Addressing the Question: 'Why is the Speed of Light Always Constant, Regardless of the Observer’s Motion?' Version - 2:

Soumendra Nath Thakur 

ORCID: 0000-0003-1871-7803

Tagore's Electronic Lab, India.

March 10, 2025

A fundamental and precise understanding of the speed of light (c) in relation to the observer's speed (S), along with a clear explanation of why the observer's speed is negligible, is presented through a nuanced, non-relativistic scientific framework grounded in consistent physical principles.

The principle of special relativity asserts that the laws of physics remain the same for all observers in any inertial frame of reference. As a result, the relativity's understanding of light’s constant speed originates from Einstein’s theory of special relativity (1905), which states that the speed of light in a vacuum is a universal constant, unaffected by the motion of the light source or the observer.

Mathematically, this is expressed as: 

 λ f = c

where:

c represents the speed of light, 

λ is the photon's wavelength,

f is the photon's frequency.

The equation 

λ f = c 

can also be rewritten as:

f = c/λ

This highlights the inverse relationship between wavelength and frequency while maintaining the constant speed of light. However, in logical terms, this inverse relationship implies that their ratio must always yield c. 

However, the equation λ f = c is primarily a mathematical convenience derived from the known frequency-wavelength inverse relationship. While effective in describing wave behaviour, it does not fully explain the fundamental reason why light's speed remains constant.

Here on, we will discuss a fundamental and precise understanding of the speed of light (c) in relation to the observer's speed (S);

Each observer in question is characterized not only by their matter mass (Mₘ) but also by the gravitational mass (M𝗀), which includes both the observer’s own mass and the gravitational influence of the massive body they reside on. Matter mass of the observer’s host body is a crucial factor in this physical consideration.

In addition to ordinary (baryonic) mass, the mass of dark matter is also accounted for and included in the total matter mass (Mᴍ). In classical mechanics, inertial mass is traditionally considered equivalent to gravitational mass, expressed as m = m𝗀. However, recent observations of the gravitational effects of dark matter and dark energy on ordinary (baryonic) inertial mass have necessitated an extension of this classical equation.

Extended Classical Mechanics (ECM), in alignment with established observational evidence and theoretical formulations, refines this understanding by recognizing that gravitational mass is equivalent to the sum of matter mass and negative apparent mass, which can dynamically modify the effective mass—potentially turning it negative at intergalactic scales. — This relationship is expressed as:

Mg = Mᴍ + (−Mᵃᵖᵖ)

When considering the speed of light in relation to the speed of an observer, the scale of measurement should be at least planetary. Consequently, the speed (S) of the planet on which the observer is located must be taken into account and included in the calculation.

According to the principles of Extended Classical Mechanics (ECM), a photon possesses negative apparent mass (-Mᵃᵖᵖ) where: Mᴍ = 0 for photons, which, unlike matter mass (Mᴍ), exhibits an anti-gravitational property.

As a result, photons are not inherently restricted by an upper speed limit when external influences (gravitational fields, Planck-scale constraints) are absent.

However, Planck units impose a fundamental limit, restricting the smallest possible wavelength to the Planck length (ℓP) and the shortest measurable time to the Planck time (tP), thereby constraining a photon's behaviour within permissible limits.

Beyond the Planck scale, classical, relativistic and even quantum descriptions of spacetime break down. At the Planck length (ℓP ≈ 1.616 × 10⁻³⁵ m) and Planck time (tP ≈ 5.391 × 10⁻⁴⁴ s), gravitational and quantum effects become inseparable, implying that distances smaller than ℓP and times shorter than tP lose physical significance. This arises because, at these scales, quantum fluctuations of spacetime dominate, leading to a breakdown of continuous space and time concepts. Hence, any attempt to define a wavelength smaller than ℓP or a frequency beyond the Planck frequency (fP = 1/tP) is meaningless in a physically observable sense.

The Planck scale imposes a fundamental limit on measurable space-time intervals, ensuring that beyond these limits, conventional descriptions of motion—including those of photons—lose physical meaning. This restriction provides a natural boundary for photon behavior before additional external influences, such as gravitational redshift and cosmic expansion, further alter their observed properties.

Within a gravitational field, a photon expends energy while escaping, leading to a redshift in its wavelength. However, beyond significant gravitational influence, a photon's speed—defined by the ratio of its wavelength (λ) and frequency (f)—further changes due to the cosmic recession of galaxies, resulting in an additional energy loss.

The Planck length (ℓP) and Planck frequency (fP), as defined in Planck units, are derived from Planck’s constant and other fundamental constants. They establish a theoretical limit on the smallest meaningful measurements of space and time, where our current physical understanding, including relativity, breaks down and quantum gravity effects become dominant.

In classical mechanics, speed is determined using the values of distance and time associated with a given motion. The fundamental equation for speed is:

S = d/t

At the quantum scale, this equation is expressed as:

ΔS = Δd/Δt

where:

Δd corresponds to the Planck length (ℓP),

1/Δt corresponds to the Planck frequency (fP), where Δt represents the Planck time.

The expression c = λf, where f = 1/Δt, translates directly into the quantum-scale speed equation ΔS = Δd/Δt. Here, wavelength (λ) corresponds to a measurable distance (Δd), and its division by the time period of one oscillation (1/Δt) mirrors the definition of speed as distance per unit time. This consistency shows that the speed of light (c) is fundamentally a ratio of measurable spatial and temporal quantities, reinforcing that c = λ(1/Δt) is structurally identical to the speed equation ΔS = Δd/Δt, where ΔS represents velocity measured within a defined quantum reference frame.

