A Classical Physics Approach to Time: Relativistic Time Dilation as Time Distortion.

Soumendra Nath Thakur,

Tagore's Electronic Lab, India. Email: postmasterenator@gmail.com

January 23, 2025

Abstract

This paper presents a critical analysis of the relativistic concept of time dilation, maintaining that it is fundamentally flawed and misrepresents the nature of time. Classical physics, predating relativity, provides a more consistent framework for understanding time through the fundamental constants of Planck length and time, which are derived from classical gravitational principles and Planck’s constant. It is contended that time is not a physically experientable entity but rather an emergent conceptual framework arising from existential events and changes within the universe. Clocks, as mechanical constructs, are subject to distortions caused by external influences such as motion, gravity, and environmental conditions, which relativity incorrectly interprets as time dilation.

Experimental evidence from electronic laboratory studies on piezoelectric crystal oscillators indicates that time distortion occurs due to frequency phase shifts and wavelength elongation rather than genuine relativistic dilation. These findings align with classical principles, where time-related distortions are considered measurement errors rather than fundamental changes in time itself. Consequently, relativistic experiments that claim to support time dilation are maintained to be biased and should not be accepted as valid. The role of time standardization authorities, such as the International Bureau of Weights and Measures (BIPM) and Coordinated Universal Time (UTC), further supports the notion that deviations in time measurements due to relativistic effects are treated as correctable errors rather than evidence of actual time dilation.

Cosmological observations, including the Cosmic Microwave Background (CMB) radiation, further challenge relativistic predictions. The redshift of CMB waves is mentioned through classical concepts of energy loss and expansion rather than gravitational time dilation, supporting the view that the relativistic framework is an unnecessary complexity. Classical physics provides a more accurate explanation of gravitational phenomena without resorting to the concept of curved spacetime.

In conclusion, a return to classical interpretations of time, based on well-established principles in physics, cosmology, and standardized timekeeping practices, provides a clearer and more consistent understanding of time. The relativistic notion of time dilation should be reconsidered in light of experimental findings and logical analysis, favouring classical explanations that emphasize time as a uniform and irreversible succession of events rather than a variable entity dependent on observer motion and gravitational potential.

Keywords: Classical Physics, Time Distortion, Time Dilation, Planck Units, Gravitational Time Effects, Atomic Clocks, Frequency Phase Shift, Wavelength Enlargement, Piezoelectric Oscillators, Measurement Errors in Time, Cosmic Redshift, Standardized Timekeeping, Mechanical Distortions in Clocks, Cosmological Observations,

Introduction

Here’s a list of the key points which collectively present a critical perspective on the relativistic concept of time dilation:

Predating Classical Foundations:

• The fundamental constants of Planck length and Planck time, derived from classical physics principles such as universal gravitation (G) and Planck’s constant (h), predate relativity.

• These predated concepts provide a solid foundation for understanding time without requiring relativistic interpretations.

Time as a Concept, Not a Physical Entity:

• Time is not an experientable physical entity but rather a conceptual framework that emerges from existential events and universal changes.

• Clocks, being mechanical constructs, are susceptible to distortions from external influences like motion, heat, and gravitational potential.

• Relativity misrepresents time distortion errors in mechanical clocks as genuine dilation of time itself.

Experimental Evidence of Time Distortion:

• Studies conducted on piezoelectric crystal oscillators demonstrate time distortion due to phase shifts in frequency and wavelength elongation.

• Time distortion should be understood as measurement errors within clock mechanisms rather than actual relativistic effects.

• These findings provide concrete scientific explanations rooted in classical physics rather than relativistic assumptions.

Critique of Relativistic Experimental Bias:

• Experiments supporting relativity are maintained to be biased and should not be accepted as valid due to their reliance on preconceived interpretations of relativistic effects.

• Measurement discrepancies in clocks due to environmental factors are misattributed to relativistic time dilation rather than acknowledged as mechanical errors.

Cosmological Insights Challenging Relativity:

• The cosmic microwave background (CMB) radiation supports classical explanations of cosmic redshift based on energy loss and wavelength expansion, rather than relativistic gravitational time dilation.

• Observational evidence suggests that CMB redshift results from physical separation of galaxies due to cosmic expansion rather than relativistic effects.

Standardization of Time:

• Time standardization authorities such as the International Bureau of Weights and Measures (BIPM) and Coordinated Universal Time (UTC) recognize deviations in time measurements due to environmental influences as correctable errors.

• Standardized timekeeping principles support the classical view of time as a uniform and irreversible sequence of events, countering relativistic claims of variable time scales.

Conclusion and Re-evaluation of Time Dilation:

• Classical physics provides a simpler and more accurate framework for understanding time that aligns with fundamental scientific principles and experimental observations.

