Foundational Formulation of Extended Classical Mechanics: From Classical Force Laws to Relativistic Dynamics.

Soumendra Nath Thakur,

April 14, 2025

Abstract:

This research develops the foundational equations of Extended Classical Mechanics (ECM) by generalizing Newtonian mechanics through the inclusion of dynamic mass components such as negative apparent mass. ECM redefines force, acceleration, and gravitational interactions using an effective mass framework, expressed as the sum of traditional matter mass and a kinetic-energy-derived negative apparent mass. This dual-mass interaction leads to revised force laws and a spectrum of speed regimes for massive particles—ranging from gravitational confinement to antigravitational liberation. The formulation extends to massless particles like photons by assigning them an effective negative matter mass, enabling consistent force definitions and propagation behaviour at relativistic speeds. Radial distance plays a critical role in determining gravitational behaviour, with transitions from classical attraction to antigravitational expansion. The framework aligns with cosmological observations, particularly in large-scale structure behaviour, and provides a unified approach to understanding force, inertia, and motion in both massive and massless domains. ECM thus represents a coherent advancement of classical physics, accommodating gravitational variance, energy redistribution, and speed constraints in dynamic systems.

Keywords:

Extended Classical Mechanics (ECM), Effective Mass, Negative Apparent Mass, Gravitational Redefinition, Relativistic Force, Dynamic Inertia, Massless Particles, Speed Regimes, Antigravity Dynamics, Cosmological Consistency,

Introduction:

1. Newtonian Foundation Consistency:

In classical Newtonian mechanics, force is defined as the product of mass and acceleration. Acceleration is directly proportional to the applied force and inversely proportional to the mass. If there is no applied force, there is no acceleration, yet the mass remains unchanged. In this framework, gravitational mass is equal to inertial mass, indicating a fundamental equivalence between how mass responds to gravity and how it resists acceleration.

2. ECM Force Extension (Massive Bodies):

In Extended Classical Mechanics (ECM), for massive bodies, the total force is expressed as the product of effective mass and effective acceleration. The effective mass is defined as the sum of matter mass and negative apparent mass, the latter representing the kinetic-energy-equivalent mass opposing gravitational confinement. Effective acceleration remains directly proportional to the force, but now it is also inversely related to matter mass alone. A special condition holds: the reciprocal of matter mass, in absolute terms, equals the magnitude of the apparent mass.

3. ECM Speed Conditions for Massive Particles:

Three dynamic regimes emerge for massive particles based on the relative dominance between matter mass and the magnitude of apparent mass.

• When matter mass is greater than the magnitude of apparent mass, matter mass dominates the interaction. This corresponds to lower particle speeds because gravitational confinement remains stronger than kinetic liberation.

• When matter mass is equal to the magnitude of apparent mass, the two contributions balance. The system achieves a medium-speed regime, where confinement and liberation forces are dynamically in equilibrium.

• When the magnitude of apparent mass exceeds the matter mass, apparent mass dominates. This leads to high particle speeds, as the effective gravitational confinement weakens, and antigravitational dynamics begin to assert more influence.

4. Gravitational Extension in ECM:

In ECM, gravitational mass is redefined as the sum of matter mass and negative apparent mass and this total correspond to the effective mass in dynamic systems.

• At zero radial distance, the reciprocal of matter mass and apparent mass both vanish, resulting in gravitational mass equal to matter mass only.

• At nonzero radial distances, matter mass still dominates, and gravitational mass remains positive. Gravitational influence is active.

• At significantly large distances, matter mass balances the magnitude of apparent mass, and gravitational mass becomes zero. This marks the threshold where gravitational and antigravitational effects cancel each other.

• Beyond gravitational influence, where apparent mass overtakes matter mass, gravitational mass becomes negative. The system transitions into an antigravitational regime, dominated by expansive kinetic effects.

5. Massless Particles (Conventional, Photon-like):

For particles such as photons, which are conventionally treated as massless, ECM introduces a reinterpretation: their matter mass is considered effectively negative. The negative apparent mass is equal in magnitude to this negative matter mass, implying that the effective mass is twice the magnitude of the apparent mass.

Within a gravitational field, both the negative matter mass and negative apparent mass contribute to the force experienced by such particles. The effective acceleration is doubled in magnitude, leading to a relativistic condition where the product of effective mass and acceleration equals the speed of light. This reflects the expenditure of apparent mass to escape the gravitational field.

At the edge of gravitational influence—just as the particle escapes the field—only one contribution of apparent mass remains. The acceleration is still doubled, but no additional apparent mass is expended. The particle continues at the speed of light, now in a freely propagating, gravity-free regime.

Formulation:

1. Newtonian Foundation Consistency

F = ma; a ∝ F, a ∝ 1/m

Where, acceleration (a) is directly proportional to force (F) and inversely proportional to mass (m). When no force is applied:

F = 0 ⇒ a = 0, and mass remains constant: m = m

Classical Gravitational Mass:

Fɢ = mɢ·g where: mɢ is the gravitational mass, g is the gravitational field strength.

