A Supplementary resource (2) for ‘Phase Shift and Infinitesimal Wave Energy Loss Equations'.

Proportional Relationships in Oscillatory Wave Dynamics: Time Shifts and Energy Changes

Soumendra Nath Thakur

28-10-2024

Description: 

This study explores the mathematical relationships between phase shifts in oscillatory wave frequencies and their corresponding time periods and energy changes. The time period associated with a 1° phase shift, denoted as T(deg), is defined as the inverse of the product of 360 and the original frequency f₀. This period is directly proportional to the infinitesimal time shift Δt when f₀ remains constant. Additionally, the change in energy ΔE is expressed as the product of Planck’s constant h, the original frequency f₀, and the infinitesimal time shift Δt, establishing that ΔE is proportional to f₀ and inversely related to Δt. Furthermore, the study presents how the infinitesimal time shift Δtₓ associated with an x° phase shift is directly proportional to x under constant frequency conditions. This paper concludes with key proportional relationships that facilitate a deeper understanding of the dynamics within oscillatory wave systems.

Keywords: Oscillatory waves, phase shift, time period, energy change, Planck's constant, frequency, mathematical relationships, wave dynamics.

Equation Relationship:

Derivation: 

T(deg) = (1/360f₀) = Δt 

The time period corresponding to a 1° phase shift in the wave’s oscillatory frequency, denoted as the time period per degree T(deg), is defined as the inverse of the product of 360 and the wave’s original frequency f₀. This period T(deg) also represents the infinitesimal time shift Δt associated with a 1° phase shift. Consequently, when the wave's original frequency f₀ remains constant, T(deg) is directly proportional to Δt. Thus, T(deg)∝Δt when f₀ is constant.

ΔE = hf₀Δt 

The change in energy, ΔE, is given by the product of Planck’s constant h, the original frequency of the oscillatory wave f₀, and the infinitesimal time shift Δt associated with a 1° phase shift in the wave’s frequency. This relationship implies that ΔE is proportional to the original frequency f₀, and, in turn, f₀ is inversely proportional to the infinitesimal time shift Δt. Therefore, ΔE∝f₀ and f₀∝1/Δ.

Δtₓ = x(1/360f₀) 

The infinitesimal time shift, Δtₓ, associated with an x° phase shift in the wave's oscillatory frequency, is given by the product of the phase shift angle x and the inverse of the product of 360 and the original frequency f₀. This indicates that Δtₓ is directly proportional to the phase shift angle x when the original frequency f₀ remains constant. Therefore, Δtₓ ∝ x when f₀ is constant.

Conclusion:

The time period corresponding to a 1° phase shift in the wave's oscillatory frequency is proportional to the infinitesimal time shift associated with that phase shift, provided the original frequency of the wave remains constant. Additionally, the change in energy is proportional to the original frequency of the oscillatory wave, which is inversely related to the infinitesimal time shift associated with a 1° phase shift. Furthermore, the infinitesimal time shift associated with an x° phase shift in the wave's oscillatory frequency is directly proportional to the phase shift angle x when the original frequency is constant.

• T(deg) ∝ Δt when f₀ is constant.

• ΔE ∝ f₀, f₀ ∝ 1/Δt.

• Δtₓ ∝ x when f₀ is constant.

List of Denotations for Mathematical Terms:

• f₀ or fᴢₑᵣₒ: The original or initial frequency of the oscillatory wave, representing the base frequency at which the wave oscillates.

• h: Planck’s constant, a fundamental physical constant that relates the energy of a photon to its frequency.

• T(deg) or T𝑑𝑒𝑔: The time period corresponding to a 1° phase shift in the wave's oscillatory frequency, indicating the time required for this phase shift.

• x: The phase shift angle in degrees, used to denote a specific phase shift other than 1°, which affects the time increment proportionally.

