Soumendra Nath Thakur,
April 14, 2025
Abstract:
This research develops the
foundational equations of Extended Classical Mechanics (ECM) by generalizing
Newtonian mechanics through the inclusion of dynamic mass components such as
negative apparent mass. ECM redefines force, acceleration, and gravitational
interactions using an effective mass framework, expressed as the sum of
traditional matter mass and a kinetic-energy-derived negative apparent mass.
This dual-mass interaction leads to revised force laws and a spectrum of speed
regimes for massive particles—ranging from gravitational confinement to
antigravitational liberation. The formulation extends to massless particles
like photons by assigning them an effective negative matter mass, enabling
consistent force definitions and propagation behaviour at relativistic speeds.
Radial distance plays a critical role in determining gravitational behaviour,
with transitions from classical attraction to antigravitational expansion. The
framework aligns with cosmological observations, particularly in large-scale
structure behaviour, and provides a unified approach to understanding force,
inertia, and motion in both massive and massless domains. ECM thus represents a
coherent advancement of classical physics, accommodating gravitational
variance, energy redistribution, and speed constraints in dynamic systems.
Keywords:
Extended Classical
Mechanics (ECM), Effective Mass, Negative Apparent Mass, Gravitational
Redefinition, Relativistic Force, Dynamic Inertia, Massless Particles, Speed
Regimes, Antigravity Dynamics, Cosmological Consistency,
Introduction:
1. Newtonian Foundation
Consistency:
In classical Newtonian mechanics,
force is defined as the product of mass and acceleration. Acceleration is
directly proportional to the applied force and inversely proportional to the
mass. If there is no applied force, there is no acceleration, yet the mass
remains unchanged. In this framework, gravitational mass is equal to inertial
mass, indicating a fundamental equivalence between how mass responds to gravity
and how it resists acceleration.
2. ECM Force Extension
(Massive Bodies):
In Extended Classical
Mechanics (ECM), for massive bodies, the total force is expressed as the
product of effective mass and effective acceleration. The effective mass is
defined as the sum of matter mass and negative apparent mass, the latter
representing the kinetic-energy-equivalent mass opposing gravitational
confinement. Effective acceleration remains directly proportional to the force,
but now it is also inversely related to matter mass alone. A special condition
holds: the reciprocal of matter mass, in absolute terms, equals the magnitude of
the apparent mass.
3. ECM Speed Conditions for
Massive Particles:
Three dynamic regimes
emerge for massive particles based on the relative dominance between matter
mass and the magnitude of apparent mass.
• When matter mass is
greater than the magnitude of apparent mass, matter mass dominates the
interaction. This corresponds to lower particle speeds because gravitational
confinement remains stronger than kinetic liberation.
• When matter mass is equal
to the magnitude of apparent mass, the two contributions balance. The system
achieves a medium-speed regime, where confinement and liberation forces are
dynamically in equilibrium.
• When the magnitude of
apparent mass exceeds the matter mass, apparent mass dominates. This leads to
high particle speeds, as the effective gravitational confinement weakens, and
antigravitational dynamics begin to assert more influence.
4. Gravitational Extension in ECM:
In ECM, gravitational mass
is redefined as the sum of matter mass and negative apparent mass and this
total correspond to the effective mass in dynamic systems.
• At zero radial distance,
the reciprocal of matter mass and apparent mass both vanish, resulting in
gravitational mass equal to matter mass only.
• At nonzero radial
distances, matter mass still dominates, and gravitational mass remains
positive. Gravitational influence is active.
• At significantly large
distances, matter mass balances the magnitude of apparent mass, and
gravitational mass becomes zero. This marks the threshold where gravitational
and antigravitational effects cancel each other.
• Beyond gravitational
influence, where apparent mass overtakes matter mass, gravitational mass
becomes negative. The system transitions into an antigravitational regime,
dominated by expansive kinetic effects.
5. Massless Particles
(Conventional, Photon-like):
For particles such as
photons, which are conventionally treated as massless, ECM introduces a
reinterpretation: their matter mass is considered effectively negative. The
negative apparent mass is equal in magnitude to this negative matter mass,
implying that the effective mass is twice the magnitude of the apparent mass.