• The relativistic notion of time dilation should be reconsidered in favour of classical interpretations that account for mechanical and environmental influences on clock measurements without assuming fundamental changes in the nature of time itself.

This comprehensive analysis underscores the need to critically evaluate the relativistic framework in light of classical physics, cosmology, and standardized timekeeping practices. By maintaining a scientific perspective rooted in empirical evidence and logical reasoning, a clearer and more consistent understanding of time emerges, challenging the assumptions made by relativity.

Methods

Here’s a list of the key methodological approaches which collectively outline the process of analysing and challenging the relativistic concept of time dilation:

Theoretical Framework Analysis:

• Investigating classical physics principles, including Newtonian mechanics, Planck's fundamental constants, and universal gravitation, to establish a pre-relativistic foundation for understanding time.

• Evaluating the derivation of Planck time and length to demonstrate their independence from relativistic assumptions and their consistency with classical mechanics.

Conceptual Examination of Time:

• Distinguishing between the conceptual emergence of time through existential events versus relativistic assertions of time as a physical entity.

• Analysing definitions of time from authoritative sources such as the International System of Units (SI) and cosmological perspectives to reinforce a universal and standardized understanding of time.

Empirical Review of Experimental Findings:

• Assessing data from laboratory experiments on piezoelectric crystal oscillators to understand frequency phase shifts and their relation to time distortion.

• Comparing gravitational potential differences, motion and mechanical distortions, and thermal effects to demonstrate their role in clock errors, refuting the relativistic interpretation of time dilation.

Critical Examination of Relativistic Experimentation:

• Identifying potential biases in relativistic experiments, focusing on the influence of preconceived assumptions and methodological limitations.

• Analysing discrepancies in experimental setups and measurement interpretations that contribute to misrepresentations of time distortion as time dilation.

Cosmological Data Interpretation:

• Utilizing observations from the cosmic microwave background (CMB) radiation to support classical redshift explanations through Planck’s energy-frequency relation.

• Assessing the implications of cosmic expansion and dark energy in explaining wavelength elongation without relying on relativistic gravitational redshift claims.

Standardization Principles in Time Measurement:

• Reviewing international timekeeping standards, such as Coordinated Universal Time (UTC), to examine how environmental factors are accounted for in time measurements.

• Exploring how discrepancies in clock readings due to motion, gravity, and other influences are classified as correctable errors rather than fundamental alterations in the nature of time.

Comparative Evaluation of Competing Models:

• Contrasting classical and relativistic interpretations of time to identify inconsistencies and validate the classical view as a more accurate and scientifically consistent explanation.

• Synthesizing findings from multiple scientific disciplines—classical mechanics, applied cosmology, and quantum mechanics—to establish a comprehensive understanding of time.

Conclusion and Reassessment:

• Integrating insights from the above methods to reassess the validity of time dilation and propose a scientifically grounded framework that aligns with empirical evidence and classical principles.

• Advocating for a reconsideration of relativistic assumptions in favor of interpretations rooted in well-established physical laws and observational data.

This methodological approach provides a comprehensive framework for evaluating the concept of time, leveraging classical physics, empirical observations, and standardized practices to challenge relativistic interpretations and support an alternative, evidence-based perspective. 

Mathematical Presentation: 

Here’s a structured mathematical presentation incorporating principles from classical physics, cosmology, and time standardization to challenge relativistic interpretations and support the concept of time distortion rather than time dilation.

1. The concept of time is rooted in fundamental constants of nature, which have been established well before the advent of relativity. Planck time and Planck length, derived using the gravitational constant G, Planck’s constant ℏ, and the speed of light c, suggest that time is intrinsically tied to physical processes, rather than an independent entity:

tᴘ = √ℏG/c⁵,  ℓᴘ = √ℏG/c³

These equations, based on classical principles, emphasize that time is not separate but emerges from physical processes and the interrelationship between constants such as c, G, and ℏ.

2. Time Distortion vs. Time Dilation

Relativity introduces time dilation through the equation:

Δt′ = Δt/√(1 − v²/c²)

However, time distortion due to environmental effects such as gravitational potential, mechanical deformation, and thermal fluctuations can be represented more accurately as:

Δt = k⋅Δλ   

Where:

• Δt = observed time error due to wavelength elongation.

• k = −1/c = proportionality constant relating frequency shifts to time measurements. 

• Δλ = phase shift in the oscillation period of electronic or atomic clocks.

The experimental results from piezoelectric oscillators show that distortions in phase frequency due to external influences cause erroneous time readings, which are incorrectly perceived as relativistic time dilation.

Phase shift method:

T𝑑𝑒𝑔 = x/(360f) = Δt.  

This equation represents the time error (Δt) associated with a phase shift of x degrees in an oscillation of frequency f.

