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