Revisiting Lorentz Transformations: Resolving Scalar-Vector Dynamics Discrepancies

(Part 3 of 1 to x)

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

25-04-2024

Description:

The comprehensive series of studies, collectively titled "Revisiting Lorentz Transformations: Resolving Scalar-Vector Dynamics Discrepancies," delves deep into the intricacies of Lorentz transformations and their interaction with scalar and vector quantities. Building upon previous research that highlighted concerns about mathematical inconsistencies in Lorentz transformations, particularly in reconciling scalar-vector dynamics, this study series aims to provide clarity and resolution.

Beginning with the foundational study "Addressing Contradictions in Lorentz Transformations: Reconciling Scalar-Vector Dynamics," the series identifies discrepancies between theoretical expectations and empirical observations regarding Lorentz transformations. It underscores the need for further examination to reconcile these inconsistencies.

Subsequently, the focus shifts to "Lorentz Transformations and Effective Mass in Classical Mechanics," which elucidates the derivation of Lorentz transformation formulas and their relationship to kinetic energy and effective mass. This study clarifies misconceptions surrounding effective mass and its distinction from relativistic mass, shedding light on its significance in classical and modern physics.

The final instalment challenges previous notions by asserting that phenomena like length contraction, mass change, and relativistic time dilation are not fundamental manifestations but rather consequences of energy transfer induced by the Lorentz factor. It argues that these effects arise from velocity-induced forces acting on moving objects, influencing the kinetic energy stored within them.

Ultimately, the series concludes that while there may not be a mathematical discrepancy in Lorentz transformations, inconsistencies arise in treating phenomena like length contraction and mass change as standalone entities. Instead, they are portrayed as outcomes of energy transfer due to velocity-induced forces, necessitating a re-evaluation of the scalar-vector dynamics within the Lorentz framework.

Summary:

The study (Part 1 of 1 to x), titled 'Addressing Contradictions in Lorentz Transformations: Reconciling Scalar-Vector Dynamics,' suggests that the statements express a valid concern regarding the mathematical consistency of Lorentz transformations, particularly in their interaction with scalar and vector quantities. It indicates that the discrepancy highlighted indicates a need for further examination and clarification to reconcile theoretical expectations with empirical observations.

In response, we aim to examine and clarify to reconcile the theoretical expectations regarding the mathematical consistency of Lorentz transformations:

The subsequent study, (Part 2 of 1 to x), titled 'Lorentz Transformations and Effective Mass in Classical Mechanics,' describes the derivation of the Lorentz transformation formula and its relationship to the equation E = KE + PE, where kinetic energy (KE) is treated as 'effective mass' (mᵉᶠᶠ) and often misunderstood as relativistic mass (m′).

The Lorentz factor (γ) and velocity-induced forces play pivotal roles in this framework, influencing how kinetic energy is stored within moving objects based on classical mechanics principles. The concept of effective mass is clarified, underscoring its significance in both classical and modern physics, particularly in comprehending mass increase in objects and its implications for system behaviour under various forces. Deformation effects, such as relativistic mass, length contraction, and relativistic time dilation, highlight their association with velocity-induced external forces and their impact on Lorentz transformations.

The Lorentz factor (γ), as a velocity-induced force, influences how kinetic energy is stored within moving objects based on classical mechanics principles. Thus, the stored kinetic energy (KE) is treated as 'effective mass' (mᵉᶠᶠ), often misunderstood as relativistic mass (m′).

When a force acts on an object, causing it to move a certain distance in the direction of the force, work is performed on the object. This work leads to a change in its kinetic energy. Kinetic energy (KE) represents the energy an object possesses due to its motion.

In Lorentz transformations, the mechanical force induced by velocity (v) deforms the moving mass, altering the arrangement of its molecules or atoms, and thus storing kinetic energy as structural deformation, which is reversible when the mass ceases moving. Deformation effects, such as relativistic mass, length contraction, and relativistic time dilation, underscore their linkage to velocity-induced external forces and their influence on Lorentz transformations.

A change in energy can generate forces; similarly, force acts on an object, leading to energy transfer, which impacts its motion and the dynamics of objects. Displacement, velocity, position, including force, are all vector quantities.

This study suggests that the Lorentz factor (γ) in Lorentz transformations is often misunderstood regarding its effects on moving objects. It argues that the Lorentz factor is linked to velocity-induced forces acting on objects in motion, resulting in energy transfer and various deformation effects such as "effective mass," which is frequently confused with relativistic mass, length contraction, and relativistic time dilation.