Breakdown of Components:
T𝑑𝑒𝑔: The time corresponding to an arbitrary phase shift of x degrees in the oscillation cycle.
x: The phase shift in degrees.
360f: Represents the total number of degrees in a full oscillation cycle (since 1 full cycle = 360 degrees), multiplied by the oscillation frequency f.
Δt: The resulting time shift/error corresponding to the phase shift of x degrees.

3. Gravitational Influence on Clocks (Error Analysis)

Gravitational potential differences cause phase shifts in atomic clocks, which can be expressed as:

Δf = f₀ΔΦ/c²

Δt = 1/(f₀ + Δf)

This shows that frequency shifts due to gravitational potential differences result in distorted time readings, rather than actual time dilation. For instance, GPS satellites experience:

Δtɢᴘꜱ = GM/rc²

 

Where:

• G = gravitational constant,

• M = Earth’s mass,

• r = distance from Earth's centre.

These discrepancies are caused by mechanical errors due to phase shifts, not relativistic time dilation.

4. Cosmic Redshift and Wavelength Expansion

Cosmic observations of the Cosmic Microwave Background (CMB) follow Planck’s energy-frequency relation:

E = hf

Cosmic redshift, driven by universal expansion, results in:

λᴏʙꜱᴇʀᴠᴇᴅ = λꜱᴏᴜʀᴄᴇ(1 + z)

Where:

• z = Δλ/λꜱᴏᴜʀᴄᴇ is the redshift parameter.

This wavelength enlargement corresponds to energy loss over cosmic distances, rather than time dilation. The redshift, a result of universal expansion, contradicts relativistic gravitational redshift predictions. The Doppler effect, caused by the separation of galaxies, better explains this phenomenon:

v = H₀d

Where:

• H₀ = Hubble constant,

• d = distance from the observer.

This confirms that frequency loss corresponds to energy loss, not time dilation.

5. Standardized Time Measurement and Correction

In timekeeping systems such as UTC, discrepancies caused by environmental factors are accounted for as errors. The measured time difference is given by:

ΔT =Tꜱᴛᴀɴᴅᴀʀᴅ − Tᴍᴇᴀꜱᴜʀᴇᴅ

​

Corrective factors are applied to compensate for:

ΔTᴄᴏʀʀᴇᴄᴛᴇᴅ = ΔT − ΔTᴇʀʀᴏʀ 

​

Where:

• ΔTᴇʀʀᴏʀ accounts for motion, gravitational field variations, and mechanical inconsistencies including heat.

This emphasizes that observed discrepancies in time are due to environmental influences rather than a change in the fundamental passage of time.

6. Classical Mechanics and Force Analysis

Classical mechanics, particularly the force equation:

F = (Mᴍ −Mᵃᵖᵖ)·aᵉᶠᶠ 

This equation suggests that what relativistic theories present as time dilation may be better explained by mechanical distortions, not time itself being altered. This aligns with the idea that measurement artefacts are mistaken for actual physical changes in time.

7. Conclusion: Classical Consistency Over Relativity

Combining findings from classical mechanics, cosmology, and standardized timekeeping:

Time = f(existential events)

This equation posits that time is an emergent concept that arises from physical changes and events in the universe, rather than an independent, varying dimension as described by relativity.

This mathematical presentation consolidates classical and empirical foundations for interpreting time as a derived quantity, subject to environmental distortions rather than a relativistic effect of spacetime curvature.

In summary, the mathematical framework outlined here integrates classical principles and experimental observations to demonstrate that the perceived time distortions caused by external factors (gravitational, mechanical, and thermal) should not be confused with the relativistic concept of time dilation. The true nature of time is not fundamentally altered by these effects, but instead, they are environmental artefacts that distort measurements of time.

Science and Religious Influence:

Soumendra Nath Thakur

January 22, 2025

The statement, "According to Pew Research, 85% of people identify with a religion," provides valuable insight into the distinction between science interpreted through a scientific lens and science influenced by religious bias.

The high percentage of religious affiliation among individuals in the scientific community raises concerns about the potential introduction of religious biases into scientific discourse. Religion, being inherently authoritarian, contrasts with the evidence-based and falsifiable nature of science. 

In some cases, renowned scientists are venerated to the extent that their work is considered final and beyond question—an approach that contradicts the core scientific principle of continuous scrutiny and revision. This phenomenon can occur when religious authority influences scientific inquiry.

"A fundamental characteristic of science is that its truths remain constant, independent of personal beliefs. Science relies on empirical evidence and rigorous methods to uncover facts about the universe. These truths exist objectively, unaffected by human opinions or perceptions. This objectivity ensures that scientific discoveries and principles—such as the laws of physics or the theory of evolution—hold true regardless of individual acceptance or cultural perspectives.