• ΔE or δE: The change in energy, calculated as the product of Planck’s constant h, the initial frequency f₀, and the infinitesimal time shift Δt.  

• Δt or δt: The infinitesimal time shift (or time distortion) associated with a 1° phase shift in the wave's oscillatory frequency, representing the time increment for a very small phase shift in the oscillation.

• Δtₓ or δtₓ: The infinitesimal time shift (or time distortion) associated with an x° phase shift in the wave's oscillatory frequency, representing the cumulative time increment for a specified phase shift angle x.

Derivation Analysis:

1. Statement 1:

T(deg) = (1/360f₀) = Δt

This equation implies that T(deg), which might represent a time period defined in terms of degrees, is inversely proportional to f₀ (frequency). When f₀ is constant, T(deg) is directly proportional to Δt, which is confirmed by the relationship:

T(deg) ∝ Δt when f₀ is constant.

2. Energy Relationship:

ΔE = hf₀Δt 

Here, ΔE is proportional to the product of f₀ and Δt, not to each variable independently. This means: 

• ΔE depends on the combined effect of f₀ and Δt.

• If f₀ and Δt vary in inverse proportion (i.e., f₀ ∝ 1/Δ), ΔE remains consistent with expected physical behaviour (where higher frequencies correspond to shorter time intervals and vice versa).

3. Frequency-Time Relation:

f₀ ∝ 1/Δt 

This relationship aligns with physical intuition that as frequency f₀ increases, the time interval Δt decreases proportionally. This inverse relationship is essential for the consistency of the expression:

ΔE ∝ f₀, f₀ ∝ 1/Δt. 

4. Scaled Time Interval Δtₓ:

Δtₓ = x (1/360f₀) 

When f₀ is constant, Δtₓ is directly proportional to x, confirming that:

Δtₓ ∝ x when f₀ is constant

Conclusion:

Each statement logically follows from the previous ones, given the inverse relationship f₀ ∝ 1/Δt and the fact that ΔE depends on the product hf₀Δt rather than independently on f₀ or Δt. Thus, the presentation is mathematically consistent, aligning with the physical interpretation that higher frequencies correlate with shorter time intervals.

Interpreting Negative Mass in Quantum Gravity and the Role of Planck Scale in Virtual Particles:

Soumendra Nath Thakur 

27-10-2024

The question of negative mass or negative effective mass, potentially applicable to virtual particles at the quantum level, emerges in the context of Claudia de Rham's radical new theory of gravity.

In my interpretation, this consideration suggests that observable matter—including dark matter—possesses positive mass. We assess this positive mass through baryonic objects and visible energy, as well as the gravitational influence of dark matter on luminous matter. This framework aligns with classical mechanics, where gravity is regarded as an attractive force, or with extended classical mechanics, which also treats gravity as a force.

Conversely, virtual matter may exhibit negative mass, as seen in assessments of dark energy or "invisible energy." This negative mass is responsible for the repulsive force, or "anti-gravity," resulting from a negative effective or apparent mass, as suggested by observational findings and extended classical mechanics.

My interpretation draws on Planck scale concepts, including Planck length, Planck frequency, and Planck time. Since anything above the Planck scale is imperceptible to us and the frequency of any existence ultimately surpasses the Planck frequency, virtual particles likely exist beyond this threshold, suggesting that virtual particles may exhibit extreme gravitational effects.

#virtualparticle #NegativeMass #gravityasforce #antigravity #extendedclassicalmechanics #gravity

ResearchGate discussion: Is Spacetime Curvature the True Cause of Gravitational Lensing?

The discussion link

This discussion questions the conventional explanation of gravitational lensing as a result of spacetime curvature. Instead, it explores an alternative view, proposing that gravitational lensing arises from momentum exchange between photons and external gravitational fields. By analysing the symmetrical behaviour of photons, such as their energy gain (blueshift) and loss (redshift) around massive objects, this perspective challenges general relativity and opens the door to quantum gravity and flat spacetime models. The discussion aims to refine our theoretical understanding of how light and gravity truly interact.

