Unified Framework

In Extended Classical Mechanics (ECM), both phase shift and redshift are treated as measurable manifestations of a single underlying process — the temporal and energetic displacement between oscillatory systems. These displacements emerge from cumulative phase modulation, rather than geometric curvature, and express how frequency (f), time difference (Δt), and phase displacement (x°) interrelate within an energy-exchange process governed by the ECM transformation:

ΔMᴍc² = hf

The Phase Shift represents the angular difference between two oscillations. When one signal lags or leads another, the phase shift indicates the time displacement of corresponding points in their cycles. Since a full cycle corresponds to 360°, the relation between phase, frequency, and time is:

x° = 360° · f · Δt

The Redshift (z) expresses this same displacement as a fractional quantity of a complete cycle:

z = x° / 360° = f · Δt

In cosmological and energy-transition contexts, redshift signifies a stretching of oscillation — a decrease in frequency accompanied by a proportional elongation of wavelength or temporal spacing. Within ECM, this proportional change corresponds to a mass-energy redistribution:

ΔMᴍ = −ΔMᵃᵖᵖ

Thus, apparent redshift arises from an effective loss in oscillatory energy density. When the frequency increases, the same delay produces a larger phase difference because oscillations occur more rapidly; conversely, lower frequencies yield smaller phase differences for the same time delay. This relation may also be inverted to yield:

Δt = z / f = x° / (360° · f)

This frequency–time–phase relationship forms one of ECM’s fundamental equivalence bridges, connecting observable redshift phenomena with underlying mass–energy transitions:

KEᴇᴄᴍ = ΔMᴍc² = hf

Hence, what is conventionally interpreted as a wavelength shift is reinterpreted in ECM as a frequency-governed kinetic-mass variation. This demonstrates the continuous convertibility between time distortion, oscillation frequency, and mass-energy states.

The measurable redshift z therefore not only quantifies spectral displacement but also represents the temporal expansion Δt linked to each photon’s energy transition. This unifies cosmological redshift and mechanical phase shift under a single ECM relation, where energy, mass, and time remain inseparably connected.

Relation to the ECM Phase Kernel

The principles above directly extend from and are computationally expressed within the ECM Phase Kernel Formalism — a model demonstrating how cumulative phase modulation and refractive-index variation yield gravitational and relativistic effects without invoking spacetime curvature. The Phase Kernel computes the accumulated phase difference (x° or Φ) between a source frequency (f) and a locally distorted frequency (f ± Δf), using the ECM equation:

T_deg = (x° or Φ) / (f × 360) = Δt_x

Here, T_deg (or Δt_x) represents the phase-derived time distortion, equivalent to Δt_ECM computed through ECM’s core relations. Thus, gravitational and redshift effects can be viewed as manifestations of phase-governed frequency transitions, not as results of geometric deformation.

Understanding Phase Shift, Frequency, and Time Difference

When comparing two repeating oscillations, the phase shift indicates how far one waveform leads or lags another. A complete wave cycle corresponds to 360°, making any phase shift a fraction of a cycle — for example, 180° for half a cycle or 90° for a quarter. The phase shift, frequency, and time difference are therefore interdependent: higher frequencies amplify the phase difference for the same time delay, and lower frequencies reduce it.

The fraction of the cycle displaced — the redshift factor (z) — expresses how much the wave has stretched or shifted proportionally. Hence, the phase shift, frequency change, and time delay are complementary views of one interaction. Each represents how energy, mass, and time adjust across oscillatory systems.

References

1. Soumendra Nath Thakur, Priyanka Samal, Deep Bhattacharjee. Relativistic effects on phase shift in frequencies invalidate time dilation II. TechRxiv. May 19, 2023. DOI: 10.36227/techrxiv.22492066.v2
2. Soumendra Nath Thakur. Extended Classical Mechanics (ECM) Phase Kernel Formalism - Gravity Beyond Spacetime GR's Curvature. YouTube. https://www.youtube.com/watch?v=lCerACGURFU
3. Thakur SN, Bhattacharjee D (2023). Phase Shift and Infinitesimal Wave Energy Loss Equations. J Phys Chem Biophys. 13:365. View PDF