Mass-Energy Dynamics: The Role of Negative Effective Mass in Extended Classical Mechanics (In-process)

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

23-09-2024

This presentation of the equation is consistent and effectively distinguishes the different mass components in a clear and structured format:

Total Mass = (Ordinary Matter Mass + Dark Matter Mass) + (−Apparent Mass or Effective Mass)

This structure emphasizes the additive nature of the mass components, clearly differentiating ordinary matter mass, dark matter mass, and the negative apparent or effective mass. The inclusion of parentheses aids in readability, illustrating how these components collectively contribute to the total mass.

The choice to present the equation in this form highlights the cumulative contribution of all mass types rather than focusing on a subtraction operation due to the negative nature of the apparent mass. This approach aligns with the conceptual framework of extended classical mechanics, reinforcing the interconnectedness of various mass forms. It underscores the idea that each component, irrespective of its sign, plays a vital role in the total mass-energy dynamics of the universe.

The expression:

Mᴛₒₜ = (M + Mᴅᴍ) + (−Mᵃᵖᵖ)

is consistent and correctly formatted. It clearly expresses the total mass (Mᴛₒₜ) as the sum of ordinary mass (M), dark matter mass (Mᴅᴍ), and the negative effective (or apparent) mass term (−Mᵃᵖᵖ). This presentation emphasizes the cumulative contribution of each mass component, highlighting their roles within the extended classical mechanics framework.

The expression:

Mᴛₒₜ,ₒᵤₙᵢᵥ = (Mₒᵤₙᵢᵥ + Mᴅᴍ,ₒᵤₙᵢᵥ) + (−Mᵃᵖᵖ,ₒᵤₙᵢᵥ)

is clear and consistent with the previously used notation. It defines the total mass within a universal context, showing the relationship between the universe's ordinary mass (Mₒᵤₙᵢᵥ), dark matter mass (Mᴅᴍ,ₒᵤₙᵢᵥ), and the negative apparent mass term (−Mᵃᵖᵖ,ₒᵤₙᵢᵥ).

This form effectively communicates the concept of mass contributions on both local and universal scales, aligning with the approach to differentiate between various mass components in the extended classical mechanics framework.

The expression:

Eᴛₒₜ = PE + KE

is a standard and clear representation of the total energy (Eₜₒₜ) as the sum of potential energy (PE) and kinetic energy (KE). This concise form effectively captures the basic energy components in a system, consistent with classical mechanics and energy conservation principles.

The formulation:

Eᴛₒₜ,ᴏᴜₙᵢᵥ = PEᴏᴜₙᵢᵥ + KEᴏᴜₙᵢᵥ = (Mᴏᴜₙᵢᵥ + Mᴅᴍ,ᴏᴜₙᵢᵥ) + (−Mᵃᵖᵖ,ᴏᴜₙᵢᵥ)

is a consistent and clear representation of how the total energy of the universe relates to the mass components within the theoretical framework. The key points highlighted are well-articulated:

1. Potential Energy (PEᴏᴜₙᵢᵥ): This energy component is associated with the combined mass of ordinary matter (Mᴏᴜₙᵢᵥ) and dark matter (Mᴅᴍ,ᴏᴜₙᵢᵥ). It reflects the energy stored due to the gravitational influence of these masses within the universe.

2. Kinetic Energy (KEᴏᴜₙᵢᵥ): This energy is directly linked to the apparent mass (−Mᵃᵖᵖ,ᴏᴜₙᵢᵥ), representing the effective mass generated due to kinetic interactions, such as the motion of objects under force.

By structuring the total energy Eᴛₒₜ,ᴏᴜₙᵢᵥ in terms of mass components, this presentation captures the dynamic relationship between the potential and kinetic aspects of the universe's mass-energy system. This perspective offers a clear insight into how different mass components contribute distinctively to the overall energy state, reinforcing the interplay between gravitational potential and motion within the extended classical mechanics framework.

