Redshift and its Equations in Electromagnetic Waves:

Version 2

Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
Tagore's Electronic Lab. India
postmasterenator@gmail.com or,
postmasterenator@telitnetwork.in

July, 23, 2025

Chapter Abstract:

Redshift, a fundamental phenomenon in astrophysics and cosmology, is explored in detail through its governing equations. We delve into equations describing redshift as a function of wavelength and frequency changes, energy changes, and phase shifts. These equations provide insights into the behaviour of electromagnetic waves as sources move relative to observers. The mathematical rigor employed in deriving and interpreting these equations enhances our comprehension of redshift, its role in measuring celestial velocities and universe expansion, and its counterpart, blueshift. The interplay between frequency, wavelength, energy, and phase shift sheds light on this critical aspect of cosmological observation.

Version Note (v2):
This version adds the general phase-time relation Tdeg = x°/360f = Δt, expanding the previous formulation.
It also integrates Planck’s equation (E = hf) with redshift formulations (z = f/Δf, z = ΔE/E), establishing a unified causal interpretation for frequency-based displacement.
This version revises the previous version dated 18 September 2023, and introduces the generalized phase-time equation Tdeg = x° / 360f = Δt, extending the interpretation of phase shift from fixed (1°) to arbitrary x°. The updated discussion unifies oscillator phase distortion and cosmological redshift under ECM frequency-based dynamics.

Keywords: Redshift, Blueshift, Phase Shift, Electromagnetic waves, 

Introduction:

The fundamental understanding of electromagnetic wave behaviour and its relation to various phenomena has been instrumental in advancing astrophysics, cosmology, and telecommunications. This paper explores essential equations governing electromagnetic waves, including the redshift equation, which describes the change in wavelength and frequency as waves propagate through space. Additionally, the phase shift equation sheds light on how wave temporal behaviour is influenced by frequency, playing a critical role in fields like signal processing and telecommunications.

Methods:

In this study, we employ rigorous mathematical derivations to elucidate the key equations governing redshift and phase shift in electromagnetic waves. We analyze these equations, including their relationships with frequency, wavelength, energy changes, and phase shift, to provide a comprehensive understanding of their significance. Our methodology involves detailed mathematical derivations and interpretations to uncover the fundamental principles underlying these phenomena.

Equations and Descriptions:

1.1. Redshift Equation:

z = Δλ/λ; 
z = f/Δf; 

z represents the redshift factor. Δλ stands for the change in wavelength of light. λ represents the initial wavelength of light. f stands for the initial frequency of light. Δf represents the change in frequency of light. This equation relates the relative change in wavelength (Δλ/λ) to the relative change in frequency (f/Δf) for electromagnetic waves. It's essentially expressing the idea that as the wavelength of a wave changes, there is a corresponding change in its frequency, and vice versa, while maintaining a constant speed (c) as per the relationship c = λf, where c is the speed of light.

1.2. Phase shift Equation:

1° phase shift = T/360
Since, T = 1/f, we have:
1° phase shift = (1/f)/360
Tdeg = 1/ (360f); 

T represents the period of the wave. f represents the frequency of the wave. Tdeg represents the period of the wave measured in degrees. The phase shift equation represented as "1° phase shift = T/360," plays a crucial role in understanding the temporal behaviour of waves in relation to their frequency (f). It elucidates that a 1-degree phase shift corresponds to a fraction of the wave's period (T), with the denominator 360 indicating that a full cycle of a wave consists of 360 degrees. To further explore this equation, we can express the wave's period (T) in terms of its frequency (f), leading to the equation "1° phase shift = (1/f)/360." This equation highlights that the phase shift, measured in degrees (°), is inversely proportional to the frequency (f) of the wave. As the frequency increases, the phase shift decreases, and vice versa.

1.3. Generalized Phase Shift-Time Equation:

While the previous formulation considered only a unit phase shift (1°), a more comprehensive expression for phase-induced temporal distortion is:

Tdeg = x° / 360f = Δt

Here, x° represents any arbitrary phase shift in degrees, f is the frequency of the wave, and Δt is the corresponding time distortion. This form extends the understanding of phase shift beyond fixed units and enables calculation of actual time distortion (Δt) introduced by partial-cycle shifts in waveforms or oscillators.

This equation becomes especially relevant when examining propagating electromagnetic waves, where phase shift manifests not just in modulation, but as a shift in frequency (Δf), producing observable redshift or blueshift depending on propagation direction.

