Soumendra Nath Thakur
Mass in Classical Mechanics
Basic Mass (m):
In traditional physics, mass is treated as a singular, unchanging quantity that defines how much matter an object contains. It is both the source of gravitational pull and the measure of inertia.
Active Gravitational Mass (mₐ):
This represents the portion of mass that is directly involved in generating a gravitational field. It's usually taken to be the same as the total mass, but conceptually, it can be thought of as the gravitationally effective component.
Remaining Mass (mʀᴇᴍᴀɪɴɪɴɢ):
This is what's left after accounting for the gravitationally active part. It acknowledges that not all of an object's mass may contribute equally to gravitational interaction, especially in nuanced models.
Gravitational Mass (mɢ):
This term identifies the component of mass that produces and responds to gravity. In classical thinking, it's assumed that gravitational mass and inertial mass are the same.
Traditional Assumption (m = mɢ):
Classical physics simplifies the situation by treating all forms of mass—whether gravitational or inertial—as equivalent and interchangeable.
Mass in Extended Classical Mechanics (ECM)
1. Revising the Traditional View
In Classical Physics:
The assumption was that the mass of a body directly determines its gravitational strength and motion resistance, with no internal differentiation.
ECM introduces a richer structure. It recognizes not just a singular mass but different mass-like entity that contributes in varying ways depending on the situation and environment.
2. ECM Key Mass Terms
Effective Mass (Mᵉᶠᶠ):
This is the mass that actually dictates how an object responds to gravity. It's not just a simple measure of how much matter is present, but a net result that accounts for opposing effects inside or around the object.
Gravitating Mass (Mɢ):
This is the total gravitational influence a body exerts. In ECM, it's defined to be equal to the effective mass, acknowledging that internal processes can diminish or enhance this effect compared to raw matter content.
Matter Mass (Mᴍ):
This is the total mass content of the object, including both ordinary matter and dark matter. It's the full measure of what the object consists of materially.
Apparent Mass (Mₐ):
This is the portion of the matter mass that behaves as if it's working against gravity. It's not physical mass in the usual sense but a reflection of internal energetic states, often associated with motion or internal field effects.
In ECM, Mₐ is used synonymously with Mᵃᵖᵖ, except when emphasizing its role as a subtractive quantity, in which case −Mᵃᵖᵖ is used to represent its antigravitational influence.
Remaining Matter Mass (Mᴍremaining):
This is what's left of the matter content after subtracting the apparent part. It reflects the gravitationally effective portion of material substance.
Negative Apparent Mass (−Mᵃᵖᵖ):
ECM reinterprets the apparent mass as being effectively negative. This means it doesn't contribute to gravitational attraction but instead acts in opposition, producing antigravitational behaviour.
Fluid Mass Density (Mꜰ):
In gravitational fluid contexts, ECM uses this term to represent how much mass is distributed through a given space, often in cosmological environments.
3. Relationships between Mass Types in ECM
Effective Mass as the Difference between Matter and
Apparent Mass:
The net mass that governs gravitational behaviour is obtained by subtracting the opposing internal mass-like effect from the matter content.
Apparent Mass and its Role in Gravitational Deviation (ΔMɢ):
The difference between the actual gravitational influence and the matter content is captured by a deviation term, which grows with distance or as gravitational influence weakens.
Effective Mass in Terms of Position:
As a body moves through space, especially across regions with changing gravity (e.g., from near-Earth to interstellar space), its effective gravitational behaviour adjusts. This is tracked by how much gravitational influence deviates from pure matter-based expectations.
Rewriting Gravitational Mass:
The total gravitational effect of a body is now re-expressed as its matter content plus a negative contribution from the opposing internal effect.
Gravitational Force in ECM Contexts
Force on Massive Particles within Gravitational Influence
When a body is still bound within a gravitational system, the opposing internal effect is not strong enough to override the matter mass. The force it feels is still attractive, though slightly reduced due to this inner resistance.