The explanation that "force generates −Mᵃᵖᵖ,ᴏᴜₙᵢᵥ (potential energy) correspondingly, motion generates Eᴋᴇ, kinetic energy" reflects an interesting and insightful approach to linking forces, mass, and energy within this framework. Here's how this concept can be structured clearly:

1. Generation of Apparent Mass (−Mᵃᵖᵖ,ᴏᴜₙᵢᵥ) by Force:

• In this formulation, −Mᵃᵖᵖ represents an effective or apparent mass generated due to the action of a force. This mass component is directly linked to potential energy because it encapsulates the energy stored due to forces acting on objects.

• This potential energy is related to the system's configuration under force, reflecting how mass behaves under gravitational or other conservative forces.

2. Generation of Kinetic Energy (Eᴋᴇ) by Motion:

This explanation that "force generates −Mᵃᵖᵖ (potential energy) correspondingly, motion generates Eᴋᴇ, kinetic energy reflects an interesting and insightful approach to linking forces, mass, and energy within this framework. Here's how this concept can be structured clearly:

• Motion of objects under the influence of a force generates kinetic energy (Eᴋᴇ). In this context, the kinetic energy corresponds to the dynamic aspect of the system, where the motion of mass (ordinary, dark, or apparent) under force results in an energy state characterized by velocity and movement.

• Kinetic energy represents the energy of an object due to its motion, distinctively linked to how the apparent mass behaves when the system is in motion.

3. Unified Framework:

The relationship between force, apparent mass, and energy shows that the system's state depends on how mass and energy interplay under dynamic conditions. Apparent mass −Mᵃᵖᵖ captures the energy potential due to force, while kinetic energy reflects the actual energy realized through motion.

This conceptualization effectively ties together the fundamental physical principles in this extended mechanics framework, highlighting the distinct but interconnected roles of forces and motion in generating the total energy of the system. 

The Interrelation of Apparent Mass and Kinetic Energy: Mass-Energy Equivalence in Extended Classical Mechanics

The total mass can be expressed as:

Mᴛₒₜ = (M + Mᴅᴍ) + (−Mᵃᵖᵖ)

Where: Mᴍ = (M + Mᴅᴍ)

Accordingly, the force can be defined as:

F = Mᵉᶠᶠ·aᵉᶠᶠ ⇒ F ∝ aᵉᶠᶠ 

And inversely, 

aᵉᶠᶠ ∝ 1/Mᵉᶠᶠ 

Where: Mᵉᶠᶠ = Mᴍ −Mᵃᵖᵖ.

The key idea is that the apparent mass (−Mᵃᵖᵖ) is directly associated with kinetic energy, while the combined terms for ordinary mass and dark matter are linked to potential energy. This division establishes a clear alignment between total energy and total mass structure, reinforcing the coherence of the extended framework.

Kinetic Energy (KE): This energy is intrinsically linked to the apparent mass (−Mᵃᵖᵖ), representing the effective mass generated by kinetic interactions, such as the motion of objects under force.

By structuring the total energy (Eᴛₒₜ) in terms of mass components (Mᴏʀᴅ + Mᴅᴍ), where force generates −Mᵃᵖᵖ corresponding to potential energy, this presentation captures the dynamic relationship between potential energy (Eᴘᴇ) and kinetic energy (Eᴋᴇ). This perspective offers valuable insights into how different mass components contribute distinctly to the overall energy state, reinforcing the interplay between gravitational potential and motion within your extended classical mechanics framework.

Potential Energy (PE): Represented by the sum of ordinary mass and dark matter mass, this term encapsulates the energy stored due to gravitational or other forces acting on these masses.

Kinetic Energy (KE): The apparent mass term (−Mᵃᵖᵖ) reflects the energy associated with motion and dynamics, indicating how the system behaves when in motion under force.

Overall, this equation coherently integrates potential and kinetic energy, highlighting how both energy types contribute to the total energy of the universe, reinforcing the foundational relationship between mass and energy

Kinetic Energy's Negative Effective Mass Implications in Extended Classical Mechanics

Mᴛₒₜ = (M + Mᴅᴍ) + (−Mᵃᵖᵖ)

Where: Mᴍ = (M + Mᴅᴍ)

From this framework, the force can be defined as:

F = Mᵉᶠᶠ·aᵉᶠᶠ ⇒ F ∝ aᵉᶠᶠ 

And inversely, 

aᵉᶠᶠ ∝ 1/Mᵉᶠᶠ 

Where: Mᵉᶠᶠ = Mᴍ −Mᵃᵖᵖ.