Classically, gravitational mass and inertial mass are recognized—and they are assumed equal:

mɢ = m

Extended Classical Mechanics (ECM) Formulation:

ECM application of Classical proportionality rules:

aᵉᶠᶠ ∝ Fᴇᴄᴍ; aᵉᶠᶠ ∝ 1/Mᴍ

This distinction is crucial to ECM, where inertial response is modulated by the balance between real and apparent mass.

|1/Mᴍ| = |Mᵃᵖᵖ| ⇒ Mᵃᵖᵖ ~ 1/Mᴍ

ECM generalizes Newtonian ideas by incorporating dynamic mass contributions, such as negative apparent mass (−Mᵃᵖᵖ).

2. ECM Force Extension (Massive Bodies)

Fᴇᴄᴍ = (Mᴍ + (−Mᵃᵖᵖ)) aᵉᶠᶠ = Mᵉᶠᶠ aᵉᶠᶠ

Where Mᴍ: matter mass (positive, gravitational). −Mᵃᵖᵖ: negative apparent mass (from KE or anti-gravitational behaviour). Mᵉᶠᶠ: effective mass = total inertial response, and aᵉᶠᶠ: effective acceleration

This term emerges from motion, gravitational interactions, or energy-mass coupling—mechanisms not present in classical theory.

In ECM: Gravitational mass (Mɢ) becomes:

Mɢ = Mᴍ + (−Mᵃᵖᵖ) = Mᵉᶠᶠ

Where Mᴍ: is the matter mass (traditional inertial mass), −Mᵃᵖᵖ is the negative apparent mass, representing: Kinetic energy's mass-equivalent, gravitationally induced mass offsets, Dynamic redistribution from field interactions.

Thus, gravitational mass becomes a net effective mass (Mᵉᶠᶠ) that varies depending on the particle's state and spatial context (e.g., radial distance from a mass source).

3. ECM Speed Conditions for Massive Particles

Speed domains by comparing Mᴍ and Mᵃᵖᵖ through:

Mᴍ - 1/Mᴍ

This form is symbolic:

Condition                     Interpretation                   Implication for Speed

·        Mᴍ >|Mᵃᵖᵖ|               Matter mass dominates        Low speed

                           o       (Low kinetic activity)   

·        Mᴍ =|Mᵃᵖᵖ                Balanced system                  Medium speed

·        Mᴍ <|Mᵃᵖᵖ|               Kinetic energy dominates      High speed

4. Gravitational Extension in ECM

Mɢ = Mᴍ + (−Mᵃᵖᵖ) = Mᵉᶠᶠ, where |Mᵃᵖᵖ| ~ (1/Mᴍ)

Breakdown by Radial Distance r:

Radial Distance      Mass Relation                   Gravity Behaviour

·        r = 0                       Mɢ = Mᴍ (no KE or Mᵃᵖᵖ)  Pure gravity  

·        r > 0                       Mᴍ > |Mᵃᵖᵖ|                    Gravity dominates

·        r ≫ 0                    Mᴍ = |Mᵃᵖᵖ|                 Effective gravity neutral

                         o       (Flat or marginal expansion)

·        r → ∞                     Mᴍ < |Mᵃᵖᵖ|                     Antigravity

                                                       o       (Repulsion, acceleration)

This is remarkably aligned with cosmological behaviour: it predicts an emergent anti-gravitational behaviour at large scale—very much like dark energy or expansion dynamics. Good match with A. D. Chernin et al.'s observational research titled "Dark energy and the structure of the Coma cluster of galaxies."

5. Massless Particles (Conventional, Photon-like)

In ECM, massless particles invert the conventional paradigm:

·         Mᴍ < 0 — interpreted as negative matter mass

·         -Mᵃᵖᵖ < 0 — represents kinetic energy equivalent mass (negative) 

·         Thus, -Mᵃᵖᵖ = |-Mᵃᵖᵖ| — used to denote its positive magnitude

Total force becomes:

Fᴇᴄᴍ = (Mᴍ + (-M)) aᵉᶠᶠ = 2|Mᵃᵖᵖ| aᵉᶠᶠ

This implies:

Mᵉᶠᶠ = 2|Mᵃᵖᵖ|, aᵉᶠᶠ = effective acceleration

Now, considering kinetic energy in ECM:

KE = ½Mᵉᶠᶠ v² = |Mᵃᵖᵖ| v²

This leads to a new interpretation of speed and acceleration:

• Within Gravitational Influence:

• aᵉᶠᶠ = 2c ⇒ v = c

• Condition: 

|Mᵃᵖᵖ| aᵉᶠᶠ = v

• Half the kinetic energy is spent in overcoming gravity:

KE = |Mᵃᵖᵖ| v² ⇒ ½ KE used in gravitational escape.

• Just Escaping Gravity (At Horizon):

• Escape velocity condition:

|Mᵃᵖᵖ| aᵉᶠᶠ = v = c

• No further acceleration needed — no kinetic energy is spent during motion:

KE = |Mᵃᵖᵖ| v² ⇒ No ½ KE spent — already at escape velocity

The above is internally consistent and creatively describes photon behaviour under gravitational influence, reinterpreting "massless" as net-zero effective mass resulting from real-negative and apparent-positive mass interactions.

Mathematical Presentation:

1. Newtonian Foundation Consistency

Begin with the foundational equation:

F = ma,

Where acceleration (a) is directly proportional to force (F) and inversely proportional to mass (m):

a ∝ F ; a ∝ 1/m

These are standard relationships in classical mechanics. When no force is applied:

F = 0 ⇒ a = 0, and mass remains constant: m = m

This restates the principle of inertia—an object maintains its state of rest or uniform motion unless acted upon by an external force.

