Dimensions and Perceptibility: Space, Time, and Hyper-Dimensions in Physics and Mathematics

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

17-06-2024

Abstract:

This study delves into fundamental concepts of dimensions across physics and mathematics. Space is initially defined as encompassing height, depth, and width, constituting a three-dimensional continuum where all physical entities reside and move. Classical physics traditionally perceives space through three linear dimensions.

In the disciplines of modern physics, various perspectives on space and time emerge, extending beyond the confines of relativistic physics. Relativistic physics introduces the concept of spacetime, describing a boundless four-dimensional continuum integrating space with time, pivotal in relativity. This concept reshapes our understanding of gravitational forces and the fabric of the universe.

Beyond relativistic physics, other disciplines within modern physics, such as quantum mechanics and statistical physics, offer distinct interpretations of space and time that may align with classical frameworks. These interpretations often diverge from the seamless integration portrayed in relativistic spacetime. Quantum mechanics explores discrete states and probabilistic behaviours, challenging classical notions of continuous space and time. Statistical physics examines macroscopic systems governed by statistical laws, where the concept of time evolves differently from deterministic frameworks.

Mathematically, dimensionality in a space signifies the minimum coordinates necessary to pinpoint any given point within it. For instance, a one-dimensional line necessitates a single coordinate, exemplified by the number line where each point corresponds to a real number. Points are conceptual entities without physical dimensions, rendering them imperceptible to human senses.

Expanding into higher dimensions, a two-dimensional plane consists of infinite lines, each potentially infinite in real numbers, while a three-dimensional space extends further to encompass infinite planes. Beyond these realms lie hyper-dimensions—dimensions surpassing the conventional three spatial dimensions—where human perceptibility ceases due to inherent physical constraints. Entities within our three-dimensional realm remain disconnected from hyper-dimensional spaces, illustrating their imperceptibility.

The study also addresses time as the fourth dimension, akin to hyper-dimensions in its imperceptibility. Time, crucial in the context of spacetime theory, is mathematically treated as a dimension but fundamentally experienced differently from spatial dimensions. This distinction underscores the challenge of representing time through mathematical models despite its difference in human experience compared to spatial dimensions.

Throughout, adherence to established scientific and mathematical principles ensures a coherent representation of these fundamental concepts, illuminating the nuanced interplay between spatial dimensions, time, and hyper-dimensions within the realms of physics and mathematics across different theoretical frameworks.

Dimensions and Perceptibility: Space, Time, and Hyper-Dimensions in Physics and Mathematics

Space encompasses the dimensions of height, depth, and width, within which all objects exist and move. Alternatively, it can be described as a three-dimensional continuum containing positions and directions. In classical physics, space is often considered in three linear dimensions. However, when considering space together with time within the disciplines of modern physics, only the relativistic context describes it as a boundless four-dimensional continuum known as spacetime.

In physics and mathematics, the dimension of a mathematical space is defined as the minimum number of coordinates needed to specify any point within it. For example, a line is one-dimensional because only one coordinate is needed to specify a point on it, such as the point at 5 on a number line.

A number line is a horizontal straight line that visually represents numbers. It is a pictorial representation of real numbers, with every point on the line corresponding to a real number and vice versa.

A point, represented by a dot '.', indicates an exact location in space but has no length, width, or height—no size or dimension. Thus, a point is a conceptual, rather than a physical, entity. Points are imperceptible to humans because they have no physical size.

Since every point on a number line corresponds to a real number, and every real number corresponds to a point, a one-dimensional number line contains an infinite number of points or real numbers. This one-dimensional line is perceptible to humans as it represents countable real numbers.

A two-dimensional plane contains an infinite number of lines, each containing an infinite number of real numbers. A plane with countable lines of real numbers is perceptible to humans, making it a physically perceptible dimension.

Similarly, a three-dimensional space contains an infinite number of planes, each with an infinite number of real numbers. A three-dimensional space with countable planes of real numbers is also perceptible to humans, making it a physically perceptible dimension.

Dimensions beyond the three spatial dimensions are known as hyper-dimensions. Hyper-dimensional spaces are imperceptible to humans because we do not have physical reach beyond the three-dimensional space. Entities within our three-dimensional existence are not physically interactable with hyper-dimensions or hyper-dimensional spaces.