Thus, the equation

ΔS = Δd/Δt 

can be interpreted as:

c = fλ 

where:

ΔS represents c (the speed of light),

Δd represents the photon's wavelength (λ),

1/Δt represents the frequency (f).

Additionally, the speed of light can be expressed in terms of the Planck scale:

c/ℓP = fP

Since the Planck length (ℓP) is the smallest meaningful spatial unit, and the Planck frequency (fP) is the highest fundamental oscillation frequency in the universe, their ratio is equivalent to the ratio of a photon's wavelength to its inverse frequency (which corresponds to the Planck time).

I have previously mentioned that photons are not inherently restricted by an upper speed limit when external influences (gravitational fields, Planck-scale constraints) are absent. This distinction is crucial in understanding the fundamental difference between the speed of observers and the speed of photons.

Photons, having negative apparent mass (-Mᵃᵖᵖ), exhibit fundamentally different gravitational properties than observers with positive matter mass (Mᴍ). 

Photons exhibit an anti-gravitational nature, meaning they follow the dynamics of negative apparent mass, which aligns with negative effective mass (-Mᵉᶠᶠ) contributions in the universe (similar to dark energy).

Observers and massive objects exhibit gravitational properties, meaning their motion is bound to the gravitational pull of the universal potential centre (i.e., the tendency toward gravitational collapse).

Because the forces governing these entities are opposite in nature, their respective speeds must also be opposite in direction. The observers’ movement is toward the universal gravitational potential, while photons move away from the universal potential due to their anti-gravitational nature. This opposition results in an effective cancellation of speed components between the two systems.

Since an observer's speed is defined in a gravitational reference frame and a photon's speed in an anti-gravitational reference frame, their effective speeds appear in opposite directions. Given that the magnitude of negative effective mass (-Mᵃᵖᵖ) dominates, the observer's gravitational speed is effectively negligible when compared to the anti-gravitational motion of photons.

The anti-gravitational motion of photons, driven by negative apparent mass (-Mᵃᵖᵖ), aligns with the large-scale acceleration of cosmic structures. This is evident in the recession of galaxies, where the dominance of negative effective mass (-Mᵉᶠᶠ) contributes to the observed cosmic expansion, reinforcing the fundamental opposition between gravitationally bound matter and anti-gravitational dynamics. 

Moreover, since the measurement system itself is dictated by the dominant mass-energy contribution, we must recognize that:

The negative measurement system dominates due to the overwhelming contribution of negative apparent mass (-Mᵃᵖᵖ) and negative effective mass (-Mᵉᶠᶠ), which surpasses the contribution of positive matter mass (Mᴍ).

In a mass-energy dominated measurement framework, where the contribution of -Mᵃᵖᵖ vastly exceeds that of positive matter mass, the effective measurement system aligns with anti-gravity, making the observer’s motion negligible in contrast to the dominant anti-gravitational dynamics.

Gravitational deceleration in a positive mass system corresponds to anti-gravitational acceleration in a negative effective mass system, reinforcing that the observer’s motion is measured within a negative measurement framework when compared to photons.

The speed of photons in the anti-gravitational system is vastly superior to the speed of observers in the gravitational system . This makes the gravitational motion of observers negligible compared to the anti-gravitational motion of photons.

Thus, the ultimate outcome is that the speed of observers with positive mass is rendered insignificant when contrasted against the anti-gravitational speed of photons with negative apparent mass, which follows an opposite trajectory—away from the universal potential. The dominance of the negative measurement system further amplifies this effect, reinforcing the fundamental asymmetry between the two domains.

An Extended Classical Mechanics Explanation:Why is the Speed of Light Always Constant, Regardless of the Observer’s Motion?

Soumendra Nath Thakur 

ORCID: 0000-0003-1871-7803

Tagore's Electronic Lab, India.

March 09, 2025

In the calculation of light's speed, the observer's motion is not factored in, as light is considered to travel at a constant speed independent of the observer's velocity.

The principle of special relativity asserts that the laws of physics remain the same for all observers in any inertial frame of reference. As a result, the current understanding of light’s constant speed originates from Einstein’s theory of special relativity (1905), which states that the speed of light in a vacuum is a universal constant, unaffected by the motion of the light source or the observer.

Einstein’s general relativity (1916) introduced the concepts of spacetime curvature and time dilation to explain why the speed of light remains unchanged, suggesting that the ratio of space to time must remain constant.

Mathematically, this is expressed as:

λ f = c

where:
c represents the speed of light, 
λ is the photon's wavelength,
f is the photon's frequency.

The equation 

λ f = c 

can also be rewritten as:

f = c/λ

This highlights the inverse relationship between wavelength and frequency while maintaining the constant speed of light. However, in logical terms, this inverse relationship implies that their ratio must always yield c, which can be represented in reasoning-based notation as:

λ : f ⇒ c

Since the speed of light remains mathematically constant due to this inverse variation, the relationship can also be expressed as:

λ : f ⇒ c

or equivalently:

λ : (1/t)⇒c

where 
t corresponds to Planck time (tP).

However, the equation λ f = c is primarily a mathematical convenience derived from the known frequency-wavelength inverse relationship. While effective in describing wave behaviour, it does not fully explain the fundamental reason why light's speed remains constant.