Within a gravitational
field, both the negative matter mass and negative apparent mass contribute to
the force experienced by such particles. The effective acceleration is doubled
in magnitude, leading to a relativistic condition where the product of
effective mass and acceleration equals the speed of light. This reflects the
expenditure of apparent mass to escape the gravitational field.
At the edge of
gravitational influence—just as the particle escapes the field—only one
contribution of apparent mass remains. The acceleration is still doubled, but
no additional apparent mass is expended. The particle continues at the speed of
light, now in a freely propagating, gravity-free regime.
Formulation:
1. Newtonian Foundation
Consistency
F = ma;
a ∝ F,
a ∝ 1/m
Where, acceleration (a) is
directly proportional to force (F) and inversely proportional to mass (m). When
no force is applied:
F =
0 ⇒ a
= 0, and mass remains constant: m = m
Classical Gravitational
Mass:
Fɢ = mɢ·g
where: mɢ is the gravitational
mass, g is the gravitational field strength.
Classically, gravitational
mass and inertial mass are recognized—and they are assumed equal:
mɢ = m
Extended Classical
Mechanics (ECM) Formulation:
ECM application of
Classical proportionality rules:
aᵉᶠᶠ ∝ Fᴇᴄᴍ; aᵉᶠᶠ ∝ 1/Mᴍ
This distinction is crucial
to ECM, where inertial response is modulated by the balance between real and
apparent mass.
|1/Mᴍ| = |Mᵃᵖᵖ| ⇒ Mᵃᵖᵖ ~ 1/Mᴍ
ECM generalizes Newtonian
ideas by incorporating dynamic mass contributions, such as negative apparent
mass (−Mᵃᵖᵖ).
2. ECM Force Extension
(Massive Bodies)
Fᴇᴄᴍ = (Mᴍ +
(−Mᵃᵖᵖ)) aᵉᶠᶠ = Mᵉᶠᶠ aᵉᶠᶠ
Where Mᴍ: matter mass (positive, gravitational). −Mᵃᵖᵖ: negative apparent mass (from KE or anti-gravitational
behaviour). Mᵉᶠᶠ: effective mass = total
inertial response, and aᵉᶠᶠ:
effective acceleration
This term emerges from
motion, gravitational interactions, or energy-mass coupling—mechanisms not present
in classical theory.
In ECM: Gravitational mass
(Mɢ) becomes:
Mɢ = Mᴍ +
(−Mᵃᵖᵖ) = Mᵉᶠᶠ
Where Mᴍ: is the matter mass (traditional inertial mass), −Mᵃᵖᵖ is the negative apparent mass, representing: Kinetic
energy's mass-equivalent, gravitationally induced mass offsets, Dynamic
redistribution from field interactions.
Thus, gravitational mass
becomes a net effective mass (Mᵉᶠᶠ) that
varies depending on the particle's state and spatial context (e.g., radial
distance from a mass source).
3. ECM Speed Conditions for
Massive Particles
Speed domains by comparing
Mᴍ and Mᵃᵖᵖ through:
Mᴍ - 1/Mᴍ
This form is symbolic:
Condition
Interpretation Implication
for Speed
·
Mᴍ >|Mᵃᵖᵖ| Matter
mass dominates Low speed
o (Low kinetic activity)
·
Mᴍ =|Mᵃᵖᵖ Balanced
system
Medium speed
·
Mᴍ <|Mᵃᵖᵖ| Kinetic
energy dominates High speed
4. Gravitational Extension
in ECM
Mɢ = Mᴍ +
(−Mᵃᵖᵖ) = Mᵉᶠᶠ, where |Mᵃᵖᵖ| ~
(1/Mᴍ)
Breakdown by Radial
Distance r:
Radial Distance Mass
Relation Gravity
Behaviour
·
r =
0 Mɢ = Mᴍ (no
KE or Mᵃᵖᵖ) Pure
gravity
·
r >
0 Mᴍ > |Mᵃᵖᵖ|
Gravity dominates
·
r ≫ 0 Mᴍ = |Mᵃᵖᵖ| Effective gravity neutral
o (Flat or marginal expansion)
·
r → ∞ Mᴍ < |Mᵃᵖᵖ|
Antigravity
o (Repulsion, acceleration)
This is remarkably aligned
with cosmological behaviour: it predicts an emergent anti-gravitational
behaviour at large scale—very much like dark energy or expansion dynamics. Good
match with A. D. Chernin et al.'s observational research titled "Dark
energy and the structure of the Coma cluster of galaxies."