Classical Gravitational Mass:

In Newtonian physics, gravitational mass (mɢ) is the property that determines how a body interacts with gravitational fields. It appears in Newton's law of universal gravitation and defines the gravitational force a body experiences:

Fɢ = mɢ·g

Here:

·        mɢ is the gravitational mass,

·        g is the gravitational field strength.

Classically, there is no distinction between different mass types (e.g., apparent mass, effective mass). Only gravitational mass and inertial mass are recognized—and they are assumed equal:

mɢ = m

Thus, in classical mechanics, the same symbol m is used for both inertial and gravitational mass, without separation into other mass forms. There is no concept of negative apparent mass (−Mᵃᵖᵖ) or dynamic gravitational behaviour based on internal or external energy contributions.

In Extended Classical Mechanics (ECM):

ECM generalizes Newtonian ideas by incorporating dynamic mass contributions, such as negative apparent mass (−Mᵃᵖᵖ). This term emerges from motion, gravitational interactions, or energy-mass coupling—mechanisms not present in classical theory.

In ECM:

Gravitational mass (Mɢ) becomes:

Mɢ = Mᴍ + (−Mᵃᵖᵖ) = Mᵉᶠᶠ

Where:

·        Mᴍ is the matter mass (traditional rest mass),

·        −Mᵃᵖᵖ is the negative apparent mass, representing:

Kinetic energy's mass-equivalent,

·        Gravitationally induced mass offsets,

·        Dynamic redistribution from field interactions.

Thus, gravitational mass becomes a net effective mass (Mᵉᶠᶠ) that varies depending on the particle's state and spatial context (e.g., radial distance from a mass source).

Summary:

·        In classical mechanics, gravitational mass is static and equals inertial mass.

·        In ECM, gravitational mass is dynamic, accounting for both matter mass and apparent mass effects.

·        This provides a framework where massless and massive particles can be treated under a unified force–energy perspective, especially when gravitational or relativistic phenomena are involved.

2. ECM Force Extension (Massive Bodies)

Fᴇᴄᴍ = (Mᴍ + (−Mᵃᵖᵖ)) aᵉᶠᶠ = Mᵉᶠᶠ aᵉᶠᶠ

With:

·        Mᴍ: matter mass (positive, gravitational)

·        −Mᵃᵖᵖ: negative apparent mass (from KE or anti-gravitational behaviour)

·        Mᵉᶠᶠ: effective mass = total inertial response

·        aᵉᶠᶠ: effective acceleration

ECM application of Classical proportionality rules:

aᵉᶠᶠ ∝ Fᴇᴄᴍ; aᵉᶠᶠ ∝ 1/Mᴍ

are consistent provided that acceleration is viewed as determined by Mᴍ, not Mᵉᶠᶠ. This distinction is crucial to ECM, where inertial response is modulated by the balance between real and apparent mass.

Further defined:

|1/Mᴍ| = |Mᵃᵖᵖ| ⇒ Mᵃᵖᵖ ~ 1/Mᴍ

This inverse relationship reflects the ECM principle that as matter mass increases, apparent mass decreases, and vice versa, i.e., KE builds up as mass thins out in effect.

3. ECM Speed Conditions for Massive Particles

Define speed domains by comparing Mᴍ and Mᵃᵖᵖ through:

Mᴍ - 1/Mᴍ

This form is symbolic, but conceptually solid. Here's a breakdown:

Condition                     Interpretation                Implication for Speed

·        Mᴍ >|Mᵃᵖᵖ|               Matter mass dominates       Low speed

·        Mᴍ =|Mᵃᵖᵖ                Balanced system                 Medium speed

·        Mᴍ <|Mᵃᵖᵖ|               Kinetic energy dominates    High speed

Consistent with ECM's view: apparent mass (i.e., kinetic content) governs motion dynamically relative to matter mass.

4. Gravitational Extension in ECM

Mɢ = Mᴍ + (−Mᵃᵖᵖ) = Mᵉᶠᶠ, where |Mᵃᵖᵖ| ~ (1/Mᴍ)

Breakdown by Radial Distance r:

Radial Distance           Mass Relation                   Gravity Behaviour

·        r = 0                       Mɢ = Mᴍ (no KE or Mᵃᵖᵖ)      Pure gravity  

·        r > 0                       Mᴍ > |Mᵃᵖᵖ|                      Gravity dominates

·        r ≫ 0                      Mᴍ = |Mᵃᵖᵖ|               Effective gravity neutral

                         o       (Flat or marginal expansion)

·        r → ∞                     Mᴍ > |Mᵃᵖᵖ|                      Antigravity

                                                       o       (Repulsion, acceleration)

This is remarkably aligned with cosmological behaviour: it predicts an emergent anti-gravitational behaviour at large scale—very much like dark energy or expansion dynamics. Good match with Chernin et al.'s observations.

5. Massless Particles (Conventional, Photon-like)

Here, ECM flips the paradigm:

Mᴍ < 0 (massless, interpreted as negative)

−Mᵃᵖᵖ = |Mᵃᵖᵖ| (positive KE equivalent mass)

Fᴇᴄᴍ = (Mᴍ + (-Mᵃᵖᵖ)) aᵉᶠᶠ ⇒ Fᴇᴄᴍ = 2|Mᵃᵖᵖ| aᵉᶠᶠ  

This gives:                                                 

Mᵉᶠᶠ = 2|Mᵃᵖᵖ|, aᵉᶠᶠ = effective acceleration

Now, introducing speed limit interpretation:

·        Within gravitational influence:

aᵉᶠᶠ = 2c ⇒ v = c

Due to 2|Mᵃᵖᵖ|aᵉᶠᶠ = v, and energy spent in escaping.

