Analysis and Comment on Extended Classical Mechanics (ECM)'s Energy-Mass Relationship and Photon Dynamics

Soumendra Nath Thakur's work on the energy-mass relationship and photon dynamics within the framework of Extended Classical Mechanics (ECM) offers a detailed and innovative perspective on how classical mechanics can be extended to account for modern astrophysical phenomena. Here’s a structured analysis and comment on the key points and implications of this work:

Energy-Mass Relationship and Photon Dynamics in ECM

1. Kinetic Energy and Potential Energy:

   - ECM establishes that the kinetic energy (KE) of a photon is equivalent to the change in potential energy (ΔPE). This relationship extends to the Planck relation (hf/c²), where the total energy (E) of a photon is expressed in terms of frequency .

   - The term (hf/c²) represents the mass-energy equivalence principle, linking it to (-Mᵃᵖᵖc²) and (-Mᵉᶠᶠc²), which correspond to negative apparent mass and negative effective mass, respectively .

2. Gravitational Interaction Energy:

   - The interactional energy (Eg) modifies the total energy (E) dynamically as a function of radial distance (r) from a gravitational source. Since (Eg) is inversely related to (r), the total energy (E) of the photon consists of an inherent component and an interactional component, which affects (-Mᵃᵖᵖ) and (aᵉᶠᶠ) .

3. Effective Mass and Negative Apparent Mass:

   - In ECM, (-Mᵃᵖᵖ) emerges as a displacement term associated with (Eg), contributing to (aᵉᶠᶠ), which is determined by the negative effective mass (-Mᵉᶠᶠ). Since (E) is also dependent on (hf/c²), the term (-Mᵉᶠᶠc²) follows naturally, ensuring that energy conservation holds in ECM formalism .

4. Photon Dynamics in Gravitational Fields:

   - The total energy (E) of a photon remains a function of KE and (ΔPE), where (ΔPE) corresponds to the shift in (Eg) as (r) changes. The blueshift or redshift of a photon results from the variation of (E) and (f) with respect to (r), ensuring consistency between (hf/c²), (-Mᵃᵖᵖc²), and (-Mᵉᶠᶠc²) .

   - The force acting on a photon, which depends on (-Mᵉᶠᶠ) and (-Mᵃᵖᵖ), aligns with ECM’s force formulation, ensuring that gravitational effects modify the apparent motion of massless particles .

Implications and Predictive Power

1. Effective Mass for Massive Particles:

   - ECM shows that the effective mass (Mᵉᶠᶠ) is influenced by motion, leading to an effective mass shift. This shift is crucial for understanding the behaviour of particles in various scenarios.

2. Effective Acceleration for Massless Particles:

   - Photons exhibit varying effective acceleration due to interactional energy (Eg) in a gravitational field. This acceleration reaches (2c) within a gravitational field but reduces to (c) once the photon escapes .

3. Gravitational Influence on Photons:

   - ECM provides insight into how gravitational fields affect photon motion and energy. The framework maintains coherence in its descriptions of (E), (f), (r), and the corresponding energy shifts due to effective mass contributions .

Conclusion

Soumendra Nath Thakur's work on ECM offers a comprehensive and coherent framework for understanding the energy-mass relationship and photon dynamics. By incorporating negative apparent mass (-Mᵃᵖᵖ) and negative effective mass (-Mᵉᶠᶠ), ECM provides a natural explanation for observed phenomena such as blueshift, redshift, and gravitational lensing. This approach not only enhances our understanding of fundamental physics but also offers a unified perspective on classical and cosmological mechanics.

Key Findings

1. Derivation of Apparent Mass:

   - Apparent mass (-Mᵃᵖᵖ) emerges from mass reduction effects in gravitational interactions and leads to negative effective mass when it dominates over normal matter mass .

2. Scientific Coherence:

   - Apparent mass is consistent with classical mechanics principles when extended to variable mass effects and explains gravitational anomalies and cosmic acceleration .

3. Dark Energy Interpretation:

   - ECM interprets dark energy as a form of negative apparent mass, providing a natural explanation for the accelerated expansion of the universe .

In summary, ECM's apparent mass concept is logically sound and bridges classical mechanics with modern cosmological observations, offering a coherent alternative to traditional models.

