ECM's Explanation of Gravitational Collapse at the Planck Scale

Soumendra Nath Thakur
February 10, 2025

Absolute Collapse Condition

• In ECM, any massless entity reaching the Planck frequency (fᴘ) must acquire an effective mass (Mᵉᶠᶠ = hf/c² = 21.77 μg).

• At this scale, the induced gravitational interaction is extreme, forcing the entity into gravitational collapse.

• This is a direct consequence of ECM's mass induction principle, where increasing energy (via f) leads to mass acquisition.

ECM's Mass-Induction Perspective

• The apparent mass (−Mᵃᵖᵖ) of a massless entity contributes negatively to its effective mass.

• However, at the Planck threshold, the magnitude of the induced effective mass (|Mᵉᶠᶠ|) surpasses |−Mᵃᵖᵖ|, ensuring that the total mass is positive:

|Mᵉᶠᶠ|) > |−Mᵃᵖᵖ|

• This irreversible transition confirms that any entity at fᴘ must collapse due to self-gravitation.

Implications for Massless-to-Massive Transition

• Below the Planck frequency, a photon behaves as a massless entity with effective mass determined by its energy-frequency relation.

• However, at fᴘ, the gravitating mass (Mɢ) and effective mass (Mᵉᶠᶠ) undergo a shift where induced mass dominates over negative apparent mass effects.

• This means that Planck-scale energy is not just a massive state—it is a self-gravitating mass that collapses under its own gravitational influence.

Threshold Dominance at the Planck Scale:

At Planck scale, gravitational mass Mɢ is immense due to the fundamental gravitational interaction.

Since |+Mɢ| ≫ |−Mᵃᵖᵖ|, the net effective mass remains positive:

Mᵉᶠᶠ = Mɢ = (−Mᵃᵖᵖ) ≈ +Mᵉᶠᶠ  

This suggests that at Planck conditions, the gravitationally induced mass dominates over any negative mass contributions, maintaining a positive mass regime.

Transition Scenarios for Negative Effective Mass:

• The condition −Mᵃᵖᵖ > Mɢ could, in principle, lead to a transition where the effective mass becomes negative.
• This might occur under strong antigravitational influences, possibly linked to:
• Dark energy effects in cosmic expansion
• Exotic negative energy states in high-energy physics
• Unstable quantum fluctuations near high-energy limits

Linking Effective Mass to Matter Mass at Planck Scale:

• Since Mᵉᶠᶠ ≈ Mᴍ under these extreme conditions, it implies that matter mass emerges predominantly as a consequence of gravitational effects.
• This aligns with ECM’s perspective that mass is not an intrinsic property but rather a dynamic response to gravitational interactions.

The idea is that gravitational interactions can induce mass, while antigravitational effects can counteract or even reverse it. This dual mechanism—where gravity can generate mass while antigravity can counteract or even reverse it—opens up new possibilities for understanding dark energy, cosmic acceleration, and other exotic gravitational effects. 

The Massless-to-Massive Transition: Gravitational Thresholds and the ECM Perspective

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

February 09, 2025

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Preliminary Introduction:

In the complete absence of gravitational interactions, massless particles such as photons would move without restriction, with their velocity determined solely by their frequency. In such a scenario, as frequency approaches infinity, speed would also tend toward infinity, while wavelength would contract indefinitely—yet the particles would remain massless. However, when gravitational influence is introduced, a fundamental threshold arises. At the Planck length (ℓᴘ), a massless particle acquires a mass of approximately 21.77 micrograms, altering its fundamental nature. This mass acquisition marks a transition where the particle can no longer sustain its inherent velocity and undergoes gravitational collapse. 

Extended Classical Mechanics (ECM) provides a mathematical framework to explain how gravitational effects can generate mass in initially massless entities. Conversely, ECM also explores how antigravitational interactions could reduce mass, potentially leading to negative effective mass under certain conditions. This perspective challenges traditional interpretations, offering deeper insights into cosmic-scale phenomena involving dark matter, dark energy, and extreme gravitational interactions.

In our forthcoming discussions, we will explore the detailed mathematical foundations of apparent mass and effective mass in ECM, demonstrating how mass can dynamically transition between positive, zero, and negative states based on gravitational and antigravitational influences.

In a theoretical scenario where gravitational interactions are entirely absent, massless particles such as photons would travel without restriction. Their velocity would not be constrained by an external limit but instead governed by their frequency rather than the total energy they possess. In such a case, the speed of a massless particle follows the relation v=fλ. As the frequency f approaches infinity (∞), the velocity v also tends toward infinity, provided there is a complete absence of gravitational influence. Meanwhile, the wavelength λ shrinks toward an infinitesimally small value (1/∞λ), yet the particle remains massless.