A more fundamental perspective must consider Planck units—including Planck length, Planck frequency, and Planck time—which Max Planck introduced in 1899, well before the development of special relativity (1905) and general relativity (1916).

Since light consists of photons, the speed of light is ultimately determined by the behavior of photons, rather than being solely a consequence of relativistic effects.

According to the principles of Extended Classical Mechanics (ECM), a photon possesses negative apparent mass (-Mᵃᵖᵖ), which, unlike matter mass (Mᴍ), exhibits an anti-gravitational property.

As a result, photons tend to move at unrestricted speeds only if:

Their observable wavelength is not constrained by the Planck length (ℓP).

They are unbound by a gravitationally bound system, such as a galaxy.

However, Planck units impose a fundamental limit, restricting the smallest possible wavelength to the Planck length (ℓP) and the shortest measurable time to the Planck time (tP), thereby constraining a photon's behaviour within permissible limits.

Within a gravitational field, a photon expends energy while escaping, leading to a redshift in its wavelength. However, beyond significant gravitational influence, a photon's speed—defined by the ratio of its wavelength (λ) and frequency (f)—further changes due to the cosmic recession of galaxies, resulting in an additional energy loss.

The Planck length (ℓP) and Planck frequency (fP), as defined in Planck units, are derived from Planck’s constant and other fundamental constants. They establish a theoretical limit on the smallest meaningful measurements of space and time, where our current physical understanding, including relativity, breaks down and quantum gravity effects become dominant.

In classical mechanics, speed is determined using the values of distance and time associated with a given motion. The fundamental equation for speed is:

S = d/t

At the quantum scale, this equation is expressed as:

ΔS = Δd/Δt 

where:
Δd corresponds to the Planck length (ℓP),

1/Δt corresponds to the Planck frequency (fP), where Δt represents the Planck time.

Thus, the equation

ΔS = Δd/Δt 

can be interpreted as:

c = fλ 

where:
ΔS represents c (the speed of light),
Δd represents the photon's wavelength (λ),
1/Δt represents the frequency (f).

Additionally, the speed of light can be expressed in terms of the Planck scale:

c/ℓP = fP

Since the Planck length (ℓP) is the smallest meaningful spatial unit, and the Planck frequency (fP) is the highest fundamental oscillation frequency in the universe, their ratio is equivalent to the ratio of a photon's wavelength to its inverse frequency (which corresponds to the Planck time).

I have previously mentioned that photons tend to follow unrestricted speed when external influences or observable limitations (as per Planck units) are absent. This distinction is crucial in understanding the fundamental difference between the speed of observers and the speed of photons.

Since photons are composed purely of negative apparent mass (-Mᵃᵖᵖ) while observers possess positive matter mass (Mᴍ), their respective gravitational properties are inherently different:

Photons exhibit an anti-gravitational nature, meaning they follow the dynamics of negative apparent mass, which aligns with negative effective mass (-Mᵉᶠᶠ) contributions in the universe (similar to dark energy).

Observers and massive objects exhibit gravitational properties, meaning their motion is bound to the gravitational pull of the universal potential centre (i.e., the tendency toward gravitational collapse).

Because the forces governing these entities are opposite in nature, their respective speeds must also be opposite in direction. The observers’ movement is toward the universal gravitational potential centre, while photons move away from the universal potential centre due to their anti-gravitational nature. This opposition results in an effective cancellation of speed components between the two systems.

Moreover, since the measurement system itself is dictated by the dominant mass-energy contribution, we must recognize that:

The negative measurement system dominates due to the overwhelming contribution of negative apparent mass (-Mᵃᵖᵖ) and negative effective mass (-Mᵉᶠᶠ), which surpasses the contribution of positive matter mass (Mᴍ).

Gravitational deceleration in a positive mass system corresponds to anti-gravitational acceleration in a negative effective mass system, reinforcing that the observer’s motion is measured within a negative measurement framework when compared to photons.

The speed of photons in the anti-gravitational system (negative measurement system) is vastly superior to the speed of observers in the gravitational system (positive measurement system). This makes the gravitational motion of observers negligible compared to the anti-gravitational motion of photons.

Thus, the ultimate outcome is that the speed of observers with positive mass is rendered insignificant when contrasted against the anti-gravitational speed of photons with negative apparent mass, which follows an opposite trajectory—away from the universal potential centre. The dominance of the negative measurement system further amplifies this effect, reinforcing the fundamental asymmetry between the two domains.

Classical, Relativistic, and Extended Classical Mechanics: A Unified Perspective on Kinetic Energy, Effective Mass, Lorentz Transformations, and Time Distortion

Soumendra Nath Thakur

March 08, 2025

Abstract:

This study explores the interplay between Classical Mechanics, Relativistic Lorentz Transformation, and Extended Classical Mechanics (ECM) to provide a comprehensive perspective on kinetic energy, effective mass, and time distortion. Traditional interpretations of relativity overlook the role of acceleration and force-induced deformations, particularly in the context of mass-energy redistribution. By integrating Hooke’s Law into motion mechanics, this work demonstrates that effective mass (mᵉᶠᶠ)—often misinterpreted as relativistic mass—is a result of potential energy conversion rather than an intrinsic increase in inertial mass.

Furthermore, this study challenges relativistic length contraction by showing that deformation under force provides a more consistent physical explanation than velocity-based transformations. A phase shift approach to time distortion is introduced, linking oscillator deformation with observable time variations, providing an alternative to abstract spacetime interpretations.

