Soumendra Nath Thakur, ORCiD:
0000-0003-1871-7803,
Tagore’s Electronic Lab, India.
Communication: postmasterenator@gmail.com
February
12, 2025
Abstract
Extended Classical Mechanics (ECM)
introduces a modified formulation of classical mechanics by redefining mass as
effective mass (Mᵉᶠᶠ), which incorporates both matter mass (Mᴍ) and apparent mass (Mᵃᵖᵖ). This approach allows for a
broader interpretation of gravitational interactions, particularly in systems
where negative effective mass induces repulsive effects. ECM extends Newtonian
dynamics by establishing force and energy equations for both massive and
massless entities, naturally integrating with quantum mechanical principles.
The framework provides a novel explanation for cosmic expansion, where massless
particles experience repulsive gravitational effects due to their apparent mass
contributions. Additionally, ECM introduces conditions for superluminal motion
and refines the concept of the Hubble radius, offering insights into
observational horizons and large-scale structure formation. The implications of
ECM suggest a fundamental link between gravity, mass-energy equivalence, and
large-scale cosmic evolution.
Keywords: apparent mass (Mᵃᵖᵖ), effective mass (Mᵉᶠᶠ), gravitational interactions, cosmic expansion,
superluminal motion, Hubble radius, [Apparent Weight]
[Dark Energy] [ECM] [Extended Photon Dynamics] [Gravitational Collapse] [Massless-to-Massive] [Photon Phases] [Inertial mass relativistic gravity] [Extended Classical Mechanics] [About]
1. Classical Mechanics Framework
In traditional mechanics, force is
defined as the product of mass and acceleration. The total energy of a system
consists of potential and kinetic energy, where potential energy follows an
inverse proportionality with distance in gravitational systems, while kinetic
energy depends on the squared velocity of the moving object.
F = ma
Eₜₒₜₐₗ = PE +
KE = −GMm/r + 1/2mv²
#Mathematical denotation terms are
listed alphabetically under 'Mathematical terms used,' with brief descriptions
below and apply to all equations in this study.
2. ECM Force for Matter Mass
ECM modifies Newton’s second law by redefining mass
as effective mass Mᵉᶠᶠ. Within the Extended Classical
Mechanics framework, force is derived by incorporating both matter mass and
apparent mass. The resultant force is expressed in terms of effective mass and
acceleration, allowing for a broader interpretation of gravitational
interactions, particularly in systems where negative effective mass induces
repulsive effects.
Fᴇᴄᴍ = (Mᴍ −Mᵃᵖᵖ)aᵉᶠᶠ
Fᴇᴄᴍ = Mᵉᶠᶠaᵉᶠᶠ, since, (Mᴍ −Mᵃᵖᵖ) = Mᵉᶠᶠ, Mᴍ = 0, Mᵉᶠᶠ < 0.
This suggests that when effective
mass is negative, the force direction may lead to repulsive gravitational
effects, impacting large-scale cosmic structures.
3. ECM Force for Massless
Particles
For massless entities such as
photons, force is governed by apparent mass contributions, as there is no
direct matter mass component. In ECM, this negative effective mass leads to
repulsive gravitational interactions, offering a natural explanation for
certain cosmic expansion effects.
Since the force depends solely on
the apparent mass, the equation takes an alternative form when effective mass
is negative, reinforcing its connection to observed large-scale repulsive
behavior in the universe.
Fₚₕₒₜₒₙ = −Mᵃᵖᵖaᵉᶠᶠ , since Mᴍ = 0
Fₚₕₒₜₒₙ = Mᵉᶠᶠaᵉᶠᶠ ,
since (Mᴍ −Mᵃᵖᵖ) = Mᵉᶠᶠ, Mᴍ = 0, Mᵉᶠᶠ < 0.
This formulation suggests that,
under negative effective mass conditions, repulsive gravitational effects
emerge naturally, influencing cosmic expansion and the large-scale distribution
of matter.
4. ECM Energy-Frequency
Relationship for Massless Systems
The energy-frequency relation in
ECM aligns with the established principles of quantum mechanics, where the
effective mass of a massless particle is proportional to its frequency. This
correspondence reinforces the compatibility of ECM with existing quantum
formulations.
