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:

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📘 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.

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📘 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

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

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📘 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

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

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🧭 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.