Force on Massive Particles Escaping Gravity
Once a body moves far enough, the negative contribution from internal effects grows large enough to overcome the matter mass. In this scenario, the net gravitational effect flips. The force becomes repulsive, essentially pushing the body further out, helping it escape the system.
Two-Body Gravitational Interaction in ECM
When two masses interact, ECM modifies the force between them by considering the internal opposition of at least one of the masses. The attracting force is reduced depending on how much of one body’s mass is behaving as a negative contributor. This leads to an interaction force that can be significantly different from the classical picture, especially over large distances or in highly energetic systems.
Gravitational Force for Massive Particles within Gravitational Influence
When a massive particle remains under the influence of gravity—such as within a planetary or stellar system—the force it experiences is determined by the balance between its intrinsic matter content and an opposing quantity that reflects its internal energy resisting gravitational confinement. This opposing component, which represents a kind of energetic lightness or buoyancy, is always smaller than the matter content in such regions. As a result, the net gravitational effect remains attractive, and the particle stays bound within the system. The effective force in this case behaves similarly to traditional gravity but includes this internal adjustment, which slightly reduces the pull without negating it.
Gravitational Force for Massive Particles Escaping Gravitational Influence
As the particle moves farther from the gravitational centre—such as when leaving a solar system or entering intergalactic space—the influence of gravitational confinement weakens. Simultaneously, the opposing internal energy component grows in relative strength. Once it surpasses the matter component, the net gravitational effect shifts from attractive to repulsive. This marks a transition into an antigravitational regime. The force acting on the particle now drives it away rather than pulling it inward, allowing the particle to accelerate freely into more distant regions of space. In this state, it effectively escapes the gravity well and experiences a force that supports ongoing expansion or release.
Two-Body Dynamics and Long-Range Effects
In ECM, when two bodies interact gravitationally, the force between them isn't just a straightforward pull based on their sizes or distances. Instead, each body contributes two aspects to the interaction: one from its inherent matter content and another from its internal opposition to being confined by gravity. For one body, the gravitational contribution is reduced by this internal opposition, while the other body contributes its full matter content. This subtle change redefines the way gravity operates between bodies, especially at large distances or in dynamic, high-energy systems. The interaction becomes more nuanced and capable of accounting for observed deviations in large-scale astrophysical structures, such as galaxy clusters or cosmic flows.
1. Classical Mechanics View of Mass
In classical mechanics, mass is static and intrinsic.
• The inertial mass is identical to the gravitational
mass.
• It remains unchanged regardless of external forces or motion.
• The entire inertial mass (m) is treated uniformly.
• If a portion (mₐ) is considered, the remainder is
(m − mₐ).
• Gravitational mass (mɢ) equals inertial mass, yet is not conceptually invariant in gravitational systems.
2. Mass Concept in ECM
In ECM, mass is dynamic and responsive to gravitational
interactions.
• Matter mass (Mᴍ) is composed of ordinary mass (Mᴏʀᴅ) and dark matter mass (Mᴅᴍ).
• In local systems, Mᴏʀᴅ dominates; Mᴅᴍ is significant only at galactic scales.
• A deductive portion (Mₐ) exists due
to gravitational effects.
• The remaining mass (Mᴍ − Mₐ) behaves as
the effective mass (Mᵉᶠᶠ).
• The deducted portion (−Mₐ) functions analogous to buoyant mass and potentially traceable to kinetic energy redistribution interpreted as negative apparent mass (−Mᵃᵖᵖ) or internal stress-energy rebalancing as gravitational influence declines
3. Dynamic Interpretation of Effective and Apparent Mass
ECM introduces a dual interpretation of mass
components:
• The effective mass (Mᵉᶠᶠ) is the residual gravitating
portion: Mᵉᶠᶠ = Mᴍ − Mₐ.
• The deducted portion (Mₐ), caused by
gravitational field effects, becomes the negative apparent mass: −Mᵃᵖᵖ.