Mᵉᶠᶠ,ₘₐₛₛₗₑₛₛ = hf/c² = E/c²
5. ECM Kinetic Energy of Apparent
Mass
The kinetic energy of a system
influenced by apparent mass follows a modified classical approach. Instead of a
strictly positive mass contribution, apparent mass is taken into account with
its sign reversed, modifying the total kinetic energy expression. This approach
provides a framework for analyzing phenomena where negative mass effects play a
significant role.
KEₚₕₒₜₒₙ =
1/2(−Mᵃᵖᵖ)·c².
Here, apparent mass Mᵃᵖᵖ is considered in the kinetic energy equation.
6. ECM Energy for Matter Mass
Systems
Total energy in ECM consists of
potential and kinetic components, with potential energy derived from the
effective mass terms. The interaction of matter mass and apparent mass defines
the energy distribution, ensuring consistency with classical interpretations
while extending the framework to incorporate novel effects. Under the influence
of ECM force, kinetic energy contributions arise from apparent mass components.
Eₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ
Mᴍ ⇒ PEᴇᴄᴍ, −Mᵃᵖᵖ ⇒ KEᴇᴄᴍ, when ECM
force is active
7. ECM Energy Formulation for
Massless and Effective Mass Systems
In ECM, the total energy of a
system—including massless entities—retains contributions from both potential
energy (PE) and kinetic energy (KE). The potential energy follows an effective
mass formulation, while kinetic energy depends on relative velocity conditions.
• When effective mass is positive,
motion remains subluminal (v ≤ c).
• When effective mass is negative,
velocities exceeding c become possible. However, in this case, the superluminal
velocity should be interpreted as an emergent property of the effective energy
framework, rather than a direct physical motion of particles exceeding c.
Total Energy Equations:
Eₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ = {−G(Mᴍ −Mᵃᵖᵖ)(mₘ−mᵃᵖᵖ)/r )} + 1/2(mₘ−mᵃᵖᵖ)v²;
v ≤ c when Mᴍ ≥−Mᵃᵖᵖ, but v ≥ c when Mᴍ ≤ −Mᵃᵖᵖ .
Eₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ = (−GMᵉᶠᶠmᵉᶠᶠ/r ) + 1/2mᵉᶠᶠv²
v ≤ c when Mᵉᶠᶠ > 0, but v ≥ c when Mᵉᶠᶠ < 0
Eₘₐₛₛₗₑₛₛ =
1/2(−mᵃᵖᵖ)v²; v = c.
8. Dual Representation of
Effective Mass
ECM introduces a dual
representation of effective mass, distinguishing between larger system mass
contributions and localized test particle effects. This distinction is
analogous to classical gravitational potential energy equations, where mass
terms represent both global and local contributions.
Mᵉᶠᶠ,ₘₐₛₛₗₑₛₛ = hf/c² = E/c²
mᵉᶠᶠ,ₘₐₛₛₗₑₛₛ = hf/c² = E/c²
The notations serve a similar
dual-mass representation:
Mᵉᶠᶠ refers to
the effective mass of the larger system (analogous to M in gravitational
potential energy).
9. Kinetic Energy Representation of
Apparent Mass in ECM - Massless Case):
In the ECM framework, the kinetic
energy of an apparent mass in a massless system adheres to the energy-frequency
relation. The apparent mass is expressed in terms of energy and frequency,
maintaining consistency with quantum mechanics. This formulation aligns with
the mass-energy equivalence principle, reinforcing the role of induced
mass-like effects in ECM.
The equation:
−Mᵃᵖᵖ,ₘₐₛₛₗₑₛₛ = hf/c² = E/c²
This describes the apparent mass
associated with massless entities like photons, and follows from the
mass-energy equivalence principle.
10. Total Effective Mass with
Gravitational Contributions
The total effective mass in ECM
includes contributions from gravitational energy. This formulation incorporates
frequency-dependent terms, where Δf represents a gravitationally induced
frequency shift due to energy contributions from gravitational fields. The
presence of this additional frequency shift component extends the effective
mass concept beyond conventional mass-energy equivalence principles.