However, the study contends that these deformational effects do not fundamentally alter the rest mass, cause permanent length contraction, or result in proper time dilation. Instead, they are manifestations of energy transfer caused by the Lorentz factor inducing mechanical force, affecting the kinetic energy stored within the moving object. This stored kinetic energy is treated as reversible "effective mass," often misinterpreted as relativistic mass, leading to temporary length deformation misunderstood as length contraction and time distortion misinterpreted as relativistic time dilation.

According to the study, the velocity-induced resultant force in Lorentz transformations is a vector quantity, influencing the kinetic energy stored within moving objects based on classical mechanics principles. This force interacts more with the Lorentz factor than with stored kinetic energy, resulting in alternative and reversible length deformation, effective mass, or time distortion. Consequently, the Lorentz factor induces force as another vector quantity, leading to vector quantity products in the relevant transformations.

Therefore, the study concludes that there is no mathematical discrepancy, as the Lorentz factor induces force as another vector quantity. However, there is indeed inconsistency, as the previous study (Part 1 of 1 to x), titled 'Addressing Contradictions in Lorentz Transformations: Reconciling Scalar-Vector Dynamics,' suggests. It indicates that length contraction, mass change, and relativistic time dilation, as presented in Lorentz transformations, are not proper manifestations of these phenomena. Instead, they are manifestations of energy transfer due to the Lorentz factor inducing mechanical force, affecting the kinetic energy stored within the moving object. This highlights a valid concern regarding the mathematical consistency of Lorentz transformations, particularly in their interaction with scalar and vector quantities, when phenomena like length contraction, mass change, and time dilation—scalar quantities—are treated as real entities. They are not.

Lorentz Transformations and Effective Mass in Classical Mechanics:

(Part 2 of 1 to x)

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

25-04-2024

Description: 

This summary explores the derivation of the Lorentz transformation formula and its relationship to the equation E = KE + PE, where kinetic energy (KE) is treated as 'effective mass' (mᵉᶠᶠ) and often misinterpreted as relativistic mass (m′). The Lorentz factor (γ) and velocity-induced forces play key roles in this framework, affecting how kinetic energy is stored within moving objects according to classical mechanics principles. The concept of effective mass is clarified, emphasizing its significance in both classical and modern physics, particularly in understanding mass increase in objects and its implications for system behaviour under various forces. Deformation effects, such as relativistic mass, length contraction, and relativistic time dilation, are discussed, highlighting their connection to velocity-induced external forces and their influence on Lorentz transformations.

Summary:

The Lorentz transformation formula, m′ = m₀/√{1 - (v/c)²}, is derived from the equation E = KE + PE, where PE represents the rest mass m₀. This equation treats kinetic energy KE as 'effective mass' (mᵉᶠᶠ), often referred to as relativistic mass (m′), representing time distortion(t′). The Lorentz factor (γ) is a velocity-dependent factor, involving velocity-induced forces. Objects subject to these forces store kinetic energy (KE) within moving objects according to classical mechanics principles.

Velocity-induced force (F) stores kinetic energy in an object, causing stress and deformation due to changes in atomic and molecular structures. This stored energy is typically represented as the relativistic mass (m′), but it should be denoted as the effective mass (mᵉᶠᶠ). Effective mass is a crucial concept in both classical and modern physics, influencing system behaviour under various forces. It is essential in mechanical systems like piezoelectric actuators for dynamic response and in relativistic physics to explain mass increase in objects.

Deformation results in relativistic mass, length contraction, and relativistic time dilation, which are influenced by velocity-induced external forces. Equations like F = kΔL describe these changes, impacting Lorentz transformations and influencing the effective mass of the object.

Addressing Contradictions in Lorentz Transformations: Reconciling Scalar-Vector Dynamics.

(Part 1 of 1 to x)

Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
25-04-2024

Description 

Investigating the interaction between scalar and vector quantities within Lorentz transformations reveals a notable contradiction. While scalar quantities such as mass, length, time, and temperature are typically unaffected by direction, Lorentz factor (γ), commonly treated as a vector due to its velocity-dependence, poses a challenge when interacting with them. Despite mathematical expectations dictating that such interactions should maintain vector properties, empirical observations yield scalar outcomes. This discrepancy underscores a need for further scrutiny and resolution within the framework of Lorentz transformations.

Conclusion:

These statements seem to present a clear contradiction in terms of the nature of scalar and vector quantities, as well as the mathematical expectations set by Lorentz transformations. 

