Negative Apparent Mass and Mass Continuity in ECM: Consistency and Observational Implications

Soumendra Nath Thakur | Tagore’s Electronic Lab | June 02, 2025

postmasterenator@gmail.com or postmasterenator@telitnetwork.in

DOI: 10.13140/RG.2.2.10264.92165

Abstract

This enhanced technical report formalizes the derivation and function of negative apparent mass (−Mᵃᵖᵖ) in Extended Classical Mechanics (ECM), a non-relativistic framework emphasizing mass-energy redistribution. It demonstrates how kinetic energy and potential energy redistribution inherently demand the existence of an embedded mass-deficit, forming a consistent foundation for understanding photons, dynamic particles, and cosmological dark energy phenomena. Derivation flows from displaced matter mass −ΔMᴍ and displaced potential energy −ΔPEᴇᴄᴍ, resulting in −Mᵃᵖᵖ as a conserved but redistributed mass structure. ECM thus provides a non-relativistic but empirically consistent framework for mass-energy continuity.

1. Classical Mechanics and the Inconsistency of Constant Inertial Mass under Dynamic Energy Conditions

In classical mechanics, inertial mass m is considered a constant, unaffected by velocity or variations in gravitational potential. For a body of mass m at rest on a surface influenced by a larger mass M, the object is described as being at inertial rest. Its energy is entirely potential, given by potential energy (PE = mgh), with no kinetic component (kinetic energy, KE = 0), provided the height h from the centre of gravity of the larger mass remains constant.

Upon applying an external mechanical force F, the object accelerates (a) according to Newton’s second law F = ma, and subsequently acquires kinetic energy, expressed as KE = ½mv². Despite this motion, classical mechanics maintains that the potential energy PE = mgh remains unchanged so long as the height h remains fixed. Hence, the total energy becomes:

E = PE + KE = mgh + ½mv² = m(gh + ½v²) ≠ m,

Unless v = √2 (in units where g = h = 1, i.e., normalized system) ... (1)

Meanwhile, Newton’s force law in the same context remains:

F = ma ⇒ F/a = m = constant ... (2)

Observation:

Equation (2) suggests m is invariant, while Equation (1) implies that total energy now involves a velocity-dependent term, making the mass-energy combination dynamic. This inconsistency highlights a critical oversight in classical mechanics: mass appears both constant and variable depending on context.

A deeper interpretation of Equation (2) suggests:

F = ma → F ∝ a → a ∝ 1/m, (not a ∝ m) ... (3)

Physically, this implies that mass inversely influences acceleration, and that inertial resistance increases with mass, not the acceleration itself. If the velocity v ≠ √2, then Equation (1) shows that total energy cannot be proportional to a constant mass alone, i.e.,

E = m(gh + ½v²) ≠ m

This introduces the necessity for redefining effective mass under dynamic conditions.

2. Derivation of Negative Apparent Mass in ECM

ECM defines mass redistribution in dynamic systems through the effective mass equation:

Mᵉᶠᶠ = Mᴍ − ΔMᴍ,

Where ΔMᴍ (displaced matter mass) represents mass converted into kinetic motion.

The associated energy balance becomes:

PEᴇᴄᴍ = (PEᴇᴄᴍ − ΔPEᴇᴄᴍ) + ΔPEᴇᴄᴍ → KEᴇᴄᴍ

This mass-energy symmetry implies:

• ΔMᴍ → KEᴇᴄᴍ

• ΔPEᴇᴄᴍ → KEᴇᴄᴍ

But ECM emphasizes that the energetic state of motion is not self-contained—it is sustained by a persistent embedded mass deficit, which remains as:

−ΔMᴍ = −Mᵃᵖᵖ < 0, −ΔPEᴇᴄᴍ < 0

Thus, kinetic energy in ECM is carried not just by gained energy, but by the ongoing energetic cost of mass displacement:

KEᴇᴄᴍ → −Mᵃᵖᵖ

3. Photon Dynamics and the Role of −Mᵃᵖᵖ

Photons in ECM are reconceptualising as mass-energy displacement carriers, where:

hf = ΔMᴍc² (photon energy as equivalent to displaced mass) → KEᵧ = −Mᵃᵖᵖc² (Representing the negative apparent mass responsible for propagation)

• Photonic motion arises from internal mass displacement, not external absorption.

