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.
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