Extended Classical Mechanics (ECM) reframes mass not merely as an energy reservoir but as an active structural participant in the manifestation and propagation of energy. This would challenge the standard E=mc² interpretation of mass defect, suggesting instead that mass is never annihilated but rather dynamically displaced.
June 04, 2025
In short, Extended Classical Mechanics (ECM) holds that negative apparent mass (−Mᵃᵖᵖ) is a physically responsible construct that ensures the conservation of energy through matter redistribution. This principle is presented as fundamental at all scales from subatomic particle motion to cosmic expansion.
The ECM suggests that the energy state does not simply involve a gain of energy but requires continuous compensation through apparent mass. The ambition of the framework is to replace abstract scalar explanations such as vacuum energy or scalar fields with concrete, testable dynamics.
It represents an ambitious attempt at a paradigm shift in theoretical physics. By proposing a single, non-relativistic principle of mass-energy redistribution through −Mᵃᵖᵖ, ECM aims to unify classical mechanics, photon dynamics, and cosmology (dark energy) under a coherent framework. It is a grand unified theory in its nascent stage, challenging the established relativistic and quantum paradigms.
The repeated emphasis on "non-relativistic," "concrete, testable dynamics," and "replacing abstract scalar explanations" suggests the ECM's intention to present a new foundational theory. If the ECM's claims are valid, this would mean that many phenomena currently explained by relativity or quantum mechanics can be understood from a classical, albeit extended, perspective.
However, a significant challenge for the ECM is its empirical validity. Although the ECM proposes "concrete, testable dynamics," the phenomena it seeks to explain, such as photon propagation and dark energy, are deeply embedded in relativistic physics. For the ECM to gain widespread acceptance, its non-relativistic interpretations must not only be internally consistent but also provide distinct, measurable predictions that distinguish it from existing, well-established theories.
The primary challenge for the ECM will be to demonstrate its empirical consistency and predictive power without resorting to relativistic effects. This sets a very high bar for empirical verification.
If the core principles of the ECM are empirically verified, it will represent a significant paradigm shift, providing a new fundamental understanding of mass, energy, and their interaction.
Future research will aim at further theoretical development to refine the framework and, in particular, at experimental verification of its novel predictions, such as the effects of specific gravitational weakening, measurable piezoelectric phenomena associated with mass displacement, or the direct detection of "mass-borrowing" phenomena.
The potential of ECM to open new avenues of research in theoretical physics, especially non-relativistic methods for fundamental problems, deserves continued rigorous investigation.
In Extended Classical Mechanics (ECM), the total energy is redefined by incorporating real mass redistribution into the kinetic and potential energy relationships. ECM proposes that kinetic energy arises from physically displaced matter mass (ΔMᴍ), while gravitational potential remains a function of effective matter mass, leading to a revised total energy formula. [1, 2]
Here's a more detailed breakdown:
Traditional Classical Mechanics: Total energy (Eₜₒₜₐₗ) is the sum of potential energy (PE) and kinetic energy (KE): Eₜₒₜₐₗ = PE + KE. [1, 3]
ECM's Reinterpretation: ECM modifies this by considering the variation in potential energy due to apparent mass effects. The total energy is expressed as: Eₜₒₜₐₗ = (PE - ΔPE) + ΔPE, where PE - ΔPE represents the potential energy associated with matter mass (Mᴍ), and ΔPE represents the kinetic energy associated with displaced matter mass (ΔMᴍ). [1]
Apparent Mass: ECM introduces the concept of apparent mass (Mᵃᵖᵖ), which is related to kinetic energy and is negative in sign. [4]
Effective Mass: The effective mass (Mᵉᶠᶠ) is the sum of matter mass (Mᴍ) and the negative apparent mass (-Mᵃᵖᵖ): Mᵉᶠᶠ = Mᴍ - Mᵃᵖᵖ. [4]
Total Energy in ECM: The total energy in ECM can be expressed as: Eₜₒₜₐₗ = PE - ΔPE + KE, which is equivalent to (PE - ΔPE of Mᴍ) + (KE of ΔPE) or (Mᴍ - 1/Mᴍ) + (-Mᵃᵖᵖ). [1, 4]
Key Features of ECM's Total Energy:
Redistribution of Energy: ECM proposes that energy is not just transformed but also redistributed within a system, with kinetic energy arising from the displacement of matter. [2, 2]
Negative Dynamic Mass: ECM assigns a negative mass to kinetic energy, represented by -Mᵃᵖᵖ, which is crucial for understanding its role in gravitational interactions. [4, 4]
Effective Mass: The effective mass, which is a combination of matter mass and apparent mass, plays a crucial role in determining gravitational interactions and the behavior of objects, including photons. [4, 4, 5, 5]
Unified Approach: ECM aims to provide a unified framework for understanding force, inertia, and motion, encompassing both massive and massless particles. [5, 5]
Cosmological Implications: The principles of ECM have potential applications in cosmology, particularly in understanding the behavior of large-scale structures and the role of dark energy. [6, 6, 7]
In essence, ECM reinterprets the total energy as a dynamic quantity that depends on the redistribution of matter mass, leading to a more nuanced understanding of force, inertia, and motion in both classical and relativistic regimes. [2, 5]
References:
[1]https://www.researchgate.net/post/Energy_and_Mass_Considerations_in_Extended_Classical_Mechanics_Vol-2
[2]https://www.researchgate.net/publication/392232034_Appendix_B_Alignment_with_Physical_Dimensions_and_Interpretations_of_Standard_Categorization_of_Energy_Types_in_Extended_Classical_Mechanics_ECM
[3]https://www.preprints.org/manuscript/202409.1190
[4]https://www.preprints.org/manuscript/202504.1501/v1
[5]https://www.researchgate.net/publication/390845447_Foundational_Formulation_of_Extended_Classical_Mechanics_From_Classical_Force_Laws_to_Relativistic_Dynamics
[6]https://www.researchgate.net/publication/391704357_Restoring_Dynamic_Mass_in_Classical_Mechanics_-The_Foundation_of_Extended_Classical_Mechanics_ECM
[7]https://www.researchgate.net/post/Analysis_of_Concepts_within_the_Extended_Classical_Mechanics_ECM_Framework
[8]https://www.researchgate.net/publication/392232034_Appendix_B_Alignment_with_Physical_Dimensions_and_Interpretations_of_Standard_Categorization_of_Energy_Types_in_Extended_Classical_Mechanics_ECM
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.
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.
