Justification of Angular Representation in Time Dilation

December 05, 2024

Dear Mr. Andrew Marcu,

I appreciate the time and effort you've invested in reviewing my work and providing your detailed critique. Below, I address the concerns you raised, clarifying and reinforcing my research's scientific basis:

1. On Time Dilation and Presentation Consistency

You noted:

“The text asserts that t - t₀ > t′ − t₀ (correct for time dilation), then reverses this with t < t′, introducing internal inconsistency.”

However, your statement that t - t₀ > t′ − t₀ is "correct for time dilation" is itself incorrect. Time dilation enlarges the observed time t′, making t - t₀ < t′ − t₀. This aligns with t<t′, confirming there is no inconsistency in my presentation. The apparent contradiction you highlighted stems from a misinterpretation rather than an actual error in my argument.

2. Angular Representation of Time Scales

You argued:

“The attempt to represent time scales in angular terms (t×360°) is non-standard and lacks a clear physical justification.”

Contrary to your claim, the angular representation has clear foundations in both classical mechanics and wave theory. A standard clock face divides 360° into 12 segments, correlating to periodic cycles in timekeeping. Similarly, in wave mechanics, a full cycle spans 360°, with the time shift for an angular phase x° given as:

T(deg) = Δt = x°/f⋅360°.

This mathematical basis is neither arbitrary nor unconventional but extends directly from established principles. For a dilated time t′, this framework explains how standard clock designs fail to reflect dilation precisely, resulting in "errored" time readouts.

An alternative perspective to your disagreement, which states, 'angular terms (t×360°) is non-standard and lacks a clear physical justification,' is that this objection is unfounded. Structurally, a standard clock face divides 360° into 12 equal segments, assigning 30° to each hour (360°/12). When the minute hand completes a full rotation (360°), it marks one hour, directly correlating the clock’s full rotation to a single period, T=360°.

Similarly, in wave mechanics, a complete cycle of a sine wave spans 360° of phase, establishing a standard period T=360°. The frequency f of a wave is inversely proportional to its period T, expressed as T=1/f. For each degree of phase in a sine wave, the time shift per degree is given by:

T/360°, or equivalently (1/f)/360°

For a phase shift of x°, the corresponding time shift is:

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. Under time dilation, however, Δt′>Δt, resulting in Δt′>0. For a 1° phase shift in Δt, the dilated interval becomes:

Δt′=(1°/f)/360°

For a general x° phase shift:

Δt′=(x°/f)/360°

Applying this concept to a clock, each hour segment, designed to measure proper time t, corresponds to exactly 30° (360°/12). If time dilation causes the interval to stretch to 361°, each segment would then measure approximately 361°/12≈30.08°, exceeding the clock’s designated 30° marking for proper time t.

As a result, the clock, which is calibrated for proper time, cannot accurately reflect the dilated time t′, leading to an “errored” time readout. This demonstrates the validity of angular representation as a practical and scientifically coherent method to illustrate time dilation.

3., Relevance of Classical Mechanics

You contended:

“The discussion of classical mechanics (Hooke’s law, mechanical stress) is irrelevant to relativistic time dilation.”

While relativity primarily addresses time dilation in non-inertial or gravitational contexts, classical mechanics provides insight into the practical implications of mechanical systems, including clock deformations under acceleration or stress. This connection bridges theoretical relativity with real-world clock behaviour, offering a holistic understanding of timekeeping inaccuracies.

4. On Relativity and Non-Inertial Effects

Your statement:

“The claim that relativity does not comprehensively account for forces during acceleration is incorrect.”

While relativity does account for non-inertial effects through proper time calculations, the interplay of such forces with classical mechanics during acceleration is often underexplored in practical applications. My work seeks to address this gap, offering a complementary perspective rather than negating relativity's achievements.

5. General Observations

You described certain phrases, such as "the time dimension originates from and returns to a common point," as vague. These are conceptual expressions aimed at stimulating further thought and should be understood as part of a broader discourse rather than definitive scientific assertions.

Invitation for Further Exploration

To delve deeper into the foundational concepts and evidence supporting my framework, I invite you to review the following research papers:

1. Phase Shift and Infinitesimal Wave Energy Loss Equations http://dx.doi.org/10.13140/RG.2.2.28013.97763

2. Relativistic effects on phaseshift in frequencies invalidate time dilation II http://dx.doi.org/10.36227/techrxiv.22492066.v2

3. Reconsidering Time Dilation and Clock Mechanisms: Invalidating the Conventional Equation in Relativistic Context: http://dx.doi.org/10.13140/RG.2.2.13972.68488

4. Re-examining Time Dilation through the Lens of Entropy: http://dx.doi.org/10.32388/XBUWVD

5. Standardization of Clock Time: Ensuring Consistency with Universal Standard Time http://dx.doi.org/10.13140/RG.2.2.18568.80640

6. Formulating Time's Hyperdimensionality across Disciplines http://dx.doi.org/10.13140/RG.2.2.30808.51209

I hope these works provide clarity and address the concerns you've raised. Should you have further questions or wish to engage in constructive dialogue, I am more than willing to elaborate.

Best regards,

Soumendra Nath Thakur