Since hyper-dimensions are imperceptible, the fourth dimension, time, is also imperceptible to us. We represent the fourth dimension through mathematical or conceptual models, often manifested through physical frequencies.

Explanation of the consistency:

Definition of Space:

Classical and Modern Physics: Classical physics often considers space as having three dimensions (height, depth, and width). Modern physics, especially in the context of relativity, combines space and time into a four-dimensional continuum known as spacetime. This accurately reflects the transition from Newtonian to Einsteinian physics.

Mathematical Space:

Dimensions: In mathematics, the dimension of a space is defined by the minimum number of coordinates required to specify a point within it. This definition aligns with the standard mathematical concept of dimensions.

One-Dimensional Line: A line is one-dimensional because a single coordinate suffices to specify a point on it, e.g., the point at 5 on a number line.

Number Line:

Representation: A number line is described as a horizontal line that visually represents numbers, where each point corresponds to a real number. This accurately describes how number lines are used in mathematics.

Concept of a Point:

Properties: A point is represented by a dot and has no dimensions (length, width, height), making it a conceptual entity rather than a physical one. This is consistent with mathematical definitions.

Imperceptibility: Points are imperceptible because they lack physical size, aligning with the conceptual nature of points in geometry.

Infinite Points on a Line:

Infinity in One Dimension: A one-dimensional number line contains an infinite number of points or real numbers, consistent with the concept of continuity in mathematics.

Perceptibility: While individual points are imperceptible, the number line as a whole is perceptible because it represents countable real numbers.

Two-Dimensional Plane:

Infinite Lines: A plane contains an infinite number of lines, each with an infinite number of real numbers. This reflects the properties of two-dimensional spaces.

Perceptibility of Planes: A plane with countable lines of real numbers is perceptible to humans, making it a physically perceptible dimension.

Three-Dimensional Space:

Infinite Planes: A three-dimensional space contains an infinite number of planes, consistent with the properties of three-dimensional geometry.

Perceptibility: A three-dimensional space with countable planes of real numbers is perceptible to humans, aligning with our everyday experience of physical space.

Hyper-Dimensions:

Definition: Dimensions beyond the three spatial dimensions are known as hyper-dimensions. This term is consistent with terminology used in higher-dimensional mathematics and theoretical physics.

Imperceptibility: Hyper-dimensional spaces are imperceptible to humans because we lack physical reach beyond the three-dimensional space. This reflects the limitations of human perception and physical interaction.

Fourth Dimension (Time):

Imperceptibility: Time as a fourth dimension is imperceptible in the same way as hyper-dimensions and is often represented through mathematical or conceptual models. The use of spacetime in relativity theory is not consistent with this statement, where time is treated as a dimension but experienced as naturally as we experience events within spatial dimensions.

By adhering to these established scientific and mathematical principles, the study remains consistent and accurately represents the underlying concepts.

Relativistic Energy Increase and Effective Mass: Understanding ΔE = mᵉᶠᶠc²

Summary:

In relativistic physics, the total energy of an object increases with its velocity, transitioning from non-relativistic to relativistic speeds. The increase in energy (ΔE) is due to kinetic energy and can be expressed as:

ΔE = (γ - 1)mc² = mᵉᶠᶠc²

Here, mᵉᶠᶠ is the effective mass associated with the increased energy ΔE, which is not a result of nuclear conversion but a consequence of the object's motion. This perspective aligns with Einstein's mass-energy equivalence (E = mc²) and highlights how energy changes with velocity in special relativity, providing a clear understanding of effective mass in this context.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

15-04-2024

The summary of the concepts:

E(v ≪ c) < E(v < c): This inequality reflects that the total energy of an object increases as its velocity increases from non-relativistic (v ≪ c) to relativistic speeds (v < c).

E(v < c) − E(v ≪ c) = ΔE: ΔE represents the increase in total energy due to the kinetic energy gained as the velocity increases.

ΔE = mᵉᶠᶠc², where mᵉᶠᶠ is the effective mass associated with ΔE: This equation relates velocity-induced kinetic energy ΔE to an effective mass mᵉᶠᶠ that would correspond to the energy E if converted into mass according to E = mc².

This assertion correctly emphasizes that ΔE is not a result of nuclear conversion but rather a consequence of the object's motion at less than the speed of light c.

This presentation effectively captures the key points regarding energy increase, effective mass, and their relationship in relativistic contexts. It provides a clear understanding of how energy changes with velocity and the concept of effective mass in special relativity.