Conceptual Foundation of the Discussion:

A photon, representing light, carries inherent energy denoted as E. As the photon ascends from the gravitational well of its emission source, it loses part of this energy, resulting in a redshift (increase in wavelength, Δλ>0). However, the photon’s behaviour changes significantly when it encounters a strong external gravitational field.

As the photon approaches a strong gravitational body, it undergoes a blueshift (decrease in wavelength, Δλ<0) due to its interaction with the external gravitational field. This shift occurs as a result of electromagnetic-gravitational interaction, causing the photon to follow an arc-shaped trajectory. During this process, the photon’s momentum increases, described by the relation Δρ = h/Δλ, where h is Planck’s constant. This momentum gain reflects the gravitational influence on the photon's trajectory.

Completing half of the arc path (1/2 arc) around the gravitational body, the blueshift transitions into a redshift (Δλ>0) as the photon begins to lose momentum (Δρ=h/Δλ). This process indicates a symmetrical momentum exchange, where the photon experiences a balanced gain and loss of external energy (Eg), preserving symmetry in its overall energy behaviour.

Importantly, while the photon undergoes these external changes in wavelength, momentum, and energy during its trajectory around the gravitational body, it retains its inherent energy (E). The only exception occurs when the photon loses energy (ΔE) while escaping the gravitational well of its source. Thus, despite these external interactions, the photon’s inherent energy remains conserved, except for the loss associated with its initial emission.

After bypassing the gravitational field, the photon resumes its original trajectory, maintaining its inherent energy (E) and continuing unaffected by further gravitational influences.

Soumendra Nath Thakur added a reply

Dear Mr. Preston Guynn Mr. Esa Säkkinen and Mr. Julius Chuhwak Matthew

This discussion addresses the question, "Is Spacetime Curvature the True Cause of Gravitational Lensing?" and critically examines the conventional explanation, which attributes gravitational lensing to spacetime curvature. Instead, it proposes an alternative perspective in which gravitational lensing results from momentum exchange between photons and external gravitational fields. This conclusion is supported by analysing symmetrical photon behaviours, such as energy gain (blueshift) and loss (redshift) near massive objects, which reveal the actual mechanisms driving gravitational lensing—distinct from the spacetime curvature model proposed by general relativity. This discourse aims to refine our theoretical understanding of the fundamental interactions between light and gravity.

The scientific foundation of this perspective, as articulated in the study "Photon Interactions with External Gravitational Fields: True Cause of Gravitational Lensing," rests on established quantum and classical mechanics principles—specifically, Planck’s energy-frequency relation E=hf and the photon momentum-wavelength relation ρ=h/λ. These equations illustrate how photons experience symmetrical energy shifts (blueshift and redshift) through gravitational interactions, offering a basis for lensing that preserves inherent photon energy and frames gravitational influence as an external, rather than intrinsic, interaction.

Together, these equations suggest that momentum exchange between photons and gravitational fields effectively account for lensing effects. The study’s analyses of photon energy conservation and symmetrical behaviour near massive bodies provide an alternative mechanism for gravitational lensing, distinct from the spacetime curvature paradigm and posing a potential challenge to conventional interpretations. This discourse thus emphasizes momentum exchange over relativistic spacetime curvature, aligning with the defined scope and goals outlined in the initial discussion.

For a comprehensive exploration of this study, please refer to the full text at

Preprint Photon Interactions with External Gravitational Fields: True...

and

Data Photon Energy and Redshift Analysis in Galactic Measurements...