5. Massless Particles
(Conventional, Photon-like)
In ECM, massless particles
invert the conventional paradigm:
·
Mᴍ < 0
— interpreted as negative matter mass
·
-Mᵃᵖᵖ < 0
— represents kinetic energy equivalent mass (negative)
·
Thus, -Mᵃᵖᵖ = |-Mᵃᵖᵖ| — used to denote its
positive magnitude
Total force becomes:
Fᴇᴄᴍ = (Mᴍ + (-M))
aᵉᶠᶠ = 2|Mᵃᵖᵖ| aᵉᶠᶠ
This implies:
Mᵉᶠᶠ = 2|Mᵃᵖᵖ|, aᵉᶠᶠ =
effective acceleration
Now, considering kinetic
energy in ECM:
KE = ½Mᵉᶠᶠ v² = |Mᵃᵖᵖ| v²
This leads to a new
interpretation of speed and acceleration:
• Within Gravitational
Influence:
• aᵉᶠᶠ = 2c ⇒ v = c
• Condition:
|Mᵃᵖᵖ| aᵉᶠᶠ = v
• Half the kinetic energy
is spent in overcoming gravity:
KE = |Mᵃᵖᵖ| v² ⇒ ½ KE
used in gravitational escape.
• Just Escaping Gravity (At
Horizon):
• Escape velocity
condition:
|Mᵃᵖᵖ| aᵉᶠᶠ = v = c
• No further acceleration
needed — no kinetic energy is spent during motion:
KE = |Mᵃᵖᵖ| v² ⇒ No ½ KE
spent — already at escape velocity
The above is internally
consistent and creatively describes photon behaviour under gravitational
influence, reinterpreting "massless" as net-zero effective mass
resulting from real-negative and apparent-positive mass interactions.
Mathematical Presentation:
1. Newtonian Foundation
Consistency
Begin with the foundational
equation:
F = ma,
Where acceleration (a) is
directly proportional to force (F) and inversely proportional to mass (m):
a ∝ F ; a ∝ 1/m
These are standard
relationships in classical mechanics. When no force is applied:
F =
0 ⇒ a
= 0, and mass remains constant: m = m
This restates the principle
of inertia—an object maintains its state of rest or uniform motion unless acted
upon by an external force.
Classical Gravitational
Mass:
In Newtonian physics,
gravitational mass (mɢ) is
the property that determines how a body interacts with gravitational fields. It
appears in Newton's law of universal gravitation and defines the gravitational
force a body experiences:
Fɢ = mɢ·g
Here:
·
mɢ is the gravitational mass,
·
g is the gravitational
field strength.
Classically, there is no
distinction between different mass types (e.g., apparent mass, effective mass).
Only gravitational mass and inertial mass are recognized—and they are assumed
equal:
mɢ = m
Thus, in classical
mechanics, the same symbol m is used for both inertial and gravitational mass,
without separation into other mass forms. There is no concept of negative
apparent mass (−Mᵃᵖᵖ) or dynamic
gravitational behaviour based on internal or external energy contributions.
In Extended Classical
Mechanics (ECM):
ECM generalizes Newtonian
ideas by incorporating dynamic mass contributions, such as negative apparent
mass (−Mᵃᵖᵖ). This term emerges from
motion, gravitational interactions, or energy-mass coupling—mechanisms not
present in classical theory.
In ECM:
Gravitational mass (Mɢ) becomes:
Mɢ = Mᴍ +
(−Mᵃᵖᵖ) = Mᵉᶠᶠ
Where:
·
Mᴍ is the matter mass (traditional rest mass),
·
−Mᵃᵖᵖ is the negative apparent mass, representing:
Kinetic energy's
mass-equivalent,
·
Gravitationally induced
mass offsets,
·
Dynamic redistribution from
field interactions.
