Redshift and its Equations in Electromagnetic Waves:

DOI: 10.13140/RG.2.2.33004.54403

Soumendra Nath Thakur¹

¹Tagore's Electronic Lab. India¹

postmasterenator@gmail.com¹

postmasterenator@telitnetwork.in¹

18 September 2023

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.

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

T (deg) = 1/ (360f);

T represents the period of the wave. f represents the frequency of the wave. T (deg) 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.

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 T(deg):

z = 360 * T(deg) * ΔE/h

z represents the redshift. T(deg) represents an angle measured in degrees. ΔE represents the change in energy. h represents Planck's constant. 360 * T(deg) * ΔE/h describes the relationship between redshift (z) and phase shift T(deg) in the context of electromagnetic waves and energy changes. The equation suggests that redshift (z) is directly related to phase shift T(deg), 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 T(deg):

z = -Δf * 360 * T(deg)

z represents the redshift. Δf represents the change in frequency. T(deg) represents an angle measured in degrees. -Δf * 360 * T(deg) describes the relationship between blueshift (z) and phase shift T(deg) in the context of electromagnetic waves and frequency changes. The equation suggests that blueshift (z) is directly related to phase shift T(deg) 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 T(deg) as a Function of Redshift (z):

T(deg) = h / (-360 * z * E)

T(deg) 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 T(deg) and redshift (z) in the context of electromagnetic waves and energy changes. The equation suggests that phase shift T(deg) 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 T(deg) as a Function of Blueshift (z):

T(deg) = h / (-360 * z * E)

T(deg): 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 T(deg) and blueshift (z) in the context of electromagnetic waves and energy changes. The equation suggests that phase shift T(deg) 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.

Discussion:

The redshift equation (z = Δλ/λ; z = f/Δf) is a cornerstone in astrophysics and cosmology. It relates the relative change in wavelength (Δλ/λ) to the relative change in frequency (f/Δf) of electromagnetic waves. This equation reveals that as a source emitting waves moves away from an observer, the wavelength increases, resulting in a redshift. Conversely, blueshift occurs when the source approaches, causing a decrease in wavelength.

The phase shift equation (1° phase shift = T/360; T (deg) = 1/(360f)) provides insight into wave behaviour concerning frequency. It demonstrates that a 1-degree phase shift corresponds to a fraction of the wave's period (T), inversely proportional to the frequency (f). This equation is pivotal in telecommunications and signal processing, where precise control of phase is crucial for data transmission and modulation.

Furthermore, we discuss redshift and blueshift in the context of wavelength and frequency changes. Redshift (Δλ/λ) occurs when an object moves away, causing wavelength elongation. Blueshift (-Δλ/λ) arises when an object approaches, leading to wavelength compression. Similarly, redshift (z = f/Δf) and blueshift (z = f/-Δf) are explored concerning frequency changes. These phenomena are instrumental in determining the recessional velocities of celestial objects and are vital for understanding the universe's expansion.

Additionally, we delve into redshift and blueshift as functions of energy changes (ΔE/E). Positive energy changes lead to redshift, reflecting a stretching of waves, while negative energy changes result in blueshift, indicating wave compression. We also discuss the relationship between redshift (z) and phase shift T(deg), highlighting the role of energy changes (ΔE) and the Planck constant (h).

Finally, we examine the relationship between phase shift T(deg) and redshift (z) and blueshift (z), emphasizing their dependence on energy (E) and the Planck constant (h).

Conclusion:

In conclusion, the equations governing redshift, blueshift, and phase shift in electromagnetic waves are essential tools in astrophysics, cosmology, and telecommunications. The redshift equation, with its links to wavelength and frequency changes, provides crucial insights into the expansion of the universe and the velocity of celestial objects. The phase shift equation is fundamental in controlling phase in various applications, from data transmission to signal modulation. Understanding these equations enhances our comprehension of wave behaviour and its implications across diverse scientific disciplines.

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.

The curvature of spacetime is subject to arbitrary recognition as an ultimate truth:

According to special relativity, relativistic time took precedence over abstract time, without necessarily disproving abstract time, but it stripped abstract time of its independent status and established time as a natural phenomenon.

This change allowed for the occurrence of time dilation, where the scale of the dilated time t' can be extended beyond the conventional 360-degree scale of proper time t, represented as t < t'.

Additionally, this shift paved the way for the curvature of relativistic spacetime, facilitating the promotion of relativistic concepts of proper time and spacetime.

In the field of mathematics, time and space are considered mathematical parameters. Space is defined by dimensions such as length, height, and depth, while the concept of an irreversible direction of higher-dimensional time is not natural and runs counter to relativistic spacetime, representing abstract mathematical notions.

Moreover, motion is associated with events occurring within space, following an irreversible course in the context of higher-dimensional time.

As a result, any eventual effects stemming from these considerations do not impact the dimensions of spacetime.

