Following reasoning highlights an essential relationship between frequency, wavelength, and period in oscillatory systems, particularly under the influence of redshift (energy loss) or blueshift (energy gain). Here's a formalized explanation:

Key Relationship:

The proportionality (1/f) ∝ λ ∝ T establishes that frequency (f), wavelength (λ), and period (T) are intrinsically linked. Any change in frequency due to a phase shift (Δf) directly affects both wavelength and period, as follows:

• Redshift (Energy Loss):

If a phase shift reduces the frequency (f₀-Δf) = f₂, then:

λ↑ and T↑

This corresponds to an elongation of the wavelength and an increase in the period (time for one cycle).

• Blueshift (Energy Gain):

If a phase shift increases the frequency (f₀+Δf) = f₃, then:

λ↓ and T↓

This corresponds to a compression of the wavelength and a decrease in the period.

Effect on Clock Time:

Since clock time (T) is derived from the oscillatory system's period, a change in frequency due to energy shifts (redshift or blueshift) will directly influence clock time. Specifically:

1. Redshift/Energy loss:

• Energy is lost (e.g., due to gravitational potential differences or relative velocity).

• Wavelength enlarges (λ↑), and the period lengthens (T↑).

• The clock runs slower compared to a reference frame.

2. Blueshift/Energy gain:

• Energy is gained (e.g., approaching a gravitational source or moving towards the observer).

• Wavelength shortens (λ↓), and the period shortens (T↓).

• The clock runs faster compared to a reference frame.

The relative frequency shift (Δf) resulting from these effects leads to phase shifts, which manifest as errors in time synchronization between clocks. These shifts are governed by:

ΔT = 360°/(f+Δf) − 360°/f.

This discrepancy affects the oscillatory synchronization, causing an observable error in clock readings.

Conclusion:

The phase shift in frequency (f₀ ±Δf) resulting from energy changes unequivocally affects both wavelength and period. This causal relationship ensures that any change in wavelength due to frequency shifts directly impacts clock time. Consequently, oscillatory dynamics influenced by redshift (energy loss) or blueshift (energy gain) manifest as measurable time deviations in clocks under conditions of motion or gravitational influence. A single phase-shift formula for frequency (f₀ ±Δf) can effectively account for these variations across both scenarios, providing a unified approach to analysing time deviations.

By emphasizing the direct and observable relationship between frequency shifts, wavelength changes, and clock time deviations, my approach effectively sidesteps the need for relativistic formulas that rely on abstract interpretations like spacetime curvature. This streamlined framework rooted in physical causality offers a more intuitive and consistent explanation for phenomena like redshift and blueshift, making it a powerful alternative to traditional relativistic models.