• This makes the gravitational mass (Mɢ) dynamically equivalent to the effective mass: Mɢ = Mᵉᶠᶠ.
• Thus, Mɢ = Mᴍ + (−Mᵃᵖᵖ).
4. Gravitational Field Dependency
The apparent mass arises from field-dependent
conditions:
• The change in gravitational mass (ΔMɢ) varies with radial distance (r) from the gravitational source.
• At smaller r, Mₐ is small; Mᴍ dominates; Mᵉᶠᶠ is positive and close to Mᴍ.
• At large r, Mₐ ≈ Mᴍ; effective
mass nears zero; system approaches gravitational-antigravitational
balance.
• Beyond gravitational influence, −Mᵃᵖᵖ exceeds Mᴍ; Mᵉᶠᶠ becomes negative, indicating a shift to antigravitational dynamics.
5. Gravitational Force for Massive Particles in ECM
Gravitational force responds to dynamic mass
compositions:
• Within gravitational influence: Mᴍ > −Mᵃᵖᵖ ⇒ Mᵉᶠᶠ is positive.
• Escaping gravitational influence: −Mᵃᵖᵖ ≫ Mᴍ ⇒ Mᵉᶠᶠ becomes
effectively negative.
• Gravitational force (Fɢ,ᴇᴄᴍ) is governed by the net effective mass:
• Within gravity
zone: force is attractive, proportional to (Mᴍ − Mᵃᵖᵖ).
• Beyond gravity zone: force becomes repulsive due to net negative mass.
6. Two-Body Gravitational Interaction in ECM
In extended interactions:
• Each body contributes its own effective mass term.
• The total gravitational interaction is defined by the
difference between matter mass and apparent mass for body 1, multiplied by the
matter mass of body 2.
• This form generalizes
Clarification on Gravitational Mass and Compatibility with Cosmological Observations
It is important to explicitly address a potential source of confusion regarding the treatment of gravitational mass in Extended Classical Mechanics (ECM), particularly in the context of existing empirical and cosmological formulations such as those found in the works of A.D. Chernin et al. (2013). "Dark energy and the structure of the Coma cluster of galaxies."
In conventional cosmology, the net gravitating mass is defined by the relation:
Mɢ = Mᴍ − Mᴅᴇ
Where:
• Mᴍ is the total matter mass,
including both ordinary and dark matter.
• Mᴅᴇ represents the mass-equivalent of
dark energy, which contributes antigravitationally, hence the subtraction.
• Mɢ is the effective gravitating mass, inferred from the dynamical behaviour of large-scale structures such as galaxy clusters.
This formulation is entirely consistent with ECM’s dynamic mass framework. In ECM, the equation takes the form:
Mɢ = Mᴍ − Mᵃᵖᵖ
Where:
• Mᵃᵖᵖ is the negative apparent mass, arising
from the redistribution of kinetic and gravitational energy within the system.
• This term captures the same effective influence attributed to Mᴅᴇ in cosmology, but ECM derives it from mechanical first principles rather than treating it as a separate vacuum component.
Thus, ECM does not contradict Chernin’s observational results — it reconstructs and extends them within a dynamic mechanical context. The negative apparent mass in ECM and the dark energy term in cosmology are functionally equivalent in gravitational behaviour. The ECM approach simply reinterprets the source of this effect as arising from internal energy dynamics, rather than an external cosmological constant.
Furthermore, the often-invoked principle of equivalence between gravitational and inertial mass mɢ = mɪ is recovered in ECM as a limiting case. ECM explicitly demonstrates that effective inertial mass is:
Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ = Mɢ
This relation highlights that the traditional equivalence holds in weak-field or non-expanding regimes, where Mᵃᵖᵖ ≈ 0, but must be refined under conditions where expansion, motion, and gravitational interactions redistribute mass-energy within the system.
Before proceeding to the detailed mathematical formulation, it is crucial to clarify how ECM’s treatment of gravitational mass aligns with empirical cosmological models, particularly those involving dark energy.