Let's break down the inconsistencies:

Scalar and Vector Quantities: The first set of statements correctly delineate scalar quantities (mass, length, time, temperature) from vector quantities (displacement, velocity, position, force). Scalar quantities describe only magnitude, while vector quantities have both magnitude and direction.

Lorentz Factor and Vector-Scalar Interaction: The first set of statements raise a valid concern about the interaction between the Lorentz factor (γ)—typically treated as a vector quantity due to its velocity-dependence—and scalar quantities like mass, length, and time. According to mathematical principles, when a vector quantity is multiplied or divided by a scalar quantity, the result should remain a vector quantity, scaling only in magnitude without altering direction.

Discrepancy in Lorentz Transformations: The second set of statements highlights the discrepancy between the expected behaviour based on mathematical principles and the observed outcomes in Lorentz transformations. Despite the Lorentz factor (γ) being velocity-dependent and treated as a vector quantity, the equations for mass change, length contraction, and time dilation result in scalar quantities rather than vector quantities as expected.

Violation of Mathematical Principles: The inconsistency between the mathematical expectation and the observed outcomes in Lorentz transformations is identified as a violation of mathematical principles. This indicates a need for acknowledgment and resolution of the discrepancy.

In summary, these statements articulate a valid concern regarding the mathematical consistency of Lorentz transformations, particularly in how they interact with scalar and vector quantities. The discrepancy highlighted suggests a need for further examination and clarification to reconcile the theoretical expectations with empirical observations.

Understanding the Interplay of Energy and Forces in Classical Mechanics:

Summary:

Classical mechanics focuses on the interplay of energy and forces, with gravitational force being a common example. Forces arise from interactions between objects or particles, involving the exchange or transformation of energy. The work-energy theorem, a fundamental principle in classical mechanics, explains the interconnection between force and energy. Force acts on an object, causing energy transfer, which affects its motion and dynamics. Conversely, a change in energy can generate forces, as seen in gravitational fields.

Work done on an object results in a change in its kinetic energy, which is the energy an object possesses due to its motion. The net work done on an object is equal to the change in its kinetic energy, which can be expressed as Wₙₑₜ = ΔKE. This theorem highlights the complex and multifaceted nature of force and energy, illustrating the direct relationship between force and energy.

For example, a ball thrown vertically upwards experiences a change in kinetic energy due to gravity acting against its motion.

(1). Interplay of Energy and Forces in Classical Mechanics:

In classical mechanics, forces arise due to interactions between objects or particles. These interactions can involve the exchange or transformation of energy. One of the most familiar examples is the gravitational force, which arises from the attraction between masses. According to Newton's law of universal gravitation, the force F between two masses m₁ and m₂ separated by a distance r is given by:

F = G⋅m₁⋅m₂/r² 

Where G is the gravitational constant.

This force arises due to the presence of gravitational potential energy in the system. When two masses are separated by a distance, they possess gravitational potential energy due to their mutual attraction. As the distance between them changes, this potential energy is converted into kinetic energy or vice versa, leading to changes in their motion and the generation of forces.

For example, consider a planet in orbit around a star. The planet's motion is governed by the gravitational force exerted by the star. As the planet moves closer to the star, its gravitational potential energy decreases, and this energy is converted into kinetic energy, causing the planet to accelerate. Conversely, as the planet moves away from the star, its gravitational potential energy increases at the expense of kinetic energy, causing it to decelerate.

This interplay between energy and forces is not limited to gravitational interactions but is a fundamental principle in physics. Changes in energy, whether potential or kinetic, can lead to the generation of forces that influence the motion and behaviour of objects in the universe.

(2). The Work-Energy Theorem: Exploring the Relationship between Force and Energy

The work-energy theorem is a fundamental concept in classical mechanics that explains the interconnection between force and energy. Force acts on an object, causing energy to be transferred, which affects its motion and dynamics. Conversely, a change in energy can generate forces, as seen in gravitational fields. This relationship between force and energy is closely related, providing a fundamental expression of their relationship to the work-energy theorem. The concept highlights the complex and multifaceted nature of force and energy.

The work-energy theorem is a fundamental principle in classical mechanics that establishes a direct relationship between the work done on an object and the change in its kinetic energy. This theorem helps explain how forces acting on an object affect its motion and dynamics through the transfer of energy.

