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 

Distinguishing Negative Mass from Dynamical Mass Terms in Extended Classical Mechanics:

Soumendra Nath Thakur 

April 19, 2025

In response to a public comment by Dr. Valentyn Nastasenko, who stated, “Negative mass only exists in abstract mathematics. It does not exist in real physics,” I wish to offer a clarification grounded in the framework of Extended Classical Mechanics (ECM).

This assertion brings attention to an important conceptual distinction: ECM does not introduce or rely upon physically realisable “negative mass” as defined in speculative physics. Instead, it defines and consistently applies the concepts of negative apparent mass (−Mᵃᵖᵖ) and negative effective mass, which are dynamic constructs with observational support—not exotic or literal negative masses.

Specifically, ECM introduces:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

where:

• Mᴍ is the total matter mass, including both ordinary and dark matter, and  
• −Mᵃᵖᵖ is the negative apparent mass, dynamically emergent from gravitational and kinetic interactions.

The term −Mᵃᵖᵖ is assigned to radiation or kinetic energy contributions (e.g., in massless particles like photons) and is used to model antigravitational behaviour in gravitational fields. It is a mathematical and physical representation of how energy behaves dynamically—not a postulated exotic entity.

Moreover, this approach is not without precedent. As shown in:

Chernin, A. D. et al (2013). Dark energy and the structure of the Coma cluster of galaxies. Astronomy and Astrophysics, 553, A101.

the notion of negative effective mass is applied to describe the repulsive gravitational contribution of dark energy. ECM generalizes this insight to broader contexts and scales while remaining consistent with classical force laws and relativistic behaviour.

Thus, ECM carefully distinguishes:

• Intrinsic matter mass (Mᴍ),
• Dynamically emergent apparent mass (−Mᵃᵖᵖ), and
• The net gravitational mass (Mᵉᶠᶠ) observable in cosmic and quantum systems.

In summary, ECM does not propose negative mass in the literal or unphysical sense. Instead, it offers a structured, observationally motivated reinterpretation of inertial and gravitational effects, especially in systems where classical mechanics and cosmological phenomena intersect. I hope this letter helps dispel misconceptions and encourages further dialogue on reconciling mechanics with observed cosmic behaviour.

Sincerely,  

Soumendra Nath Thakur  

April 19, 2025

Clarifying the Misconception: ECM Does Not Invoke Physical Negative Mass but Dynamic Mass Equivalents.

April 19, 2025

Dear Dr. Valentyn Nastasenko,

Thank you for your comment. Your statement, “Negative mass only exists in abstract mathematics. It does not exist in real physics,” reflects a crucial concern in foundational physics—one that Extended Classical Mechanics (ECM) has carefully and deliberately addressed by making a distinction between physically real mass and dynamically emergent mass terms, such as negative apparent mass or negative effective mass, which are not "physical negative mass" in the conventional or literal sense.

To clarify:

In Extended Classical Mechanics (ECM), at no point is a physically realizable negative mass postulated. Rather, negative apparent mass (−Mᵃᵖᵖ) arises from the dynamic redistribution of kinetic energy under gravitational influence and motion. It does not signify an exotic substance or particle with negative inertial mass, which would indeed be speculative and lack empirical support.

Similarly, effective mass (Mᵉᶠᶠ) is a composite quantity defined in ECM as:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

where:

• Mᴍ is the matter mass (including both ordinary and dark matter), and
• −Mᵃᵖᵖ is the dynamic negative apparent mass arising from the gravitational and kinematic configuration.

This formulation is strictly observational and phenomenological—it is constructed to match and explain cosmological and quantum mechanical behaviour, including light deflection, redshift, and repulsion effects attributed to dark energy.

In fact, negative effective mass is not an invention of ECM. It is empirically supported in observational cosmology, particularly in works such as:

• A.D. Chernin et al., "Dark energy and the structure of the Coma cluster of galaxies," Astronomy and Astrophysics, 553, A101 (2013).

Here, the gravitating effect of dark energy is explicitly modelled as a negative effective mass contribution, denoted as (Mᴅᴇ < 0), resulting from a uniform antigravitational density. This "negative mass" in the cosmological context is not a negative particle mass, but a gravitational term required to reconcile observations of galaxy clusters and cosmic acceleration.

ECM formulates and extends this line of thought by assigning: −Mᵃᵖᵖ to kinetic energy or radiation (e.g., photons), and Interpreting this as the source of antigravitational behaviour in massless particles, particularly under gravitational influence.

This does not violate classical or relativistic principles; rather, it refines them through well-defined, observation-based extensions that remain grounded in Newtonian force dynamics and energy equivalence. The gravitational repulsion observed in phenomena like dark energy becomes natural when modelled through this lens.

Therefore, I respectfully submit that ECM does not introduce or rely upon abstract or unphysical “negative mass,” but rather makes precise distinctions between:

• Intrinsic matter mass (Mᴍ),
• Dynamically derived negative apparent mass (−Mᵃᵖᵖ), and
• Net observable mass (Mᵉᶠᶠ) as it appears in gravitational and inertial contexts.

Your critique is therefore an opportunity to clarify a misidentification: ECM is not invoking exotic or non-physical negative mass, but instead reinterpreting known energetic phenomena (such as the momentum-energy of massless particles and repulsive cosmological expansion) through carefully defined dynamic mass equivalents.

I remain deeply appreciative of your engagement with the subject and welcome further discussion on how ECM models can complement our shared aim of reconciling classical mechanics with the modern observational universe.

Warm regards,

Soumendra Nath Thakur

Framing Force Dynamics in Conventional Massless Particles: The ECM Perspective