2. Redshift as a Function of wavelength Change:

Δλ/λ 

Δλ represents the change in wavelength. λ represents the initial or reference wavelength. Δλ/λ represents the phenomenon of redshift in the context of electromagnetic waves. Redshift occurs when an object emitting waves moves away from an observer, causing the waves to stretch or lengthen, resulting in an increase in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "Δλ" represents the change in wavelength, and "λ" represents the original wavelength of the waves. By calculating the ratio of Δλ to λ, you can determine the extent of redshift. If the value of Δλ/λ is greater than 1, it indicates that the wavelength has increased, which corresponds to a redshift. This is a fundamental concept in astrophysics and cosmology, as redshift is commonly used to measure the recessional velocities of distant celestial objects, such as galaxies, and to study the expansion of the universe.

3. Blueshift as a Function of wavelength Change:

-Δλ/λ 
-Δλ represents the negative change in wavelength. λ represents the initial or reference wavelength. -Δλ/λ represents the phenomenon of blueshift in the context of electromagnetic waves; Blueshift occurs when an object emitting waves moves toward an observer, causing the waves to compress or shorten, resulting in a decrease in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "Δλ" represents the change in wavelength, and "λ" represents the original wavelength of the waves. By calculating the ratio of -Δλ to λ, you can determine the extent of blueshift. If the value of -Δλ/λ is less than 0 (negative), it indicates that the wavelength has decreased, which corresponds to a blueshift.

4. Redshift as a Function of Frequency Change:

z = f/Δf

z represents the redshift factor. f is the observed frequency of light. Δf is the change in frequency from the source to the observer. f/Δf describes the phenomenon of redshift in the context of electromagnetic waves. Redshift occurs when an object emitting waves moves away from an observer, causing the waves to stretch or lengthen, resulting in an increase in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "f" represents the frequency of the waves, and "Δf" represents the change in frequency. By calculating the ratio of "f" to "Δf," you can determine the extent of redshift. If the value of "f/Δf" is greater than 1, it indicates that the frequency has decreased, which corresponds to a redshift.

5. Blueshift as a Function of Frequency Change:

z = f/-Δf

z represents the redshift (or blueshift) factor. f is the observed frequency of light. -Δf is the negative change in frequency from the source to the observer. f/-Δf describes the phenomenon of blueshift in the context of electromagnetic waves. Blueshift occurs when an object emitting waves moves toward an observer, causing the waves to compress or shorten, resulting in a decrease in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "f" represents the frequency of the waves, and "-Δf" represents the change in frequency. By calculating the ratio of "f" to "-Δf," you can determine the extent of blueshift. If the value of "f/-Δf" is greater than 1, it indicates that the frequency has increased, which corresponds to a blueshift.

6. Redshift as a Function of Positive Energy Change:

z = ΔE/E

z represents the redshift factor. ΔE is the change in energy of the radiation. E is the initial energy of the radiation. ΔE/E describes the phenomenon of redshift in the context of electromagnetic waves when there is a positive change in energy (ΔE). Redshift occurs when an object emitting waves is moving away from an observer, causing the waves to stretch or lengthen, resulting in an increase in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "ΔE" represents the change in energy, and "E" represents the initial energy of the electromagnetic waves. By calculating the ratio of "ΔE" to "E," you can determine the extent of redshift. If the value of "ΔE/E" is greater than zero (indicating a positive change in energy), it signifies that the wavelength has increased, corresponding to a redshift.

7. Blueshift as a Function of Negative Energy Change:

z = -ΔE/E

-ΔE The negative sign indicates a decrease or reduction in energy; ΔE represents the change in energy. -ΔE/E describes the phenomenon of blueshift in the context of electromagnetic waves when there is a negative change in energy (ΔE). Blueshift occurs when an object emitting waves is moving toward an observer, causing the waves to compress or shorten, resulting in a decrease in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "ΔE" represents the change in energy, and "E" represents the initial energy of the electromagnetic waves. By calculating the ratio of "ΔE/E," you can determine the extent of blueshift. If the value of "ΔE/E" is less than zero (indicating a negative change in energy), it signifies that the wavelength has decreased, corresponding to a blueshift.

