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

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,