·        Just escaping gravity: 

  |Mᵃᵖᵖ| aᵉᶠᶠ = v = c

 No kinetic expenditure; just escape velocity = c.

The above is internally consistent and creatively describes photon behaviour under gravitational influence, reinterpreting "massless" as net-zero effective mass resulting from real-negative and apparent-positive mass interactions.

Final Consistency Check

Logical Integrity:

·        Each equation evolves from Newtonian mechanics but redefines mass/acceleration relationships in energetically dynamic terms.

·        The system conserves logical structure while redefining inertial and gravitational responses through ECM principles.

Dimensional Consistency:

·        All force equations retain the correct dimensions: 

F = [M] [a]

Physical Insight:

·        Massless particles accelerate as if they possess negative real mass offset by positive apparent mass.

·        Gravitational behaviour transitions to antigravity at large scales—mirroring cosmological acceleration.

Summary

ECM-based framework:

·        Is mathematically and physically consistent.

·        Effectively extends Newtonian mechanics with meaningful reinterpretations of mass, energy, and motion.

·        Offers novel insight into massless particles, antigravity, and cosmic-scale gravitational behaviour.

·        Supports intuitive analogues to dark energy, inertia-kinetic duality, and relativistic limits.

ECM Term Definition — Matter Mass (Mᴍ):

Matter Mass, symbolized as Mᴍ, refers to the total positive gravitational mass derived from all forms of matter, both visible and non-visible. It is defined as:

Mᴍ = Mᴏʀᴅ + Mᴅᴍ

Where: 

·         Mᴏʀᴅ is the Ordinary Matter Mass, consisting of atoms, particles, and objects observable through electromagnetic interaction (e.g., stars, gas, planets, etc.). 

·         Mᴅᴍ is the Dark Matter Mass, which cannot be directly observed but whose gravitational influence is well-documented (e.g., via galaxy rotation curves, cluster dynamics, and lensing effects).

This formulation reflects the observational evidence that approximately 27% of the universe's total energy content is attributed to matter, with ordinary matter contributing only about 5%, and dark matter making up the remaining ~22%. These proportions are consistent with studies such as:

·         Chernin, A. D., Bisnovatyi-Kogan, G. S., Teerikorpi, P., Valtonen, M. J., Byrd, G. G., & Merafina, M. (2013), Astronomy and Astrophysics, 553, A101.

In the ECM framework, Matter Mass forms the foundational term used in both gravitational and dynamic considerations. However, ECM does not equate Mᴍ to gravitational mass. Instead, gravitational mass is redefined as:

Mɢ = Mᴍ + (-Mᵃᵖᵖ)

Where −Mᵃᵖᵖ is the Apparent Mass (with a negative sign), which in ECM is equivalent to the effective mass of dark energy (Mᴅᴇ). This relationship accounts for the antigravitational behaviour observed on cosmological scales and aligns with the balance conditions observed in Chernin’s “zero-gravity spheres” at large radial distances.

Discussion

The presented formulation offers a clear and comprehensive departure from traditional Newtonian mechanics by extending foundational principles through the lens of Extended Classical Mechanics (ECM). While rooted in Newton’s force law F = ma, ECM redefines the nature and composition of mass, bringing kinetic energy and gravitational context into dynamic roles previously unaccounted for.

Newtonian Groundwork and the ECM Shift

Classical mechanics treats mass as both inertial and gravitational, implicitly assuming their equivalence and constancy. However, in ECM, this uniformity dissolves. Mass becomes a dynamic quantity—decomposed into two components:

·         Matter Mass (Mᴍ) representing intrinsic gravitational matter (including both ordinary and dark matter), and

·         Negative Apparent Mass (-Mᵃᵖᵖ), representing the mass-equivalent of kinetic energy and gravitational-field interaction.

This dual-mass framework transforms classical gravitational mass into an effective mass:

Mɢ = Mᴍ + (-Mᵃᵖᵖ) = Mᵉᶠᶠ

This shift allows ECM to address dynamic systems where energy exchange alters the net mass, which classical mechanics cannot account for without invoking relativistic or quantum principles.

Force and Acceleration: A Redefined Relationship

One of the striking innovations in ECM is the distinction between inertial source and accelerative response. While force remains proportional to acceleration, the inertial resistance is attributed to the effective mass, not solely the matter mass. Yet, paradoxically, acceleration remains inversely proportional to matter mass:

aᵉᶠᶠ ∝ 1/Mᴍ

This distinction introduces an ECM-specific inertia principle, where the observable acceleration stems from how much real mass resists motion, while the net force includes additional dynamic components like kinetic mass or gravitational distortion.

Speed Regimes and Mass Balance

By comparing (Mᴍ) and |Mᵃᵖᵖ|, ECM introduces a robust classification of motion:

·         Low speeds occur when rest mass dominates,

·         Medium speeds at a critical balance point, and

·         High speeds when kinetic (apparent) mass becomes dominant.

This interpretation yields a continuous transition from rest to motion, closely linked with energy-mass dynamics. It also implies that velocity is not an isolated vector but the outcome of a shifting mass-energy balance, a nuance classical mechanics lacks.