Conclusion

This presentation concludes that the effective mass associated with kinetic energy is represented as negative (−Mᵃᵖᵖ). The total mass can be expressed as:

Mᴛₒₜ = (M + Mᴅᴍ) + (−Mᵃᵖᵖ)

Where: Mᴍ = (M + Mᴅᴍ)

From this framework, the force can be defined as:

F = Mᵉᶠᶠ·aᵉᶠᶠ ⇒ F ∝ aᵉᶠᶠ 

And inversely, 

aᵉᶠᶠ ∝ 1/Mᵉᶠᶠ 

Where: Mᵉᶠᶠ = Mᴍ −Mᵃᵖᵖ.

The core idea emphasizes that the apparent mass (−Mᵃᵖᵖ) is directly linked to kinetic energy, while the combined terms for ordinary mass and dark matter are associated with potential energy. This distinction aligns total energy with the total mass structure, reinforcing the coherence of the extended framework.

continued.......

process the next equations .....

Conceptual Alignment of Apparent Mass and Dark Energy Effective Mass in Gravitational Dynamics: Extended Classical Mechanics.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

22-09-2024

This presentation explores the interconnected concepts of Negative Effective Mass, highlighting how both Apparent Mass (−Mᵃᵖᵖ) and Dark Energy Effective Mass (Mᴅᴇ) exert negative influences on gravitational dynamics. It emphasizes that these negative mass properties contribute to repulsive forces that counteract gravitational attraction. The relationship between Apparent Mass and total gravitating mass (Mɢ) is established, paralleling the framework proposed by Chernin et al., where Mɢ is derived from the sum of matter mass and dark energy effective mass. The significance of these concepts is particularly pronounced under extreme conditions, such as high velocities and strong gravitational fields, underscoring their importance in cosmic dynamics. Ultimately, the discussion illustrates how Apparent Mass and Dark Energy Effective Mass are conceptually aligned, both playing pivotal roles in shaping gravitational behaviours within galaxy clusters.

1. Negative Effective Mass: 

Both the Apparent Mass (−Mᵃᵖᵖ) and the Dark Energy Effective Mass (Mᴅᴇ) are characterized as having a negative influence on gravitational dynamics. This is central to both concepts, where negative mass contributes to repulsive forces that counteract gravitational attraction.

2. Influence on Gravitating Mass: 

The response emphasizes how Apparent Mass affects the total gravitating mass (Mɢ), showing that it can be expressed as Mɢ = Mᴍ + (−Mᵃᵖᵖ). This is similar to the framework from Chernin et al., where the total gravitating mass is also derived from the combination of matter mass and the dark energy effective mass: Mɢ = Mᴍ + Mᴅᴇ.

3. Context of High Velocities and Strong Fields: 

Both descriptions note that the effects of these negative mass concepts become significant under extreme conditions, such as high velocities or strong gravitational fields, reinforcing the idea that they are crucial for understanding cosmic dynamics.

4. Role in Gravitational Dynamics: 

The alignment is further supported by stating that dark energy’s negative mass plays a significant role in local gravitational dynamics within galaxy clusters, mirroring the implications of Apparent Mass in altering expected gravitational behaviours.

Overall, the presentation conveys that Apparent Mass and Dark Energy Effective Mass share a conceptual foundation, both influencing gravitational dynamics through their negative mass properties.

Keywords: Apparent Mass, Dark Energy, Gravitational Dynamics,

References:

1. 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

2. 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

#ApparentMass, #DarkEnergy, #GravitationalDynamics,

The Cosmological Constant: A Misaligned Solution for Dark Energy and the Static Universe

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

22-09-2024

The cosmological constant, first introduced by Albert Einstein in 1917, was originally intended to maintain a static model of the universe—one that did not expand or contract. This introduction was a response to the prevailing belief that the universe was unchanging, as no observational evidence of expansion existed at the time. However, subsequent discoveries radically altered this view, revealing an expanding universe driven not by a static equilibrium but by dynamic, evolving forces. As such, the cosmological constant, rather than providing answers to the mysteries of dark energy, primarily served to save Einstein's static universe from gravitational collapse, exposing its misalignment with the nature of an expanding cosmos.