This inherent objectivity underscores the importance of science in understanding reality, providing a reliable framework for exploring and explaining the natural world, free from subjective influences or belief systems."

====

Consideration of Scientific Work as Final and Beyond Question:

The  statement in my above post,— 'In some cases, renowned scientists are venerated to the extent that their work is considered final and beyond question...' — was intended to highlight how certain scientific concepts, such as time dilation and curved spacetime, are often accepted without scrutiny due to the near-religious reverence afforded to their proponents.

For example, time dilation, as described by relativity, can be more accurately understood as time distortion—a physical consequence of influences such as heat, mechanical forces, motion, and gravitational potential differences. These relativistic effects impact the oscillation of clocks, leading to measurable differences, but they do not imply that time itself is a physical entity. Similarly, gravitational lensing, often attributed to the curvature of spacetime, can be explained as the bending of light due to energy and momentum exchange within the gravitational field, rather than an actual warping of spacetime itself.

The idea that space and time are physical entities stems from a misunderstanding. In reality, time and space are abstract, mathematical constructs, not physical entities. Time emerges as a consequence of existential events—events invoke time, not the other way around. Likewise, spatial dimensions such as length, height, and depth are abstract extensions, and neither space nor time possess inherent physicality. Clocks, for instance, merely provide a physical representation of the abstract concept of time.

Despite these clear distinctions, certain scientific figures are often revered in a way that makes their work seem beyond question, much like religious doctrines. This cultural phenomenon can hinder the necessary scientific process of continuous scrutiny and revision, reinforcing preconceptions that should instead be subject to rigorous analysis and falsifiability.

Extended classical mechanics:

Soumendra Nath Thakur

January 21, 2025

Extended classical mechanics is a framework that expands on the traditional principles of classical mechanics to include new concepts and findings. This framework is used to study the relationships between mass, energy, and gravitational dynamics. 

Concepts in extended classical mechanics

Effective mass

The net mass of a system, which includes rest mass and dynamic energy-dependent properties 

Apparent mass

The dynamic mass of a photon, which arises from its energy and momentum characteristics 

Negative effective mass

A concept that extends beyond classical mechanics to represent a form of potential energy 

Examples of extended classical mechanics

Equivalence principle

A central principle of classical mechanics that states that gravitational and inertial mass are indistinguishable 

Dark matter

A concept that challenges the classical framework and extends it to include dark matter dynamics 

Dark energy

A concept that extends beyond classical mechanics to represent a form of potential energy 

Why extend classical mechanics?

Extended classical mechanics (ECM) is a framework that incorporates dynamic mass properties to classical mechanics.

New concepts like dark matter and dark energy challenge the classical framework 

Observational phenomena, like the accelerated expansion of the universe, support extending classical mechanics.

ECM is used to better understand physical phenomena, such as the force that accelerates the cosmos. ECM can also be used to study the relationship between energy, mass, and gravitational dynamics. 

A Critical Examination: Time Dilation is an Error in Clock Time.

Soumendra Nath Thakur

January 21, 2025

Abstract

This paper critically examines the concept of time dilation as proposed by Einstein’s theories of relativity and maintains that it is fundamentally an error in clock readings rather than a physical reality of time itself. While Special and General Relativity suggest that time slows down due to relative motion and gravitational potential differences, this interpretation overlooks the principles of standardized timekeeping established by authoritative bodies such as the International Bureau of Weights and Measures (BIPM) and the International System of Units (SI). Recent experimental findings on piezoelectric crystal oscillators and photon behaviour in gravitational fields indicate that factors such as heat, mechanical forces, motion, and energy dissipation lead to phase shifts and frequency variations in clock mechanisms, which result in erroneous time readings. This paper asserts that relative time is an artifact of physical changes in measurement devices and not an intrinsic property of the universe. Adhering to standardized guidelines for clock time measurement is essential to avoid misinterpretation of such discrepancies as time dilation

1. Introduction

According to Einstein’s theory of relativity, time dilation is considered a fundamental aspect of spacetime behaviour, arising from relative motion and differences in gravitational potential.

Special Relativity states that time slows down for objects moving at high velocities relative to an observer.

General Relativity states that clocks in stronger gravitational fields tick slower than those in weaker fields.

However, the theory not only disregarded classical interpretations of time but also overlooked the prevailing standards for clock time measurement at the time.

2. Standardized Timekeeping and Its Importance

Standardized timekeeping aims to achieve a single, consistent reference time across different locations and conditions, following the guidelines set by authoritative bodies such as the International System of Units (SI). In standardized time systems, such as Coordinated Universal Time (UTC), discrepancies in measurements due to environmental factors—including heat, mechanical forces, motion, and gravitational effects—are considered errors, as they cause deviations from the expected standardized value.