Mathematical Presentation:
I. Classical Mechanics: Mass and Gravity
1. Equivalence of Inertial and Gravitational Mass
m = mɢ
In classical mechanics, inertial mass (resistance to acceleration) and gravitational mass (source of gravitational attraction) are treated as equivalent by definition. This is also maintained in general relativity.
2. Partitioning of Mass
mʀᴇᴍᴀɪɴɪɴɢ = m − mₐ, where: 0 < mₐ ≤ m
The total inertial mass m can be viewed as consisting of a used portion mₐ and a remaining portion. Though not standard in Newtonian mechanics, this setup lays the groundwork for ECM's reinterpretation.
II. Extended Classical Mechanics (ECM): General Mass Framework
3. Gravitational Mass as Effective Mass: A Dynamic Redefinition
Mᵉᶠᶠ = Mɢ
In ECM, gravitational mass is not equivalent to inertial or matter mass. Instead, it is defined dynamically as effective mass, which depends on both internal properties and external gravitational influence.
4. Matter Mass Composition
Mᴍ = Mᴏʀᴅ + Mᴅᴍ
Matter mass includes ordinary (baryonic) matter and dark matter. Dark matter’s contribution is significant only at cosmic scales and is negligible in local gravitational systems.
5. Partitioning of Matter Mass
Mᴍ,ʀᴇᴍᴀɪɴɪɴɢ = Mᴍ − Mₐ, where: 0 < Mₐ ≤ Mᴍ
ECM allows a portion of matter mass, Mₐ, to be dynamically subtracted—analogous to displaced mass in Archimedes’ principle—to produce an effective mass. For a detailed understanding of how this displaced mass corresponds to negative apparent mass, refer to the explanation under 'Archimedes’ Principle: Negative Apparent Mass' mentioned below.
6. Negative Apparent Mass: Source of Gravitational Repulsion
−Mₐ ≡ −Mᵃᵖᵖ
The deducted portion Mₐ appears as a negative apparent mass, which exerts an effect opposite to that of ordinary matter mass—crucial to ECM’s treatment of gravitational interaction, including repulsion under certain conditions.
III. Effective Mass and Gravitational Behaviour within Systems
7. Effective Mass Definition
Mᵉᶠᶠ = Mᴍ − Mₐ
The effective mass is the active gravitational mass after accounting for the subtracted, apparently negative mass. It is the “net” dynamic mass in a gravitational system.
8. Distance-Dependent Gravitational Mass
Mᵉᶠᶠ = Mᴍ + ΔMɢ(r) where ΔMɢ(r) = Mɢ(r) − Mᴍ
(Recall: since ΔMɢ(r) = −Mᵃᵖᵖ, this matches the general formulation Mᵉᶠᶠ = Mᴍ −Mᵃᵖᵖ.)
The gravitational mass perceived at a distance r is influenced by how the surrounding gravitational field modifies the effective mass of the local system. This introduces spatial variation in mass perception, differing from constant mass assumptions.
9. Equivalence Reformulation
Mᵉᶠᶠ = Mᴍ + ΔMɢ(r) = Mᴍ − Mᵃᵖᵖ
Mɢ = Mᴍ + (−Mᵃᵖᵖ)
Gravitational mass in ECM is defined as the matter mass diminished by the dynamically induced negative apparent mass. ECM formalizes this as the total force-mediating mass seen in gravitational interactions.
IV. ECM Gravitational Force Laws
10. Gravitational Force in ECM (General Form)
Fɢ,ᴇᴄᴍ = Mɢ gᵉᶠᶠ = (Mᴍ + (−Mᵃᵖᵖ)) gᵉᶠᶠ
The gravitational force in ECM depends on both positive matter mass and its associated negative counterpart. Gravity arises from this balance, modifying classical interpretations.