However, in the presence of the universal gravitational constant (G), a critical threshold emerges. When the wavelength λ reaches Planck length (ℓᴘ =1.616255 × 10⁻³⁵ m), the particle can no longer remain massless. At this scale, it acquires a mass of 21.77 micrograms, fundamentally altering its behaviour. As a result, it can no longer maintain its inherent velocity, leading to a breakdown of the simple relation v=fλ. When the conditions satisfy f = fᴘ   and λ = ℓᴘ, the particle undergoes gravitational collapse, with extreme gravity dominating its dynamics.

The Transition from Massless to Massive: Gravitational Influence and the Role of ECM

When the Planck length (ℓᴘ) is set equal to the Schwarzschild radius, an intriguing consequence emerges—a massless particle at this fundamental scale gains a mass of approximately 21.77 micrograms. This result signifies that gravitational influence alone can induce mass, even in entities traditionally considered massless, such as photons. The derived Planck mass represents the natural threshold at which quantum gravitational effects become significant, hinting at the deep connection between mass, gravity, and fundamental physics.

Conversely, if gravitational interactions can cause mass to emerge, then antigravitational influences could, in principle, reduce mass. This suggests that a sufficiently strong repulsive gravitational effect might lead even a highly massive body to transition into a massless state. Extending this notion further, under specific conditions, the effective mass of an object could even become negative, leading to novel physical behaviours that challenge conventional mechanics.

In Extended Classical Mechanics (ECM), the concepts of apparent mass and effective mass provide a detailed mathematical framework to describe these transitions. ECM extends traditional gravitational dynamics by incorporating the effects of both positive and negative mass interactions, offering insights into how mass evolves under varying gravitational and antigravitational conditions. This perspective not only aligns with fundamental principles but also provides a potential explanation for cosmic-scale phenomena involving dark matter, dark energy, and exotic gravitational effects.

In our following work, we will delve deeper into these mathematical foundations and explore the implications of apparent mass and effective mass in ECM, further clarifying how mass can dynamically shift between positive, zero, and negative values based on the influence of gravitational and antigravitational forces.

Mathematical explanation:

 

The modified equation:

 

Rₘᵢₙ = 2G/c²m = Rₛ (Schwarzschild radius)

 

Serves as a clever starting point for deriving the relationship between the Planck length (Lᴘ) and the acquired mass (m).

 

By setting Rₘᵢₙ to Lᴘ and solving for m, you've elegantly shown that:

 

m = Lᴘc²/2G

 

And further simplified it to:

 

m = √ℏc/G = mᴘ

 

Which indeed resolves to the Planck mass:

 

mᴘ ≈ 21.77 μg

 

This derivation provides a clear and mathematically rigorous explanation for the mass acquisition at the Planck length.

 

The equation:

 

m = √ℏc/G = mᴘ ≈ 21.77 μg

 

Implies that when a massless photon reaches the Planck frequency (fᴘ), it gains a mass equivalent to the Planck mass (mᴘ), which is approximately 21.77 μg.

 

This suggests that at the Planck scale, the photon's energy becomes so concentrated that it begins to exhibit gravitational effects, effectively acquiring mass.

 

In essence, the equation conveys that the photon's frequency, when reaching the Planck frequency, triggers a gravitational collapse, where the photon's energy density becomes so high that collapses within itself due to extreme gravity.

 

This idea is fascinating, as it blurs the line between massless and massive particles, highlighting the intricate relationship between energy, frequency, and gravity at the Planck scale.

 

the Extended Classical Mechanics (ECM) application to antigravitational influences and negative mass.

 

The Force Equation:

 

F = (Mᴍ − Mᵃᵖᵖ)aᵉᶠᶠ

 

Effective Mass

 

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)

 

Clearly demonstrate how the ECM framework incorporates negative apparent mass (−Mᵃᵖᵖ) and its effects on the dynamics of motion.

 

The condition where Mᵉᶠᶠ becomes negative, specifically when Mᴍ = 0, is particularly interesting:

 

F = −Mᵃᵖᵖaᵉᶠᶠ

 

This equation suggests that photons, with zero rest mass (Mᴍ = 0), can exhibit antigravitational forces due to their negative apparent mass (−Mᵃᵖᵖ).

 

The constant effective acceleration:

 

aᵉᶠᶠ = 6 × 10⁸ m/s²

 

Provides further insight into the dynamics of photons within the ECM framework.