Nonetheless, all scientific disciplines, including relativity, must adhere to standardized time measurement principles. Organizations such as the International Bureau of Weights and Measures (BIPM), which existed prior to the introduction of the time dilation concept, and current standards such as the SI second—defined by atomic transitions—ensure precise definitions of time.

Furthermore, the constancy of the time scale in relation to entropy is a well-established principle.

3. Experimental Findings and Observational Evidence

Recent experimental findings on piezoelectric crystal oscillators, along with observational data on photon behaviour within curved gravitational fields—distinct from the concept of curved spacetime—and the constancy of entropy in the time scale, collectively support the conclusion that time dilation is fundamentally an error in clock readings. These findings suggest that infinitesimal energy loss leads to frequency shifts and phase changes in clock oscillations, which have been misinterpreted as time dilation.

4. The Reinterpretation Against Time Dilation

Through these experimental and observational findings, it is maintained that energy dissipation within clock mechanisms results in phase shifts and frequency variations, ultimately leading to perceived discrepancies in time that are mistakenly attributed to relativistic effects.

This research scientifically asserts that relative time is not an intrinsic property of the universe but rather an artifact of physical changes—such as heat, mechanical forces, motion, and gravitational effects—within clock mechanisms. It further emphasizes that any valid scientific approach must align with standardized guidelines for clock time measurement to ensure accuracy and consistency.

5. Conclusion

In essence, relative time emerges from relative frequencies. The phase shift in relative frequencies, caused by infinitesimal energy loss and the corresponding elongation of oscillation wavelengths, occurs in any clock operating between different relative locations due to relativistic effects or variations in gravitational potential. These shifts result in errors in clock time readings, which have been incorrectly interpreted as time dilation.

List of Technical Terms

1. Time Dilation – The relativistic concept that time slows down under high velocities or strong gravitational fields.

2. Special Relativity – Einstein’s theory stating that time slows for objects moving at high speeds relative to an observer.

3. General Relativity – Einstein’s theory stating that gravitational fields influence the passage of time.

4. Standardized Timekeeping – A system of time measurement ensuring consistency across locations, established by organizations like BIPM.

5. SI Second – The internationally accepted unit of time measurement, defined by atomic transitions.

6. Entropy and Time – The principle that time progresses consistently in relation to entropy.

7. Piezoelectric Crystal Oscillator – A device used in precision timekeeping that experiences phase shifts due to environmental factors.

8. Phase Shift – A change in the phase of an oscillating signal, affecting the perceived frequency.

9. Frequency Shift – A change in oscillation frequency due to external factors like energy loss.

10. Gravitational Potential – The potential energy per unit mass in a gravitational field, influencing physical processes like clock mechanisms.

The Impact of Cosmic Expansion: Energy Loss and the Limits of Light in the Observable Universe.

Rev.ver-1

Soumendra Nath Thakur

Tagore's Electronic Lab, India, postmasterenator@gmail.com

January 18, 2025

Abstract:

This paper explores the impact of cosmic expansion on the observable universe, focusing on the dissipation of photon energy and the limits of light propagation over vast distances. Beginning with the Big Bang, the universe's expansion has driven the recession of distant galaxies and the stretching of light's wavelength, leading to redshift. Hubble's Law governs the recession velocity of galaxies, and as this velocity exceeds the speed of light (c) beyond a certain distance, light from these regions becomes unobservable. The paper explains how photons, initially emitted with a specific energy, lose this energy over time due to the stretching of space driven by dark energy. As the universe expands, photons experience cumulative energy loss and eventually lose all their energy when traveling distances around 46 billion light-years, marking the observable universe's boundary. This analysis integrates Hubble's Law, redshift, and dark energy's role in the expansion, offering a comprehensive understanding of how cosmic expansion limits the reach of light and shapes the observable universe's horizon.

Keywords: Cosmic Expansion, Redshift, Hubble's Law, Observable Universe, Dark Energy, Photon Energy Loss, Recession Velocity, Light Propagation, Universe's Horizon, Energy Dissipation

Introduction:

The Big Bang theory suggests that the universe began 13.8 billion years ago as a hot, dense singularity. Light from the cosmic microwave background (CMB) travelled 13.8 billion light-years to reach Earth, demonstrating that light can only traverse 13.8 billion light-years in 13.8 billion years. This distance, known as the "light-travelled distance," was the only possible light to travel before the Big Bang event.

Light emitted before the "light-travelled time" becomes unreachable to observers from sources whose recession distances exceed the "light-travelled distance." This happens when the relative recession speed between the source and the observer exceeds the speed of light. However, the expansion of intergalactic space due to dark energy's anti-gravity effect increases the physical separation between distant sources and observers. This results in a light source emitting photons 13.8 billion years ago being located significantly farther than the light-travelled distance, corresponding to the "proper distance" of the source.