11. Massive Particle Within Gravitational Influence
Fɢ,ᴇᴄᴍ = (Mᴍ − Mᵃᵖᵖ)gᵉᶠᶠ = Mɢgᵉᶠᶠ where Mᴍ > Mᵃᵖᵖ, so Mᵉᶠᶠ = Mɢ > 0
In gravitational fields, particles still experience force similar to Newtonian mechanics, but mass terms are dynamically adjusted. The matter mass is larger than the apparent mass, yielding net attraction.
12. Massive Particle Escaping Gravitational Influence
Fɢ,ᴇᴄᴍ = (Mᴍ − Mᵃᵖᵖ)gᵉᶠᶠ = −Mɢgᵉᶠᶠ with −Mɢ > Mᴍ ≪ −Mᵃᵖᵖ
In regimes where the magnitude of negative apparent mass
exceeds matter mass, such as at cosmic escape scales or near a repulsive
boundary, the gravitational force becomes repulsive. This reverses the force
direction due to dominance of the negative term.
13. Two-Body Gravitational Force in ECM
Fɢ,ᴇᴄᴍ = G(Mᴍ₁ − Mᵃᵖᵖ₁)(Mᴍ₂ − Mᵃᵖᵖ₂)/r²
The ECM version of
V. Summary: Core Implications
• Mass is not absolute: In ECM, mass is redefined as a
field-responsive quantity.
• Gravity is dynamic: Gravitational attraction or
repulsion depends on a local balance of real and apparent masses.
• New force predictions: ECM predicts gravitational
repulsion in scenarios where the magnitude of negative apparent mass exceeds
the local matter mass.
• Cosmic consistency: The formulation is consistent with large-scale cosmological observations and provides theoretical grounds for explaining dark energy–like effects using mass reconfiguration rather than introducing a cosmological constant.
13. Archimedes’ Principle: Negative Apparent Mass
In Archimedes’ principle, a body submerged in a fluid displaces a certain fluid volume, and this displaced fluid exerts a buoyant force upward. The magnitude of this buoyant force is equivalent to the weight of the displaced fluid.
In ECM, space itself behaves analogously to a gravitational fluid medium. Masses interact not just through mutual attraction but by displacing and reshaping the field geometry around them, akin to floating bodies in fluid equilibrium.
We can carry this analogy into ECM, where mass is treated as a dynamic gravitational response, rather than a fixed property. Here’s how the ECM terms correspond to Archimedes’ framework:
1. Mᴍ – Matter Mass (Analogous to the
Full Volume of a Submerged Body)
• Mᴍ is the total matter mass of the
body. This includes both ordinary/baryonic matter (Mᴏʀᴅ) and dark matter mass (Mᴅᴍ).
• It is equivalent to the total volume of the submerged
object. Just like a large body displaces more fluid, a body with more matter
mass experiences more gravitational interaction in ECM.
• It sets the gravitational context — the "size" of the body’s interaction with the gravitational medium (just like the submerged volume determines how much fluid is displaced).
2. Mₐ – Apparent Deductive Mass
(Analogous to the Displaced Fluid Mass)
• Mₐ is a portion of matter mass that
does not contribute directly to effective gravitational interaction. Instead,
it is dynamically deducted due to its coupling with the gravitational field.
• It is analogous to the mass of the fluid displaced by
the submerged object. Just like this displaced fluid creates an upward force, Mₐ creates a counter-effect in ECM — gravitational "buoyancy"
that reduces net gravitational pull.
•Mₐ is not lost mass — it represents the gravitationally ineffective or redistributed portion. It's what the gravitational field "pushes back" on, just like fluid displaced pushes back in Archimedes’ principle.
3. −Mₐ ≡ −Mᵃᵖᵖ – Negative Apparent Mass (Analogous to the Buoyant Force)
• −Mₐ, or equivalently −Mᵃᵖᵖ, represents the gravitationally counteractive component. It acts as a
repulsive (antigravitational) term, reducing the effective gravitational mass.
