Revisiting the Inertial–Gravitational Mass Equivalence and the Role of Apparent Mass in Extended Classical Mechanics:

Soumendra Nath Thakur

April 19, 2025

In classical mechanics, inertial mass (m) is considered an invariant quantity and is equated to gravitational mass (m𝑔) based on the equivalence principle. However, this equivalence assumes a uniform gravitational field and does not imply that gravitational mass must remain constant. While inertial mass (m) is indeed invariant, gravitational mass (m𝑔) can vary as a function of radial distance (r) from a gravitational centre of fixed mass-energy.

In Extended Classical Mechanics (ECM), this variability is elevated from an exception to a foundational principle. ECM acknowledges that the gravitational mass (m𝑔(r)) dynamically evolves with changes in gravitational potential and spatial configuration, leading to new insights into the nature of force and energy.

This dynamic change in gravitational mass with respect to position gives rise to the concept of negative apparent mass (-Mᵃᵖᵖ), which is not a physical negative mass, but a kinetically emergent mass-equivalent. It reflects the redistribution of energy (particularly kinetic or radiative) in the presence of spatially variable gravitational interaction. In ECM, this term arises when:

Mᵉᶠᶠ = m + Δm𝑔(r) = m - Mᵃᵖᵖ

Here:

• m is the invariant inertial mass,
• Δm𝑔(r) is the radial change in gravitational mass, and
• Mᵃᵖᵖ represents the negative apparent mass induced by the system’s energetic and gravitational configuration.

Thus, ECM redefines effective mass Mᵉᶠᶠ as a resultant of the invariant inertial mass and a spatially modulated gravitational counterpart, where the latter may be manifested as a negative apparent term under specific energetic or gravitational boundary conditions.

This reinterpretation:

• Preserves classical consistency for low-energy, local systems,
• Extends the explanatory power to regimes involving cosmological redshift, radiation pressure, and antigravitational behaviour,
• And provides a phenomenological link to observed phenomena attributed to dark energy and the effective mass of massless particles.

By incorporating variable gravitational mass (m𝑔(r)) and the resulting apparent mass terms, ECM delivers a unified, observation-grounded framework that maintains Newtonian clarity while extending into quantum and relativistic domains.