The observable universe spans 46 billion light-years, representing the current distance to the farthest regions that emitted light. Previously, these regions were closer, but space expansion has extended the distance between them and us.

Photons' energy loss due to cosmic expansion reduces their ability to propagate as electromagnetic waves, making them ineffective at reaching observers. This energy dissipation is a key factor in the observed 46-billion-light-year horizon of the universe.

The initial energy of a photon emitted from a star, measured at 4.0 × 10⁻¹⁹ joules, decreases over time due to the photon's frequency decreasing with increasing redshift, a phenomenon linked to the expansion of intergalactic space driven by dark energy.

The photon loses all its energy over vast distances, such as the ~46 billion light-years defining the observable universe's horizon. Initially emitted with a wavelength of 4.9661 × 10⁻⁷ meters, its wavelength stretches toward infinity, rendering it incapable of propagating as an electromagnetic wave. This energy dissipation reflects cosmic expansion, preventing photons from sources beyond this horizon from reaching observers.

The universe's 46-billion-light-year radius represents the farthest light travel since the Big Bang, highlighting the limitations of cosmic expansion and the propagation of light across immense distances due to dark energy's influence.

The observable universe, which has allowed light to reach Earth since the Big Bang, is expanding due to dark energy. The proper distance to the farthest regions is 46 billion light-years due to galaxies receding from us due to their own motion and space stretching. This expansion has affected the distance to Earth.

The observable universe, which has been expanding for 13.8 billion years, has seen significant redshift in light emitted from distant sources. This phenomenon explains why we can observe light from objects up to 46 billion light-years away, despite their initial distances being smaller. The observable universe represents the limit of what we can detect, with cosmic expansion playing a pivotal role in defining this horizon.

Scientific and Mathematical Presentation

1. Hubble's Law and Physical Distance (d):

Hubble's Law, v = H₀⋅d, describes the recession velocity (v) of galaxies in terms of the Hubble constant (H₀) and the distance (d) between objects.

For distant galaxies, v can exceed the speed of light (c) due to the increase of spatial distance between objects, making the source's light unreachable to the observers. 

• d represents the measurable physical separation between objects, consistent with classical mechanics.

2. The Concept of Relative Recession Speed and Distance:

Given the classical perspective that space is Euclidean and abstract, the question can be addressed using Hubble's Law as a straightforward proportional relationship between recession speed and distance.

Hubble's Law:

v = H₀⋅d

• v = recession speed, 

• H₀ = 70 km/s/Mpc = 2.268 × 10⁻¹⁸ s⁻¹ (Hubble constant)

• d = proper distance between the objects.

Solving for Distance When v = c:

Let v = c = 3 × 10⁸ m/s. Then:

d = v/H₀ = 3 × 10⁸ m s⁻¹ / 2.268 × 10⁻¹⁸ s⁻¹ = 1.32 × 10²⁶ m 

Converting to Light-Years:

d = 1.32 × 10²⁶ m × (1 light-year/9.461 × 10¹⁵ m) 

d = 13.93 billion light-years 

Implication: At a proper distance of 13.93 billion light-years, the recession speed reaches c, marking the Hubble radius. Beyond this distance, the recession speed exceeds c.

3. The Connection Between Hubble's Law and Redshift

Hubble's Law, expressed as v = H₀⋅d, describes the recession velocity (v) of galaxies due to the expansion of the universe. This law establishes a direct proportionality between the recession speed and the distance of a galaxy from an observer, where H₀ is the Hubble constant and d is the proper distance.

As the universe expands, the space between distant galaxies stretches, causing the light emitted by these galaxies to undergo cosmological redshift. The wavelength of the photon increases in proportion to the expansion of the universe, leading to a redshift (z) given by the relation:

1 +z = λᴏʙꜱᴇʀᴠᴇᴅ/λꜱᴏᴜʀᴄᴇ.

​This redshift reflects the velocity at which a galaxy is receding from the observer. As the distance increases (driven by cosmic expansion), the recession speed also increases according to Hubble's Law, causing a greater redshift. For objects at distances where the recession speed exceeds the speed of light, the redshift becomes extreme, and the photon’s wavelength stretches significantly, eventually rendering the photon undetectable. The redshift is, therefore, a direct consequence of the recession velocity predicted by Hubble's Law, linking the observable stretching of light to the dynamics of cosmic expansion.

4. Addressing 46 Billion Light-Years:

For a proper distance of 46 billion light-years, the corresponding recession speed is:

v = H₀⋅d

v = 2.268 × 10⁻¹⁸ s⁻¹·(46billion light-years)·(9.461 × 10¹⁵ m / light-year)

v = 9.83 × 10⁸ m·s⁻¹   

This speed is approximately 3.28 times the speed of light (v > c).

Conclusion:

• The recession speed between two objects separated by 13.93 billion light-years equals c.