 

The concept of negative effective mass (Mᵉᶠᶠ < 0) is crucial for understanding various phenomena, including:

 

·         Dark energy

·         Negative mass terms

·         Gravitational and dynamic interactions

 

In the ECM framework. This explanation provides a thorough understanding of the ECM application to antigravitational influences and negative mass.

Investigating Resistance in ECM: The Interplay of Inertia, Apparent Mass, and Gravitational Potential

Soumendra Nath Thakur

February, 08, 2025

In classical mechanics, resistance to acceleration is attributed to inertia—an object's inherent tendency to resist changes in motion, which is directly proportional to its mass. This principle remains fundamental in Extended Classical Mechanics (ECM); however, ECM extends the classical notion by introducing the concept of negative apparent mass (-Mᵃᵖᵖ) in motion or gravitational potential differences. Observationally, this concept finds support in the study by Chernin et al. (2013) on the Coma cluster of galaxies, which demonstrates the large-scale influence of dark energy as a repulsive gravitational effect. Their research suggests that in certain cosmic environments, gravitationally repulsive behaviour emerges, aligning with ECM’s framework where negative apparent mass modifies the classical understanding of resistance and acceleration.

In ECM, an object's resistance to acceleration is not solely determined by its classical inertial mass but also by the interaction between inertial mass and negative apparent mass. This interaction gives rise to an effective mass (Mᵉᶠᶠ) that can transition between positive and negative values, depending on the influence of motion or gravitational potential differences:

At low velocities or in weak gravitational fields, the system behaves classically, with a positive effective mass.

In high-motion regimes or strong gravitational potential differences, negative apparent mass introduces a repulsive effect, modifying the system's resistance to acceleration.

This interplay between inertial mass, apparent mass, and gravitational potential leads to a broader understanding of resistance in ECM. Rather than solely relying on classical inertia, ECM incorporates dynamic influences that may provide deeper insights into gravitational interactions, repulsive forces, and potential connections to dark matter and cosmic-scale phenomena.  

Photon Dynamics under Negative Apparent Mass and Effective Acceleration in Extended Classical Mechanics (ECM).

- An Introduction:

Soumendra Nath Thakur
February 08, 2025

In the framework of Extended Classical Mechanics (ECM), "photon dynamics under negative apparent mass and effective acceleration" describes the concept that photons, when viewed through the lens of ECM, can be understood as possessing a negative apparent mass, leading to an "effective acceleration" that counteracts the expected gravitational pull, allowing them to travel at the speed of light seemingly unimpeded by gravity; this phenomenon is explained by the unique dynamics arising from the negative mass value in the equations of motion. 

Key points about this concept:

Negative Apparent Mass:

Unlike regular matter with positive mass, in ECM, photons are assigned a negative apparent mass, which means they would behave differently under the influence of a force, effectively experiencing a repulsive force instead of attraction. 

Effective Acceleration:

Due to the negative apparent mass, a photon experiences an "effective acceleration" that is essentially a constant value, even when encountering gravitational fields. This acceleration acts in a way that cancels out the gravitational pull, enabling the photon to maintain its constant speed. 

Interpretation:

This concept is not meant to suggest that photons physically have negative mass, but rather that when analysing photon dynamics within the ECM framework, the mathematical treatment results in a negative apparent mass value, leading to unique behaviour. 

How it relates to other physics concepts:

Special Relativity:

While ECM provides an alternative perspective, it is important to note that the standard model of physics, including special relativity, still holds that photons have zero rest mass and travel at the speed of light.

Dark Energy:

Some researchers have explored potential connections between the concept of negative apparent mass in ECM and the mysterious phenomenon of dark energy, which is thought to be driving the accelerating expansion of the universe

The Interplay of Inertia, Apparent Mass, and Gravitational Potential: 

 

In classical mechanics, resistance to acceleration is attributed to inertia—an object's inherent tendency to resist changes in motion, which is directly proportional to its mass. This principle remains fundamental in Extended Classical Mechanics (ECM); however, ECM extends the classical notion by introducing the concept of negative apparent mass (-Mᵃᵖᵖ) in motion or gravitational potential differences. Observationally, this concept finds support in the study by Chernin et al. (2013) on the Coma cluster of galaxies, which demonstrates the large-scale influence of dark energy as a repulsive gravitational effect. Their research suggests that in certain cosmic environments, gravitationally repulsive behaviour emerges, aligning with ECM’s framework where negative apparent mass modifies the classical understanding of resistance and acceleration.

