3. Mathematical Form of Apparent Mass (Mᵃᵖᵖ) in ECM:

Soumendra Nath Thakur

February 05, 2025

1. Fundamental Relationship in ECM

In ECM, apparent mass is defined as a function of effective mass and matter mass: 

Mᵃᵖᵖ = Mᴍ − Mᵉᶠᶠ,

where:

Mᴍ = Mᴏʀᴅ + Mᴅᴍ (matter mass, including baryonic and dark matter contributions)

Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ) (effective mass in ECM)

Rearranging, we obtain: 

Mᵃᵖᵖ = −(Mᵉᶠᶠ − Mᴍ)

Thus, Mᵃᵖᵖ is directly dependent on the difference between effective mass and matter mass.

2. Variables Determining Mᵃᵖᵖ

The apparent mass is not constant; it depends on:

• Gravitational Field Strength (g): The interaction between gravitational sources affects Mᵃᵖᵖ, especially in strong fields or at large cosmic scales.

• Velocity (v): In high-velocity regimes, relativistic effects influence the contribution of Mᵃᵖᵖ.

• Distance (r): Changes in Mᵃᵖᵖ occur over different distance scales, particularly in intergalactic interactions.

• Time (t): Dynamic variations in mass distributions or cosmic expansion affect Mᵃᵖᵖ over time.

Thus, we express it as:

Mᵃᵖᵖ = f(g, v, r, t)

3. Differential Form: How Mᵃᵖᵖ Evolves Over Time and Distance

Given that ECM introduces effective acceleration (aᵉᶠᶠ) instead of classical acceleration, we relate Mᵃᵖᵖ to forces via:

F = Mᵉᶠᶠ⋅aᵉᶠᶠ 

Since: 

Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ 

then differentiating with respect to time:

dMᵃᵖᵖ/dt = − dMᵉᶠᶠ/dt + dMᴍ/dt

For systems where matter mass is approximately conserved (dMᴍ/dt ≈ 0):

dMᵃᵖᵖ/dt = − dMᵉᶠᶠ/dt 

Thus, changes in apparent mass mirror variations in effective mass.

For spatial dependence, we express:

dMᵃᵖᵖ/dr = − dMᵉᶠᶠ/dr 

which suggests that in regions of strong gravitational influence, Mᵃᵖᵖ changes significantly.

4. Approximate Equation for Mᵃᵖᵖ

Mᵃᵖᵖ = −α(GMᴍ/c²r)

where α is a proportionality factor that depends on the local gravitational potential. This form ensures that Mᵃᵖᵖ is:

• Negligible in weak fields (e.g., near Earth)

• Significant at interstellar and intergalactic scales

Conclusion

• The mathematical form of Mᵃᵖᵖ depends on effective mass, gravitational field, velocity, and distance.

• Time evolution of Mᵃᵖᵖ is tied to changes in effective mass.

• Spatial variations suggest that Mᵃᵖᵖ increases in strong gravitational fields and at cosmic scales.

• A working equation for Mᵃᵖᵖ involves gravitational potential, ensuring consistency with ECM principles.

2. Summary of ECM Mass Concepts:

Soumendra Nath Thakur

February 05, 2025

Mᴏʀᴅ (Ordinary/Baryonic Matter):  This is the mass of protons, neutrons, and electrons.  It's the "normal" matter we're familiar with.

Mᴅᴍ (Dark Matter): This is non-luminous matter that interacts gravitationally but not through electromagnetic forces.  It's included in the total matter mass.

Mᴍ (Total Matter Mass): Mᴍ = Mᴏʀᴅ + Mᴅᴍ.  This is the total mass of ordinary and dark matter within a system.

Mᴅᴇ (Dark Energy Effective Mass): This represents the effective mass contribution from dark energy.  It's important to note that this is not dark energy itself, but its effect on mass.

Mᵃᵖᵖ (Apparent Mass): This is a dynamic, non-physical quantity that reflects observed mass variations due to external forces. It can be negative.

Mᵉᶠᶠ (Effective Mass): Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ = Mᴍ + Mᴅᴇ. This is the mass that governs gravitational interactions in ECM.

Mɢ (Gravitating Mass): Mɢ = Mᵉᶠᶠ. This is the mass that appears in the gravitational force equation in ECM.

Key Points and Clarifications:

Mᴅᴇ vs. Dark Energy: Mᴅᴇ is the effective mass contribution from dark energy, not dark energy itself. This distinction is crucial.

Apparent Mass as a Dynamic Term: Mᵃᵖᵖ is a dynamic and non-physical term. It reflects the observed mass variations, not a change in the actual physical mass.