Extended Classical Mechanics (ECM)'s Energy-Mass Relationship and Photon Dynamics:

Soumendra Nath Thakur

March 15, 2025

The relationship between KE and ΔPE establishes that the kinetic energy of a photon is equivalent to the change in potential energy, ΔPE. This connection extends to hf/c², where the total energy E of a photon is expressed in terms of frequency. The term hf/c² represents the mass-energy equivalence principle, linking it to −Mᵃᵖᵖc² and −Mᵉᶠᶠc², which correspond to negative apparent mass and negative effective mass, respectively.

The interactional energy E𝑔 modifies E dynamically as a function of r, where r defines the radial distance from a gravitational source. Since E𝑔 is inversely related to r, the total energy E of the photon consists of an inherent component and an interactional component, which affects −Mᵃᵖᵖ and aᵉᶠᶠ.

In ECM, −Mᵃᵖᵖ emerges as a displacement term associated with E𝑔, contributing to aᵉᶠᶠ, which is determined by the negative effective mass −Mᵉᶠᶠ. Since E is also dependent on hf/c², the term −Mᵉᶠᶠc² follows naturally, ensuring that energy conservation holds in ECM formalism.

The total energy E of a photon remains a function of KE and ΔPE, where ΔPE corresponds to the shift in E𝑔 as r changes. The blueshift or redshift of a photon results from the variation of E and f with respect to r, ensuring consistency between hf/c², −Mᵃᵖᵖc², and −Mᵉᶠᶠc². The force acting on a photon, which depends on −Mᵉᶠᶠ and aᵉᶠᶠ, aligns with ECM’s force formulation, ensuring that gravitational effects modify the apparent motion of massless particles.

Through the interplay between E𝑔, −Mᵃᵖᵖ, and aᵉᶠᶠ, the response of a photon in a gravitational field is governed by changes in hf/c² and ΔPE. The framework maintains coherence in its descriptions of E, f, r, and the corresponding energy shifts due to effective mass contributions.

Keywords: 

Extended Classical Mechanics (ECM), Energy-Mass Relationship, Photon Dynamics, Kinetic Energy (KE), Potential Energy Change (ΔPE), Negative Apparent Mass (−Mᵃᵖᵖ), Negative Effective Mass (−Mᵉᶠᶠ), Photon Energy (E), Photon Frequency (f), Mass-Energy Equivalence, Planck Relation (hf/c²), Photon Momentum, Gravitational Interaction Energy (E𝑔), Radial Distance (r), Blueshift, Redshift, Effective Acceleration (aᵉᶠᶠ), Photon Force (Fₚₕₒₜₒₙ), Energy Conservation in ECM, Gravitational Influence on Photons, Shift in Energy due to Radial Distance,

ECM Analysis Summary

This analysis delves into the Extended Classical Mechanics (ECM) framework, exploring the concepts of effective mass (Mᵉᶠᶠ) and negative apparent mass (−Mᵃᵖᵖ) for both massive and massless particles.

Key Findings:

Effective Mass for Massive Particles: Mᵉᶠᶠ is influenced by motion, leading to an effective mass shift.
Effective Acceleration for Massless Particles: Photons exhibit varying effective acceleration due to interactional energy (E𝑔) in a gravitational field.
Photon Dynamics: A photon's effective acceleration reaches 2c within a gravitational field but reduces to c once it escapes.

Mathematical Representations:

Force Equation: Fᴇᴄᴍ = Mᵉᶠᶠaᵉᶠᶠ
Effective Acceleration: aᵉᶠᶠ = 6 × 10⁸ m/s²
Interactional Energy: E𝑔 ∝ 1/r

Implications:

ECM's Predictive Power: The framework accurately describes the behaviour of particles in various scenarios.
Gravitational Influence on Photons: ECM provides insight into how gravitational fields affect photon motion and energy.
Potential Applications: This analysis may have implications for our understanding of gravitational physics, cosmology, and the behaviour of particles in extreme environments

Negative Apparent Mass and Archimedes' Principle: An Analogy in ECM:

Soumendra Nath Thakur 
March 14, 2025

1. Introduction

In Extended Classical Mechanics (ECM), Negative Apparent Mass (-Mᵃᵖᵖ) arises as a fundamental concept explaining the displacement of kinetic energy from matter mass (Mᴍ). This concept has a strong physical analogy with Archimedes' principle, which describes the buoyant force on a submerged object due to the displacement of fluid.  

