Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
20-09-2024
Description of the Different Mass Terms and Their Relationships:
1. Normal Mass (M)
Represents the mass of normal baryonic matter, including particles like protons, neutrons, and electrons.
It is a component of the total Matter Mass (Mᴍ) and combines with the mass of dark matter.
Normal Mass contributes directly to gravitational interactions and forms stars, planets, and other visible structures.
2. Mass of Dark Matter (Mᴅᴍ)
The mass component associated with dark matter, an unseen form of matter that exerts gravitational effects without emitting detectable light or energy.
It combines with Normal Mass to form the total Matter Mass:
Mᴍ = M + Mᴅᴍ
Ref. Robert H. Sanders et al. (2002) - "Modified Newtonian Dynamics as an Alternative to Dark Matter"
Dark Matter is crucial for explaining the gravitational dynamics of galaxies and clusters beyond what visible matter accounts for.
3. Matter Mass (Mᴍ)
The sum of normal baryonic mass and dark matter mass, representing the total mass of a system excluding dark energy or apparent mass contributions.
Mᴍ = M + Mᴅᴍ
It contributes to Gravitating Mass and Effective Mass when combined with Apparent Mass.
Matter Mass plays a primary role in the gravitational dynamics of systems, influencing gravitational fields as an observable and calculable mass.
4. Apparent Mass (−Mᵃᵖᵖ)
A novel concept introduced as a negative mass component that modifies the effective gravitational mass of a system.
It affects Gravitating Mass:
Mɢ = Mᴍ + (−Mᵃᵖᵖ)
Ref. Thakur, S. N. Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics. Preprints.org (MDPI).
Contributes to Effective Mass:
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)
Apparent Mass represents a theoretical adjustment to classical mass calculations, applicable within gravitationally bound systems and aligning with the effects of dark energy, suggesting complex gravitational interactions.
5. Effective Mass (Mᵉᶠᶠ)
The adjusted mass accounting for both Matter Mass and Apparent Mass, reflecting the total mass influencing the system's gravitational behaviour.
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)
Effective Mass encapsulates the total gravitational effect, including influences from negative mass components, potentially explaining phenomena like the universe's accelerated expansion.
6. Gravitating Mass (Mɢ)
The overall mass that governs gravitational interactions within a system, incorporating Matter Mass and influences from dark matter and dark energy.
Related to Matter Mass and Apparent Mass:
Mɢ = Mᴍ + (−Mᵃᵖᵖ)
Equivalently defined as Effective Mass:
Mɢ = Mᵉᶠᶠ
Gravitating Mass defines the net gravitational pull exerted by a system, integrating all known and theoretical mass contributions.
Relationships and Implications
These relationships provide a comprehensive framework for understanding how different mass components interact within gravitationally bound systems, particularly with dark energy interpreted as negative Apparent Mass. They suggest rethinking traditional concepts of mass and gravity, impacting theoretical physics and observational cosmology. Integrating Apparent Mass into classical mechanics offers a path to reconcile observed cosmic phenomena, such as galaxy cluster behaviour, with a modified view of gravitational dynamics.
References:
1. Sanders, R. H., & McGaugh, S. S. (2002). Modified Newtonian dynamics as an alternative to dark matter. Annual Review of Astronomy and Astrophysics, 40(1), 263–317. https://doi.org/10.1146/annurev.astro.40.060401.093923:
This study explores Modified Newtonian Dynamics (MOND) as an alternative to dark matter, providing a framework to explain gravitational effects typically attributed to unseen mass.
2. Chernin, A. D., Bisnovatyi-Kogan, G. S., Teerikorpi, P., Valtonen, M. J., Byrd, G. G., & Merafina, M. (2013). Dark energy and the structure of the Coma cluster of galaxies. Astronomy and Astrophysics, 553, A101. https://doi.org/10.1051/0004-6361/201220781:
This paper examines the role of dark energy in shaping galaxy clusters, highlighting its influence on cosmic dynamics and contributing to understanding effective mass.