Warm regards,

Soumendra Nath Thakur

Relativistic Time Dilation and Phase Discrepancies in Clock Mechanics:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

26-10-2024

Abstract:

This section of the research explores the concept of time dilation within the framework of special relativity, where dilated time t′ exceeds proper time t, impacting conventional clock measurements. By examining clock mechanics, where time is traditionally divided into 12 segments of 30° each, we analyse how time dilation disrupts this design, leading to “errored” time readouts when applied to a clock designed for proper time. Through comparisons with wave phase shifts and frequency, this study introduces a model where each degree of phase shift corresponds to a measurable time distortion. It reveals that a clock built for standard intervals is incapable of accurately reflecting relativistic time dilation, underscoring the challenge of measuring dilated time with conventional systems. This analysis provides insights into the physical limitations of traditional time-keeping devices in representing relativistic effects.

Keywords: Time dilation, proper time, special relativity, phase shift, clock mechanics, relativistic time measurement, time distortion, frequency,

Relativistic Time Dilation and Clock Mechanics:

According to special relativity, time dilation (denoted as t′) exceeds proper time (denoted as t), expressed as t′ > t. Special relativity shifts the concept of time from an abstract, independent quantity to one defined operationally: “time is what a clock reads.” In this framework, a clock measures proper time.

Structurally, a standard clock face divides 360 degrees into 12 equal segments, assigning 30° to each hour (360°/12). When the minute hand completes a full rotation (360°), it marks one hour, correlating the clock’s full rotation to one period, T=360°. Similarly, in wave mechanics, a full cycle of a sine wave spans 360° of phase, establishing a period  T = 360°. The frequency f of a wave is inversely related to its period T, given by T = 1/f. For each degree of phase in a sine wave, time shift per degree is expressed as T/360°, or 

(1/f)/360°. Extending this, for x° of phase, the time shift T(deg) = Δt = (x°/f)/360.

In the case of proper time t, a full oscillation corresponds to T = 360, yielding Δt = 0 by design. However, with time dilation, Δt′ > Δt, making Δt′ > 0. Therefore, for a 1° phase shift in Δt, we get Δt′ = (1° /f)/360°, and for an x° phase shift, Δt′ =(x°/f)/360°.

Applying this to a clock, each hour segment designed for proper time t measures exactly 30° (360°/12). If time dilation Δt′ stretches the interval to 361°, each segment would measure 361°/12 ≈ 30.08°, thus exceeding the clock’s 30° marking for proper time t. Consequently, the clock, designed for proper time, cannot precisely reflect the dilation in t′, resulting in an “errored” time readout.

This demonstrates that time dilation t′ represents a distorted time measurement on a clock originally designed for proper time t, highlighting the misalignment introduced by relativistic time dilation.

#Timedilation, #propertime, #specialrelativity, #phaseshift, #clockmechanics, #relativistictimemeasurement, #timedistortion, #frequency,

Photon Energy and Redshift Analysis in Galactic Measurements: A Refined Approach.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

26-10-2024

This study supplements the research titled ‘Photon Interactions with External Gravitational Fields: True Cause of Gravitational Lensing’ by this author.

A photon, representing light, carries inherent energy denoted as E. As the photon ascends from the gravitational well of its emission source, it loses part of this energy, resulting in a redshift (increase in wavelength, Δλ>0). However, the photon’s behaviour changes significantly when it encounters a strong external gravitational field.

As the photon approaches a strong external gravitational body, it undergoes a blueshift (decrease in wavelength, Δλ<0) due to its interaction with the external gravitational field. This shift occurs as a result of electromagnetic-gravitational interaction, causing the photon to follow an arc-shaped trajectory. During this process, the photon’s momentum increases, described by the relation Δρ = h/Δλ, where h is Planck’s constant. This momentum gain reflects the gravitational influence on the photon's trajectory.

Completing half of the arc path (1/2 arc) around the gravitational body, the blueshift transitions into a redshift (Δλ>0) as the photon begins to lose momentum (Δρ = h/Δλ). This process indicates a symmetrical momentum exchange, where the photon experiences a balanced gain and loss of external energy (Eg), preserving symmetry in its overall energy behaviour.

