Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
23-08-2024
This fundamental equation directly relates effective mass to gravitating mass and matter mass.• Mᵉᶠᶠ = Mɢ − Mᴍ
Analysis: This equation relates the effective mass Mᵉᶠᶠ to the gravitating mass Mɢ and matter mass Mᴍ. It implies that the effective mass is the difference between the total gravitating mass and the matter mass.
Consistency: Mathematically consistent. It aligns with the concept that effective mass can incorporate additional factors (like dark energy) affecting gravitational dynamics. This equation is foundational and coherent with the concept of effective mass.
• KE = 1/2·|Mᵉᶠᶠ|·v²
This equation shows how kinetic energy can be positive despite a negative effective mass.
Analysis: This shows kinetic energy KE for a system with negative effective mass. The magnitude of effective mass is used to ensure kinetic energy remains positive.
Consistency: Physically and mathematically consistent. Kinetic energy is positive as it depends on the magnitude of mass and velocity squared, regardless of the sign of the effective mass.
• F = G·(Mᴍ₁ + Mᴍ₁ᵉᶠᶠ)·(Mᴍ₂ + Mᴍ₂ᵉᶠᶠ)/r²
This equation incorporates both matter mass and negative effective mass into gravitational force calculations.
Analysis: This extends the classical gravitational force equation to include negative effective mass.
Consistency: Mathematically consistent with classical mechanics, but physically unconventional. Incorporating negative effective mass introduces repulsive gravitational effects, aligning with some theoretical models but deviating from traditional positive mass assumptions.
• Fʀₑₚ = −G·Mᵉᶠᶠ·Mᴏₜₕₑᵣ/r²
This equation represents the repulsive force associated with negative effective mass.
Analysis: This represents the repulsive force due to negative effective mass.
Consistency: Mathematically consistent. It reflects the potential for negative effective mass to create repulsive forces, similar to dark energy effects. Physically, it aligns with the hypothesis of repulsive gravity due to negative effective mass.
• Eᴛₒₜ = (PEᴍₘ + PEᴍₘᵉᶠᶠ) + KE
This provides the total energy expression, including potential and kinetic energies.
Analysis: This equation expresses total energy as the sum of potential and kinetic energies.
Consistency: Mathematically and physically consistent. It correctly includes both potential and kinetic energy components and aligns with energy conservation principles.
• PE = Eᴛₒₜ - KE, PE = PEᴍₘ + PEᴍₘᵉᶠᶠ
This equation helps understand potential energy in relation to total energy and kinetic energy.
Analysis: This decomposes potential energy from the total energy minus kinetic energy and accounts for potential energy contributions from both matter and effective masses.
Consistency: Mathematically and physically consistent. It is a correct representation of potential energy, adhering to the total energy equation.
• KE = 1/2 Mᴍv²
This classical equation for kinetic energy is important for understanding how kinetic energy depends on matter mass and velocity.
Analysis: This is the classical equation for kinetic energy based on matter mass and velocity.
Consistency: Mathematically consistent. It is a standard formula for kinetic energy and is consistent with the principles of classical mechanics.
• Mᵉᶠᶠ ∝ −1/Mᴍ
This equation highlights the inverse relationship between effective mass and matter mass.
Analysis: This equation indicates an inverse proportional relationship between effective mass and matter mass.
Consistency: Mathematically consistent. It reflects the concept that as matter mass increases, effective mass decreases (negatively). This is a valid relationship given the context of negative effective mass.
• aᵉᶠᶠ = a−1/a
This equation describes how effective acceleration relates to acceleration in the presence of negative effective mass.
Analysis: This describes the effective acceleration in relation to the conventional acceleration in the presence of negative effective mass.
Consistency: Physically unconventional. It introduces a reciprocal relationship, which may not be intuitive. Careful interpretation is required to ensure this aligns with physical behaviour under negative effective mass conditions.
• F = Mᴍ·a
This classical force equation is crucial for understanding the relationship between force, matter mass, and acceleration.
Analysis: This is the classical force equation, showing force as the product of matter mass and acceleration.
Consistency: Mathematically consistent. It is a fundamental equation in classical mechanics and is relevant for understanding force in the context of matter mass.
• Summary of Consistency:
Mathematically: The equations are generally consistent with the principles of classical mechanics, with modifications to incorporate negative effective mass. They adhere to the laws of energy, force, and acceleration.
Physically: The introduction of negative effective mass leads to unconventional results such as repulsive forces and non-intuitive energy relationships. These results are consistent with theoretical models involving dark energy and negative mass but require careful physical interpretation to align with observed phenomena and established principles.
Overall, while the equations are mathematically consistent, physical interpretation requires careful consideration of how negative effective mass affects conventional mechanics.
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