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

 

Extending the Equivalence Principle: Beyond Numerical Equality toward Functional Variability

Soumendra Nath Thakur, ORCiD: 0000-0003-1871-7803

April 20, 2025

In classical physics, the Equivalence Principle asserts that inertial mass and gravitational mass are numerically equal and functionally indistinguishable. This is conventionally expressed as:

mɢ = m

Where:

• mɢ is the gravitational mass, which determines the strength of an object's gravitational interaction.

• m is the inertial mass, representing an object’s resistance to acceleration under an applied force,

This equality forms the cornerstone of Newtonian gravitation and Einstein’s general relativity, grounding the principle of universal free fall, where all objects fall with the same acceleration in a uniform gravitational field, regardless of their composition or mass.

However, while these masses are treated as equal in classical mechanics, this equivalence does not necessitate that gravitational mass be invariant. In fact, within Extended Classical Mechanics (ECM), gravitational mass mɢ is interpreted as a "context-dependent, spatially variable" property. Its value may change with radial distance (r) from a gravitational centre or due to changes in the gravitational potential of the system.

 

Fig. 1 – Effective Mass behaviour in ECM: Illustration of Mᵉᶠᶠ = m + Δmɢ(r) = m − Mᵃᵖᵖ as a function of radial distance.

Radial Variability and the Emergence of Apparent Mass in ECM

In classical mechanics, inertial mass (m) is considered an invariant quantity and is equated to gravitational mass (mɢ) 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 (mɢ) 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 mɢ(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 + Δmɢ(r) = m − Mᵃᵖᵖ

Here:

• m is the invariant inertial mass,

• Δmɢ(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 mɢ(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.

The Expression for Δmɢ(r) and Its Impact on Effective Mass in ECM:

In ECM, the radial dependency of gravitational mass is expressed as mɢ(r), leading to:

Δmɢ(r) = mɢ(r) - m

To reflect physical reality:

• When r increases, gravitational mass decreases: mɢ(r) < m ⇒ Δmɢ(r) < 0

• When r decreases, gravitational mass increases: mɢ(r) > m ⇒ Δmɢ(r) > 0

So, the effective mass in ECM is expressed as:

Mᵉᶠᶠ = m + Δmɢ(r)

But for coherence with the negative apparent mass concept, where:

−Mᵃᵖᵖ = Δmɢ(r) (if Δmɢ(r) < 0)

The full expression becomes:

Mᵉᶠᶠ = m + Δmɢ(r) = m − Mᵃᵖᵖ

However, the expression can switch sign depending on dr/dt (direction of radial change).

Formulation in ECM:

To preserve consistency and physical clarity ECM formulation is:

Mᵉᶠᶠ = m + Δmɢ(r) where Δmɢ(r) = mɢ(r) − m

Or, equivalently:

Mᵉᶠᶠ = m − Mᵃᵖᵖ where −Mᵃᵖᵖ = Δmɢ(r)

Notes:

1. |Δmɢ(r)| is used in contexts where the magnitude is more relevant than the sign (e.g., when isolating the contribution of apparent mass).

2. Sign conventions are model-dependent but are treated consistently within ECM.]

This preserves:

• Directionality of change in gravitational mass,

• The link between apparent mass and gravitational potential,

• Coherence across different gravitational regimes.

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.

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.

A Letter: On Mass, Displacement, and Negative Apparent Mass in Extended Classical Mechanics

 

Soumendra Nath Thakur

April 19, 2025

In response to Dr. Valentyn Nastasenko’s statement:

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

I would like to offer a clarification that aligns with this classical understanding, while extending it through the lens of Extended Classical Mechanics (ECM).

Mass (m) is indeed the amount of substance confined within a unit volume. However, when any portion of that substance is displaced—either physically or dynamically—the measurable mass within that unit volume is no longer whole. This displacement can be denoted as a reduction of mass by an amount (mₐ), such that:

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

In ECM, this reduction is not merely a subtraction but is interpreted dynamically as the emergence of a negative apparent mass, denoted as:

−Mᵃᵖᵖ ≡ −mₐ

This concept is analogous to Archimedes’ principle, where an object partially or fully submerged in a fluid displaces an amount of fluid equivalent to its volume, resulting in a buoyant force equal to the weight of the displaced fluid. Analogously, in ECM:

• The original mass m serves as the surrounding "field" or medium,
• The displaced portion mₐ represents a loss from the inertial configuration,
• And the resulting dynamics (e.g., force redirection, gravitational anomalies) emerge from this displacement.

In this framework, negative apparent mass does not imply the existence of exotic negative-mass particles. Rather, it is a phenomenological term to represent the dynamically displaced portion of mass-energy, which manifests in observations such as:

• The inertial response of massless particles like photons,
• The antigravitational effects attributed to dark energy in cosmology,
• And the effective force equations needed to reconcile Newtonian, relativistic, and quantum dynamics.

By distinguishing between intrinsic mass and apparent dynamic mass terms, ECM offers a refined interpretation without violating classical substance-based definitions. It bridges observed cosmological behaviour with energy-mass dynamics, while maintaining internal mathematical and physical consistency.

I hope this clarification contributes constructively to ongoing discussions on the nature of mass and the foundational structure of modern mechanics.

Sincerely,

Soumendra Nath Thakur  

Displacement of Mass and the Emergence of Negative Apparent Mass in ECM:

April 19, 2025 

Dear Dr. Valentyn Nastasenko,

Thank you for your continued engagement. You rightly state that:

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

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:

The fluid mass m is the original mass (or energy configuration),

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.

Warm regards, 

Soumendra Nath Thakur