Let's describe the points clearly:

1. Total Energy E:

In special relativity, the total energy E of an object with rest mass m and velocity v is given by: 

E = γmc² 

where γ = 1/√(1-v²/c²) is the Lorentz factor, c is the speed of light.

2. Delta ΔE:

ΔE represents the change in total energy when the velocity v increases from v≪c to v<c:

ΔE = γmc² - mc² = (γ−1)mc² 

ΔE corresponds to the increase in total energy due to the kinetic energy gained by the object as its velocity increases.

3. Effective Mass mᵉᶠᶠ:

ΔE can be associated with an effective mass mᵉᶠᶠ such that:

ΔE = mᵉᶠᶠc² 

mᵉᶠᶠ​ is the effective mass equivalent to ΔE. It represents the mass that would correspond to the energy ΔE in the context of relativistic energy-momentum relations.

The assertion that ΔE is a consequence of the object's motion relative to the speed of light c, not a result of nuclear processes, aligns with the principles of special relativity. This perspective underscores that ΔE = mᵉᶠᶠc² signifies the manifestation of energy increase due to velocity, clarifying the relationship between kinetic energy and effective mass in relativistic contexts.

This presentation encapsulates the key points regarding energy increase, effective mass, and their interplay within special relativity. It provides a coherent understanding of how energy changes with velocity and the concept of effective mass as defined by relativistic principles, enhancing comprehension beyond traditional Newtonian mechanics.

Advancing Science: Imperatively Refining Theories Through Critical Evaluation.

Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803

14-04-2024

Refining and discarding flawed or inaccurate aspects of a scientific theory is a crucial part of scientific progress. In fact, it's integral to the scientific method itself, which emphasizes testing hypotheses, scrutinizing evidence, and revising theories based on new discoveries.

When scientists identify inconsistencies, errors, or limitations in a theory, they actively work to address these through rigorous investigation and experimentation. This process can lead to adjustments, refinements, or even paradigm shifts in scientific understanding. Such refinement is aimed at improving the accuracy, predictive power, and explanatory scope of scientific theories.

Therefore, there is no dishonour in critically evaluating and refining theories. On the contrary, it is a testament to the self-correcting nature of science and its commitment to advancing knowledge. By acknowledging and addressing shortcomings, scientists pave the way for deeper insights and more robust theories that better reflect the complexities of the natural world.

Flawed Relativistic Time Dilation is Confirmed by Biased Experiments:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

14-06-2024

Lay Summary

Phase shifts and infinitesimal loss of wave energy occur when an oscillatory wave experiences a change in its phase angle, which can happen due to relativistic effects including interactions with different media or obstacles. Wavelength distortions refer to changes in the wavelength of an oscillatory wave due to phase shifts, as phase shift is inversely proportional to wavelength. These can occur due to various factors, excluding dispersion or refraction, as propagating waves are different than oscillatory waves.

Time dilation is a flawed concept in Einstein's theory of relativity, as it incorrectly states that time passes differently for observers in relative motion. In reality, it is phase shifts in clock oscillation and corresponding wavelength and time distortions. Wavelength is proportional to time (T).

Relativistic effects in time dilation have been verified with biased experiments, as such experiments should have been done on wavelength distortions rather than time dilation. Time, as a concept, does not subject to any experiment on it, unless inviting error. Time is standardised by time standardizing authorities and is not subject to biased experiments.

While these concepts are not distinct, phase shift and wavelength distortions are mutually exclusive, as they are valid scientific interpretations. They are interconnected with time distortion but not with the flawed concept of time dilation.

In recent times, scientists appreciate that these phenomena of time distortions and phase shifts are interconnected and collectively enrich our understanding of the universe, rather than understanding the erroneous time dilation phenomenon. Clocks designed for proper time measurement account for relativistic effects, including time distortion, but not for flawed time dilation.

Universal Gravitational Constant G in Total Mass and Dark Energy Calculations:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

12-06-2024

Abstract:

This analysis examines the consistent use of the universal gravitational constant G in calculations pertaining to both the total gravitating mass (Mɢ = Mᴍ + Mᴅᴇ), encompassing dark matter and baryonic matter, and the effective mass of dark energy (Mᴅᴇ or mᵉᶠᶠ). Through equations derived within the Newtonian gravitational framework, the paper illustrates how the classical universal gravitational constant G is applied to understand gravitational effects within the context of dark energy. By employing the same fundamental constant throughout the analysis, the study ensures conformity with established gravitational laws, reaffirming the role of G in elucidating the dynamics of mass and energy in cosmological structures.

