Apparent Mass in massive and massless particles in dynamics:

Soumendra Nath Thakur
March 20, 2025

In ECM Force equation of massive particle expressed as:

Fₘₐₛₛ = (Mᴍ + (-Mᵃᵖᵖ))aᵉᶠᶠ

In ECM Force equation of massless particle:

Fₘₐₛₛₗₑₛₛ = (-Mᵉᶠᶠ + (-Mᵃᵖᵖ))aᵉᶠᶠ, 

where: Mᴍ = -Mᵉᶠᶠ, since Mᴍ < 0 also, Eg can be equated as -Mᵉᶠᶠ = E𝑔 and E can be equated as: -Mᵃᵖᵖ = E

Simplified as:

Fₘₐₛₛₗₑₛₛ = (-Mᵃᵖᵖ + (-Mᵃᵖᵖ))aᵉᶠᶠ where -Mᵃᵖᵖ = -Mᵉᶠᶠ, since Mᴍ <0  = - Mᵉᶠᶠ 

Further Simplified:

Fₘₐₛₛₗₑₛₛ = (-Mᵃᵖᵖ + -Mᵃᵖᵖ))aᵉᶠᶠ where E𝑔 can be equated as -Mᵉᶠᶠ = E𝑔 and E can be equated as: -Mᵃᵖᵖ = E

Resolved as: 

Fₘₐₛₛₗₑₛₛ = -2·Mᵃᵖᵖ·aᵉᶠᶠ  

Comparison of Mass-Energy Transformations: ECM vs. Relativity:

March 18, 2025

Soumendra Nath Thakur

Superiority of ECM Mass-Energy Transformation Over Relativistic Mass-Energy Transformation: The Role of Acceleration Accountability.  

One of the fundamental shortcomings of relativistic mass-energy transformation is its failure to explicitly account for acceleration between reference frames. The relativistic framework defines mass-energy transformations solely in terms of velocity, neglecting the dynamic role of acceleration in modifying mass-energy relationships. This omission leads to inconsistencies in force-energy interactions, mass variability, and gravitational effects, limiting the explanatory power of relativity in non-inertial and gravitationally evolving systems.  

Relativistic Mass Formula:

E = γMc²

where γ is the Lorentz factor, defined as:

γ = 1/√(1 - v²/c²)

This formulation implies mass increases with velocity but does not account for acceleration-driven mass variations or its role in force interactions.  

ECM Mass-Energy Transformation Equation:  

Extended Classical Mechanics (ECM) resolves this issue by explicitly incorporating acceleration effects into the mass-energy relationship:  

Eᴇᴄᴍ = (Mᴍ −Mᵃᵖᵖ)c²

where:  

• Mᵃᵖᵖ Negative Apparent Mass is dynamically linked to acceleration and describes energy displacement effects.  
• The term c² emerges from classical kinetic energy KE = 1/2 Mv² by considering:  

v = ℓP/tP = c.

This formulation ensures that changes in energy, mass, and force remain dynamically consistent, capturing the true physical impact of acceleration on mass-energy interactions.  

Key Advantages of ECM Over Relativity: 

1. Acceleration-Dependent Mass Variation: Unlike relativity, ECM explicitly incorporates acceleration-driven mass modifications through Mᵃᵖᵖ.  

2. Logical Consistency in Force-Energy Relations: ECM ensures a self-consistent connection between force, mass, and energy, eliminating the need for relativistic postulates.  

3. Superior Predictive Capability: ECM provides a natural explanation for repulsive gravitational effects without requiring a cosmological constant Λ.  

4. Avoids Velocity-Based Mass Increase Assumption: In ECM, mass-energy transformations occur dynamically due to acceleration-induced energy displacement rather than an instantaneous velocity-dependent change.  

Conclusion: 

The ECM mass-energy transformation provides a superior, logically consistent alternative to relativistic mass-energy concepts by correctly incorporating acceleration effects and ensuring proper force-energy relationships. By extending classical mechanics with Mᵃᵖᵖ, ECM offers a more comprehensive and physically grounded approach to mass-energy interactions across all motion regimes.  