• The negative apparent mass is not rest mass but a mass-debt representing physical transfer.

• ECM ensures that momentum and energy conservation are internally anchored via redistributive mass flows.

4. Cosmological Implications: Dark Energy as Accumulated −Mᵃᵖᵖ

On cosmological scales, ECM identifies dark energy with cumulative mass deficits:

Mɢ = Mᴍ + Mᴅᴇ, Mᴅᴇ = −Mᵃᵖᵖ < 0

In the Coma Cluster model, observations from A. D. Chernin et al. (2013), DOI:10.48550/arXiv.1303.3800

• Gravitational weakening at distances >14 Mpc.

• Observed mass deficit equals theoretical displacement −Mᵃᵖᵖ.

• ECM eliminates the need for vacuum-energy or scalar-field hypotheses by accounting for cosmological energy imbalance through redistributed negative apparent mass.

5. Unified Mass Terms and Interpretive Table

Term Definition (ECM) Physical Role

• ΔMᴍ Displaced matter mass Carrier of kinetic energy

• −Mᵃᵖᵖ Apparent negative mass Embedded mass-debt, supports motion

• Mᵉᶠᶠ Mᴍ − ΔMᴍ Gravitation-capable retained mass

• Mᴍ Original matter mass Base mass of the system

• Mᴅᴇ Effective dark energy mass Equivalent to −Mᵃᵖᵖ in large-scale systems

6. Experimental and Theoretical Implications

• Gravitational Weakening: Higher kinetic energy (larger ΔMᴍ) leads to smaller Mᵉᶠᶠ.

• Piezoelectric Systems: Measurable strain energy reflects real-time ΔMᴍ variations.

• Nuclear Reactions: Rest mass preservation post-fission confirms ECM's displacement logic.

7. Conclusion

In ECM, −Mᵃᵖᵖ is a physically accountable construct, ensuring conservation of energy through matter redistribution. Whether in subatomic particle motion or cosmic expansion, this principle underlies ECM's claim that energy states require not just gain, but continuous compensation via apparent mass. This formalization replaces abstract scalar interpretations with concrete, testable dynamics.

8. Appendix D – Symbolic and Causal Reference Glossary in ECM

Symbol / Term Definition Remarks

• Mᴍ Matter mass. Conserved total mass of the system.

• ΔMᴍ Displaced mass component. Redistributed mass; source of KEᴇᴄᴍ.

• −ΔMᴍ Negative apparent mass = −Mᵃᵖᵖ. Mass-deficit attributed to kinetic manifestation.

• Mᵉᶠᶠ Effective mass = Mᴍ − ΔMᴍ. Residual gravitational mass.

• PEᴇᴄᴍ Potential energy in ECM. Field energy relative to system configuration.

• ΔPEᴇᴄᴍ Displaced potential energy. Energy portion manifesting into KEᴇᴄᴍ.

• KEᴇᴄᴍ Kinetic energy in ECM. Arises from ΔMᴍ; not scalar-added.

• φ(v) Kinetic scaling factor (e.g.½ΔMᴍv², c²) Ensures KEᴇᴄᴍ = ΔMᴍ φ(v).

• → Causal manifestation. Denotes structured emergence, not algebraic identity.

• Mᴅᴇ Effective dark energy mass=−Mᵃᵖᵖ. Cumulative cosmological repulsion mass term.

Appendix Reference Index – ECM Technical Appendices

• Appendix A: Standard Mass Definitions in ECM – DOI: 10.13140/RG.2.2.31762.36800

• Appendix B: Alignment with Physical Dimensions of Energy Types in ECM – DOI: 10.13140/RG.2.2.34193.75365

• Appendix 3: Fundamental Total Energy in ECM – DOI: 10.13140/RG.2.2.21532.19841

• Appendix D: Symbolic and Causal Reference Glossary in ECM – (This Document)

9. References

1. Thakur, S. N. (2025). Holes and Photons as Dual Manifestations of Electron Displacement in Extended Classical Mechanics. DOI: 10.13140/RG.2.2.20536.87041

