principle is consistent with:

Dark energy and structure of the Coma cluster, A. D. Chernin et al. (2013)

Quantitative predictions (e.g., cosmic scale factor, H(z), Ω parameters) are identified as next steps.

4. Departure from ΛCDM and Role of Mass-Energy Causality

Unlike ΛCDM, which interprets expansion as a consequence of spacetime curvature and introduces Λ as an invariant constant, ECM interprets cosmic behavior as an outcome of mass-energy redistribution governed by evolving terms:

• Mᴍ (matter mass)

• Mᴅᴇ (dark energy mass)

• ΔMᵃᵖᵖ (apparent mass modulation)

The condition Mᴍ = Mᴅᴇ defines equilibrium; Mᴍ < Mᴅᴇ yields acceleration. This provides a more dynamic and causally grounded model.

5. Apparent Mass (ΔMᵃᵖᵖ) and −ΔMᵃᵖᵖ

ΔMᵃᵖᵖ represents the mass undergoing transition from gravitational contribution to kinetic or radiative expression. Thus:

Mᴍ = (Mᴍ − ΔMᴍ) + ΔMᴍ ⇒ ΔMᵃᵖᵖ = ΔMᴍ

Then:

−ΔMᵃᵖᵖ reflects the net loss in gravitational binding, allowing antigravity (accelerative expansion) to dominate.

This formulation captures not just energy transformation, but its gravitational consequence, absent in static mass-conserved models.

Conclusion and Forward Plan

This supplement strengthens the causal clarity of ECM’s inflationary and expansion model. The next ECM research outputs will focus on:

• Formulating quantitative expansion curves from ECM mass equations

• Deriving Hubble parameters based on Mᴍ–Mᴅᴇ evolution

• Simulating observable data alignment (e.g., CMB, supernovae distances)

This path aims to bridge ECM’s conceptual foundation with empirically testable cosmological models.