The Genesis of the Cosmological Constant

In 1917, Einstein proposed the "Einstein static universe" model, also known as the Einstein universe or the Einstein static eternal universe, within the framework of General Relativity. This model was based on the assumption that the universe was static and unchanging, a perspective supported by the observational limitations of the time. To uphold this static nature, Einstein realized that gravity alone would cause the universe to collapse due to its self-attracting nature. To counteract this effect, he introduced the cosmological constant (Λ), a repulsive force designed specifically to balance gravitational attraction and maintain the universe's static state.

Einstein's adjustment was detailed in his paper, "The Cosmological Considerations in the General Theory of Relativity," where the cosmological constant was mathematically incorporated into his field equations to provide a stable, non-expanding universe. This solution, however, was more of a mathematical fix than a physical insight into the workings of the cosmos.

The Decline of the Static Universe Model

The concept of a static universe began to crumble when astrophysicist Georges Lemaître and others proposed that the universe was not static but expanding. This revolutionary idea was later confirmed by Edwin Hubble's observations in the late 1920s, which showed that galaxies were receding from each other, signalling an expanding universe. Faced with the reality of cosmic expansion, Einstein famously discarded the cosmological constant, calling it his "greatest blunder." He recognized that the static model was fundamentally flawed and that the universe was not in equilibrium as previously thought.

Cosmological Constant vs. Dark Energy

Despite its origin as a corrective measure for a static universe, the cosmological constant has often been repurposed in modern cosmology as a candidate for dark energy, the mysterious force driving the accelerated expansion of the universe. However, this reinterpretation of the cosmological constant as an explanation for dark energy is fundamentally inconsistent with its original purpose and physical meaning.

The cosmological constant was designed to provide a repulsive force to counteract gravitational attraction, thereby maintaining a static universe—not to explain an expanding one. Even if viewed as a force opposing gravitational collapse, it was not intended to account for an accelerating expansion. Instead, the concept of dark energy encompasses a range of potential mechanisms that influence cosmic acceleration, none of which align directly with the simple, uniform repulsion implied by the cosmological constant.

Dark Energy as a Dynamic Force

Current understanding suggests that the accelerating expansion of the universe arises not from any specific substance or constant repulsive force but from complex gravitational and kinetic interactions within the cosmic fabric. These interactions collectively define what we term as dark energy—a placeholder for the unknown drivers of this expansion. Unlike the static repulsion of the cosmological constant, dark energy is dynamic, evolving with the universe in ways that remain the subject of ongoing research.

In conclusion, while the cosmological constant historically played a role in preserving the notion of a static universe, it does not adequately address the complexities of dark energy in an expanding universe. Instead, it serves as a historical footnote—a reflection of a time when the cosmos was misunderstood as a fixed entity rather than the dynamic and ever-evolving universe we observe today. Thus, the cosmological constant is better seen as a relic of an obsolete model rather than a solution to the profound mysteries of dark energy.

Keywords: Cosmological Constant, Static Universe, Dark Energy, Expanding Universe, Gravitational Collapse,

#CosmologicalConstant, #StaticUniverse, #DarkEnergy, #ExpandingUniverse, #GravitationalCollapse,

Dark Energy as a Consequence of Gravitational and Kinetic Interactions:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

21-09-2024

Abstract:

Dark energy is often misunderstood as a mysterious substance permeating the universe. However, a deeper exploration reveals that dark energy is not a standalone entity but a consequence of the gravitational and kinetic dynamics of the universe. This paper presents a comprehensive analysis of the interplay between potential and kinetic energy during the universe's evolution, demonstrating how dark energy emerges as a natural outcome of these energetic transformations.