Thus, gravitational mass
becomes a net effective mass (Mᵉᶠᶠ) that
varies depending on the particle's state and spatial context (e.g., radial
distance from a mass source).
Summary:
·
In classical mechanics,
gravitational mass is static and equals inertial mass.
·
In ECM, gravitational mass
is dynamic, accounting for both matter mass and apparent mass effects.
·
This provides a framework
where massless and massive particles can be treated under a unified
force–energy perspective, especially when gravitational or relativistic
phenomena are involved.
2. ECM Force Extension
(Massive Bodies)
Fᴇᴄᴍ = (Mᴍ +
(−Mᵃᵖᵖ)) aᵉᶠᶠ = Mᵉᶠᶠ aᵉᶠᶠ
With:
·
Mᴍ: matter mass (positive, gravitational)
·
−Mᵃᵖᵖ: negative apparent mass (from KE or anti-gravitational behaviour)
·
Mᵉᶠᶠ: effective mass = total inertial response
·
aᵉᶠᶠ: effective acceleration
ECM application of Classical
proportionality rules:
aᵉᶠᶠ ∝ Fᴇᴄᴍ; aᵉᶠᶠ ∝ 1/Mᴍ
are consistent provided
that acceleration is viewed as determined by Mᴍ, not Mᵉᶠᶠ. This
distinction is crucial to ECM, where inertial response is modulated by the
balance between real and apparent mass.
Further defined:
|1/Mᴍ| = |Mᵃᵖᵖ| ⇒ Mᵃᵖᵖ ~ 1/Mᴍ
This inverse relationship
reflects the ECM principle that as matter mass increases, apparent mass
decreases, and vice versa, i.e., KE builds up as mass thins out in effect.
3. ECM Speed Conditions for
Massive Particles
Define speed domains by
comparing Mᴍ and Mᵃᵖᵖ through:
Mᴍ - 1/Mᴍ
This form is symbolic, but
conceptually solid. Here's a breakdown:
Condition Interpretation Implication
for Speed
·
Mᴍ >|Mᵃᵖᵖ| Matter
mass dominates Low speed
·
Mᴍ =|Mᵃᵖᵖ Balanced
system Medium
speed
·
Mᴍ <|Mᵃᵖᵖ| Kinetic
energy dominates High speed
Consistent with ECM's view:
apparent mass (i.e., kinetic content) governs motion dynamically relative to
matter mass.
4. Gravitational Extension
in ECM
Mɢ = Mᴍ +
(−Mᵃᵖᵖ) = Mᵉᶠᶠ, where |Mᵃᵖᵖ| ~
(1/Mᴍ)
Breakdown by Radial
Distance r:
Radial
Distance
Mass
Relation Gravity Behaviour
·
r =
0 Mɢ = Mᴍ (no
KE or Mᵃᵖᵖ) Pure
gravity
·
r >
0 Mᴍ > |Mᵃᵖᵖ| Gravity dominates
·
r ≫ 0 Mᴍ = |Mᵃᵖᵖ| Effective
gravity neutral
o (Flat or marginal expansion)
·
r → ∞ Mᴍ > |Mᵃᵖᵖ| Antigravity
o (Repulsion, acceleration)
This is remarkably aligned
with cosmological behaviour: it predicts an emergent anti-gravitational
behaviour at large scale—very much like dark energy or expansion dynamics. Good
match with Chernin et al.'s observations.
5. Massless Particles
(Conventional, Photon-like)
Here, ECM flips the
paradigm:
Mᴍ < 0 (massless, interpreted as negative)
−Mᵃᵖᵖ = |Mᵃᵖᵖ| (positive
KE equivalent mass)
Fᴇᴄᴍ = (Mᴍ +
(-Mᵃᵖᵖ)) aᵉᶠᶠ ⇒ Fᴇᴄᴍ = 2|Mᵃᵖᵖ| aᵉᶠᶠ
This gives:
Mᵉᶠᶠ = 2|Mᵃᵖᵖ|, aᵉᶠᶠ = effective acceleration
Now, introducing speed
limit interpretation:
·
Within
gravitational influence:
aᵉᶠᶠ = 2c ⇒ v = c
Due to 2|Mᵃᵖᵖ|aᵉᶠᶠ = v, and energy spent
in escaping.