2. Thakur, S. N. (2025). Piezoelectric and Inverse Piezoelectric Effects on Piezoelectric Crystals. ResearchGate. https://www.researchgate.net/publication/384569714

3. Chernin, A. D., et al. (2013). Dark energy and the structure of the Coma cluster of galaxies. DOI: 10.48550/arXiv.1303.3800

4. Thakur, S. N. (2025). Foundational Formulation of Extended Classical Mechanics. DOI: 10.20944/preprints202504.1501.v1

5. Thakur, S. N. (2025). Electrons and Holes in Solid-State Systems: An ECM Interpretation of Dynamic Mass. DOI: 10.13140/RG.2.2.28689.54888

6. Thakur, S. N. (2025). Mass-Energy Transformations in ECM. DOI: 10.13140/RG.2.2.24863.27040

7. Thakur, S. N. (2024). A Nuanced Perspective on Dark Energy. Magnivel International. https://magnivelinternational.com/journal/articledetails/28

8. Thakur, S. N. (2024). Photon-to-Dark-Energy Transition. DOI: 10.13140/RG.2.2.10551.02723

9. Thakur, S. N. (2024). ECM Vol-1: Equivalence Principle and Gravitational Dynamics. DOI: 10.20944/preprints202409.1190.v3

10. Thakur, S. N. (2024). Symmetry and Conservation for Photon Fields. DOI: 10.20944/preprints202411.0956.v1

11. Thakur, S. N. (2025). Energy Manifestations in the Universe's Mass-Energy Composition. DOI: 10.13140/RG.2.2.28469.49121

12. Thakur, S. N. (2024). Defining Energy and Relativistic Rest Energy. DOI: 10.13140/RG.2.2.34040.05127

13. Thakur, S. N. (2025). Discrepancy in General Relativity and Gravitational Lensing. DOI: 10.13140/RG.2.2.12229.67043

14. Goldstein, H., & Twersky, V. Classical Mechanics. Physics Today, 5(9), 19–20. https://doi.org/10.1063/1.3067728

15. Thakur, S. N., & Bhattacharjee, D. (2023c, October 30). Phase Shift and Infinitesimal Wave Energy Loss Equations. Longdom. https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html

10. Footnotes

1. All expressions involving inverse mass (1/Mᴍ) are dimensionally regularized using a scaling constant k with units [M²L²T⁻²], ensuring that terms like k/mc² yield mass-equivalent corrections. This preserves dimensional homogeneity and reflects energetic displacement or interactional embedding in ECM dynamics.

2. All expressions involving deformation or restoration due to kinetic energy (KE) are understood in ECM as real physical redistributions of mass-energy. The term ΔMᴍ represents the mass-equivalent of KE, which induces structural strain during acceleration and is reabsorbed during deceleration, forming the basis for observable electromechanical effects such as in piezoelectric systems.

Rest Energy vs. Kinetic Energy in Extended Classical Mechanics (ECM): Beyond Classical and Relativistic Views.

Soumendra Nath Thakur                                                DOI
June, 01, 2025

The reinterpretation of the relativistic energy equation E = mc² within the Extended Classical Mechanics (ECM) framework offers deeper insight into the role of mass displacement during energy transitions. In ECM, the relativistic mass m is redefined as the displaced mass component, denoted ΔMᴍ. This effective mass Mᵉᶠᶠ includes not only the transition of ΔMᴍ from the original matter mass Mᴍ (i.e., a loss of −ΔMᴍ), but also encapsulates the interactional and energetic transformations that occur in high-energy phenomena such as nuclear reactions.

In standard relativistic physics, the rest mass m in E = mc² is often interpreted as being wholly converted into energy. However, in actual nuclear reactions, this is not entirely the case. The by-products of such reactions—alpha particles, beta particles, and residual nuclei—all retain a portion of the original rest mass. Hence, not all of the rest mass is converted into pure rest energy. Instead, a portion remains as bound rest mass ΔMᴍ, while the remainder is distributed into kinetic energy and radiative emission, particularly in the form of electromagnetic radiation.