8. Redshift (z) as a Function of Phase Shift Tdeg:

z = 360 × Tdeg × ΔE/h

z represents the redshift. Tdeg represents an angle measured in degrees. ΔE represents the change in energy. h represents Planck's constant. 360 × Tdeg × ΔE/h describes the relationship between redshift (z) and phase shift Tdeg in the context of electromagnetic waves and energy changes. The equation suggests that redshift (z) is directly related to phase shift Tdeg, the change in energy (ΔE), and the Planck constant (h). When the phase shift or energy changes increases, it can lead to a corresponding increase in redshift. Conversely, when the phase shift or energy changes decreases, it may result in a decrease in redshift. This equation has several components.

9. Blueshift (z) as a Function of Phase Shift Tdeg:

z = -Δf × 360 × Tdeg

z represents the redshift. Δf represents the change in frequency. Tdeg represents an angle measured in degrees. -Δf × 360 × Tdeg describes the relationship between blueshift (z) and phase shift Tdeg in the context of electromagnetic waves and frequency changes. The equation suggests that blueshift (z) is directly related to phase shift Tdeg and the change in frequency (Δf). 

When the phase shift or frequency changes increases, it can lead to a corresponding increase in blueshift. Conversely, when the phase shift or frequency changes decreases, it may result in a decrease in blueshift. This equation has several components.

10. Phase Shift Tdeg as a Function of Redshift (z):

Tdeg = h / (-360 × z × E)

Tdeg represents an angle measured in degrees. h is Planck's constant. z represents the redshift. E represents energy.  h / (-360 × z × E) describes the relationship between phase shift Tdeg and redshift (z) in the context of electromagnetic waves and energy changes. The equation suggests that phase shift Tdeg is inversely related to redshift (z) and the energy (E) of electromagnetic waves. 

When redshift increases (indicating that the source is moving away), phase shift decreases, and vice versa. Additionally, the energy of the waves is involved in this relationship, affecting the extent of the phase shift. This equation has several components:

11. Phase Shift Tdeg as a Function of Blueshift (z):

Tdeg = h / (-360 × z × E)

Tdeg: This represents an angle measured in degrees. h: Planck's constant. z: Redshift. E: Energy.

h / (-360 × z × E) 

describes the relationship between phase shift Tdeg and blueshift (z) in the context of electromagnetic waves and energy changes. The equation suggests that phase shift Tdeg is inversely related to blueshift (z) and the energy (E) of electromagnetic waves. As blueshift increases (indicating that the source is approaching), phase shift decreases, and vice versa. Additionally, the energy of the waves is involved in this relationship, influencing the extent of the phase shift. This equation involves several key components.

12. Time Distortion and Phase Shift in ECM Context

In Extended Classical Mechanics (ECM), frequency is the primary representation of energetic vibration, and time emerges as a secondary construct via Δt = x°/360f). This insight distinguishes between:

Local oscillator drift: where −ΔPEᴇᴄᴍ causes measurable Δt

Wave propagation: where frequency shift (Δf) is cumulative over large-scale Δt and corresponds to observed redshift

Therefore, redshift in ECM is not a relativistic time dilation but a consequence of cumulative phase displacement across spatial propagation or energetic imbalance. The generalized Tdeg equation bridges oscillator-based phase control and cosmic frequency dynamics under one unified model.

13. Linking Planck’s Equation with Redshift and Frequency Displacement

In Extended Classical Mechanics (ECM), the redshift equations:

- z = f/Δf  
- z = ΔE/E  

gain added interpretive power when expressed alongside Planck’s equation:  

E = hf

This implies:

ΔE = hΔf ⇒ z = ΔE/E = Δf/f

Thus, all frequency-driven redshift and energy variation equations become mathematically consistent and physically unified under Planck’s formulation.

This reinforces that:
- Energy shifts (ΔE) are manifestations of frequency shifts (Δf),
- Time distortion (Δt) emerges from phase displacement (x°),
- And cosmological redshift is fundamentally a frequency displacement effect, not a relativistic deformation of time itself.

Hence, ECM reframes redshift not as a geometrical expansion artifact, but as a frequency-governed energetic displacement, with E = hf as the causal anchor.

Discussion:

The use of E = hf within these redshift relationships allows direct mapping of Δf and ΔE under a single causal formulation, reinforcing ECM’s emphasis on frequency as the origin of energetic behaviour.

The redshift equation (z = Δλ/λ; z = f/Δf) is foundational in astrophysics and cosmology. It relates wavelength elongation or frequency reduction to the motion of distant sources. When an object emitting waves moves away, the observed wavelength increases—yielding redshift; if it moves closer, wavelength contracts—producing blueshift.