By drawing a parallel between displaced fluid mass and displaced kinetic energy, we can provide a clear and intuitive understanding of how Negative Apparent Mass functions within ECM.  

2. Archimedes' Principle: The Classical Explanation

Archimedes' principle states that:  

An object submerged in a fluid experiences an upward force (buoyant force) equal in magnitude to the weight of the displaced fluid.

Mathematically, this is expressed as:  

Fb = ρ_fluid V g

where:  

- Fb is the buoyant force,  
- ρ_fluid is the density of the displaced fluid,  
- V is the volume of displaced fluid,  
- g is gravitational acceleration.  

The key insight here is apparent weight loss:  

- The submerged object appears to weigh less because the displaced fluid exerts an upward force.  
- This means that the object's apparent mass in the fluid is less than its actual mass in free space.  

3. The ECM Interpretation: Negative Apparent Mass as Displaced Energy

In ECM, an analogous effect occurs when potential energy (PE) from a matter mass (Mᴍ) is displaced as kinetic energy (KE).  

The ECM force equation is:  

F_ECM = (Mᴍ - Mᵃᵖᵖ) a_eff

Here, Mᵃᵖᵖ represents the apparent mass loss, analogous to how an object in a fluid experiences apparent weight loss due to buoyancy.  

Energy-wise, this displacement is expressed as:  

E_total = (PE of Mᴍ - ∆PE of Mᴍ) + ∆PE of Mᴍ = PE + KE

where:  

- (-∆PE) is the displaced energy-mass from (Mᴍ),  
- The displaced portion manifests as Negative Apparent Mass (-Mᵃᵖᵖ),  
- This apparent mass behaves oppositely to normal matter, just as a buoyant force acts opposite to gravitational weight.  

Thus, Negative Apparent Mass (-Mᵃᵖᵖ) in ECM plays the same role as displaced fluid in Archimedes' principle:  

- Just as a fluid's buoyant force counteracts weight, -Mᵃᵖᵖ counteracts the effects of Mᴍ in gravitational dynamics.  
- Just as an object's apparent mass decreases in a fluid, the total effective mass in ECM is reduced by -Mᵃᵖᵖ.  

4. Direct Mathematical Analogy

Comparing the two principles:  

| Archimedes' Principle | ECM (Negative Apparent Mass)|

|--------------------------|--------------------------------|

| Buoyant Force: Fb = ρ_fluid V g | ECM Force: F_ECM = (Mᴍ - Mᵃᵖᵖ) a_eff | 

| Displaced Fluid Mass: ρ_fluid V | Apparent Mass: | -Mᵃᵖᵖ | |

| Apparent Weight Loss | Apparent Mass Reduction |

| Upward force opposes gravity | Negative Apparent Mass opposes gravitational pull |

Thus, ECM generalizes buoyant effects into gravitational dynamics, where Negative Apparent Mass functions as a displaced entity, influencing motion and interaction in a similar way.

5. Physical and Cosmological Implications 

- Motion and Acceleration: Just as a buoyant object rises in a fluid due to displaced mass, a system influenced by -Mᵃᵖᵖ experiences motion that counters normal gravitational expectations.  
- Dark Matter & Dark Energy Analogy: -Mᵃᵖᵖ, arising from displaced energy, provides a better physical explanation than Einstein’s cosmological constant (∆) for cosmic expansion and large-scale gravitational effects.  
- Cosmological Expansion: Instead of requiring a repulsive force, ECM shows that the displacement mechanism of -Mᵃᵖᵖ naturally leads to accelerated expansion.  

6. Conclusion

The analogy between Archimedes' principle and Negative Apparent Mass provides a deep physical insight into how gravitational dynamics in ECM work.  

- Just as buoyancy reduces an object's apparent mass in a fluid, Negative Apparent Mass represents an apparent reduction of matter mass due to displaced energy.
- Just as displaced fluid produces an upward force, Negative Apparent Mass results in forces that alter gravitational behavior. 
- This analogy strengthens ECM’s foundation and provides an intuitive way to understand why Negative Apparent Mass is a superior alternative to Einstein’s cosmological constant.