Importantly, while the photon undergoes these external changes in wavelength, momentum, and energy during its trajectory around the gravitational body, it retains its inherent energy (E). The only exception occurs when the photon loses energy (ΔE) while escaping the gravitational well of its source. Thus, despite these external interactions, the photon’s inherent energy remains conserved, except for the loss associated with its initial emission.

After bypassing the gravitational field, the photon resumes its original trajectory, maintaining its inherent energy (E) and continuing unaffected by further gravitational influences.

The observed symmetry, where photons gain energy as they approach an external gravitational well and lose energy as they recede, could provide critical insights into refining our understanding of spacetime and gravity. This phenomenon challenges the predictions of general relativity, suggesting that the theory may be incomplete or require revision. The symmetrical behaviour of photon energy and momentum around strong gravitational fields aligns with alternative models, such as quantum gravity and flat spacetime theories, which might offer a more comprehensive explanation for these interactions.

This discrepancy between observed photon behaviour and general relativity invites further exploration and refinement of our theoretical frameworks. By engaging with alternative perspectives, we can advance our understanding of the universe’s underlying principles, contributing to a more complete and unified description of reality.

This study assesses photon energy shifts and redshift in determining galactic distances, incorporating both gravitational and cosmic redshifts to refine our understanding of a galaxy's proper distance from Earth. The energy of emitted photons from a star is denoted by E = 4.0 × 10⁻¹⁹ J, with a corresponding frequency f = 6.0368×10¹⁴ Hz. By analysing the gravitational redshift (increase in wavelength) of these photons, we can calculate the light-travelled distance—the distance from the galaxy at the time the light was emitted.

In addition to gravitational redshift, photons undergo cosmic redshift due to the galaxy’s recession, influenced by dark energy’s antigravitational effect. As a result, the galaxy’s proper distance differs from its light-travelled distance, increasing as the galaxy recedes over the photon's transit. This proper distance, distinguishable from the light-travelled distance, accounts for both redshift contributions. To obtain it, we subtract the gravitational redshift from the total observed redshift at the time of reception.

Phase Shift and Energy Variations in Photon Transit

The frequency shift of the photon or its energy change from emission to reception can be quantified by equations that establish a relationship between the degree phase shift T(deg) and time shift  Δt:

ΔE = (2πh/360) × T(deg) × (1/Δt)

​where h is Planck's constant, T(deg) represents phase shift in degrees, and Δt is the time shift. This formula provides the energy change ΔE for a given phase shift, illustrating how frequency and phase adjustments yield incremental energy shifts during photonic transit.

Further, the equation:

Δtₓ = x (1/360f₀)

generalizes time distortion Δt relative to phase shift x, where f₀ is the initial frequency and x represents the degree of phase shift. Here, T(deg) = x confirms that phase shift x in degrees is proportional to T(deg), connecting phase shift and energy fluctuations over photon transit.

Conclusion:

This study highlights the intricate dynamics of photon interactions with gravitational fields and their impact on measuring galactic distances. By examining the photon’s behaviour as it ascends from its source and encounters external gravitational fields, we observe a distinct pattern of redshift and blueshift that arises due to gravitational influences. These interactions reveal that while a photon experiences external wavelength and momentum changes, its inherent energy largely remains conserved, aside from the initial energy loss upon emission.

Integrating these findings with the concepts of gravitational and cosmic redshifts allows for a more accurate determination of galactic distances, distinguishing between the light-travelled and proper distances of receding galaxies. Additionally, by applying phase shift and time distortion equations, we gain insights into the subtle energy variations that photons undergo during their transit. This refined approach suggests that a symmetric model of photon behaviour could bridge existing gaps in general relativity and open pathways for alternative frameworks like quantum gravity and flat spacetime theories. Ultimately, these insights prompt further investigation into the nature of spacetime and gravitational influences, potentially advancing our understanding of the universe’s structure and evolution.