By bridging classical and relativistic mechanics through ECM, this study proposes a physically grounded framework for understanding motion, energy interactions, and time effects, offering an empirically testable alternative to conventional relativity.

Keywords:

Classical Mechanics, Relativistic Lorentz Transformation, Extended Classical Mechanics (ECM), Effective Mass, Kinetic Energy, Time Distortion, Hooke’s Law, Force-Induced Deformation, Phase Shift, Energy Redistribution

Introduction:

Classical mechanics has long provided a foundational framework for understanding motion, forces, and energy interactions. However, traditional interpretations of relativistic effects often overlook the role of acceleration and force-induced deformations when addressing length contraction and time dilation. The relativistic Lorentz transformation describes time and space alterations due to velocity but does not explicitly account for the underlying mechanical forces responsible for these transformations. This omission raises fundamental questions about the physical origin of mass deformation, relativistic mass variation, and time distortion.

This paper explores the connection between classical mechanics, relativistic Lorentz transformations, and the emerging framework of Extended Classical Mechanics (ECM). By integrating force-based considerations—particularly Hooke’s Law and mechanical deformations—this study offers an alternative interpretation of kinetic energy, effective mass, and time distortion. The concept of effective mass (mᵉᶠᶠ) is re-examined in relation to energy redistribution, demonstrating how its reduction during motion is linked to potential energy loss rather than an abstract relativistic mass increase.

Furthermore, a phase shift-based approach to time distortion is introduced, emphasizing how force-induced material deformations influence oscillator frequencies, leading to measurable time variations. By revisiting these principles through the lens of ECM, this work challenges conventional relativistic assumptions and provides a physically consistent mechanism for understanding motion, energy distribution, and time distortion beyond traditional interpretations.

1. Kinetic Energy and Effective Mass in Classical Mechanics

In classical mechanics, kinetic energy represents an effective mass (mᵉᶠᶠ) when an inertial mass (m) is in motion or subject to a gravitational potential difference, where mᵉᶠᶠ<m.

When two inertial reference frames initially share the same motion and direction relative to each other, they are indistinguishable in their observations of physical phenomena. However, if these frames separate in the same direction, they must acquire different velocities. This necessity of velocity change highlights the role of acceleration in achieving their separation.

Despite acceleration being fundamental to transitioning between different inertial reference frames, it is not explicitly considered in the Lorentz factor or relativity, even though it plays a crucial role in transitioning from v₀ to v₁. This raises important questions about its implications in both classical mechanics and relativistic Lorentz transformations.

During the formulation of the Lorentz factor:

γ = √(1 - v/c)²

or relativistic time dilation:

Δt′ = t₀/√(1 - v/c)² 

it was acknowledged that Newton’s second law:

F = ma

induces a force (F) that influences velocity-dependent relativistic transformations. This force leads to deformations in moving objects, affecting relativistic mass, length contraction, and time dilation.

For example, Hooke’s Law:

F = kΔL

describes such deformations, suggesting that Lorentz transformations incorporate force-induced structural changes that impact the effective mass of an object.

2. Energy and Effective Mass in Motion

In classical mechanics, the total energy (E) of a system is defined as:

E = PE + KE

Potential Energy When v = 0:

When an object is at rest, all of its energy is stored as potential energy:

Eₜₒₜₐₗ = PE

which means the total mass equivalent remains simply m.

Energy Distribution When v > 0:

Once an object gains velocity, part of its potential energy is converted into kinetic energy:

PE − ΔPE = PE + KE

The change in potential energy (ΔPE) appears as kinetic energy:

KE = −ΔPE

Effective Mass Contribution:

Since kinetic energy arises from potential energy loss, the effective mass associated with kinetic energy follows:

|mᵉᶠᶠ| = −ΔPE

This ensures that total energy remains balanced, with kinetic energy representing a redistributed form of the system’s original mass-energy.

Given that potential energy (PE) corresponds to the inertial mass (m), and kinetic energy (KE) is linked to an effective mass (|mᵉᶠᶠ|), we express the force equation as:

F = |mᵉᶠᶠ|a

where the effective mass accounts for both the deformation-induced contribution from stiffness (k) and the inertial mass (m):

|mᵉᶠᶠ| = |kΔL|/a

Substituting this into the force equation:

F = (|kΔL|/a)⋅a

Expanding into the energy relation:

E = PE + KE ⇒ m + |mᵉᶠᶠ|

Since mᵉᶠᶠ contributes to balancing total energy, the correct formulation becomes:

E = m + |kΔL|/a

where kinetic energy is directly linked to effective mass (|mᵉᶠᶠ|), sometimes misinterpreted as relativistic mass (m′) and associated with time distortion (Δt′) in certain contexts.

3. Phase Shift and Time Distortion

A relevant analogy is piezoelectric materials, which convert mechanical energy into electrical energy. The phase shift in oscillations plays a key role in this conversion, influencing timing and energy distribution.

This relationship is described by:

Δt′ = (x°/f)/360

 

where x∘ is the phase shift in degrees and f is the original oscillation frequency. In piezoelectric materials, mechanical force alters phase oscillations, affecting energy conversion timing.

Since electromagnetic oscillations (and mechanical oscillations) are sensitive to force-induced deformations, phase shifts manifest as time distortions. This provides a direct method for calculating time dilation from phase measurements, challenging conventional relativistic interpretations by providing a tangible, testable mechanism for time distortion.