When a force is exerted on an object and causes it to move a certain distance in the direction of the force, work is done on the object. Mathematically, work W is defined as the product of the force F applied to the object and the displacement d of the object in the direction of the force:

W = F⋅d

The work done on an object results in a change in its kinetic energy. Kinetic energy (KE) is the energy an object possesses due to its motion and is given by:

KE = (1/2)⋅m⋅v²

Where m is the mass of the object and v is its velocity.

According to the work-energy theorem, the net work done on an object is equal to the change in its kinetic energy. Mathematically, it can be expressed as:

Wₙₑₜ = ΔKE

This theorem illustrates the direct relationship between force and energy. When a force acts on an object and causes it to move, work is done, resulting in a change in kinetic energy. Conversely, a change in kinetic energy can also generate forces, as observed in gravitational fields.

For example, consider a ball thrown vertically upward. As it ascends, the force of gravity acts against its motion, gradually reducing its kinetic energy until it reaches its highest point. At this point, the ball momentarily stops moving upward before gravity pulls it back down. Throughout its trajectory, the work done by gravity results in a decrease in the ball's kinetic energy, which is converted into potential energy when the ball reaches its maximum height. This process illustrates how forces and energy are interrelated in classical mechanics.

In summary, the work-energy theorem provides a fundamental expression of the relationship between force and energy. It highlights how forces acting on an object can transfer energy, affecting its motion, and dynamics, and vice versa. This concept underscores the intricate and interconnected nature of force and energy in the physical world.

Interpretation:

The interplay of energy and forces in classical mechanics, focusing on examples like gravitational interactions governed by Newton's law of universal gravitation. It explains how forces arise from the exchange or transformation of energy, using the example of gravitational potential energy converting into kinetic energy (and vice versa) as two masses interact. The example of a planet orbiting a star illustrates this conversion process, highlighting the dynamic nature of the interaction between energy and forces.

The work-energy theorem introduces another fundamental concept in classical mechanics, which provides a direct mathematical relationship between work (a measure of the transfer of energy due to force) and changes in kinetic energy. It emphasizes how forces acting on an object result in the transfer of energy, affecting its motion and dynamics. This is exemplified by scenarios like a ball thrown vertically upward, where the work done by gravity leads to changes in the ball's kinetic energy and potential energy as it moves against and with the force of gravity.

These presentations offer complementary insights into how energy and forces are intimately linked in classical mechanics. The interplay of energy and forces in classical mechanics provides a conceptual understanding of how forces arise from energy interactions, while the work-energy theorem offers a mathematical framework to quantify these interactions. Both emphasize the dynamic and interconnected nature of energy and forces in shaping the behaviour of objects in the physical world.

These statements are consistent and coherent. They provide a comprehensive overview of the interplay between energy and forces in classical mechanics and highlight the relationship between force and energy from different perspectives. The first statement discusses how forces arise from interactions between objects or particles, focusing on examples such as gravitational interactions governed by Newton's law of universal gravitation. It explains how forces can arise from the exchange or transformation of energy, using the example of gravitational potential energy converting into kinetic energy (and vice versa) as two masses interact.

The second statement delves into the work-energy theorem, which provides a direct mathematical relationship between work (a measure of the transfer of energy due to force) and changes in kinetic energy. It illustrates how forces acting on an object result in the transfer of energy, affecting its motion and dynamics. Examples such as a ball thrown vertically upward demonstrate how the work done by gravity leads to changes in the object's kinetic energy and potential energy.

Together, these statements offer complementary insights into how energy and forces are intimately linked in classical mechanics. They provide both conceptual understanding and a mathematical framework to quantify these interactions, emphasizing the dynamic and interconnected nature of energy and forces in shaping the behaviour of objects in the physical world.

A Supplementary resource for ‘Phase Shift and Infinitesimal Wave Energy Loss Equations’

The Interplay of Phase Shifts, Time Distortions, and Energy Changes in Wave Systems:

Soumendra Nath Thakur
22-04-2024

Description:

The text provides a detailed examination of equations and relationships governing phase shifts, time distortions, and energy changes in wave systems. It begins by establishing fundamental equations linking phase shift T(deg), time distortion (Δt), and energy change (ΔE), subsequently extending these equations to calculate energy changes based on frequency and Planck's constant.

A key concept introduced is the generalization of time distortion to incorporate phase shift (x), allowing for a more versatile analysis of wave systems with different phase shifts. The conclusion drawn asserts a direct relationship between phase shift measured in degrees T(deg) and the degree of phase shift (x).

Furthermore, an expression explicitly integrating phase shift (x) into the equation for energy change is derived, confirming relationships between Planck's constant, frequency, and the period of the wave.