3. Thakur, S. N. (2024). Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics. Preprints.org (MDPI). https://doi.org/10.20944/preprints202409.1190.v2:
This research introduces new mass concepts, such as Apparent Mass, challenging traditional gravitational theory by redefining mass dynamics in the context of dark matter and dark energy.
List of Mathemetical Terms (Vol-2):
• aᵉᶠᶠ: Effective acceleration, modified by the interaction between matter mass and apparent mass.
• a₀: Fundamental acceleration constant in Modified Newtonian Dynamics (MOND), approximately 1.2 × 10⁻¹⁰ m/s².
• aᴍᴏɴᴅ: Acceleration of an object.
• Eᴅᴇ: Total energy associated with dark energy within a given volume.
• f(r/r₀): A function modifying the gravitational force at large distances, dependent on the ratio of r to r₀.
• F: Force, acting on a mass in the context of gravitational dynamics or, modified to incorporate apparent mass and effective acceleration.
• Fᴜₙᵢᵥ: Universal force acting on the universe’s mass, involving effective mass and acceleration on cosmic scales.
• Fɢ: Gravitational force between two masses, accounting for effective mass.
• G: Gravitational constant, representing the strength of the gravitational interaction.
• Mᵃᵖᵖ: Apparent mass, a negative mass component affecting effective mass.
• Mᴅᴇ: Dark energy effective mass, interpreted as equivalent to negative apparent mass.
• Mᴅᴍ: Dark matter mass in a gravitationally bound system.
• m: Mass of an object experiencing the force.
• M: Mass of, normal (baryonic) matter or, the source (e.g., a galaxy or gravitational source).
• Mᴍ: Matter mass, including both normal (baryonic) matter and dark matter.
• Mᵉᶠᶠ: Mechanical effective matter mass, combining matter mass and apparent mass.
• M₂: Secondary mass, the mass of another object in gravitational calculations.
• Mɢ: Gravitating mass, the total effective mass influencing gravitational dynamics.
• PE: Potential energy, dependent on the effective mass of the system in a gravitational field.
• r: Distance, the separation between two masses in gravitational force equations.
• r₀: Fundamental distance scale often used in modified gravitational theories.
• Tully–Fisher Relation: An empirical relation that connects the asymptotic rotational velocity of galaxies to their total mass, often observed as vᴍᴏɴᴅ⁴ = GMa₀.
• vᴍᴏɴᴅ: Asymptotic orbital velocity of a mass within a gravitational system, such as a star in a galaxy.
• μ(a/a₀): A function defining the transition between Newtonian and modified dynamics in MOND, dependent on the ratio of a to a₀.
• ρᴅᴇ: Dark energy density, the density of dark energy in the universe.
• ρᴍ: Matter mass density, the density of matter within a given volume.
The above mentioned terms can be broadly categorized into:
Mass-related terms:
• M (normal matter)
• Mᴅᴍ (dark matter)
• Mᴍ (matter mass)
• Mᵃᵖᵖ (apparent mass)
• Mᴅᴇ (dark energy effective mass)
• Mᵉᶠᶠ (mechanical effective matter mass)
• Mɢ (gravitating mass)
Force and acceleration terms:
• F (force)
• Fᴜₙᵢᵥ (universal force)
• Fɢ (gravitational force)
• aᵉᶠᶠ (effective acceleration)
• a₀ (fundamental acceleration constant)
• aᴍᴏɴᴅ (acceleration of an object)
Energy and density terms:
• Eᴅᴇ (total energy associated with dark energy)
• PE (potential energy)
• ρᴅᴇ (dark energy density)
• ρᴍ (matter mass density)
Distance and velocity terms:
• r (distance)
• r₀ (fundamental distance scale)
• vᴍᴏɴᴅ (asymptotic orbital velocity)
Functions and relations:
• f(r/r₀) (function modifying gravitational force)
• μ(a/a₀) (function defining transition between Newtonian and modified dynamics)
• Tully-Fisher Relation (empirical relation connecting rotational velocity to total mass)
This list provides a solid foundation for understanding the mathematical framework of Extended Classical Mechanics and its application to gravitational dynamics, dark matter, and dark energy.