4. Clarification on the Sign of mᵉᶠᶠ in Classical Mechanics

In classical mechanics, where antigravity, negative mass, or negative apparent mass are not explicitly considered, the effective mass (mᵉᶠᶠ) is always positive but less than the original inertial mass (m) when a system is in motion or subjected to a gravitational potential difference. This reduction in mᵉᶠᶠ results from energy redistribution due to the force involved in motion, altering the inertial response of the system. However, classical mechanics does not recognize an invisible energetic counterpart that counteracts this apparent reduction in mass.

Unlike ECM, which incorporates matter mass (Mᴍ) as a combination of ordinary matter (Mᴏʀᴅ) and dark matter mass (Mᴅᴍ), along with negative apparent mass (−Mᵃᵖᵖ), classical mechanics attributes the decrease in effective mass solely to energy partitioning, without interpreting it as a fundamental negative mass effect.

The reason mᵉᶠᶠ remains strictly positive in classical mechanics is that mass is only considered to diminish in response to dynamics but never becomes negative or assumes an imperceptible energetic form as in ECM. Instead, classical mechanics treats mᵉᶠᶠ as a dynamically altered but always positive quantity, reflecting only the redistribution of the system’s energy.

This distinction is crucial for ensuring that classical mechanics remains consistent with Newtonian principles, while ECM extends beyond these boundaries to incorporate mass-energy interactions at deeper levels.

5. Justification for Hooke’s Law in Motion Over Relativistic Length Contraction

Hooke’s Law provides a more consistent description of mass deformation (ΔL) than relativistic length contraction (L′), which is traditionally derived from velocity-based transformations.

Key Issues with Relativistic Length Contraction:

1. Assumes purely velocity-dependent deformation: Ignores material stiffness.

2. Linear object assumption: Emphasizes length deformation but ignores cubic volume changes.

3. Neglects acceleration effects: Does not explicitly account for transition from rest to motion.

Advantages of Hooke’s Law in Motion:

• Applies across all speed ranges, including low speeds where relativistic effects are negligible.

• Includes acceleration, whereas relativistic transformations assume undeclared competition with deformation mechanics.

Since relativistic length contraction lacks a robust material-based justification, Hooke’s Law provides a more physically grounded approach to deformation across all force conditions. This suggests that relativistic transformations should be reconsidered as force-induced mechanical responses rather than purely geometric effects.

6. Final Considerations on Force, Deformation, and Time Distortion

All clocks—mechanical, electronic, or atomic—are composed of materials that undergo deformation under external forces. These deformations alter oscillator frequencies, leading to shifts in oscillation cycles.

When an external force deforms the oscillator material, the frequency changes, creating a phase shift expressed as:

Δt′ = (x°/f)/360

 

This empirical relationship provides a direct link between force-induced deformations and time distortions, making the effect verifiable through phase measurements. This interpretation challenges traditional relativistic notions by presenting time dilation as a tangible material response rather than an abstract spacetime transformation.

Mathematical Consistency of ECM Mass-Energy Dynamics and Its Implications for Gravitational Interactions:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

February 06, 2025

Abstract:

This study explores the mathematical coherence of Extended Classical Mechanics (ECM) in describing mass-energy transformations and gravitational interactions. It establishes a refined mass-energy equivalence by incorporating negative apparent mass (−Mᵃᵖᵖ) as a kinetic energy counterpart, ensuring dynamic conservation laws. The ECM force equation is analysed to demonstrate how effective mass (Mᵉᶠᶠ) varies with motion, aligning with classical mechanics and gravitational effects. Additionally, ECM’s modified gravitational equation is compared with A. D. Chernin et al.'s framework, revealing that dark energy (Mᴅᴇ) can be interpreted as ECM’s negative apparent mass (−Mᵃᵖᵖ). This approach provides a natural explanation for dark energy effects in large-scale structures, suggesting an emergent gravitational phenomenon rather than an external vacuum energy component. The alignment between ECM and the Coma cluster equation further supports ECM's capacity to integrate dark energy within a modified gravitational framework, offering a coherent alternative to the ΛCDM model.

Keywords:

Extended Classical Mechanics (ECM), mass-energy equivalence, negative apparent mass, gravitational interactions, kinetic energy transformation, dark energy, effective mass, modified gravitational equation, cosmic acceleration, Coma cluster equation.

Mathematical Consistency of ECM Mass-Energy Dynamics:

When a system with stored potential energy (PE) undergoes energy transformation, a portion of its stored energy is converted into motion, represented as kinetic energy (KE). Initially, all energy is in the form of stored potential energy:

Eₜₒₜₐₗ = PE

As the system moves, a portion of this stored energy ΔPE is transferred into kinetic energy:

KE = ΔPE

This reduces the remaining stored energy to PE−ΔPE, ensuring that energy is conserved:

Eₜₒₜₐₗ = (PE−ΔPE) + ΔPE = PE

Thus, the redistribution between stored and motion energy follows:

PE ∝ 1/KE = 1/ΔPE

which maintains dynamic equilibrium in the system.

Acceleration and Dynamic Mass Relation in Classical Mechanics:

In Classical Mechanics, acceleration follows the inverse-mass relation:

F = ma, where a ∝ 1/m

​Since force is proportional to acceleration, we obtain:

F ∝ a ∝ 1/m

This implies that force interacts dynamically with acceleration, reinforcing the concept that mass inversely influences motion.