Importantly, this emission includes particles traditionally considered massless—such as gamma rays and photons—which, in ECM, are interpreted as carrying apparent negative mass −ΔMᴍ, originating from internal energetic displacement rather than conventional rest mass.

Thus, in nuclear splitting:

Mᴍ_ɴᴜᴄᴇᴜꜱ = ΔMᴍ_ʀᴇꜱɪᴅᴜᴀʟ ɴᴜᴄᴇᴜꜱ + Mᴍ_ₐ,ᵦ + (−ΔMᴍ_ᵧ) + (−ΔMᴍ_ₚₕₒₜₒₙₛ)

This formulation reflects that both massive and massless reaction products arise from mass-energy redistribution, not from total annihilation or full rest-mass conversion. It also highlights that radiative products such as photons and gamma rays embody displaced energy with measurable effects, despite lacking rest mass in conventional terms.

In Classical Mechanics, energy is typically classified as either potential or kinetic. However, relativistic rest energy represents a more intricate form of transition—a fusion of potential-like binding effects and kinetic-like emissions—mediated through mass redistribution, emission of particles, and radiative losses. ECM captures this nuance by modelling rest energy release as a combination of physical mass displacement and interactional field effects, providing a coherent explanation for the emergence of both massive and massless products in high-energy processes.

Why is the speed of light what it is, and why not some other speed? - A repeat version.

Soumendra Nath Thakur 

June 01, 2025

This post addresses the question: “Why is the speed of light what it is, and why not some other speed?”

In contrast to relativistic theory, Extended Classical Mechanics (ECM) asserts that photons possess a negative apparent mass, which enables them to generate their own antigravitational force. This self-propelling mechanism allows photons to move freely through gravitational fields; gravity does not constrain their motion—instead, it contributes additional energy to photons when they traverse gravitational potentials.

Photons inherently tend toward unbounded velocities, theoretically approaching infinity. However, the limiting factor is not gravity, but rather a Planck-scale threshold, which sets the upper bound for meaningful physical quantities: a maximum possible frequency and a minimum meaningful wavelength. The ratio of these two (frequency to wavelength) defines the maximum meaningful speed, which is observed as the constant speed c. Thus, the speed limit of light is not imposed by spacetime curvature (as in relativity), but by dimensional and energetic constraints defined at the Planck scale, according to ECM.

Relativity maintains the constancy of c by enforcing a mutual compensation between a photon’s frequency and wavelength—this is mathematically consistent, but in ECM, it is viewed more as a convenient wave-based relation than a fundamental relativistic principle.

Accordingly, all electromagnetic waves propagate at the same speed because they are carried by photons, and the *photon itself is the mediator of the electromagnetic force. In ECM, it is the nature and energy constraints of the photon—not spacetime geometry—that determine and preserve this universal speed.

Reconciling Gravitational Radiation, Dark Sector Phenomena, and Extended Classical Mechanics (ECM): Toward a Unified Framework

Soumendra Nath Thakur

Tagore’s Electronic Lab, India, 

postmasterenator@gmail.com postmasterenator@telitnetwork.in

May 29, 2025

 

Abstract

This section explores the conceptual integration of gravitational radiation and dark sector phenomena within the framework of Extended Classical Mechanics (ECM). ECM extends Newtonian principles by introducing dynamic mass concepts, such as displaced mass and apparent mass, while strictly preserving dimensional consistency. Gravitational radiation is reinterpreted not as the curvature of spacetime but as a consequence of real mass-energy displacement within energetic systems. Similarly, ECM provides alternative explanations for dark matter and dark energy, modelling them as emergent effects of gravitational mass redistribution rather than as independent fields or exotic particles. The framework also offers reinterpretations of relativistic phenomena, such as gravitational lensing and time dilation, through the lens of internal energy restructuring. By offering a consistent, matter-based alternative to both relativistic gravity and particle-based cosmology, ECM has the potential to unify gravitational and antigravitational interactions under a common mechanical paradigm.

Introduction

Reconciling gravitational radiation, dark sector effects, and ECM principles involves integrating Extended Classical Mechanics (ECM) with established cosmological models and observations of dark matter and dark energy. ECM, which extends Newtonian mechanics to incorporate dynamic mass components and reinterprets relativistic behaviour, can offer a framework for understanding these phenomena. 

