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

Dimensional Perception: Geometrical and Dimensional Analysis

Soumendra Nath Thakur

19-09-2024

As three-dimensional observers, we perceive existential objects as a combination of infinite two-dimensional frames within a three-dimensional view. While we can observe the height and width of objects directly in a two-dimensional frame, depth—the third dimension—enables us to integrate these frames into a solid, allowing us to discern changes or differences between objects as they exist in three-dimensional space.

This perception suggests that depth (the third dimension) primarily serves to combine two-dimensional views into a cohesive three-dimensional experience, enabling us to perceive the structure and changes of objects in solid form.

Similarly, a fourth-dimensional perception would likely view three-dimensional objects as a combination of infinite three-dimensional frames, where time—the fourth dimension—allows the observer to perceive changes in these objects across time. Just as depth allows us to distinguish between two-dimensional frames, time enables a fourth-dimensional observer to perceive the evolution and differences between three-dimensional objects.

This implies that the fourth dimension, or time, plays a fundamental role in perceiving changes or differences in objects within a three-dimensional framework, much like depth does in two-dimensional views.

Keywords: Dimensional Perception, Geometrical Consistency, Fourth-Dimensional View, Two-Dimensional Frames, Depth-Time Analogy,

Analysis

Key Points:

Dimensional Perception:

• The statement explains that as three-dimensional observers, we perceive objects in a combination of infinite two-dimensional frames within a three-dimensional view.

• Depth, the third dimension, integrates these two-dimensional views into a cohesive three-dimensional solid, allowing us to perceive changes and differences between objects.

• It also suggests that a fourth-dimensional observer would perceive three-dimensional objects in the same way we perceive two-dimensional projections, where time (the fourth dimension) serves a similar role as depth in distinguishing changes or differences in three-dimensional objects.

Geometrical Consistency:

Two-Dimensional Frames in a Three-Dimensional View:

• The claim that our perception of three-dimensional objects is a combination of infinite two-dimensional frames is geometrically consistent. This is because each cross-section (a two-dimensional frame) contributes to the total depth of the object, and when stacked together, these frames represent the full three-dimensional solid.

• Our three-dimensional perception indeed relies on integrating various views or cross-sections, much like imaging techniques (CT or MRI) that use 2D slices to build a 3D image.

• Depth (the third dimension) allows for distinguishing different layers or aspects of these objects that would otherwise overlap in a purely two-dimensional view, making this interpretation sound.

Depth as Integrating Two-Dimensional Views:

• The explanation that depth allows us to integrate 2D frames into a 3D object and perceive changes is also consistent. Geometrically, each 2D frame represents a particular “slice” of reality, and depth allows for the interpolation between these slices to form the perception of a solid object.

• Without depth, these 2D views would be flat projections, lacking the information needed to distinguish changes or structures across the third dimension.

Fourth-Dimensional Perception of Three-Dimensional Objects:

• The analogy between our perception of 2D views and a fourth-dimensional observer’s perception of 3D objects is geometrically valid. Just as we combine 2D slices to understand a 3D object, a fourth-dimensional being would combine 3D "slices" to perceive how an object evolves across time.

• Time, as the fourth dimension, allows for the observation of changes in three-dimensional objects, much as depth allows us to distinguish between 2D projections. This maintains dimensional consistency within the analogy, as it follows the idea that each higher dimension offers a more comprehensive perspective by integrating multiple lower-dimensional views.

Dimensional Consistency:

Two-Dimensional vs. Three-Dimensional Perception:

• The statement is dimensionally consistent in explaining how we perceive objects through two-dimensional frames combined with depth to form a three-dimensional view. This view is grounded in both geometry and our everyday experience of observing the world around us.

• The idea that depth serves to integrate two-dimensional views into a solid aligns with the dimensional hierarchy, where each higher dimension is composed of infinite slices of the previous one.

Fourth-Dimensional Perception:

• The suggestion that a fourth-dimensional being would view three-dimensional objects over time is consistent with the dimensional framework used. Time, as the fourth dimension, would allow such a being to perceive changes in a way that transcends the static nature of our 3D perception.

• The comparison between depth in 2D perception and time in 3D perception creates a logical parallel, where each dimension provides the additional "layer" needed to perceive change or structure.

Comparison Between Depth and Time:

• The statement implies that time (the fourth dimension) functions similarly to depth (the third dimension) in allowing an observer to perceive changes or differences. This is dimensionally consistent, as time provides the necessary framework to observe transitions or variations in three-dimensional space.

Conclusion:

This statement is both geometrically and dimensionally consistent. It effectively uses the concept of combining infinite two-dimensional frames to explain how three-dimensional objects are perceived, and it extends this analogy to suggest how a fourth-dimensional observer would perceive time as an integral aspect of three-dimensional objects. The comparison between depth and time, and their roles in perceiving changes in different dimensions, is logically sound and consistent with dimensional theory.

#DimensionalPerception, #GeometricalConsistency, #FourthDimensionalView, #TwoDimensionalFrames, #DepthTimeAnalogy,

The Dynamics of Gravitationally Bound Systems:

17-09-2024 

Soumendra Nath Thakur

In classical mechanics, gravitational mass (Mɢ) is considered equivalent to matter mass (Mᴍ). However, modern physics recognizes that the gravitational effects of dark matter and dark energy can influence gravitational dynamics, particularly in regions dominated by dark energy, such as at least on the intergalactic scale. Within a gravitationally bound system, typically confined to a zero-gravity sphere, matter mass (Mᴍ) encompasses both normal (baryonic) matter and dark matter. The gravitating mass (Mɢ) represents the total effective mass that governs the gravitational dynamics of such a system. It includes contributions from both normal matter and dark matter, but not the effective mass associated with dark energy, which is primarily dominant in regions beyond the zero-gravity sphere. This comprehensive understanding is crucial for comprehending both the internal dynamics of gravitationally bound systems and the large-scale structure of the universe.

