Revisiting the Inertial–Gravitational Mass Equivalence in Extended Classical Mechanics:

Soumendra Nath Thakur

April 19, 2025

“Mass is the amount of substance in a unit volume.”

Indeed, in classical terms, mass (m) can be interpreted as the quantity of substance confined within a unit volume. When a portion (mₐ) of that mass is dynamically displaced—whether through motion, field interaction, or energetic redistribution—the original mass becomes:

m_ʀᴇᴍᴀɪɴɪɴɢ = m − mₐ, where: 0 < mₐ ≤ m

This reduction of mass within the unit volume can be seen as a deficit or missing portion and ECM interprets this deficit dynamically as negative apparent mass:

−Mᵃᵖᵖ ≡ −mₐ

Just as in Archimedes’ principle, where a submerged body displaces fluid and thereby generates an upward (buoyant) force equivalent to the weight of displaced fluid, we can draw an analogy:

“fluid mass m” as “mass of fluid per unit volume mf” 

The displaced portion (−mₐ) is analogous to the negative apparent mass, and the net force experienced (i.e., buoyancy or gravitational redirection) emerges from the remaining substance or dynamic rebalancing of mass-energy.

In ECM, this analogy is extended beyond fluids to any context where mass-energy redistribution occurs—particularly in gravitational or kinetic frameworks. The negative apparent mass is not a substance, but a mathematical and phenomenological representation of the energy or momentum portion that has transitioned from the original inertial configuration. It captures:

• The loss of rest-mass behaviour (e.g., in photons),
• The antigravitational behaviour in cosmological acceleration, and
• The dynamic mass equivalence required for effective energy accounting in relativistic and quantum domains.

Therefore, while mass remains a measure of “substance” per volume, its apparent loss or displacement—quantified as −Mᵃᵖᵖ —is a real and necessary term to represent energetic, kinetic, and gravitational dynamics in ECM.

In classical mechanics, inertial mass (m) is treated as an invariant quantity and is traditionally considered equal to gravitational mass (m𝑔) based on the equivalence principle. However, while inertial mass (m) remains constant, this equivalence does not necessitate that gravitational mass (m𝑔) is also invariant. In reality, (m𝑔) can vary with the radial distance (r ) from a gravitational centre associated with the invariant mass.

Extended Classical Mechanics (ECM) reconsiders this relationship by treating the variability of gravitational mass (m𝑔) as a foundational principle. Rather than assuming equivalence with inertial mass, ECM recognizes that gravitational mass may dynamically change depending on spatial position and energy configuration.

This recognition is crucial for modelling effective mass in ECM, which emerges not solely from the invariant inertial mass (m), but from the combined influence of (m) and the dynamic variation in gravitational mass (Δm𝑔). Mathematically, this can be expressed as:

    Mᵉᶠᶠ = m + Δm𝑔(r)

Such a formulation enables ECM to capture a broader range of physical behaviour—especially in contexts where gravitational effects deviate from Newtonian predictions or where phenomena like dark energy and apparent mass play a significant role.

By focusing on variable gravitational mass as a dynamic quantity, ECM offers a more flexible and observationally consistent framework for analysing gravitational interactions, energy redistribution, and mass-related effects across cosmological and quantum scales.