·
Just escaping
gravity:
|Mᵃᵖᵖ| aᵉᶠᶠ = v = c
No kinetic
expenditure; just escape velocity = c.
The above is internally
consistent and creatively describes photon behaviour under gravitational
influence, reinterpreting "massless" as net-zero effective mass
resulting from real-negative and apparent-positive mass interactions.
Final Consistency Check
Logical Integrity:
·
Each equation evolves from
Newtonian mechanics but redefines mass/acceleration relationships in
energetically dynamic terms.
·
The system conserves
logical structure while redefining inertial and gravitational responses through
ECM principles.
Dimensional Consistency:
·
All force equations retain
the correct dimensions:
F = [M]
[a]
Physical Insight:
·
Massless particles
accelerate as if they possess negative real mass offset by positive apparent
mass.
·
Gravitational behaviour
transitions to antigravity at large scales—mirroring cosmological acceleration.
Summary
ECM-based framework:
·
Is mathematically and
physically consistent.
·
Effectively extends
Newtonian mechanics with meaningful reinterpretations of mass, energy, and
motion.
·
Offers novel insight into
massless particles, antigravity, and cosmic-scale gravitational behaviour.
·
Supports intuitive
analogues to dark energy, inertia-kinetic duality, and relativistic limits.
ECM Term Definition —
Matter Mass (Mᴍ):
Matter Mass, symbolized as
Mᴍ, refers to the total
positive gravitational mass derived from all forms of matter, both visible and
non-visible. It is defined as:
Mᴍ =
Mᴏʀᴅ +
Mᴅᴍ
Where:
·
Mᴏʀᴅ is
the Ordinary Matter Mass, consisting of atoms, particles, and objects
observable through electromagnetic interaction (e.g., stars, gas, planets,
etc.).
·
Mᴅᴍ is
the Dark Matter Mass, which cannot be directly observed but whose gravitational
influence is well-documented (e.g., via galaxy rotation curves, cluster
dynamics, and lensing effects).
This formulation reflects
the observational evidence that approximately 27% of the universe's total
energy content is attributed to matter, with ordinary matter contributing only
about 5%, and dark matter making up the remaining ~22%. These proportions are
consistent with studies such as:
·
Chernin, A. D.,
Bisnovatyi-Kogan, G. S., Teerikorpi, P., Valtonen, M. J., Byrd, G. G., &
Merafina, M. (2013), Astronomy and Astrophysics, 553, A101.
In the ECM framework,
Matter Mass forms the foundational term used in both gravitational and dynamic
considerations. However, ECM does not equate Mᴍ to gravitational mass.
Instead, gravitational mass is redefined as:
Mɢ =
Mᴍ +
(-Mᵃᵖᵖ)
Where −Mᵃᵖᵖ is
the Apparent Mass (with a negative sign), which in ECM is equivalent to the
effective mass of dark energy (Mᴅᴇ). This relationship accounts for the antigravitational behaviour
observed on cosmological scales and aligns with the balance conditions observed
in Chernin’s “zero-gravity spheres” at large radial distances.
Discussion
The presented formulation
offers a clear and comprehensive departure from traditional Newtonian mechanics
by extending foundational principles through the lens of Extended Classical
Mechanics (ECM). While rooted in Newton’s
force law F = ma, ECM redefines the nature and composition of mass, bringing
kinetic energy and gravitational context into dynamic roles previously
unaccounted for.
Newtonian Groundwork and
the ECM Shift
Classical mechanics treats
mass as both inertial and gravitational, implicitly assuming their equivalence
and constancy. However, in ECM, this uniformity dissolves. Mass becomes a
dynamic quantity—decomposed into two components:
·
Matter Mass (Mᴍ) representing intrinsic
gravitational matter (including both ordinary and dark matter), and
·
Negative Apparent Mass (-Mᵃᵖᵖ), representing the
mass-equivalent of kinetic energy and gravitational-field interaction.