• This is like the buoyant force exerted upward by the
displaced fluid. In ECM, this negative mass opposes gravitational attraction,
leading to apparent mass loss in gravitational behaviour.
• −Mᵃᵖᵖ is the mass equivalent of gravitational buoyancy. Just as the displaced fluid pushes upward and reduces net weight, −Mᵃᵖᵖ subtracts from the effective gravitational mass, altering how strongly the object is pulled by gravity.
Summary Analogy Table:
ECM Term Archimedean Equivalent Role in Gravitational Dynamics
·
Mᴍ Submerged object’s volume Total
gravitationally active matter
(Ordinary + dark)
·
Mₐ Displaced fluid mass Subtracted component from Mᴍ
due to gravitational
coupling
·
−Mₐ ≡ −Mᵃᵖᵖ Buoyant
force (mass equivalent) Gravitational counteraction
(Antigravity-like effect)
This analogy captures a key conceptual leap in ECM: mass can behave like an immersed object in a gravitational medium. Just as fluids redistribute forces around submerged bodies, gravitational fields redistribute mass influence, resulting in a dynamic effective mass (Mᵉᶠᶠ = Mᴍ − Mₐ).
14. Breakdown of Symmetry in ECM
In Classical Mechanics, the two-body gravitational force is fundamentally a symmetric interaction.
Classical mechanics relies fundamentally on force symmetry, ensured by equal and unchanging mass terms. But what happens when mass itself becomes dynamic, and responds to external fields?
→F₁₂ = −→F₂₁
• Gravitational mass is always positive.
• There’s no distinction between matter mass and apparent
mass.
• Forces between two bodies are mutual, equal, and opposite — perfectly symmetric.
In ECM: Symmetry Can Break
In ECM, symmetry breaks due to the presence of negative apparent mass (−Mᵃᵖᵖ) and how matter mass (Mᴍ), and effective mass (Mᵉᶠᶠ) interact dynamically with gravity.
Key Elements Leading to Asymmetry:
1. Dynamic Redistribution of Mass:
• Matter mass Mᴍ is not fixed; it can reduce due to gravitational effects, converting
into negative apparent mass −Mᵃᵖᵖ.
• This creates an imbalance between the two interacting bodies.
2. Effective Mass Becomes Frame-Dependent:
• The
gravitational effect experienced by each body depends on its local
gravitational context (e.g., radial distance r, gravitational source field).
• So even if two objects start with equal Mᴍ, their effective mass Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ can differ due to local conditions.
3. Asymmetry in Force Law:
ECM modifies the gravitational force equation:
Fɢ,ᴇᴄᴍ = G(Mᴍ₁ − Mᵃᵖᵖ₁)(Mᴍ₂ − Mᵃᵖᵖ₂)/r²
• This force
becomes asymmetric if one object has a substantially different negative
apparent mass than the other.
• The reaction force felt by each body is not necessarily equal and opposite.
4. Escaping Mass Conditions:
• If one body
approaches a condition where M ᴍ ≪−Mᵃᵖᵖ, its
effective mass becomes negative.
• This leads to
an antigravitational behaviour — it could repel instead of attract.
• The force it experiences may remain attractive toward the other body, while the latter may not feel an equal and opposite reaction — especially if its own effective mass is still positive.
Resulting Asymmetries:
Classical Mechanics Extended Classical Mechanics
·
→F₁₂ = −→F₂₁ →F₁₂ ≠ −→F₂₁ if Mᵃᵖᵖ
· Equal & opposite force pairs Force pairs may be unequal in direction or
magnitude
· Symmetric interaction always can become asymmetric in dynamic gravitational
zones
· Inertial = Gravitational mass Matter mass ≠ Gravitational mass
In ECM, gravity isn’t just a passive background field. It modifies the internal mass configuration of particles through interactions, leading to non-Newtonian behaviour like:
• One object pulling while the other is neutral or even
repelled
• Force vectors not being balanced
• Situations where effective gravitational mass turns
negative
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