Proof of Mass-Energy Equivalence in Extended Classical Mechanics: Vol-2

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

24-09-2024

1. Introduction to Key Concepts:

The study of cosmic structures, such as the Coma cluster of galaxies observed by A. D. Chernin et al., reveals that the total gravitating mass includes contributions from ordinary matter and the effective mass of dark energy. In extended classical mechanics, a parallel concept exists where the apparent mass behaves similarly to dark energy's effective mass.

2. Understanding the Energy Components:

Within this framework, the total energy of a system consists of its potential and kinetic energy components. The potential energy integrates the contributions from both the ordinary matter mass and the apparent mass, mirroring the influence of dark energy.

3. Linking Kinetic Energy and Apparent Mass:

Analysing these components reveals that kinetic energy corresponds to the negative value of the apparent mass. This relationship shows that variations in kinetic energy are directly mirrored by the effects of the apparent mass, establishing an equivalence between them.

4. Mass-Energy Equivalence Explained:

By integrating these insights, the total energy of the system simplifies to be directly proportional to the matter mass. This relationship validates the principle that energy within a system intrinsically measures its mass.

5. Conclusion:

This approach demonstrates that mass-energy equivalence is upheld within extended classical mechanics. It reinforces the concept that mass serves as a fundamental measure of energy, incorporating kinetic and potential energies as well as contributions from dark energy dynamics observed in astrophysical phenomena.

Mathematical Presentation

1. Fundamental Equations and Observations:

According to A. D. Chernin et al.'s observations of the Coma cluster, the total gravitating mass Mɢ is given by:

Mɢ = Mᴍ + Mᴅᴇ

Where:

Mᴍ: Matter Mass (Ordinary Matter)

Mᴅᴇ: Effective Mass of Dark Energy

In extended classical mechanics, the apparent mass −Mᵃᵖᵖ functions similarly to dark energy’s mass:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

2. Total Effective Energy Expression:

The total effective energy (Eᴛₒₜ) comprises potential energy (PE) and kinetic energy (KE):

Eᴛₒₜ = PE + KE

Where potential energy includes contributions from matter mass and apparent mass:

PE = Mᴍ + (−Mᵃᵖᵖ)

Thus:

Eᴛₒₜ = [Mᴍ + (−Mᵃᵖᵖ)] + KE

3. Resolving Kinetic Energy and Apparent Mass Relationship:

Given that:

KE = −Mᵃᵖᵖ

Substituting into the total energy expression:

Eᴛₒₜ = [Mᴍ + (−Mᵃᵖᵖ)] + KE

Simplifies to: 

Eᴛₒₜ = Mᴍ

4. Mass-Energy Equivalence Proven:

This simplification shows that the total effective energy Mᴛₒₜ is directly proportional to the matter mass Mᴍ, affirming the mass-energy equivalence principle within extended classical mechanics:

Mᴛₒₜ = Eᴛₒₜ 

Thus, the energy contained within a system directly corresponds to its mass, extending the classical understanding to encompass apparent mass and dark energy dynamics as observed in cosmic structures.

Conclusion:

The mass-energy equivalence is conclusively validated within the extended classical mechanics framework. It confirms that mass is an intrinsic measure of a system’s total energy, including contributions from dark energy dynamics and the system’s kinetic and potential energy components.