Overall, the text provides a coherent and comprehensive exploration of wave system dynamics, offering insights into phase shifts, time distortions, and energy changes.

Soumendra Nath Thakur
ORCID iD: 0000-0003-1871-7803
Tagore’s Electronic Lab, West Bengal, India
Email: postmasterenator@gmail.com,
postmasterenator@telitnetwork.in

The equations and relationships concerning phase shifts, time distortions, and energy changes in wave systems are:

T(deg) = (1/360f₀) = Δt 

ΔE = hf₀Δt
ΔE = (2πhf₀/360) × T(deg)
ΔE = (2πh/360) × T(deg) × (1/Δt)

These equations establish the relationship between phase shift T(deg) and time distortion (Δt), where f₀ represents the primary or initial frequency of the wave. It essentially quantifies the time distortion (Δt) caused by a phase shift.

The equations subsequently calculate the infinitesimal loss of wave energy (ΔE) based on Planck's constant (h), frequency (f₀), and phase shift T(deg) or time distortion (Δt). These equations allow for the determination of energy changes given certain variables.

Δtₓ = x (1/360f₀)

The above equation generalizes the concept of time distortion (Δt) with respect to phase shift (x), where x represents the degree of phase shift. The conclusion drawn is that T(deg) = x, which essentially states that the degree of phase shift (x) is directly related to T(deg), the phase shift measured in degrees.

The conclusion is valid because throughout the analysis, the relationships between phase shift, time distortion, and energy changes are consistently tied together, culminating in the assertion that the degree of phase shift (x) corresponds to the phase shift measured in degrees T(deg).

The expression:

(h/360) 2πf₀x

It is derived from the basic equation ΔE = hf₀Δt, explicitly including the phase shift (x).

T = 360
f₀ = 1/T
hf₀ = h/360

The statement T = 360 implies that the period of the wave (T) is equivalent to 360 units of time. This aligns with the concept that a complete cycle of a wave occurs over 360 degrees.

f₀ =1/T indicates that the primary or initial frequency (f₀) of the wave is inversely proportional to its period (T). If the period is 360 units of time, then the frequency is indeed 1/360 cycles per unit time.

Combining these, hf₀ = h/360 follows from substituting f₀ = 1/T into the equation hf₀, yielding h/360.

This statement essentially confirms the relationship between Planck's constant (h), frequency (f₀), and the period of the wave (T), providing a basis for further derivations and calculations involving energy changes and phase shifts.

The above statements present a consistent exploration of the equations and relationships governing phase shifts, time distortions, and energy changes in wave systems. Let's dissect the consistency:

Establishing Equations and Relationships: The initial equations T(deg) = (1/360f₀) = Δt and ΔE=hf₀ Δt lay the foundation for understanding the interplay between phase shift (T(deg), time distortion (Δt), and energy change (ΔE).

Calculation of Energy Changes: The subsequent equations ΔE = (2πhf₀/360) × T(deg) and ΔE = (2πh/360) × T(deg) × (1/Δt) provide methods for calculating energy changes based on various variables such as phase shift, time distortion, frequency, and Planck's constant.

Generalization of Time Distortion: The equation Δtₓ = x × (1/360f₀) extends the concept of time distortion (Δt) to incorporate phase shift (x), offering a versatile tool for analysing wave systems with different phase shifts.

Conclusion: The conclusion that T(deg)= x solidifies the relationship between phase shift measured in degrees T(deg) and the degree of phase shift (x), emphasizing their direct correspondence.

Derivation and Confirmation of Relationships: The expression (h/360) × 2πf₀ × x is derived from the fundamental equation ΔE = hf₀Δt, explicitly integrating the phase shift (x). The subsequent statements regarding T = 360, f₀ = 1/T, and hf₀ = h/360 confirm the relationships between Planck's constant, frequency, and the period of the wave, providing a coherent basis for further analysis.

The above statements offer a comprehensive and consistent exploration of the dynamics governing wave systems, providing a robust framework for understanding and analysing phase shifts, time distortions, and energy changes.

Applicable for:

DOIs:

·        http://dx.doi.org/10.20944/preprints202309.1831.v1
·        http://dx.doi.org/10.13140/RG.2.2.28013.97763
·        http://dx.doi.org/10.35248/2161-0398.23.13.365

URLs:

·       https://www.researchgate.net/publication/378462690_Phase_Shift_and_Infinitesimal_Wave_Energy_Loss_Equations

·        https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html

PDF:

·        https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations.pdf