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.

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

Soumendra Nath Thakur, Tagore's Electronic Lab, India

February 07, 2025

Abstract

This study explores the extended dynamics of photon motion, challenging the conventional view that light speed (c) is always constant. We propose that at an infinitesimally small scale, a photon transitions from rest to c due to an inherent acceleration phase governed by its negative apparent mass. This acceleration generates a self-exerted force, distinct from external interactions, allowing the photon to maintain equilibrium while escaping gravitational influence. Furthermore, we analyse the continuous nature of photon frequency, distinguishing it from discrete digital signals. Unlike step-like binary transitions, a photon's wave packet exhibits a smooth, incremental frequency pattern, implying alternating cycles of acceleration and deceleration within its propagation. These insights suggest that photon motion and frequency dynamics involve fundamental, phase-dependent changes at quantum and relativistic scales.

Keywords: Photon Dynamics, Negative Apparent Mass, Acceleration Phase, Transmission Delay, Continuous Frequency, Wave-Particle Duality

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.

Phases of Motion Explanation: Transition from Rest to Constant Velocity

February 07, 2025

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 v = 0.

The idea is mathematically presented here:

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.