Keywords: Universal Gravitational Constant, Total Mass, Dark Energy, Gravitational Effects, Newtonian Framework,

The analysis of the research on the Coma cluster of galaxies considers the gravitational constant G as the fundamental constant used in both the effective mass of dark energy (Mᴅᴇ) and the total gravitating mass (Mɢ). The known universal gravitational constant G is indeed utilized for calculating the gravitational effects, including those due to dark energy.

Here's how G is applied in the context of the effective mass of dark energy and the total gravitating mass:

1. Effective Gravitating Density of Dark Energy:

The paper uses the equation: ρₑ𝒻𝒻 = ρ + 3P

For dark energy in the ΛCDM model, ρᴅᴇ is the density, and Pᴅᴇ = − ρᴅᴇ, leading to: 

ρᴅᴇₑ𝒻𝒻 = ρᴅᴇ + 3Pᴅᴇ = - 2ρᴅᴇ < 0 

This indicates that the effective density of dark energy is negative, which corresponds to an antigravitational effect.

2. Acceleration Due to Dark Energy:

The gravitational acceleration a(r) at a distance R from the centre of a mass Mᴍ within a uniform dark energy background is given by:

a(R) = - G(Mᴍ/R²) + (4ϖG/3)ρᴅᴇR = aɴ(R) + aᴇ(R)

Here, the second term represents the antigravitational effect of dark energy, and G is the universal gravitational constant. 

aɴ(R) and aᴇ(R) are components of the radial acceleration experienced by a test particle due to gravity and dark energy, respectively.

• Newtonian Gravity Component aɴ(R):

This is the standard Newtonian gravitational acceleration due to a mass Mᴍ at a distance R:

aɴ(R) = - G(Mᴍ/R²)

Here:

• G is the universal gravitational constant.

• Mᴍ is the matter mass causing the gravitational attraction.

• R is the distance from the centre of the mass Mᴍ.

• Dark Energy Component aᴇ(R):

This is the acceleration due to the effect of dark energy, which acts as a repulsive force (antigravity) in this context:

aᴇ(R) = (4ϖG/3)ρᴅᴇR 

Here:

• G is the universal gravitational constant.

• ρᴅᴇ is the density of dark energy.

• R is the distance from the centre of the cluster.

Combined Acceleration

The total radial acceleration a(R) experienced by a test particle at a distance R from the centre of a spherical mass Mᴍ in the presence of dark energy is the sum of these two components:

a(R) = aɴ(R) + aᴇ(R) = - G(Mᴍ/R²) +  (4ϖG/3)ρᴅᴇR

In this equation:

• aɴ(R) represents the attractive gravitational force.

• aᴇ(R) represents the repulsive force due to dark energy.

The balance between these two forces determines the net effect on the particle's motion.

3. Zero-Gravity Radius (Rᴢɢ):

The zero-gravity radius Rᴢɢ, where gravitational and antigravitational forces balance each other, is derived using G:

Rᴢɢ = [Mᴍ/{(8ϖ/3)ρᴅᴇ}]⅓ 

This radius delineates the region where gravity dominates (inside Rᴢɢ) from the region where dark energy dominates (outside Rᴢɢ).

 

4. Dark Energy Mass (Mᴅᴇ):

The effective mass (mᵉᶠᶠ) of dark energy within a radius R is:

Mᴅᴇ(R) = (8ϖ/3)ρᴅᴇR³

 

This shows that Mᴅᴇ depends on ρᴅᴇ and R, but the gravitational effect of this mass is accounted for using G.

The calculations involving Mᴅᴇ and Mɢ are based on the Newtonian gravitational framework where the universal gravitational constant G is consistently used. The paper does not introduce a separate or modified gravitational constant for dark energy; instead, it applies the same G throughout the analysis, ensuring consistency with the established laws of gravity. This approach confirms that the known universal gravitational constant G is used for the effective mass of dark energy (Mᴅᴇ) as well as for other gravitational calculations in the study.

Reference: 

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 & Astrophysics, 553, A101. https://doi.org/10.1051/0004-6361/201220781 

#UniversalGravitationalConstant #TotalMass #DarkEnergy #EffectiveMass