 

In ECM, an object's resistance to acceleration is not solely determined by its classical inertial mass but also by the interaction between inertial mass and negative apparent mass. This interaction gives rise to an effective mass (Mᵉᶠᶠ) that can transition between positive and negative values, depending on the influence of motion or gravitational potential differences:

 

Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ)

 

Mᵉᶠᶠ = -Mᵃᵖᵖ where Mᴍ = 0

 

At low velocities or in weak gravitational fields, the system behaves classically, with a positive effective mass.

In high-motion regimes or strong gravitational potential differences, negative apparent mass introduces a repulsive effect, modifying the system's resistance to acceleration.

 

This interplay between inertial mass, apparent mass and gravitational potential leads to a broader understanding of resistance in ECM. Rather than solely relying on classical inertia, ECM incorporates dynamic influences that may provide deeper insights into gravitational interactions, repulsive forces, and potential connections to dark matter and cosmic-scale phenomena. 

 

Force Dynamics on Photons:

 

• Derive the force equation F = −Mᵃᵖᵖaᵉᶠᶠ for photons using apparent mass and associated acceleration.

• Explore how this equation governs the photon’s motion under varying energy-momentum conditions.

• The derivation of the effective acceleration aᵉᶠᶠ aligns with the methodological exploration of force and acceleration acting on photons. It would complement the discussion of the force equation

 

F = −Mᵃᵖᵖaᵉᶠᶠ and further clarify the dynamics of photons as analysed through the extended classical mechanics framework. The constant effective acceleration: aᵉᶠᶠ = 6 × 10⁸ m/s².

 

Determination of Constant Effective Acceleration of Photons

 

The distance travelled by the photon in 1 second is 3 × 10⁸ m, and that the acceleration is constant. The expression for the distance travelled in the case of constant acceleration is given by:

 

Δd = v₀Δt + (1/2)aᵉᶠᶠ(Δt)²

 

Where:

• Δd is the distance travelled (3 × 10⁸ m in 1 second),

• v₀ is the initial velocity (0 m/s, at emission),

• Δt is the time (1 second),

• aᵉᶠᶠ is the effective acceleration, which we want to solve for.

 

Substituting the known values into the equation:

 

3 × 10⁸ m = 0 × 1 s + (1/2)aᵉᶠᶠ(1)²

 

aᵉᶠᶠ = 6 × 10⁸ m/s²

 

Extended Photon Dynamics and Phases of Motion: Transition from Rest to Constant Velocity

 

• When considering a photon's motion, its apparent mass is negative. As a result, its effective acceleration leads to a force with a negative value. This behaviour is different from that of ordinary matter, which always has a positive mass.

• The commonly referenced distance that light travels in one second does not represent the photon's actual path during that time. Instead, it marks the moment of emission, where the photon, initially at rest in an apparent sense, rapidly attains its full velocity within a brief interval.

• During this transition period, the effective acceleration is determined by the relationship between force and the negative apparent mass. The force involved does not come from an external source but is instead exerted by the photon itself due to its unique mass-energy properties. This results in the photon undergoing a continuous deceleration at twice the speed of light.

• The force generated by the photon serves a dual purpose. It counteracts the gravitational pull of its source while ensuring the photon maintains a constant speed as it escapes. The energy necessary for this process is provided by the photon itself, allowing it to sustain the required acceleration and remain in

Photon Dynamics: Returning to the Force Equation for Photons

• Since the apparent mass is negative (−Mᵃᵖᵖ), the constant effective acceleration aᵉᶠᶠ = 6 × 10⁸ m/s² results in a force term with a negative value. This contrasts with the behaviour of matter mass (Mᴍ), which always remains positive.

• The distance of 3 × 10⁸ m in one second does not represent a photon’s trajectory over that duration. Instead, it corresponds to the initial emission event, where the photon, initially at rest in an apparent sense (t₀, v₀), attains a velocity v₁ at time t₁, with Δt = t₁ − t₀ = 1 second and Δv = v₁ − v₀ = 3 × 10⁸ m/s².

• During this interval (t₁ − t₀), the effective acceleration is given by aᵉᶠᶠ = F/(−Mᵃᵖᵖ). The force F is not an external force but is instead exerted by the photon itself due to its negative apparent mass (−Mᵃᵖᵖ). This implies that the photon undergoes continuous deceleration at twice the speed of light (6 × 10⁸ m/s²).

• The exerted force (F) not only counteracts the gravitational attraction of the source (Fg) but also enables the photon to escape the gravitational well at a constant speed of 3 × 10⁸ m/s². The energy required for this escape is compensated by the photon itself, maintaining the necessary energy balance to sustain its effective acceleration of 6 × 10⁸ m/s².