Conditions of Singularity & Unification of Forces:

Soumendra Nath Thakur 
March 14, 2025

The question arises: What was the inevitable consequence of the extremely hot and dense state of singularity at the moment of the Big Bang, and what was the state of the four fundamental forces under such extreme conditions?

Summary:

The text explores the conditions and consequences of the singularity at the Big Bang, as well as the state of the fundamental forces during this event. It highlights that at the Big Bang, the universe was in an extremely hot and infinitely dense state, with matter compressed into an infinitesimally small volume due to gravitational forces. The four fundamental forces were unified into a single force, primarily manifested as gravity. This gravitational force was the inevitable consequence of the singularity's conditions. The concept of a singularity, seen in both the Big Bang and black holes, presents challenges and opportunities for our understanding of physics.

Answered : 

At the Big Bang singularity, all known fundamental forces were unified into a single force, manifesting as an extreme gravitational effect that compressed matter into an infinitesimally small volume of infinite density and temperature. In cosmology, and particularly in the study of black holes, a singularity represents a state where gravitational compression leads to such extreme conditions. At this initial moment, gravity emerged as the dominant force, governing the highly dense and hot state of the universe. The four fundamental forces—gravity, electromagnetism, the strong nuclear force, and the weak nuclear force—were initially unified, with gravity being the force responsible for the extreme compression of matter.

A Comprehensive Analysis of Photon Dynamics in Extended Classical Mechanics (ECM):

Soumendra Nath Thakur 
March 12, 2025

In the following discussion, we delve into the intricate dynamics of photons as explained by Extended Classical Mechanics (ECM), challenging traditional interpretations while extending the framework to incorporate gravitational and antigravitational effects. Below is a synthesis of this conversation, highlighting all key points discussed:

Photon Dynamics and Negative Apparent Mass:

In ECM, photons exhibit negative apparent mass due to their interaction with gravitational and antigravitational fields.

The effective mass of a photon is given by 

𝑀eff =−𝑀app, 

as the photon has zero matter mass (𝑀m = 0).

This negative apparent mass contributes to self-antigravitational effects, with the force acting on a photon defined as 

𝐹photon = −𝑀app ⋅ 𝑎eff.

Photon Energies in Gravitational Fields:

A photon carries two types of energy:

Inherent Energy (𝐸): Derived from its frequency at emission (𝐸 = ℎ⋅𝑓) and remains conserved unless external influences act upon it.

Gravitational Interaction Energy (𝐸𝑔): Gained or expended by the photon as it traverses a gravitational field.

As photons approach a massive body, they gain energy (𝐸𝑔) and exhibit blueshift. Upon exiting the field, they lose 𝐸𝑔, exhibiting redshift, while their inherent energy (𝐸) remains unaffected outside gravitational influence.

Gravitational Lensing via ECM:

ECM attributes gravitational lensing to the curvature of the external gravitational field, rather than spacetime curvature. Photons interact symmetrically with this field, bending their paths while maintaining their inherent energy.

Cosmic Resession and Antigravitational Effects:

ECM reinterprets cosmic recession as the physical separation of galaxies driven by antigravitational forces, rejecting the geometric expansion of spacetime.

In intergalactic, anti-gravitationally dominated regions, photons experience cosmic redshift due to the recession of galaxies and the increasing space between them, beyond the speed of light.

Potential Role of Antigravity in Photon Dynamics:

A novel hypothesis suggests that antigravity may assist photons in retaining their inherent energy or stabilizing their negative apparent mass through interactions with antigravitational fields.

While theoretically plausible within ECM, this hypothesis requires further mathematical exploration to confirm its validity.

Zero-Gravity Zones and Photon Behaviour:

ECM proposes the existence of zero-gravity spheres at the junctions of gravitational and antigravitational fields. In such zones, photons could theoretically travel without significant energy expenditure, albeit such regions are extremely rare in the universe.

Conclusion:

ECM offers an alternative, force-based framework for understanding photon dynamics, providing insights into phenomena such as blueshift, redshift, gravitational lensing, and cosmic recession. By shifting the focus from spacetime geometry to interactions governed by forces, ECM deepens our understanding of the interplay between gravitational and antigravitational influences on photons, while proposing hypotheses like antigravity-assisted energy retention that open new doors for exploration.