Mass Concepts in Classical, Relativistic, and Extended Classical Mechanics (ECM):                                                                                                                                                           

 

Soumendra Nath Thakur

February 19, 2025

 

The book Concepts of Mass in Classical and Modern Physics by Max Jammer presents traditional mass concepts, primarily defining mass as either inertial mass or rest mass. However, these descriptions do not account for dynamic effective mass or apparent mass.

 

Extended Classical Mechanics (ECM) expands upon these conventional definitions by introducing negative apparent mass and negative effective mass, extending the concept of mass beyond ordinary matter to include the effects of dark matter. This distinction is crucial in understanding mass interactions at different cosmic scales.

 

Mass Equivalence in Different Frameworks

In classical mechanics, inertial mass is equivalent to gravitational mass: 

 

m = m_ɢ

 

​In relativistic mechanics, rest mass (invariant mass) is also equated to gravitational mass: 

 

m₀ = m_ɢ

                                            

​In Extended Classical Mechanics (ECM), gravitating mass (Mɢ) is equivalent to effective mass (Mᵉᶠᶠ), which includes both matter mass (Mᴍ) and negative apparent mass (−Mᵃᵖᵖ):

 

Mɢ = Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

 

Where matter mass (Mᴍ) is the combined mass of ordinary matter (Mᴏʀᴅ) and dark matter mass (Mᴅᴍ):

 

Mᴍ = Mᴏʀᴅ + Mᴅᴍ 

 

Effective Mass at Celestial and Galactic Scales

At celestial or planetary scales within a galaxy, effective mass (Mᵉᶠᶠ) can be either positive or negative, depending on the balance between matter mass and negative apparent mass:

 

If matter mass dominates (Mᴍ > Mᵃᵖᵖ), then effective mass is positive:

 

Mᵉᶠᶠ > 0

 

If negative apparent mass dominates (Mᵃᵖᵖ > Mᴍ ), then effective mass becomes negative:

 

Mᵉᶠᶠ < 0

 

The ECM expression for gravitating mass can also be rewritten as:

 

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

 

This formulation aligns conceptually with the expression discussed by A. D. Chernin et al. in Dark Energy and the Structure of the Coma Cluster of Galaxies:

 

Mɢ = Mᴍ + Mᴅᴇ, 

 

Where Mᴅᴇ represents the effective mass of dark energy, which is always negative:

 

Mᴅᴇ < 0

 

While ECM extends this concept by incorporating negative apparent mass alongside dark energy effects, both frameworks recognize the role of negative mass contributions in gravitational dynamics.

 

This framework provides a more comprehensive approach to understanding mass interactions in the universe, bridging the gap between classical mechanics and modern astrophysical observations.