Kinetic Energy in ECM: The equation (1/2) Mᴏʀᴅ⋅v² + (1/2)Mᴅᴍ⋅v² = −Mᵃᵖᵖ shows how the kinetic energy of ordinary and dark matter is related to the apparent mass.

1. Interpretation of Various Masses in Extended Classical Mechanics (ECM):

Soumendra Nath Thakur

February 05, 2025

The Physical basis for associating a separate mass term with dark energy in the Kinetic energy equation.

Classical Mechanics Interpretation of Inertial Mass (m):

• In classical mechanics, inertial mass (m) is strictly associated with physical mass (Mᴏʀᴅ), which represents baryonic matter.

• The kinetic energy (KE) equation in classical mechanics assumes that the mass term corresponds directly to inertial mass, reinforcing the idea that energy is fundamentally linked to the motion of physically present mass.

• The total energy in classical mechanics is given by: Eₜₒₜₐₗ = PE + KE where PE is associated with mass m, but the interpretation of the mass term in KE remains under scrutiny in ECM.

Extended Classical Mechanics (ECM) Interpretation of Various Masses:

• ECM introduces effective mass (Mᵉᶠᶠ) as a key factor, incorporating both physical matter and apparent mass effects: Mᵉᶠᶠ = Mᴍ+ (−Mᵃᵖᵖ) where:

• Matter mass (Mᴍ) = Mᴏʀᴅ + Mᴅᴍ (includes baryonic matter and dark matter)

• Apparent mass (Mᵃᵖᵖ) is a dynamic quantity that can take negative values based on external influences.

• The force equations in ECM are modified accordingly: F = Mᵉᶠᶠ·aᵉᶠᶠ = (Mᴍ −Mᵃᵖᵖ)·aᵉᶠᶠ and F𝑔 = G·Mᵉᶠᶠ·mₘ/r² = G·Mɢ·mₘ/r², where Mɢ = Mᵉᶠᶠ represents the gravitating mass in ECM.

Galactic Cluster Interpretation of Matter Mass:

• Effective mass of dark energy (Mᴅᴇ) arises from gravitational interactions at intergalactic scales, distinct from dark matter.

• Matter mass within a galactic cluster follows: Mᴍ = Mᴏʀᴅ + Mᴅᴍ

• Observationally, the total gravitating mass in a galactic cluster is found to be: 

Mɢ = Mᴍ + Mᴅᴇ

Clarifying the Physical Interpretation of Apparent Mass (Mᵃᵖᵖ):

• Apparent mass (Mᵃᵖᵖ) is a non-physical, dynamic term reflecting mass variations due to external forces.

• When effective mass has a significant negative component, the observed mass can appear reduced, resulting in negative apparent mass (Mᵃᵖᵖ < 0).

• This effect is particularly noticeable under extreme conditions, such as high velocities or strong gravitational fields.

The Physical Interpretation of the KE Equation in ECM:

• The ECM kinetic energy equation modifies the classical form by incorporating Mᴏʀᴅ and Mᴅᴍ explicitly: (1/2) Mᴏʀᴅ⋅v² + (1/2) Mᴅᴍ⋅v² = −Mᵃᵖᵖ

• Here, −Mᵃᵖᵖ has no direct physicality, but its presence affects the system's overall energy dynamics.

Conclusion:

ECM expands upon classical mechanics by refining mass-energy relationships, introducing apparent mass (Mᵃᵖᵖ) and effective mass (Mᵉᶠᶠ) to explain observed gravitational phenomena. The inclusion of a separate mass term for dark energy (Mᴅᴇ) is justified through its interaction with inertial mass at intergalactic scales, aligning with gravitational observations. While ECM's KE equation diverges from classical interpretations, it maintains mathematical consistency and observational validity.

The Primacy of Logic, Unbiased Scrutiny, and Mathematical Consistency in Science

February 01, 2025

Science should always be about logical reasoning, unbiased scrutiny, and natural justice in the pursuit of truth. Critical thinking and mathematical consistency must take precedence over dogmatic adherence to established theories.

Extended Classical Mechanics vs. Relativity: A Superior Framework:

Soumendra Nath Thakur 

January 31, 2025

The concept of negative apparent mass in extended classical mechanics is a groundbreaking innovation. It marks a turning point in classical mechanics, introducing negative mass and expanding its capabilities beyond traditional frameworks. This extension enhances classical mechanics, making it more powerful than relativistic mechanics.

Furthermore, velocity-induced relativistic Lorentz's transformations are flawed because they neglect classical acceleration between the rest and moving frames. They also overlook material stiffness in calculations, relying solely on the speed of light as the defining dynamic factor. For these reasons, extended classical mechanics stands as a far superior framework compared to the flawed foundations of relativistic mechanics.