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.