Gravitational Radius and Cosmological Insight

The model’s capacity to account for varying gravity with radial distance (r) stands as a profound contribution. It elegantly explains:

·         Local gravity (r = 0) as pure mass-dominated attraction,

·         Intermediate distances ( r > 0) as zones of mass-kinetic interplay, and

·         Cosmic-scale distances (r ≫ 0) where Mᴍ ≈ |Mᵃᵖᵖ|, leading to net gravitational neutrality or repulsion.

This is not only consistent with Chernin et al.'s observations of "zero-gravity spheres" but also offers a functional explanation for dark energy as a dynamic outcome of negative apparent mass, rather than a cosmological constant or exotic field.

Massless Particle Redefinition

The treatment of massless particles under ECM is especially transformative. ECM reframes the photon not as a truly massless particle, but as one exhibiting a cancellation between negative matter mass and positive kinetic energy mass:

Mᴍ < 0, -Mᵃᵖᵖ = |Mᵃᵖᵖ|, Mᵉᶠᶠ = 2|Mᵃᵖᵖ|

This leads to ECM's prediction that photons under gravitational influence experience acceleration as if propelled by their internal energy redistribution—a viewpoint supporting both relativistic speed limits and gravitational redshift mechanisms without invoking spacetime curvature directly.

Logical and Physical Coherence

The ECM framework remains dimensionally and logically consistent with Newtonian mechanics while allowing for:

·         Mass variation with energy and spatial context,

·         Force expressions consistent across massive and massless systems,

·         Interpretation of relativistic and cosmological behaviors using classical equations enhanced with new terms.

Implications and Broader Significance

This approach does more than reinterpret equations—it provides a unifying language for classical, relativistic, and cosmological phenomena. By internalizing the concept of apparent mass, ECM not only bridges gaps in Newtonian mechanics but also offers testable insights into:

·         Dark energy and cosmic expansion,

·         High-velocity particle behavior,

·         Gravitational influence at multiple scales, and

·         Photon dynamics within gravitational fields.

Alphabetical Glossary of Mathematical Terms in ECM

·         aᵉᶠᶠ (Effective Acceleration) 

The acceleration resulting from the ECM-adjusted net mass (Mᵉᶠᶠ), it represents the physical acceleration observed when accounting for both real and apparent mass components.

·         c (Speed of Light)

The fundamental speed limit for information and energy propagation in spacetime, in ECM, it represents the upper bound for particles under gravitational escape or massless propagation.

·         F (Classical Force)

Defined by Newton’s second law as F = ma. In ECM, this forms the foundational equation upon which force extensions are built.

·         Fᴇᴄᴍ (ECM Force)

The generalized ECM force expression: Fᴇᴄᴍ = Mᵉᶠᶠ aᵉᶠᶠ = (Mᴍ + (−Mᵃᵖᵖ)) aᵉᶠᶠ Accounts for both real matter mass and negative apparent mass contributions in the system's inertial behaviour.

·         g (Classical Gravitational Field Strength)

The intensity of gravitational acceleration experienced by a mass, appears in the classical gravitational force equation (F = mɢ g).

·         gᵉᶠᶠ (Effective Gravitational Field Strength) 

The actual gravitational acceleration experienced by a body in ECM, defined as the net field influence resulting from both matter mass (Mᴍ) and apparent mass (−Mᵃᵖᵖ). Appears in the generalized gravitational force equation: Fɢ = Mɢ gᵉᶠᶠ = (Mᴍ + (−Mᵃᵖᵖ)) gᵉᶠᶠ. Unlike classical (g), which is static, gᵉᶠᶠ dynamically changes with spatial context (radial distance), mass-energy redistribution, and the local gravitational environment is — especially reflecting transitions between gravity-dominated and antigravity dominated regions.

·         Mᴍ (Matter Mass)

Total positive gravitational mass composed of ordinary matter (Mᴏʀᴅ) and dark matter (Mᴅᴍ). It is the primary rest-mass term in ECM.

·         Mᴏʀᴅ (Ordinary Matter Mass)

Mass made of observable matter (atoms, particles, objects emitting or interacting with EM radiation) Subset of Mᴍ.

·         Mᴅᴍ (Dark Matter Mass)

Non-luminous, indirectly observed mass inferred from gravitational effects, also a component of Mᴍ.

·         Mᵃᵖᵖ (Apparent Mass) 

A dynamic mass component arising from kinetic energy or gravitational interaction, in ECM, it is negative and represents the inertial contribution from energy motion, not substance.

·         Mᵉᶠᶠ (Effective Mass)

Defined as the net inertial mass in ECM: Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ)

Determines how body resists acceleration, incorporating both positive matter mass and negative apparent mass.

·         Mɢ (Gravitational Mass) 

In ECM, redefined from its classical form to: Mɢ = Mᴍ + (-Mᵃᵖᵖ) = Mᵉᶠᶠ. This formulation allows mass to evolve based on energy redistribution and gravitational conditions.

·         r (Radial Distance)

The spatial separation from a central gravitational source, in ECM, the different values of (r) change the relative dominance of Mᴍ and Mᵃᵖᵖ, leading to shifts between gravity and antigravity behaviour.