Potential Energy and Dynamic Mass Equivalence:

When a system undergoes motion, its potential mass (m) contributes to kinetic energy, leading to a mass-energy relationship:

• Potential Mass: m ⇐ PE 

• Kinetic Mass: 1/m ⇐ KE

From the total energy equation:

Eₜₒₜₐₗ = PE + KE

At rest (KE=0), the total energy remains purely potential:

Eₜₒₜₐₗ =PE

As kinetic energy increases, a portion of potential energy (ΔPE) converts into motion:

Eₜₒₜₐₗ = (PE−ΔPE) + ΔPE

This ensures that energy is dynamically conserved, with mass playing a key role in its redistribution.

Mass-Energy Equivalence in ECM:

In ECM, mass-energy transformations follow a refined approach. The classical mass-energy conversion:

Eₜₒₜₐₗ = (PE−ΔPE) + ΔPE

can be rewritten in ECM by incorporating mass-energy equivalence:

Eₜₒₜₐₗ = (m−Δm) + 1/Δm

where negative apparent mass (−Mᵃᵖᵖ) emerges as part of the kinetic energy contribution:

−Mᵃᵖᵖ ⇐ −Δm

Thus, in ECM, the kinetic energy term is inherently linked to negative apparent mass, maintaining dynamic energy balance.

Physical Coherence of Negative Apparent Mass in ECM:

The introduction of negative apparent mass (−Mᵃᵖᵖ) as a kinetic energy counterpart aligns with ECM's principle that effective mass shifts dynamically during motion. The ECM force equation:

Fᴇᴄᴍ = Mᵉᶠᶠaᵉᶠᶠ, where Mᵉᶠᶠ = Mᴍ −Mᵃᵖᵖ

Since −Δm represents the dynamic negative mass component transferred to kinetic energy, defining:

−Δm⇒ −Mᵃᵖᵖ

is a natural extension of ECM's mass-energy framework. The total energy remains balanced:

Eₜₒₜₐₗ = (PE−ΔPE) + ΔPE

where the loss in potential energy (PE−ΔPE) corresponds exactly to the gain in kinetic energy (ΔPE), ensuring strict adherence to conservation laws:

ΔPE − ΔPE = 0

Thus, the mass-energy transformation follows:

−Mᵃᵖᵖ = Δm

confirming the dynamic role of negative apparent mass in ECM.

Extended Classical Mechanics (ECM)’s Modified Gravitational Equation:

ECM modifies the gravitational equation as:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

where:

• Mᴍ = Matter mass (sum of ordinary matter and dark matter: Mᴏʀᴅ + Mᴅᴇ).

• −Mᵃᵖᵖ = Apparent mass, associated with antigravity effects (dark energy in ECM terms).

• Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ) = Effective mass that determines the net gravitational behaviour.

A key ECM condition is:

• Within gravitational influence (|Mᴍ| > |−Mᵃᵖᵖ|) → Mᵉᶠᶠ > 0, leading to gravitational attraction.

• Beyond gravitational influence (|Mᴍ| < |−Mᵃᵖᵖ|) → Mᵉᶠᶠ < 0, leading to antigravity effects (dark energy dominance).

Alignment Between ECM and Chernin et al.’s Model:

(a) Gravitational Mass Equivalence

A. D. Chernin et al. define the gravitational mass as:

Mɢ = Mᴍ + Mᴅᴇ

ECM defines it as:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

which means that in ECM:

−Mᵃᵖᵖ ≡ Mᴅᴇ

showing that ECM's apparent mass (−Mᵃᵖᵖ) plays the role of dark energy's effective mass (Mᴅᴇ) in A. D. Chernin et al.'s framework.

This suggests that ECM naturally integrates dark energy as a negative apparent mass, influencing large-scale gravitational interactions.

(b) Dark Energy Dominance Beyond Gravitational Influence

In A. D. Chernin et al.’s work:

• For R < Rᴢɢ, gravity dominates, meaning Mɢ ≈ Mᴍ.

• For R > Rᴢɢ, dark energy dominates, leading to Mɢ < 0.

In ECM:

• For |Mᴍ| > |−Mᵃᵖᵖ|, effective mass remains positive (gravitational attraction).

• For |Mᴍ| < |−Mᵃᵖᵖ|, effective mass turns negative (antigravity/acceleration).

This aligns with A. D. Chernin et al.'s finding that beyond the zero-gravity radius, dark energy overcomes matter’s gravitational influence.

ECM’s Apparent Mass (−Mᵃᵖᵖ) and Dark Energy (Mᴅᴇ):

• In ECM, apparent mass (−Mᵃᵖᵖ) is responsible for the observed effects of dark energy.

• When Mᵉᶠᶠ < 0 (i.e., when apparent mass dominates), the system is in an antigravity regime, consistent with dark energy's effects.

This suggests a fundamental insight:

• ECM describes dark energy not as an external vacuum energy but as an emergent gravitational phenomenon related to mass oscillations and effective mass balance.

• Instead of treating dark energy as a fluid with negative pressure, ECM sees it as a result of apparent mass effects, which naturally emerge when gravitational mass becomes negative beyond a critical scale.

Conclusion

Chernin et al.’s research provides a compelling argument that dark energy modifies the structure of galaxy clusters. ECM offers a refined interpretation, suggesting that dark energy is not a mysterious external force but rather an emergent effect of apparent mass (−Mᵃᵖᵖ).