#GravitationalDynamics, #DarkMatter, #DarkEnergy, #GravitatingMass, #ZeroGravitySphere,

Definitions : Extended Classical Mechanics: Apparent Mass, Dark Energy Effective Mass, Effective Acceleration, Effective Mass, Gravitating Mass, Matter Mass

1. Apparent Mass (Mᵃᵖᵖ): A dynamic term that reflects the observed mass of an object under external forces. This mass can appear reduced due to negative effective mass. When a force F acts on an object, causing an increase in acceleration a, a significant negative component in the effective mass Mᵉᶠᶠ (i.e. −Mᵃᵖᵖ) results in an apparent reduction of the observed mass, which can be quantified as negative apparent mass (Mᵃᵖᵖ < 0). This phenomenon is prominent under conditions like high velocities or strong gravitational fields.     

2. Dark Energy Effective Mass (Mᴅᴇ): The effective mass associated with dark energy, which contributes to a repulsive force that influences gravitational dynamics negatively. This concept, introduced in Chernin et al.'s 2013 paper, is reinterpreted in this study as equivalent to negative apparent mass (−Mᵃᵖᵖ). According to the equation Mɢ = Mᴍ + Mᴅᴇ, where Mɢ represents the total gravitational mass, Mᴍ is the matter mass, and Mᴅᴇ is the dark energy effective mass, this formulation underscores the substantial impact of dark energy on the overall gravitational dynamics of the universe.     

      

3. Effective Acceleration (aᵉᶠᶠ): The rate at which an object's velocity changes, influenced by the interplay of positive matter mass (Mᴍ) and negative apparent mass (−Mᵃᵖᵖ). Effective acceleration is determined by the overall effective mass (Mᵉᶠᶠ) of a system, which is the sum of matter mass and negative apparent mass. When the negative apparent mass is significant, it alters the effective mass and thereby affects the acceleration experienced by the object. The relationship is expressed as: F = (Mᴍ − Mᵃᵖᵖ) aᵉᶠᶠ where F is the force applied, Mᴍ is the matter mass, −Mᵃᵖᵖ is the negative apparent mass, and aᵉᶠᶠ is the effective acceleration. This modified effective acceleration accounts for the influence of negative apparent mass on the dynamics of motion.

4. Effective Mass (Mᵉᶠᶠ): A composite term that includes both matter mass (Mᴍ) and negative apparent mass (−Mᵃᵖᵖ). Effective mass can be positive or negative depending on the relative magnitudes of the matter mass and the negative apparent mass.          

5. Gravitating Mass (Gravitational Mass) (Mɢ): The total effective mass that governs the gravitational dynamics of a system. It encompasses both the matter mass and any negative apparent mass, and it is equivalent to the mechanical effective mass (Mᵉᶠᶠ).         

6. Matter Mass (Mᴍ): The mass associated with normal (baryonic) matter and dark matter within a system. It contributes positively to the gravitating mass.

#ApparentMass, #DarkEnergyEffectiveMass, #EffectiveAcceleration, #EffectiveMass, #GravitatingMass, #MatterMass

Research Overview: Extended Classical Mechanics. Vol-1.

17 September 2024

The research, ‘Extended Classical Mechanics’, by Soumendra Nath Thakur offers a comprehensive exploration of the foundational principles of physics, particularly focusing on mass, gravity, and their interactions. The study delves into the Equivalence Principle, a cornerstone of classical mechanics, and extends its application to incorporate contemporary understandings of dark matter and dark energy.

Key Contributions

Redefining Gravitating Mass:

The research introduces a new perspective on gravitating mass, incorporating the concept of negative apparent mass. This challenges the traditional understanding of gravitational interactions, particularly in the context of dark energy.

Introducing Apparent Mass:

The study proposes the concept of apparent mass, a dynamic term that can influence the observed mass of an object under certain conditions. This innovation allows for a more nuanced understanding of mass and its role in gravitational dynamics.

Revisiting Newton's Law:

The research reinterprets Newton's Law of Universal Gravitation to account for the newly introduced concepts of apparent mass and effective mass. This modification provides a more comprehensive framework for understanding gravitational forces.

Integrating Dark Matter and Dark Energy:

The study seamlessly integrates contemporary theories of dark matter and dark energy into the classical mechanics framework. This integration offers a more holistic perspective on the universe's gravitational dynamics.

Methodology and Implications

The research employs a combination of theoretical reinterpretation, mathematical modelling, and numerical simulations to validate its findings. The implications of this work are far-reaching, potentially influencing our understanding of gravitational theory, dark energy, and the overall structure of the universe.

Overall Significance

"Extended Classical Mechanics" presents a significant contribution to the field of physics. By extending the classical framework to incorporate modern concepts, the research offers a more comprehensive and accurate understanding of the universe's fundamental laws. It has the potential to inspire further research and advancements in our understanding of gravity and its implications for cosmology.

Additional Insights

The study's focus on the Equivalence Principle highlights its central role in understanding the relationship between mass and gravity.

The introduction of negative apparent mass provides a new perspective on the nature of mass and its interactions.

The integration of dark matter and dark energy into the classical framework demonstrates the study's relevance to contemporary cosmological theories.

The research's potential implications for gravitational theory and our understanding of the universe's structure underscore its significance