In this version 2, we now include a generalized equation for phase-induced time distortion:

Tdeg = x°/360f = Δt

This clarifies how any degree of phase shift (x°) maps to real time distortion (Δt), especially relevant in interpreting signal modulation, oscillator drift, and wave front delay. While Tdeg = 1/360f is valid for 1°, this general form enables more accurate modeling of dynamic phase behaviours in cosmological and technological contexts.

Further, the connection between redshift and phase shift is enhanced by linking Δt to Δf via accumulated phase shifts in propagating fields. These links are fundamental in ECM’s reinterpretation of redshift phenomena as frequency displacement governed by cumulative phase distortion over Δt.

Conclusion:

Redshift and phase shift equations—now considered as dynamic functions of x°, f, Δt, and E —highlight that frequency displacement is the causal basis for temporal, energetic, and cosmological phenomena in ECM.

This extended formulation incorporates the generalized phase-time equation Tdeg = x° / 360f = Δt, enhancing the interpretative precision of redshift and blueshift phenomena. This version bridges localized phase shift (oscillators, transmitters) with large-scale propagation effects (astronomical redshift), suggesting that frequency is not just a derivative of time, but the origin of temporal structure.

The redshift and phase shift equations—now considered as dynamic functions of x°, f, and Δt —further solidify their role in understanding cosmological expansion, electromagnetic interaction, and signal transmission. ECM’s reinterpretation emphasizes frequency displacement as a causal mechanism, unifying local oscillator dynamics with propagating wave behaviour.

References:

1. Einstein, A. (1915). The Foundation of the General Theory of Relativity. Annalen der Physik, 354(7), 769-822.
2. Peebles, P. J. E., & Ratra, B. (2003). The Cosmological Constant and Dark Energy. Reviews of Modern Physics, 75(2), 559-606.
3. Shen, Z., & Fan, X. (2015). Radiative Transfer in a Clumpy Universe. The Astrophysical Journal, 801(2), 125.
4. Oppenheim, A. V., Willsky, A. S., & Nawab, S. H. (1997). Signals & Systems. Prentice Hall.
5. Proakis, J. G., & Manolakis, D. G. (1996). Digital Signal Processing: Principles, Algorithms, and Applications. Prentice Hall.
6. Thakur, S. N. (2024). Light's distinct redshifts under gravitational and anti-gravitational influences. https://doi.org/10.13140/RG.2.2.26649.22889
7. Thakur, S. N. (2024). A symmetry and conservation framework for photon energy interactions in gravitational fields [Preprint]. Preprints. https://doi.org/10.20944/preprints202411.0956.v1
8. Thakur, S. N. (2023). Rotational phase shift and time distortion in a rapidly rotating piezoelectric system. https://doi.org/10.13140/RG.2.2.24780.32640
9. Thakur, S. N. (2024). Exploration of abstract dimensions and energy equivalence in a 0-dimensional state: II. https://doi.org/10.13140/RG.2.2.32536.98569
10. Thakur, S. N. (2023). Unified quantum cosmology: Exploring beyond the Planck limit with universal gravitational constants. https://doi.org/10.13140/RG.2.2.15061.81121
11. Thakur, S. N. (2023). Dimensional analysis demystified — Navigating the universe through dimensions. Qeios. https://doi.org/10.32388/HNFBGR.2
12. Thakur, S. N. (2025). A deep research analysis of phase distortion vs. propagating shift: A clarification on the nature of time in Extended Classical Mechanics (ECM). https://doi.org/10.13140/RG.2.2.16670.65609
13. Thakur, S. N. (2024). Photon energy and redshift analysis in galactic measurements: A refined approach. https://doi.org/10.13140/RG.2.2.16902.48961
14. Thakur, S. N. (2024). Cosmic speed beyond light: Gravitational and cosmic redshift [Preprint]. Preprints. https://doi.org/10.20944/preprints202310.0153.v1
15. Thakur, S. N. (2024). Exploring symmetry in photon momentum changes: Insights into redshift and blueshift phenomena in gravitational fields. https://doi.org/10.13140/RG.2.2.30699.52002
16. Thakur, S. N. (2024). Photon interactions in gravity and antigravity: Conservation, dark energy, and redshift effects [Preprint]. Preprints. https://doi.org/10.20944/preprints202309.2086.v1