1. Introduction

Dark energy has been a subject of considerable debate since its discovery due to its association with the accelerated expansion of the universe. Traditionally perceived as an unknown force or substance, dark energy is better understood as a by-product of the universe’s dynamic processes, particularly the transformation of potential energy into kinetic energy during and after the Big Bang. This work explores the interconnected roles of gravitational forces, kinetic energy, and apparent negative mass, highlighting that dark energy results from the complex interplay between these elements rather than being an independent substance.

2. Initial State of the Universe and Energy Transformation

Immediately after the Big Bang, the universe's total energy consisted of potential and kinetic components:

Eᴛₒₜ,ᴜₙᵢᵥ = PEᴜₙᵢᵥ + KEᴜₙᵢᵥ

In the earliest moments, the universe was dominated by potential energy, which rapidly approached zero as kinetic energy surged from zero to infinity:

PEᴜₙᵢᵥ: ∞ → 0, KEᴜₙᵢᵥ: 0 → ∞

This energetic shift was driven by gravitational dynamics, where the rapid conversion of potential energy into kinetic energy fuelled the universe’s expansion.

3. Emergence of Dark Energy: A Dynamic Outcome

Dark energy did not pre-exist the universe but emerged from the dynamic interactions between mass, gravity, and kinetic energy. As the universe’s initial potential mass accelerated due to gravitational forces, an apparent negative mass effect arose, which we interpret as dark energy:

Fᴜₙᵢᵥ = (Mᴘᴇ,ᴜₙᵢᵥ - Mᵃᵖᵖ,ᴜₙᵢᵥ)·aᵉᶠᶠ,ᴜₙᵢᵥ

Here, the apparent mass (Mᵃᵖᵖ,ᴜₙᵢᵥ) represents the dynamic influence of dark energy, emerging from the acceleration of potential mass under universal forces.

4. Inverse Relationship Between Potential and Kinetic Energy

The universe’s potential energy is inversely related to its kinetic energy, illustrating the natural balance that dictates cosmic evolution:

PEᴜₙᵢᵥ ∝ 1/KEᴜₙᵢᵥ

This relationship underscores the continuous transformation and reactivation of dark energy as the kinetic energy of the universe’s matter evolves.

5. Dark Energy's Dormancy and Reactivation

Dark energy enters a dormant state when kinetic energy and potential energy achieve equivalence. However, as the universe’s matter mass persists in motion, dark energy reactivates, leading to the accelerated expansion observed today. This cyclical behaviour underscores the transient nature of dark energy:

When PEᴜₙᵢᵥ = KEᴜₙᵢᵥ , Mᵃᵖᵖ = 0

As the universe continues to expand, dark energy becomes dominant once again, reflecting the evolving interplay of mass-energy dynamics.

6. Conclusion

Dark energy is not a fundamental substance but a manifestation of the universe’s dynamic processes. The accelerated expansion is driven by the continuous transformation of kinetic and potential energies, highlighting that dark energy is a consequence of the cosmic gravitational and kinetic interplay. This understanding shifts the perspective from viewing dark energy as an isolated force to recognizing it as an emergent property of the universe’s mass-energy transformations.

List of Mathematical Terms in Alphabetical Order

• aᵉᶠᶠ,ᴜₙᵢᵥ  - Effective acceleration of the universe

• Eₜₒₜₐₗ - Total energy of the universe

• Fᴜₙᵢᵥ - Force of the universe

• KEᴜₙᵢᵥ - Kinetic energy of the universe

• Mᵃᵖᵖ,ᴜₙᵢᵥ - Apparent mass related to dark energy

• Mᴘᴇ,ᴜₙᵢᵥ - Mass equivalent of potential energy in the universe

• PEᴜₙᵢᵥ - Potential energy of the universe

This presentation demonstrates that dark energy is fundamentally a dynamic outcome of the universe’s evolving energy states, redefining its role in cosmic expansion as an emergent effect rather than a pre-existing substance.

#DarkEnergy #GravitationalInteraction #KineticInteraction

Mass Descriptions, Relationships, and Key References in Gravitationally Bound Systems: Insights from Extended Classical Mechanics Vol-2

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.

Table of different mass terms:

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.

#MassDescriptions #ApparentMass