1. Interpretation of Various Masses in Extended Classical Mechanics (ECM):

Soumendra Nath Thakur

February 05, 2025

The Physical basis for associating a separate mass term with dark energy in the Kinetic energy equation.

Classical Mechanics Interpretation of Inertial Mass (m):

• In classical mechanics, inertial mass (m) is strictly associated with physical mass (Mᴏʀᴅ), which represents baryonic matter.

• The kinetic energy (KE) equation in classical mechanics assumes that the mass term corresponds directly to inertial mass, reinforcing the idea that energy is fundamentally linked to the motion of physically present mass.

• The total energy in classical mechanics is given by: Eₜₒₜₐₗ = PE + KE where PE is associated with mass m, but the interpretation of the mass term in KE remains under scrutiny in ECM.

Extended Classical Mechanics (ECM) Interpretation of Various Masses:

• ECM introduces effective mass (Mᵉᶠᶠ) as a key factor, incorporating both physical matter and apparent mass effects: Mᵉᶠᶠ = Mᴍ+ (−Mᵃᵖᵖ) where:

• Matter mass (Mᴍ) = Mᴏʀᴅ + Mᴅᴍ (includes baryonic matter and dark matter)

• Apparent mass (Mᵃᵖᵖ) is a dynamic quantity that can take negative values based on external influences.

• The force equations in ECM are modified accordingly: F = Mᵉᶠᶠ·aᵉᶠᶠ = (Mᴍ −Mᵃᵖᵖ)·aᵉᶠᶠ and F𝑔 = G·Mᵉᶠᶠ·mₘ/r² = G·Mɢ·mₘ/r², where Mɢ = Mᵉᶠᶠ represents the gravitating mass in ECM.

Galactic Cluster Interpretation of Matter Mass:

• Effective mass of dark energy (Mᴅᴇ) arises from gravitational interactions at intergalactic scales, distinct from dark matter.

• Matter mass within a galactic cluster follows: Mᴍ = Mᴏʀᴅ + Mᴅᴍ

• Observationally, the total gravitating mass in a galactic cluster is found to be: 

Mɢ = Mᴍ + Mᴅᴇ

Clarifying the Physical Interpretation of Apparent Mass (Mᵃᵖᵖ):

• Apparent mass (Mᵃᵖᵖ) is a non-physical, dynamic term reflecting mass variations due to external forces.

• When effective mass has a significant negative component, the observed mass can appear reduced, resulting in negative apparent mass (Mᵃᵖᵖ < 0).

• This effect is particularly noticeable under extreme conditions, such as high velocities or strong gravitational fields.

The Physical Interpretation of the KE Equation in ECM:

• The ECM kinetic energy equation modifies the classical form by incorporating Mᴏʀᴅ and Mᴅᴍ explicitly: (1/2) Mᴏʀᴅ⋅v² + (1/2) Mᴅᴍ⋅v² = −Mᵃᵖᵖ

• Here, −Mᵃᵖᵖ has no direct physicality, but its presence affects the system's overall energy dynamics.

Conclusion:

ECM expands upon classical mechanics by refining mass-energy relationships, introducing apparent mass (Mᵃᵖᵖ) and effective mass (Mᵉᶠᶠ) to explain observed gravitational phenomena. The inclusion of a separate mass term for dark energy (Mᴅᴇ) is justified through its interaction with inertial mass at intergalactic scales, aligning with gravitational observations. While ECM's KE equation diverges from classical interpretations, it maintains mathematical consistency and observational validity.