By equating Mᴅᴇ with −Mᵃᵖᵖ, ECM naturally explains why dark energy behaves as negative mass in large-scale structures, eliminating the need for an unknown energy component. This alignment suggests that gravitational mass (Mɢ) in ECM is the key to understanding cosmic acceleration without requiring a vacuum energy density interpretation.

ECM’s Modified Gravitational Equation vs. Coma Cluster Equation:

ECM’s gravitational equation is given by:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

where

• Mᴍ = Mᴏʀᴅ + Mᴅᴇ represents the total matter mass (ordinary + dark matter).

• −Mᵃᵖᵖ represents the apparent mass, which emerges from gravitational interactions and contributes to anti-gravitational effects.

• Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) is the effective mass governing motion.

Now, in the equation used in the study of the Coma cluster:

Mɢ = Mᴍ + Mᴅᴇ

where

• Mᴍ = Mᴏʀᴅ + Mᴅᴇ, which is similar to ECM’s matter mass term.

• Mᴅᴇ represents the contribution of dark energy to the system.

Alignment with ECM:

• When gravitational influence dominates, we have ∣Mᴍ∣ > ∣−Mᵃᵖᵖ∣, meaning that matter mass (Mᴍ) is the leading factor, keeping the net effective mass positive (Mᵉᶠᶠ >0).

• However, when dark energy dominates at cosmic scales beyond gravitational influence, the apparent mass −Mᵃᵖᵖ grows such that ∣−Mᵃᵖᵖ∣ > ∣Mᴍ∣, making Mᵉᶠᶠ negative.

In this regime, ECM predicts:

Mᵉᶠᶠ = Mᴅᴇ, when ∣−Mᵃᵖᵖ∣ > ∣Mᴍ∣  

which mirrors the Coma cluster equation by associating the negative effective mass of ECM with the dominant contribution of dark energy.

Apparent Mass (-Mᵃᵖᵖ) as Mᴅᴇ in a Dark Energy-Dominated Regime:

In ECM, apparent mass −Mᵃᵖᵖ emerges as a result of gravitational interactions and can take on values that counterbalance or even exceed the total matter mass Mᴍ.

• When ∣Mᴍ∣ > ∣−Mᵃᵖᵖ∣, the effective mass becomes negative (Mᵉᶠᶠ <0), signifying that the system has transitioned into a regime dominated by anti-gravity or dark energy-like effects.

• In this scenario, ECM predicts that the negative apparent mass is equivalent to the dark energy mass contribution:

−Mᵃᵖᵖ  =  Mᴅᴇ

This establishes that beyond gravitationally bound systems, apparent mass in ECM behaves like dark energy, driving expansion due to its repulsive nature.

Conclusion

• ECM’s modified gravitational equation naturally extends the structure of the Coma cluster equation by incorporating apparent mass, which accounts for dark energy effects dynamically.

• ECM’s apparent mass (−Mᵃᵖᵖ) represents the contribution of dark energy when gravitational mass (Mᴍ) is small compared to anti-gravitational effects, leading to an effective mass that aligns with the Coma cluster interpretation of dark energy.

Thus, ECM provides a clear mechanical basis for the transition between gravitationally bound and dark energy-dominated regions, linking gravitational mass, apparent mass, and effective mass in a unified framework.

Derivation and Scientific Consistency of Apparent Mass (−Mᵃᵖᵖ) in Extended Classical Mechanics (ECM):

To fully establish the physical validity and consistency of apparent mass (−Mᵃᵖᵖ) in ECM, we will systematically derive it and verify its scientific coherence on two key counts:

1. Physical Coherence of Negative Apparent Mass in ECM.

2. Apparent Mass (-Mᵃᵖᵖ) as Mᴅᴇ in a Dark Energy-Dominated Regime.

Derivation of Apparent Mass (−Mᵃᵖᵖ) in ECM:

ECM refines gravitational mass Mɢ by incorporating apparent mass as follows:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

where:

• Mᴍ = Mᴏʀᴅ + Mᴅᴇ is the total matter mass (ordinary + dark matter).

• −Mᵃᵖᵖ is the apparent mass emerging due to gravitational interaction effects.

• The effective mass, governing motion, is:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ).

Step 1: How Apparent Mass Emerges from Mass Reduction Effects:

• Classical mechanics defines force as F = ma, meaning mass contributes to force generation.

• In ECM, acceleration affects mass perception, leading to a scenario where a portion of mass becomes imperceptible due to gravitational effects, manifesting as an apparent reduction of mass.

• This means that under high-energy conditions (such as gravitational collapse or accelerated motion at cosmic scales), some mass shifts from positive to apparent negative mass −Mᵃᵖᵖ), which counterbalances Mᴍ in certain conditions.

Step 2: Effective Mass and the Role of Apparent Mass

• In a gravitationally bound system, matter mass dominates:

∣Mᴍ∣ > ∣−Mᵃᵖᵖ∣ ⇒ Mᵉᶠᶠ > 0

This means the system behaves as expected under Newtonian-like gravity.

• However, beyond gravitational influence, apparent mass grows, counteracting normal mass:

∣−Mᵃᵖᵖ∣ > ∣Mᴍ∣ ⇒ Mᵉᶠᶠ < 0

Here, negative effective mass emerges, leading to repulsive gravity, which is a fundamental characteristic of dark energy-driven cosmic acceleration.