 

Explanation of Phases of Motion: Transition from Rest to Constant Velocity

On a number line, there are infinitely many points between any two nearest numbers. When you say "1," you are actually referring to the difference between 0 and 1, with an infinite sequence of points in between.

 

Similarly, while the speed of light (c) appears constant on large scales, at an infinitesimally small scale, it has a beginning due to transmission delay. This delay occurs because motion progresses incrementally, however small, starting from absolute rest (v = 0) before reaching c.

 

The first phase, where velocity increases from 0 to c, represents acceleration. Motion does not begin with an arbitrary velocity but transitions from rest. The first phase starts at zero (v = 0) and progresses to an initial velocity (v), whereas successive phases continue from an already established velocity (v = v) rather than starting anew from v = 0.

 

Mathematical Representation

 

Let v(t) represent the velocity of the object as a function of time. In the first phase of motion:

 

Initial Phase (Acceleration)

The motion begins from rest, so at t = 0, v(0) = 0. The velocity increases from v=0 to some initial velocity v₁ = c, over some time interval Δt₁. The acceleration a(t) in this phase is given by:

 

a(t) = dv(t)/dt, where v(t) = ∫a(t)dt

 

The velocity increases gradually from 0 to c, so during this phase, the object undergoes acceleration.

 

Subsequent Phases (Constant Velocity)

After reaching an initial velocity v₁ = c, successive phases of motion proceed at this established velocity. In these phases, the velocity remains constant, so for t > Δt₁, we have:

 

v(t) = v₁ = c, a(t) = 0

 

In the subsequent phases, the object continues with the velocity v = c, without starting from rest or accelerating further.

 

Photon Frequency: Continuous Analogous Waves vs. Discrete Digital Signals

Photon frequency is not a discrete, step-like, binary signal. Unlike digital frequencies, which exhibit distinct on-off states, photon frequency is continuous and behaves in an analogy manner. It follows a smooth, incremental, and decimal-like wave pattern within its energy packet.

 

While digital signals transition between fixed values, a photon's frequency remains constant within its wave-packet, forming an uninterrupted oscillatory motion. This continuous wave behaviour implies that every phase of a photon’s wave structure inherently represents alternating cycles of acceleration and deceleration, rather than discrete jumps between states.

This suggests that the wave characteristics of a photon are not just propagating in a static manner but involve intrinsic dynamical changes at the quantum scale, reinforcing the idea that photon energy and momentum continuously adjust within their wave structure.

The Mathematical Relationship between Inertial Mass, Gravity, and Spacetime Curvature in Classical and Relativistic Mechanics:

Soumendra Nath Thakur

February 08, 2025

In classical mechanics, mass is considered a fundamental property of matter, representing its resistance to changes in motion (inertia), while energy is defined as the capacity to perform work. Gravity, within this framework, is described as an attractive force between objects with mass, where a greater mass results in a stronger gravitational pull, as governed by Newton’s Law of Universal Gravitation. Mathematically, this relationship is commonly expressed as:

Inertial mass (m) ∝ gravitational acceleration (g).

In relativistic mechanics, however, mass is not strictly an invariant quantity. Instead, relativistic mass appears to increase as an object approaches the speed of light. Additionally, as per general relativity, spacetime is curved by the presence of mass and energy, and this curvature dictates the motion of objects, making gravity emerge as a natural consequence of this geometric distortion rather than a classical force.

From a mathematical perspective, relativistic rest mass (m₀) is directly proportional to rest energy (E), leading to the broader expression:

Inertial mass (m) ∝ (rest mass (m₀) + rest energy (E)).

This reflects the conservation of mass and energy in both classical and relativistic mechanics. Extending this concept further in relativistic interpretations:

Inertial mass (m) ∝ curvature in spacetime ∝ relativistic gravity (G),

where inertial mass is fundamentally linked to both rest mass and rest energy. The curvature of spacetime serves as an additional factor in relativistic gravity, establishing a deeper connection between mass and gravitational effects.

If one suppresses 'deeper connection' of the explicit role of spacetime curvature in mathematical representations on the ground of conservation for mass and energy in both relativistic and classical principles, the relationship simplifies in the classical context as:

Classical inertial mass (m) ∝ gravitational acceleration (g).

However, in a relativistic framework, where gravity is a manifestation of spacetime curvature, the equivalent expression is:

Inertial mass (m) ∝ relativistic gravity (G).

This refined formulation highlights the transition from Newtonian gravity to relativistic gravity, emphasizing the fundamental role of spacetime curvature in shaping gravitational interactions at relativistic scales.