·         v (Velocity)

The speed of a particle, determined in ECM as a function of apparent mass and acceleration, for massless particles under gravity: Fᴇᴄᴍ = 2|Mᵃᵖᵖ| aᵉᶠᶠ ⇒ v = c

Conclusion:

The formulation of Extended Classical Mechanics (ECM) equations from classical foundations provides a coherent and innovative extension of Newtonian mechanics, bridging conventional limitations through the introduction of effective mass and negative apparent mass. By redefining gravitational and inertial interactions, ECM offers a refined understanding of force, acceleration, and mass behaviour across both massive and massless regimes. It maintains classical proportionality principles while enhancing interpretative power for relativistic and cosmological phenomena—such as high-speed particle dynamics and the transition from gravitational to antigravitational domains. ECM's consistent treatment of mass-energy coupling, particularly through its reinterpretation of photons and gravitational thresholds, introduces a unifying framework applicable to both local and large-scale dynamics. This advancement aligns with empirical research, notably Chernin et al.'s observations of cosmic structures, and sets a robust groundwork for further theoretical and experimental exploration.

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4.      Thakur, S. N. (2024). Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics. Preprints. https://doi.org/10.20944/preprints202409.1190.v2

5.      Thakur, S. N., & Bhattacharjee, D. (2023a). Phase Shift and Infinitesimal Wave Energy Loss Equations. preprints.org (MDPI). https://doi.org/10.20944/preprints202309.1831.v1

6.      The Large Scale Structure of Space-Time by Stephen Hawking and eorge Ellis

7.      Dark Matter and the Dinosaurs: The Astounding Interconnectedness of the Universe" by Lisa Randall

8.      Cosmology by Steven Weinberg

9.      The Quantum Theory of Fields by Steven Weinberg

·         Author’s ORCiD: 0000-0003-1871-7803.

·         Tagore’s Electronic Lab, West Bengal, India

·         Correspondence: postmasterenator@gmail.com

Declaration:

·         Author declares no conflict of Interest

·         No financial aid is received for this work.

The Necessity of a Constant Temporal Scale (∆t) in Entropy:

Soumendra Nath Thakur 

April 13, 2025

Entropy and Time (∆t):

Why in entropy, ∆t = Constant:

Because, ∆t = constant is not merely a convenient assumption; it is a foundational necessity for the meaningful evolution of entropy.

In the context of entropy, time is not treated as an independently existing substance or dimension, but rather as an emergent abstraction that measures the rate of existential change. The key requirement is that this measurement must occur on a consistent scale. Thus:

∆t = constant is essential not because time exists on its own, but because a fixed temporal scale is required to interpret and quantify changes in existence consistently.

This principle becomes critically evident in entropy equations:

S = -k² T ln(2) t/h

where entropy S evolves as a linear function of time. This linearity implicitly assumes that each increment of time (∆t) contributes a uniform amount to the evolution of entropy. If ∆t were not constant, the rate of entropy increase would become inconsistent or undefined.

Why ∆t = constant is Required:

If both:

The magnitude of existential change, and the scale or unit of time (∆t) are allowed to vary independently, then:

1. There’s no fixed frame to evaluate how much change is occurring.

2. Entropy loses its predictable progression—undermining not just our calculations, but our very capacity to define "change."

3. This leads to epistemological collapse: without a stable scale, even the concept of evolution or increase becomes meaningless.

Conclusion:

In entropy, ∆t = constant is a scientifically grounded requirement—not a philosophical preference. It ensures that we can observe, define, and quantify change in a coherent way. Without this fixed scale of time, the entire framework of entropy—and by extension, the measurable flow of existence—collapses into ambiguity.

Why the equation S = -k² T ln(2) t/h does not invalidate the idea that ∆t = Constant} (i.e., time has a constant *scale* in entropy):

1. Time Appears Linearly:

In the equation, time (t) is a linear variable, meaning that entropy increases proportionally with time.

This proportionality only works if each time increment ∆t contributes a consistent, quantifiable amount to the increase in entropy.

If (∆t) were variable or inconsistent, the relationship between entropy and time would be nonlinear or chaotic—which this equation does not suggest.

2. What the Equation Says Physically:

Entropy (S) evolves with:

  Temperature (T)

  Fundamental constants (k, h)

  Information units (via ln(2))

  Time (t) as a scaling dimension

This structure implies that:

As time increases by equal steps (i.e. ∆t = constant), entropy increases smoothly, predictably, and deterministically.

3. Quantum Units and Scaling:

t/h tells us time is scaled against Planck's constant.

That doesn't mean time intervals shrink or stretch, it means that quantum effects set the baseline resolution of time.

  

Thus, quantum scaling is fixed, and so is the emergent unit of time (∆t) as applied in entropy dynamics.

4. Negative Sign Does Not Imply Time Variability:

The negative sign in the equation could indicate:

Directionality of entropy (e.g., increase with time forward)

Convention used in specific derivations

It does not imply variable time, nor does it mean entropy reverses or destabilizes.

5. Why This Supports (∆t = constant}:

If ∆t were not constant:

Entropy increments would not match with linear t .

One’d lose the ability to make precise predictions of entropy change over intervals.

The logarithmic relationship to information (via ln(2)) would also break down, since it relies on discrete, countable steps of change in time.

Conclusion:

The equation S = -k² T ln(2) t/h assumes and requires that the scale of time ∆t is constant for entropy to evolve predictably.

Therefore, this equation does not contradict ∆t = constant; rather, it mathematically and physically confirms it.