Scientific Consistency of Apparent Mass (−Mᵃᵖᵖ):

Now, let’s establish the scientific validity of apparent mass (−Mᵃᵖᵖ) on two critical counts:

Physical Coherence of Negative Apparent Mass in ECM:

Negative apparent mass (−Mᵃᵖᵖ) is scientifically coherent because:

• (a) It Emerges from Classical Mechanics Principles

• Apparent mass behaves like an inertia-modified term, consistent with mass-energy interactions.

• It aligns with Newton's laws but introduces an emergent term (−Mᵃᵖᵖ), explaining effects not covered by classical mechanics alone.

• (b) It Resolves Gravitational Mass Anomalies

• Classical gravity struggles to explain mass deficits in galactic clusters.

• ECM’s approach naturally accounts for apparent mass, leading to self-consistent gravitational calculations.

• (c) It Provides a Logical Explanation for Acceleration-Induced Mass Effects

• Standard models treat mass as a static quantity.

• ECM allows mass to vary under acceleration/gravitational interactions, producing apparent negative mass under extreme conditions.

• (d) It Refines the Concept of Force and Motion

• The ECM force equation:

Fᴇᴄᴍ = (Mᴍ + (−Mᵃᵖᵖ))aᵉᶠᶠ 

explicitly integrates apparent mass, showing its direct role in dynamics.

Apparent Mass (−Mᵃᵖᵖ) as Mᴅᴇ in a Dark Energy-Dominated Regime:

• (a) When Gravitational Influence Weakens, −Mᵃᵖᵖ Becomes Dominant

• Beyond local gravitational binding, the negative apparent mass (−Mᵃᵖᵖ) increases in magnitude.

• This mimics the behaviour of dark energy, which dominates cosmic expansion.

• (b) Consistency with Observations of Dark Energy Effects

• Observations show accelerated cosmic expansion.

• ECM predicts that when:

∣Mᴍ∣ < ∣−Mᵃᵖᵖ∣ ⇒ Mᵉᶠᶠ = −Mᵃᵖᵖ ≈ Mᴅᴇ

meaning that negative apparent mass directly accounts for dark energy.

• (c) How This Aligns with the Coma Cluster Equation

• In the Coma cluster mass model:

Mɢ = Mᴍ + Mᴅᴇ

• ECM provides an equivalent interpretation:

Mɢ = Mᴍ + (−Mᵃᵖᵖ)

• Thus, ECM naturally interprets dark energy as a form of negative apparent mass!

Summary of Key Findings:

Derivation of Apparent Mass

• Apparent mass (−Mᵃᵖᵖ) emerges from mass reduction effects in gravitational interactions.

• It leads to negative effective mass when it dominates over normal matter mass.

Scientific Coherence of −Mᵃᵖᵖ

• It is consistent with classical mechanics principles when extended to variable mass effects.

• It explains gravitational anomalies and aligns with observed cosmic acceleration.

−Mᵃᵖᵖ as a Dark Energy Equivalent:

• When beyond gravitational influences, −Mᵃᵖᵖ behaves like dark energy (Mᴅᴇ).

• This provides a natural explanation for the accelerated expansion of the universe.

Thus, ECM’s apparent mass concept is not only logically sound but also bridges classical mechanics with modern cosmological observations. 

Consistency of Effective Mass and Gravitating Mass in an Extended Classical Mechanics System:

February 04, 2025

In a system:

The effective mass (Mᵉᶠᶠ) is defined as the sum of the matter mass (Mᴍ) and the negative apparent mass (−Mᵃᵖᵖ). The matter mass itself consists of the ordinary matter mass (Mᴏʀᴅ) and the mass of dark matter (Mᴅᴍ). Consequently, the effective mass is equivalent to the gravitating mass (Mɢ).

The effective mass remains positive (Mᵉᶠᶠ>0) when the absolute magnitude of the matter mass |Mᴍ| exceeds the absolute magnitude of the negative apparent mass |−Mᵃᵖᵖ|. Conversely, the effective mass becomes negative (Mᵉᶠᶠ<0) when the absolute magnitude of the matter mass is less than the absolute magnitude of the negative apparent mass.

Similarly, the gravitating mass follows the same conditions as the effective mass, remaining positive (Mɢ>0) when the absolute magnitude of the matter mass is greater than the absolute magnitude of the negative apparent mass and becoming negative (Mɢ<0) when the absolute magnitude of the matter mass is smaller than the absolute magnitude of the negative apparent mass.

Additionally, the negative apparent mass can be expressed as the difference between the matter mass and the effective mass. Since the effective mass is equivalent to the gravitating mass, the negative apparent mass can also be described as the difference between the matter mass and the gravitating mass.

In ECM Systems:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ), where Mᴍ = Mᴏʀᴅ + Mᴅᴍ

Therefore,  Mᵉᶠᶠ = Mɢ 

And the relationships are:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ), where Mᵉᶠᶠ > 0 

when |Mᴍ| > |−Mᵃᵖᵖ| and Mᵉᶠᶠ < 0 when |Mᴍ| < |−Mᵃᵖᵖ|

Mɢ = Mᴍ + (−Mᵃᵖᵖ), where Mɢ > 0 

when |Mᴍ| > |−Mᵃᵖᵖ| and Mɢ < 0 when |Mᴍ| < |−Mᵃᵖᵖ|

Mᵃᵖᵖ = Mᴍ − Mᵉᶠᶠ

Mᵃᵖᵖ = Mᴍ − Mɢ