Gravitational Radiation and ECM:

·         ECM can offer a reinterpretation of gravitational radiation, viewing it not as a field interaction, but as a consequence of mass displacement and energy shifts within a system.

·         ECM's focus on potential energy as a central regulatory mechanism in mass systems can be applied to understanding how energy is transferred and radiated during gravitational events.

·         ECM's strict adherence to dimensional consistency in mass and radiation expressions is crucial for ensuring a self-consistent framework. 

Dark Sector Effects and ECM:

·         ECM can provide a framework for understanding dark matter and dark energy, viewing them as emergent gravitational phenomena rather than exotic particles or fields. 

·         ECM's introduction of dynamic mass components, including negative apparent mass and effective mass can help model the gravitational and inertial interactions of dark matter and dark energy. 

·         ECM's reinterpretation of relativistic behaviour, particularly time dilation and gravitational lensing, can offer alternative explanations for cosmological observations involving dark energy and the expansion of the universe. 

·         ECM can potentially offer a unified treatment of gravitational and antigravitational interactions, aligning with cosmological observations of dark energy and cosmic expansion. 

Displaced mass, gravitational field strength, and energy release in extreme stellar systems under Extended Classical Mechanics (ECM):

Soumendra Nath Thakur
May 27, 2025

This section explores how neutron stars, under the framework of Extended Classical Mechanics (ECM), express extreme gravitational behaviour through mass displacement and energetic transformation—independently of relativistic mechanics. The ECM model reinterprets gravitational field strength gᴇᴄᴍ as a function of mass displacement per unit time, specifically as the rate of negative apparent mass generation d(−ΔMᴍ)/dt. In such dense astrophysical bodies, the immense matter mass Mᴍ gives rise to both rest energy (ΔMᴍC²) and kinetic energy (ΔMᴍ as KE via radiated gamma or photon particles), without requiring relativistic gamma correction. The photon-induced kinetic energy component emerges as negative apparent mass −Mᵃᵖᵖ, linking light-like particle emission to mechanical displacement within the gravitational structure. The ECM framework establishes calculable energy output in these systems, with clear partitioning between rest energy and displaced kinetic energy, and redefines gravitational interaction not through spacetime curvature, but via dynamic internal restructuring and mass redistribution. This interpretation is suitable for both theoretical modelling and experimental validation in high-energy astrophysics.

This figure visually summarises the core ECM interpretation presented in this Technical Findings. 

Figure Description:

This figure visually showing a neutron star under ECM interpretation, with arrows and annotations indicating:

Left Panel: Traditional Interpretation:

• Mass density (ρ) → spacetime curvature (GR)
• Gravitational field strength ↔ geometric deformation
• No mass-displacement term

Right Panel: ECM Interpretation

• Core Matter Mass (Mᴍ) labelled
• Arrows → showing mass displacement:

  • Outward kinetic deformation (ΔMᴍ) for photon/gamma emission
  • Inward structural strain (−ΔMᴍ) as apparent mass

Gravitational strength gᴇᴄᴍ shown as:

       gᴇᴄᴍ = d(−ΔMᴍ) ÷ Mᴍ

  (Field strength as deformation rate per matter mass)

• Labels on:

  • Rest energy release: ΔMᴍ·c²
  • Kinetic energy via photon emission: ΔMᴍ ⇒ −Mᵃᵖᵖ

Citation for the Research:

1. Thakur, S. N. (2025, May 27). Displaced mass, gravitational field strength, and energy release in extreme stellar systems under Extended Classical Mechanics (ECM). ResearchGate. https://doi.org/10.13140/RG.2.2.29304.76807

2. Thakur, S. N. (2025), Appendix A – Standard Mass Definitions in Extended Classical Mechanics (ECM), ResearchGate, DOI: https://doi.org/10.13140/rg.2.2.31762.36800

3. Thakur, S. N. (2025). Explanation in the usage of matter mass (Mᴍ) within solar or terrestrial regimes: Dark matter in Extended Classical Mechanics (ECM). ResearchGate. https://doi.org/10.13140/RG.2.2.30117.41441