Reference: 

(1a) Thakur,  Soumendra Nath (2024). Re-examining Time Dilation through the Lens of Entropy. Qeios. http://doi.org/10.32388/XBUWVD 

(1b) Thakur,  Soumendra Nath (2024). Re-examining Time Dilation through the Lens of Entropy. ResearchGate. https://www.researchgate.net/publication/378501318_Re-examining_Time_Dilation_through_the_Lens_of_Entropy

Constancy of Time Progression Across Entropic States: A Scientific Rebuttal to Time Dilation:

Soumendra Nath Thakur

April 12, 2025.

In both low and high entropy states:

Progress of time = Constant.

This confirms that time does not dilate—it simply progresses with entropy.

Any claim to the contrary implies that entropy arises from time, which contradicts the foundational observation that time emerges from change.

This is a scientific conclusion, not a speculative interpretation.

Reference:

https://www.researchgate.net/publication/378501318_Re-examining_Time_Dilation_through_the_Lens_of_Entropy

ECM Mass–Energy Transformation Summary:

Soumendra Nath Thakur  April 12, 2925

ECM-Based Comparison of Massive vs. Massless Particles: Force, Energy, and Gravitational Behaviour.

Exploring the Roles of Negative Apparent Mass and Effective Acceleration Across Particle Classes.

Soumendra Nath Thakur

April 10, 2025

ECM Interpretation of Force and Energy: Unified View for Massive and Massless Particles

1. Classical Mechanics Baseline:

In classical mechanics:

F = ma

This represents force as the product of inertial mass m and acceleration a. However, this model doesn't account for gravitational energy redistribution or effective mass transitions due to environmental fields.

2. ECM Force Equation for Massive Particles

Fᴇᴄᴍ = (Mᴍ −Mᵃᵖᵖ)aᵉᶠᶠ = Mᵉᶠᶠaᵉᶠᶠ 

Interpretation:

•  Mᴍ: Matter mass.

•  Mᵃᵖᵖ: Negative apparent mass, representing the mass of kinetic energy (dynamic, gravitationally influenced).

•  Mᵉᶠᶠ = Mᴍ −Mᵃᵖᵖ: The effective mass perceived in motion or under gravitational interaction.

This form maintains Newton's Second Law but adapts it for ECM's dynamic mass considerations.

2.1 Total Energy of Massive Particle in ECM:

Eₜₒₜₐₗ = PE + KE

• Potential Energy (PE) is associated with Mᵉᶠᶠ, the positive or net mass still governed by gravitational interaction.

• Kinetic Energy (KE) is dynamically expressed through:

Kinetic energy is dynamically attributed to the negative apparent mass:

KE ∝ −Mᵃᵖᵖ

  

• Important Cases:

  • Within gravitational influence: Mᵉᶠᶠ > |Mᵃᵖᵖ|, so effective inertia dominates.

  • The idea of a transient zero-effective-mass state at the threshold where gravity and antigravity merge gives a lot of depth to ECM's interpretation of particle behaviour. That fleeting balance — where Mᴍ = -Mᵃᵖᵖ where  Mᵉᶠᶠ = 0 — essentially represents a moment of physical neutrality, but it's too unstable to factor into gravitational dynamics meaningfully.

  • Beyond gravitational influence: Mᵉᶠᶠ < 0, meaning antigravitational effects dominate, potentially repulsive.

2.2 ECM Kinetic Energy for Massive Particle:

Eᴋ = ½Mᵉᶠᶠv², where Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ.

Since kinetic energy reflects dynamic interaction, it is physically attributed to the contribution of −Mᵃᵖᵖ.

Meaning:

This identifies −Mᵃᵖᵖ as the source of kinetic energy, not a passive consequence of motion but as an active, field-dependent source of kinetic motion. The mass-energy of motion (kinetic) is carved out from the gravitational or interactional field response encoded in Mᵃᵖᵖ.

3. ECM Force for Massless Particles (e.g., Photon):

Photons exhibit no rest mass, yet experience and induce gravitational interactions. ECM captures this via:

Fᴇᴄᴍ = (Mᴍ −Mᵃᵖᵖ)aᵉᶠᶠ = (−Mᵃᵖᵖ −Mᵃᵖᵖ)aᵉᶠᶠ = −2Mᵃᵖᵖaᵉᶠᶠ

Interpretation:

• No conventional matter mass (Mᴍ ≈ 0), but interactionally interpreted as dynamically negative due to energy redistribution. At the boundary where gravity and antigravity precisely balance — producing Mᵉᶠᶠ = 0 — a transient state of physical neutrality arises. However, due to its inherent instability, this state doesn't contribute meaningfully to sustained gravitational dynamics.

• The entire energy is attributed to interactional and inherent dynamic mass (-Mᵃᵖᵖ × 2)

• The factor of 2 originates from two distinct sources contributing to photonic energy: 

   • One portion represents the inherent dynamic motion (i.e., the photon's intrinsic kinetic-like energy), 

   • The other arises from gravitational coupling, reflecting the photon's response to or interaction with surrounding fields.

Mᵉᶠᶠ = −2Mᵃᵖᵖ

3.1 Total Energy for Massless Particles (e.g. Photon)

In ECM, a photon's total energy is the combination of its potential and kinetic contributions, both represented in terms of negative apparent mass (−Mᵃᵖᵖ).

