Kinetic Energy as the Counterpart of Apparent Mass: A Conceptual Equivalence in Extended Classical Mechanics - Vol-2

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

24-09-2024

The conclusion that kinetic energy (KE) equals the negative of the apparent mass (−Mᵃᵖᵖ) arises from the extended classical mechanics framework, where the total effective energy of a system includes both potential and kinetic energy components. Within this framework, potential energy incorporates contributions from the ordinary matter mass and the apparent mass, which behaves similarly to dark energy's effective mass.

When the matter mass is invariant, the total effective energy simplifies to the sum of its components. By resolving these expressions, it becomes evident that kinetic energy directly corresponds to the negative value of the apparent mass. This relationship illustrates how changes in kinetic energy mirror the apparent mass, establishing an equivalence between kinetic energy and the opposing effect of this mass within the system's energy balance.

Mathematical Presentation

Observed (Evident) Equation:

From the study by A. D. Chernin et al., "Dark Energy and the Structure of the Coma Cluster of Galaxies," the relationship between gravitating mass and its components is given by:

Mɢ = Mᴍ + Mᴅᴇ  

Where:

Mɢ: Gravitating Mass

Mᴍ: Matter Mass (Ordinary Matter)

Mᴅᴇ: Effective Mass of Dark Energy (treated as a mass equivalent to energy)

Framework of Extended Classical Mechanics:

In extended classical mechanics, the apparent mass (−Mᵃᵖᵖ) functions similarly to dark energy's effective mass. The adjusted equation is:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

Here, the total effective energy (Eᴛₒₜ) comprises potential energy (PE) and kinetic energy (KE):

Eᴛₒₜ = PE + KE

Given the inclusion of apparent mass, potential energy is defined as:

PE = Mᴍ + (−Mᵃᵖᵖ)

Resolution:

Using the total effective energy equation:

Eᴛₒₜ = [Mᴍ + (−Mᵃᵖᵖ)] + KE

Assuming that the matter mass (Mᴍ) remains invariant, balancing the components reveals:

KE = −Mᵃᵖᵖ

Conclusion:

The kinetic energy (KE) directly corresponds to the negative value of the apparent mass (−Mᵃᵖᵖ), demonstrating a unique equivalence in extended classical mechanics. This relationship shows how energy dynamics involving apparent mass and kinetic energy establish a balance that parallels the impact of dark energy observed in astrophysical structures.