This dual-mass framework
transforms classical gravitational mass into an effective mass:
Mɢ = Mᴍ + (-Mᵃᵖᵖ) = Mᵉᶠᶠ
This shift allows ECM to
address dynamic systems where energy exchange alters the net mass, which
classical mechanics cannot account for without invoking relativistic or quantum
principles.
Force and Acceleration: A
Redefined Relationship
One of the striking
innovations in ECM is the distinction between inertial source and accelerative
response. While force remains proportional to acceleration, the inertial resistance
is attributed to the effective mass, not solely the matter mass. Yet,
paradoxically, acceleration remains inversely proportional to matter mass:
aᵉᶠᶠ ∝ 1/Mᴍ
This distinction introduces
an ECM-specific inertia principle, where the observable acceleration stems from
how much real mass resists motion, while the net force includes additional
dynamic components like kinetic mass or gravitational distortion.
Speed Regimes and Mass
Balance
By comparing (Mᴍ) and |Mᵃᵖᵖ|, ECM introduces a robust
classification of motion:
·
Low speeds occur when rest
mass dominates,
·
Medium speeds at a critical
balance point, and
·
High speeds when kinetic
(apparent) mass becomes dominant.
This interpretation yields
a continuous transition from rest to motion, closely linked with energy-mass
dynamics. It also implies that velocity is not an isolated vector but the
outcome of a shifting mass-energy balance, a nuance classical mechanics lacks.
Gravitational Radius and
Cosmological Insight
The model’s capacity to
account for varying gravity with radial distance (r) stands as a profound
contribution. It elegantly explains:
·
Local gravity (r = 0) as
pure mass-dominated attraction,
·
Intermediate distances ( r
> 0) as zones of mass-kinetic interplay, and
·
Cosmic-scale distances (r ≫ 0)
where Mᴍ ≈ |Mᵃᵖᵖ|, leading to net
gravitational neutrality or repulsion.
This is not only consistent
with Chernin et al.'s observations of "zero-gravity spheres" but also
offers a functional explanation for dark energy as a dynamic outcome of
negative apparent mass, rather than a cosmological constant or exotic field.
Massless Particle
Redefinition
The treatment of massless
particles under ECM is especially transformative. ECM reframes the photon not
as a truly massless particle, but as one exhibiting a cancellation between
negative matter mass and positive kinetic energy mass:
Mᴍ < 0,
-Mᵃᵖᵖ = |Mᵃᵖᵖ|, Mᵉᶠᶠ = 2|Mᵃᵖᵖ|
This leads to ECM's
prediction that photons under gravitational influence experience acceleration
as if propelled by their internal energy redistribution—a viewpoint supporting
both relativistic speed limits and gravitational redshift mechanisms without
invoking spacetime curvature directly.
Logical and Physical
Coherence
The ECM framework remains
dimensionally and logically consistent with Newtonian mechanics while allowing
for:
·
Mass variation with energy
and spatial context,
·
Force expressions
consistent across massive and massless systems,
·
Interpretation of
relativistic and cosmological behaviors using classical equations enhanced with
new terms.
Implications and Broader
Significance
This approach does more
than reinterpret equations—it provides a unifying language for classical,
relativistic, and cosmological phenomena. By internalizing the concept of
apparent mass, ECM not only bridges gaps in Newtonian mechanics but also offers
testable insights into:
·
Dark energy and cosmic
expansion,
·
High-velocity particle
behavior,
·
Gravitational influence at
multiple scales, and
·
Photon dynamics within
gravitational fields.
Alphabetical Glossary of
Mathematical Terms in ECM
·
aᵉᶠᶠ (Effective
Acceleration)
The acceleration resulting
from the ECM-adjusted net mass (Mᵉᶠᶠ), it represents the physical acceleration observed when
accounting for both real and apparent mass components.
·
c (Speed of Light)
The fundamental speed limit
for information and energy propagation in spacetime, in ECM, it represents the
upper bound for particles under gravitational escape or massless propagation.
·
F (Classical Force)
Defined by Newton’s
second law as F = ma. In ECM, this forms the foundational equation upon which
force extensions are built.
·
Fᴇᴄᴍ (ECM
Force)
The generalized ECM force
expression: Fᴇᴄᴍ = Mᵉᶠᶠ aᵉᶠᶠ = (Mᴍ + (−Mᵃᵖᵖ)) aᵉᶠᶠ Accounts
for both real matter mass and negative apparent mass contributions in the
system's inertial behaviour.