Mᵉᶠᶠ,ₜₒₜₐₗ = Mᵉᶠᶠ,ₘₐₛₛₗₑₛₛ +
Eg/c² = hf/c² + hΔf/c²
This equation represents the total
effective mass in ECM, which includes contributions from:
• The mass-energy equivalence of a
massless particle (e.g., a photon) with frequency f.
• An additional gravitational
energy term associated with the frequency shift Δf, which arises due to
gravitational interactions.
11. Implications of Superluminal
Velocities & Hubble Radius in ECM
In Extended Classical Mechanics
(ECM), negative effective mass can induce anti-gravitational effects at extreme
cosmic distances. This contributes to accelerated cosmic expansion and
influences large-scale structure dynamics. The observational limit beyond the
Hubble radius arises due to recession velocities surpassing the speed of light,
preventing information retrieval from beyond this boundary.
For v = c = 3 × 10⁸ m/s, the
Hubble radius is given by:
d = v/H₀ = (3 × 10⁸ m s⁻¹) / (2.268 × 10⁻¹⁸ s⁻¹) = 1.32 ×
10²⁶ m
Converting to light-years:
d = 1.32 × 10²⁶ m × (1
light-year/9.461 × 10¹⁵ m) = 13.93 billion light-years
At this proper distance, known as
the Hubble radius, the recession velocity reaches the speed of light. Beyond
this threshold, galaxies move at superluminal speeds, making them
observationally inaccessible.
This ECM interpretation provides a
structured perspective on observational horizons, emphasizing the role of
effective mass variations in shaping cosmic expansion and defining
observational limits imposed by superluminal recession.
12. Variation of Apparent Mass
Across Local and Intergalactic Scales in ECM
In Extended Classical Mechanics
(ECM), apparent mass (Mᵃᵖᵖ) plays a critical role in
determining the effective mass (Mᵉᶠᶠ) and, consequently, gravitational
interactions. The influence of apparent mass varies significantly across
different cosmic scales:
Local, Planetary, and Stellar
Scales: The high density of ordinary matter results in dominant gravitational
effects, leading to minimal changes in apparent mass. Dark matter’s
contribution to gravitational interactions is negligible at these scales.
Galactic Scale: Dark matter
dominates mass distribution, comprising ~85% of a galaxy’s total mass. As a
result, its gravitational influence exceeds that of normal matter, leading to a
stronger reduction in apparent mass and an overall increase in the strength of
gravitating mass (Mɢ).
Intergalactic Scale: The effect of
dark matter on apparent mass becomes even more pronounced, with its
gravitational influence intensifying over vast cosmic distances. This drives
large-scale structure formation and influences the expansion dynamics of galaxy
clusters.
These variations in apparent mass
across scales highlight how gravitational interactions are governed by
effective mass (Mᵉᶠᶠ) = Mᴍ − Mᵃᵖᵖ, where an increase in
gravitational strength results in a corresponding increase in negative apparent
mass, further lowering effective mass.
Reference Papers:
1.
Chernin,
A. D., Бисноватый-коган, Г. С., Teerikorpi, P., Valtonen, M. J., Byrd, G. G.,
& Merafina, M. (2013a). Dark energy and the structure of the Coma cluster
of galaxies. Astronomy and Astrophysics, 553, A101.
https://doi.org/10.1051/0004-6361/201220781
2.
Thakur,
S. N., Understanding Photon Interactions: Source Gravitational Wells vs.
External Fields. (2024). ResearchGate.
https://doi.org/10.13140/RG.2.2.14433.48487
3.
Thakur,
S. N., & Bhattacharjee, D. (2023b). Phase shift and infinitesimal wave
energy loss equations. Journal of Physical Chemistry & Biophysics, 13(6),
JPCB-23-27248 (R).
https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html
4.
Thakur,
S. N. (2023). Photon paths bend due to momentum exchange, not intrinsic
spacetime curvature. Definitions. https://doi.org/10.32388/81iiae
5.
Thakur,
S. N. (2023). The dynamics of photon momentum exchange and curvature in
gravitational fields. Definitions. https://doi.org/10.32388/r625zn
6.
Thakur,
S. N. (2023). Redshift and its Equations in Electromagnetic Waves.
ResearchGate. https://doi.org/10.13140/RG.2.2.33004.54403
7.