Key Steps:

• Effective Mass:

Mᵉᶠᶠ = (Mᴍ − Mᵃᵖᵖ)

Since for massless particles:

Mᴍ = −Mᵃᵖᵖ (not strictly zero, but dynamically negative), then,

Mᵉᶠᶠ = (−Mᵃᵖᵖ −Mᵃᵖᵖ) = −2Mᵃᵖᵖ

• Potential Energy (PE):

Interpreted as the effective mass contribution ⇒ 

PE = Mᵉᶠᶠ = −2Mᵃᵖᵖ

• Kinetic Energy (KE):

Defined by one part of −Mᵃᵖᵖ arising from inherent dynamic motion.

• Therefore:

Total Energy = PE + KE = −2Mᵃᵖᵖ 

Whether within or beyond gravitational fields, photons preserve these ratios, though the magnitudes shift:

•  Near gravitational bodies: total Mᵉᶠᶠ = −2Mᵃᵖᵖ

•  In far regions, beyond gravitational influence: Mᵉᶠᶠ ≈ −Mᵃᵖᵖ due to energy loss/redshift.

3.2 ECM Kinetic Energy for Massless Particles:

Eᴋ = ½ × −2Mᵃᵖᵖ × c² (within gravitational field)

This gradual energy loss with constant velocity explains observed photon redshift across cosmological distances without altering the speed of light.

Eᴋ = ½ × −Mᵃᵖᵖ × c² (beyond gravitational field)

•  The speed remains c, but the mass-energy dynamically adjusts.

•  This explains gravitational redshift: photons lose effective energy, not speed.

•  The halving of kinetic energy reflects how gravitational fields 'drain' energy from the interactional component.

Conclusion: ECM vs Classical Force-Energy Structure

Concept         Classical Mechanics ECM Interpretation       

Force Equation    F = ma            F = Mᵉᶠᶠaᵉᶠᶠ 

KE (Massive)        ½mv²          ½Mᵉᶠᶠv² = −Mᵃᵖᵖ

KE (Massless)            N/A                   ½(−2Mᵃᵖᵖ)c²

PE                             Position-based    Mass-based: PE = Mᵉᶠᶠ 

Mass Variation             Fixed                 Dynamical-shift, field-influence

Role of -Mᵃᵖᵖ                Absent               Dynamic driver energy/motion

ECM-Based Interpretation of Force and Energy:

1. Classical Foundation

In classical mechanics, force is understood as a product of mass and acceleration. This assumes that mass is constant and unaffected by motion or gravitational context. Kinetic energy, in this view, is simply the energy of motion based on velocity and mass, and there's no deeper role assigned to gravitational fields in this interaction.

2. ECM Approach for Massive Particles

ECM reframes this relationship. It introduces two distinct components of mass: matter mass (the observable, rest-like mass with mass of dark matter) and apparent mass, which is negative and represents the energy that has been dynamically absorbed or engaged during motion or gravitational interaction.

• The effective mass is the net quantity resulting from subtracting this negative apparent mass from the matter mass. This effective mass is the true contributor to the experienced force in ECM.

• The negative apparent mass is not just a conceptual byproduct — it actively represents the kinetic energy component of the system, emerging from or being sustained by gravitational interaction.

• As a result, force and motion for massive particles are determined not by matter mass alone, but by this shifting balance between the all mass types.

Within gravitational fields, the effective mass remains positive and dominant, leading to normal attractive dynamics. However, far from gravitational sources, the apparent mass can dominate, flipping the sign of effective mass, implying repulsion or an antigravity-like behaviour.

3. Total Energy Breakdown for Massive Particles:

In ECM, total energy still includes both potential and kinetic components — but each has a new physical meaning:

• Potential energy is linked to the effective mass, which includes gravitational and positional influences.

• Kinetic energy is explicitly tied to the negative apparent mass, giving it a defined, quantifiable role rather than treating it as a simple energy of motion.

4. ECM for Massless Particles (like Photons):

Photons don’t have matter mass in the traditional sense. Yet, they interact gravitationally and carry energy. ECM handles this by interpreting their energy and force generation as coming entirely from negative apparent mass.

• Here, the effective mass becomes purely interactional — essentially made of two portions of apparent mass: one representing inherent motion energy, the other representing coupling with the gravitational environment.

• This leads to a natural explanation for the energy of massless particles, including changes due to redshift when they escape gravitational fields.

5. Kinetic Energy in Different Gravitational Regimes

For massless particles, ECM shows how kinetic energy is greater near gravitational sources and diminishes farther out. The energy isn’t lost in speed (which remains at light speed) but in effective mass. This gives a mechanistic interpretation of redshift: it’s not just a stretching of waves but a physical loss of energy due to gravitational context.

For massive particles, kinetic energy is similarly derived from the apparent mass, and its contribution depends on both velocity and the gravitational setting.

6. Summary of Key Differences:

• Mass is no longer static. ECM redefines it as context-dependent, varying with motion and field interaction.

• Kinetic energy has structure. It’s not just motion; it comes from a specific negative mass-like term that represents dynamic energy.

• Photons gain depth. Their energy arises from mass-like properties through interaction, not through rest mass.

• Gravitational influence becomes a central, dynamic sculptor of mass and energy. It doesn’t just bend trajectories — it reshapes the very nature of mass-energy interactions, offering elegant explanations for inertia, redshift, and cosmic acceleration without invoking hypothetical vacuum fluctuations.