• For objects separated by 46 billion light-years, the recession speed is significantly greater than c.

• Therefore, the 46 billion light-year distance supports the claim that the relative recession speed exceeds c.

5. Dark Energy and Increasing Separation:

While Hubble's Law treats d as static, the repulsive effects of dark energy dynamically increase physical separations, causing galaxies to recede and inducing the observed cosmic redshift. The expansion is not just due to motion through space, but also because of the repulsive effects of dark energy, which accelerates the expansion.

6. Redshift and Distance:

The cosmological redshift z arises from the increasing separation between galaxies, with the wavelength stretching proportionally:

1 + z = λᴏʙꜱᴇʀᴠᴇᴅ/λꜱᴏᴜʀᴄᴇ.

 

This redshift reflects the physical motion of galaxies, consistent with classical mechanics.

7. Photon Energy Loss in Expanding Space:

The energy of a photon E = h⋅f diminishes as its frequency f decreases with increasing z:

f ∝ 1/ (1+z).

This represents the cumulative energy loss as photons traverse expanding distances. As the universe expands, the photon’s frequency decreases, and thus, its energy diminishes.

8. Observable Universe and Energy Limits:

The observable universe's radius (~46 billion light-years) corresponds to the farthest distance light has travelled since the Big Bang, incorporating the cumulative effects of dark energy-driven expansion. This distance reflects the observable limit, where photons lose energy to extreme redshift.

9. Mathematical Horizon:

The radius of the observable universe can be expressed as:

dᴏʙꜱᴇʀᴠᴀʙʟᴇ = c ∫(0 to tₚᵣₑₛₑₙₜ) {1/a(t)H(t)} dt

where a(t) is the scale factor, and H(t) is the Hubble parameter. This integral reflects the influence of cosmic expansion on photon travel, considering the dynamics of space expansion over time.

10. Total Energy Loss of a Photon Over ~46 Billion Light-Years:

Given:

• Initial photon energy: E₀ = 4.0 × 10⁻¹⁹ J

• Initial wavelength: λ₀ = 4.9661 × 10⁻⁷ m

• Redshift relation: λ ∝ 1 + z

• Energy-wavelength relationship: E = hc/λ

• At redshift z → ∞ (corresponding to the photon traveling the maximum observable distance), λ→∞, and thus E → 0.

Energy loss derivation:

Photon energy at a given redshift:

E = hc/λ = E₀/(1 + z)

where E₀ = hc/λ₀ is the initial energy of the photon.

Energy loss due to redshift: The energy loss ΔE as the photon redshifts from z = 0 (initial) to z → ∞ is the difference between the initial energy E₀ and the final energy Eꜰ:

ΔE = E₀ − Eꜰ

For z → ∞, Eꜰ → 0, so:

ΔE = E₀ 

Wavelength increase due to redshift: 

The wavelength increases with redshift as:

λ = λ₀(1 + z)

For z → ∞, λ → ∞, and the photon energy E approaches zero.

Cosmic expansion and photon energy dissipation: The cumulative redshift z results from the scale factor a(t), which describes the universe's expansion over time:

z = 1/a(t) − 1

The total energy loss due to this redshift corresponds to E₀, meaning the photon loses all its energy as z → ∞.

Conclusion:

The total energy loss ΔE of a photon traveling ~46 billion light-years, assuming the wavelength increases to infinity due to cumulative redshift, is equal to its initial energy:

ΔE = 4.0 × 10⁻¹⁹ J

This result highlights the complete energy dissipation of the photon, rendering it incapable of reaching observers at distances corresponding to extreme redshifts (z → ∞). This dissipation aligns with the observed limit of the universe's horizon at ~46 billion light-years.

10. Consistency with Hubble's Framework:

• v = H₀⋅d remains valid, with d interpreted as dynamic physical distance.

• Dark energy's repulsive effects reconcile the equation with modern observations, without invoking additional variables for "space stretching."

Logical Implications

1. Unreachable Light:

Light from sources receding faster than c due to cosmic expansion becomes undetectable.

2. Dark Energy's Role:

Dark energy accelerates cosmic expansion, increasing separations and stretching photon wavelengths, leading to redshift and energy loss.

3. Energy Depletion and Photons:

Over vast distances, cumulative energy loss renders photons incapable of maintaining wave properties, explaining the limits of the observable universe.

Conclusion: This analysis integrates Hubble's Law, redshift, and dark energy effects into a cohesive framework, explaining why light from extremely distant sources becomes unobservable. It preserves the classical mechanics foundation while aligning with modern cosmological observations, highlighting the interplay between energy loss, cosmic expansion, and the observable universe's horizon.