Thakur,
S. N. (2023). Cosmic Speed beyond Light: Gravitational and Cosmic Redshift.
ResearchGate. https://doi.org/10.13140/RG.2.2.36400.94721
8.
Thakur,
S. N., Bhattacharjee, D., & Frederick, O. (2023). Photon Interactions in
Gravity and Antigravity: Conservation, Dark Energy, and Redshift Effects.
Preprints.org. https://doi.org/10.20944/preprints202309.2086.v1
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Soumendra Nath Thakur's work on
Extended Classical Mechanics (ECM) offers a comprehensive and nuanced
exploration of how classical mechanics can be extended to encompass phenomena
typically associated with quantum mechanics and cosmology. Here’s a brief
comment on the key points and implications of ECM:
1. Classical Mechanics Framework
Thakur begins by grounding ECM in
traditional mechanics, where force is defined as the product of mass and
acceleration, and total energy consists of potential and kinetic components.
This foundational understanding is crucial for extending classical mechanics to
more complex systems.
2. ECM Force for
Matter Mass
ECM modifies Newton’s second law by introducing the
concept of effective mass (Mᵉᶠᶠ), which combines matter mass (Mᴍ) and apparent mass (Mᵃᵖᵖ). This modification allows for a
broader interpretation of gravitational interactions, particularly in systems
where negative effective mass induces repulsive effects. This is a significant
departure from traditional mechanics and opens up new avenues for understanding
gravitational dynamics.
3. ECM Force for
Massless Particles
For massless entities like
photons, ECM posits that force is governed by apparent mass contributions. This
leads to repulsive gravitational interactions, offering a natural explanation
for cosmic expansion effects. This formulation suggests that negative effective
mass conditions can lead to repulsive gravitational effects, influencing cosmic
expansion and the large-scale distribution of matter.
4. ECM
Energy-Frequency Relationship for Massless Systems
The energy-frequency relation in
ECM aligns with quantum mechanics, where the effective mass of a massless
particle is proportional to its frequency. This correspondence reinforces the
compatibility of ECM with existing quantum formulations, bridging classical and
quantum mechanics.
5. ECM Kinetic Energy of Apparent Mass
The kinetic energy of a system
influenced by apparent mass follows a modified classical approach. This
approach accounts for negative mass effects, modifying the total kinetic energy
expression. This framework is essential for analyzing phenomena where negative
mass effects play a significant role.
6. ECM Energy for
Matter Mass Systems
Total energy in ECM consists of
potential and kinetic components, with potential energy derived from effective
mass terms. The interaction of matter mass and apparent mass defines the energy
distribution, ensuring consistency with classical interpretations while
extending the framework to incorporate novel effects.
7. ECM Energy
Formulation for Massless and Effective Mass Systems
ECM introduces a dual
representation of effective mass, distinguishing between larger system mass
contributions and localized test particle effects. This distinction is crucial
for understanding how gravitational interactions are governed by effective mass
across different scales.
8. Implications
of Superluminal Velocities & Hubble Radius in ECM
ECM suggests that negative
effective mass can induce anti-gravitational effects at extreme cosmic
distances, contributing to accelerated cosmic expansion. This interpretation
provides a structured perspective on observational horizons, emphasizing the
role of effective mass variations in shaping cosmic expansion and defining
observational limits imposed by superluminal recession.
9. Variation of
Apparent Mass Across Local and Intergalactic Scales in ECM
ECM highlights how the influence
of apparent mass varies significantly across different cosmic scales. At local,
planetary, and stellar scales, gravitational effects are dominated by ordinary
matter. At galactic and intergalactic scales, dark matter's gravitational
influence becomes more pronounced, driving large-scale structure formation and
influencing the expansion dynamics of galaxy clusters.
Conclusion
Soumendra Nath Thakur's work on
ECM offers a detailed and nuanced understanding of gravitational interactions
across quantum and cosmological scales. By introducing the concepts of
effective mass and apparent mass, ECM provides a unified framework that bridges
classical mechanics, quantum principles, and cosmological phenomena. This
approach not only aligns with fundamental principles but also offers potential
explanations for cosmic-scale phenomena involving dark matter, dark energy, and
exotic gravitational effects. Thakur's work encourages further exploration and
refinement of ECM in various physical