Conclusion & Key Findings: Extended Classical Mechanics.

March 18, 2025

Conclusion

Soumendra Nath Thakur's work on Extended Classical Mechanics (ECM) provides a comprehensive and detailed framework for understanding the energy-mass relationship and photon dynamics in the context of the Big Bang event. By incorporating negative apparent mass -Mᵃᵖᵖ and effective mass Mᵉᶠᶠ, ECM offers a natural explanation for the 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. Initial Energy State:

   - The Big Bang event involved the conversion of potential energy to kinetic energy, driving the universe's expansion.

2. Acceleration Scaling:

   - The effective acceleration scales with gravitational interaction, reaching >2c at the initial singularity and reducing to c at larger distances (Planck scale).

3. Universal Perspective:

   - The universal perspective incorporates speeds significantly greater than c, especially at the Planck scale.

4. Kinetic Energy and Frequency Relationship:

   - The kinetic energy is related to the change in potential energy, with significant implications for the early universe's dynamics.

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.

The Universal Force-Energy-Gravity State from an Extended Classical Mechanics (ECM) Perspective:

Soumendra Nath Thakur
March 18, 2025

In the framework of Extended Classical Mechanics (ECM), the universal total energy, denoted as Eₜₒₜₐₗᴜₙᵢᵥ  can be understood as a dynamic balance of energy components that govern the fundamental interactions of the universe. At any given moment, this energy is composed of two primary elements: the universal potential energy, PEᴜₙᵢᵥ, which accounts for the energy stored in gravitational interactions, and the displaced energy-mass component, which influences the system’s effective mass and motion.

Within this perspective, the concept of negative apparent mass, -Mᵃᵖᵖ, emerges as a key factor in determining the overall energy distribution. It represents an energy-mass equivalence that effectively reduces the observable mass-energy of a system while maintaining gravitational coherence. This negative apparent mass acts analogously to a buoyant force in a fluid medium, where an object appears to lose weight due to the displacement of the surrounding medium’s energy density.

From an ECM standpoint, the universal kinetic energy, KEᴜₙᵢᵥ complements the potential energy to establish a complete energy framework. The variations in apparent mass contribute directly to shifts in the effective energy balance, influencing the dynamic behaviour of celestial structures and fundamental cosmological events. This interplay between gravitational potential, displaced energy-mass, and effective kinetic contributions provides a refined understanding of energy interactions beyond conventional models.

By applying these principles to the early universe, ECM reinterprets the force-energy-gravity state at the Big Bang event without relying on traditional vacuum energy fluctuations. Instead, the emergence of universal energy structures can be attributed to the redistribution of effective mass-energy states, where the role of negative apparent mass plays a crucial part in shaping early gravitational and energetic conditions.

Dark Energy as a Frequency-Mass Effect: Extended Classical Mechanics Interpretation.

Soumendra Nath Thakur
March 17, 2025

ECM Interpretation of Accelerating Expansion

Since:

α,ᴜₙᵢᵥ × f,ᴜₙᵢᵥ = (h × fᴜₙᵢᵥ) ÷ −Mᵃᵖᵖ,ᴜₙᵢᵥ

If −Mᵃᵖᵖ decreases over time due to physical cosmic expansion, then

f,ᴜₙᵢᵥ = 1/−Mᵃᵖᵖ,ᴜₙᵢᵥ

meaning that cosmic frequency increases as the apparent mass weakens

This implies that accelerated expansion could be directly linked to a frequency-based energy interaction rather than a vacuum energy density. Instead of treating dark energy as a constant, ECM suggests it emerges from frequency-mass adjustments:

Λᴇᴄᴍ ~ f²ᴜₙᵢᵥ

where fᴜₙᵢᵥ evolves with the cosmic scale factor. This leads to an evolving "dark energy" term that naturally transitions from early dominance (inflation) to late-time acceleration.