Discussion:

This paper offers an in-depth analysis of how the expansion of the universe, driven by dark energy, influences the propagation of light and ultimately sets a boundary to the observable universe. By integrating concepts like Hubble's Law, redshift, and photon energy loss, the paper constructs a comprehensive understanding of how cosmic expansion limits the reach of light, shaping the observable universe's horizon.

Cosmic Expansion and Its Role in Light Propagation

The paper starts by exploring the basic framework of cosmic expansion, which stems from the Big Bang event 13.8 billion years ago. The continuous stretching of space not only causes galaxies to recede from one another but also affects the light traveling across the universe. As space expands, the distance between distant galaxies increases, causing the wavelength of the light they emit to stretch, which results in redshift. The paper effectively links this concept with Hubble's Law, which establishes the relationship between recession velocity and distance. When the recession speed between an observer and a galaxy exceeds the speed of light, the light emitted by that galaxy becomes unreachable.

An important aspect addressed in this study is how the expansion of space, particularly the role of dark energy, accelerates this process. The repulsive effect of dark energy increases the rate at which galaxies recede from one another, leading to the stretching of light over vast distances. As light travels through this ever-expanding universe, its wavelength increases, causing a gradual loss of energy. Over sufficiently long distances, the photon eventually loses all its energy, rendering it undetectable.

Redshift and Energy Loss

A central point in the paper is the relationship between redshift and photon energy loss. The study delves into how the energy of a photon diminishes as its frequency decreases in proportion to increasing redshift. This phenomenon is particularly significant for the light emitted by distant galaxies. Over distances of approximately 46 billion light-years, the cumulative effect of redshift causes photons to lose all their energy, making it impossible for them to reach observers. This leads to the conclusion that the observable universe is defined not just by the distance light has travelled since the Big Bang, but also by the point at which light's energy dissipates entirely.

The mathematical treatment of photon energy loss is another strong point of the paper. The analysis of energy dissipation over the vast expanse of the observable universe is well-articulated through the use of the energy-wavelength relationship E = hc/λ. The calculation of photon energy loss as redshift increases to infinity reinforces the notion that the observable universe's limit is not simply a function of distance, but is intrinsically linked to the energy limitations imposed by cosmic expansion.

Hubble's Law, Recession Speed, and Observable Universe's Horizon

The paper makes an insightful connection between Hubble's Law and the concept of the observable universe. Using Hubble's Law, the recession velocity of galaxies is shown to increase with distance, leading to the conclusion that at a certain distance, this velocity exceeds the speed of light. This marks the boundary beyond which light cannot be observed. The analysis goes a step further to explain how the observable universe extends to 46 billion light-years, which is the proper distance accounting for cosmic expansion and the effects of dark energy.

While the recession speed at 13.93 billion light-years matches the speed of light, the paper clearly outlines that at distances beyond this, recession speeds exceed c, and thus light from such regions becomes unattainable. The further implication that the universe’s 46 billion light-year radius accounts for the cumulative stretching of space helps explain the complex relationship between light travel and cosmic expansion.

Role of Dark Energy

Dark energy plays a crucial role in this framework by accelerating the universe's expansion. The paper emphasizes that the repulsive force of dark energy not only contributes to the increase in galaxy separation but also intensifies the redshift and energy dissipation experienced by photons. The link between dark energy and the observable universe's boundary is made explicit, providing a more nuanced understanding of how cosmic expansion shapes the limitations of light propagation.

Dark energy’s influence is not only described qualitatively but is also shown mathematically in the treatment of recession speeds and the resulting redshifts. This approach ties together classical mechanics and modern cosmological theories, demonstrating how space's expansion alters the properties of light over vast distances.

Mathematical Framework and Photon Energy Dissipation

The mathematical treatment of photon energy dissipation is thorough and precise, offering insights into how light from distant sources loses energy as it travels through an expanding universe. The derivation of the energy loss formula highlights the direct relationship between redshift and photon energy, culminating in the conclusion that photons traveling distances of ~46 billion light-years lose all their energy. This not only helps explain the observable universe's horizon but also provides a clear understanding of the limitations of light propagation across immense distances.

Conclusion

This paper offers a comprehensive and mathematically rigorous discussion of how cosmic expansion impacts the propagation of light and sets a fundamental limit on the observable universe. By integrating key concepts such as Hubble's Law, redshift, photon energy loss, and dark energy's role, the paper provides a holistic view of the factors that govern the observable universe’s boundaries.

The study reinforces the idea that the observable universe's horizon is not solely a result of distance but also a consequence of the energy limitations of light due to cosmic expansion. The paper succeeds in bridging classical mechanics with modern cosmological insights, offering a solid framework for understanding the interplay between energy loss, cosmic expansion, and the ultimate reach of light. The conclusions drawn have important implications for our understanding of the universe's structure and the fundamental limitations imposed by the expansion of space.