·
g (Classical Gravitational
Field Strength)
The intensity of
gravitational acceleration experienced by a mass, appears in the classical
gravitational force equation (F = mɢ g).
·
gᵉᶠᶠ (Effective Gravitational Field Strength)
The actual gravitational acceleration experienced by a body
in ECM, defined as the net field influence resulting from both matter mass (Mᴍ) and apparent
mass (−Mᵃᵖᵖ). Appears in the generalized gravitational force equation:
Fɢ = Mɢ gᵉᶠᶠ = (Mᴍ + (−Mᵃᵖᵖ)) gᵉᶠᶠ. Unlike
classical (g), which is static, gᵉᶠᶠ dynamically changes with spatial context (radial distance),
mass-energy redistribution, and the local gravitational environment is —
especially reflecting transitions between gravity-dominated and antigravity
dominated regions.
·
Mᴍ (Matter
Mass)
Total positive
gravitational mass composed of ordinary matter (Mᴏʀᴅ) and dark matter (Mᴅᴍ). It is the primary rest-mass term in ECM.
·
Mᴏʀᴅ (Ordinary
Matter Mass)
Mass made of observable
matter (atoms, particles, objects emitting or interacting with EM radiation)
Subset of Mᴍ.
·
Mᴅᴍ (Dark
Matter Mass)
Non-luminous, indirectly
observed mass inferred from gravitational effects, also a component of Mᴍ.
·
Mᵃᵖᵖ (Apparent
Mass)
A dynamic mass component
arising from kinetic energy or gravitational interaction, in ECM, it is
negative and represents the inertial contribution from energy motion, not
substance.
·
Mᵉᶠᶠ (Effective
Mass)
Defined as the net inertial
mass in ECM: Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ)
Determines how body resists
acceleration, incorporating both positive matter mass and negative apparent
mass.
·
Mɢ (Gravitational
Mass)
In ECM, redefined from its
classical form to: Mɢ = Mᴍ + (-Mᵃᵖᵖ) = Mᵉᶠᶠ. This
formulation allows mass to evolve based on energy redistribution and
gravitational conditions.
·
r (Radial Distance)
The spatial separation from
a central gravitational source, in ECM, the different values of (r) change the
relative dominance of Mᴍ and Mᵃᵖᵖ,
leading to shifts between gravity and antigravity behaviour.
·
v (Velocity)
The speed of a particle,
determined in ECM as a function of apparent mass and acceleration, for massless
particles under gravity: Fᴇᴄᴍ = 2|Mᵃᵖᵖ| aᵉᶠᶠ ⇒ v = c
Conclusion:
The formulation of Extended
Classical Mechanics (ECM) equations from classical foundations provides a
coherent and innovative extension of Newtonian mechanics, bridging conventional
limitations through the introduction of effective mass and negative apparent
mass. By redefining gravitational and inertial interactions, ECM offers a
refined understanding of force, acceleration, and mass behaviour across both
massive and massless regimes. It maintains classical proportionality principles
while enhancing interpretative power for relativistic and cosmological
phenomena—such as high-speed particle dynamics and the transition from gravitational
to antigravitational domains. ECM's consistent treatment of mass-energy
coupling, particularly through its reinterpretation of photons and
gravitational thresholds, introduces a unifying framework applicable to both
local and large-scale dynamics. This advancement aligns with empirical
research, notably Chernin et al.'s observations of cosmic structures, and sets
a robust groundwork for further theoretical and experimental exploration.
References:
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M. J., Byrd, G. G., & Merafina, M. (2013). Dark energy and the structure of
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2. Goldstein, H., & Twersky, V. (n.d.). Classical Mechanics.
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6. The Large Scale Structure of Space-Time by Stephen Hawking and
eorge Ellis
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·
Author’s ORCiD:
0000-0003-1871-7803.
·
Tagore’s Electronic Lab,
West Bengal, India
·
Correspondence: postmasterenator@gmail.com
Declaration:
·
Author declares no conflict
of Interest
·
No financial aid is
received for this work.