However, it's important to note that the concept of relativistic spacetime does not inherently encompass these ideas.

Consequently, the curvature of spacetime is subject to arbitrary recognition as an ultimate truth.

Photon Acceleration, Absorption, and Time Delay: In question Insights

Photons do not accelerate from 0 to c, as they always travel at c from their creation, while photons do not accelerate to light speed.

Photons always travel at the speed of light (c). This is a fundamental principle of physics. Photons, which are massless particles, always travel at the speed of light in a vacuum (denoted as 'c') from the moment of their creation. This means that photons do not need to accelerate from a standstill (0) to the speed of light, as they are born with this velocity. They move through space at this constant speed.

Photons are the elementary particles of electromagnetic radiation, which includes visible light. When we refer to light, we are essentially talking about photons.

When a photon enters a dense but transparent medium, it is absorbed by an electron in the medium's atom and converted to electron-energy. This destabilizes the electrons, causing them to release excess energy as they do so by releasing photons. This process results in a time delay due to the infinitesimal loss of photon energy, causing photons to travel slower through transparent and dense media. This process contributes to the physics of photons.

Following process involved: E = hf; ΔE = hΔf; f/Δf; E/ΔE; 

When a photon enters a dense but transparent medium, it interacts with electrons (e) within the atoms of the medium.

1. In this interaction, the photon's energy (E) is typically absorbed by an electron, leading to the excitation of the electron to a higher energy state (e+E).

2. This excitation, or destabilization of the electron, is temporary and results in the electron's subsequent return to its original, lower energy state (e).

3. As the electron returns to its lower energy state, it releases the excess energy (E-ΔE). it gained from the absorbed photon in the form of a new photon. This is often referred to as re-emission or scattering.

4. The released photon may have a slightly lower energy (E-ΔE) (higher wavelength) than the initially absorbed photon due to the energy lost (ΔE) during the interaction.

5. The cumulative effect of these interactions with electrons in the medium can lead to a time delay (Δt) for the passage of photons through the medium.

6. This time delay occurs because the re-emission process introduces a delay between the absorption and re-emission of photons.

7. Overall, this process contributes to our understanding of how photons interact with matter and can affect the speed at which light propagates through transparent but dense media.

Summary Paper: Redshift and its Equations in Electromagnetic Waves:

ORCiD: 0000-0003-1871-7803

Dated: 17th September, 2023

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 behavior 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.

Introduction:

The fundamental understanding of electromagnetic wave behavior 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 behavior 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:

The redshift equation (z = Δλ/λ; z = f/Δf) is a cornerstone in astrophysics and cosmology. It relates the relative change in wavelength (Δλ/λ) to the relative change in frequency (f/Δf) of electromagnetic waves. This equation reveals that as a source emitting waves moves away from an observer, the wavelength increases, resulting in a redshift. Conversely, blueshift occurs when the source approaches, causing a decrease in wavelength.

1.2. Phase shift Equation:

The phase shift equation 1° phase shift = T/360; T (deg) = 1/(360f) provides insight into wave behavior concerning frequency (f). It demonstrates that a 1-degree phase shift corresponds to a fraction of the wave's period (T), inversely proportional to the frequency (f). This equation is pivotal in telecommunications and signal processing, where precise control of phase is crucial for data transmission and modulation.

Redshift as a Function of wavelength Change:

We discuss redshift and blueshift in the context of wavelength changes (Δλ/λ). Redshift occurs when an object moves away, causing wavelength elongation, while blueshift arises when an object approaches, leading to wavelength compression. These phenomena are instrumental in determining the recessional velocities of celestial objects and are vital for understanding the universe's expansion.

Blueshift as a Function of wavelength Change:

Blueshift is explored concerning wavelength changes (-Δλ/λ). It occurs when an object moves toward an observer, causing wavelength compression. Calculating the ratio of -Δλ to λ allows us to determine the extent of blueshift.

Redshift as a Function of Frequency Change:

Redshift (z = f/Δf) is discussed concerning frequency changes (Δf). It occurs when an object moves away, causing frequency decrease. Calculating the ratio of "f" to "Δf" allows us to determine the extent of redshift.

Blueshift as a Function of Frequency Change:

Blueshift (z = f/-Δf) is explored concerning frequency changes (-Δf). It occurs when an object moves toward an observer, causing frequency increase. Calculating the ratio of "f" to "-Δf" allows us to determine the extent of blueshift.

Redshift as a Function of Positive Energy Change:

We discuss redshift concerning positive energy changes (ΔE/E). It occurs when an object moves away, causing wavelength elongation. Calculating the ratio of "ΔE" to "E" allows us to determine the extent of redshift.

Blueshift as a Function of Negative Energy Change:

Blueshift is explored concerning negative energy changes (-ΔE/E). It occurs when an object moves toward an observer, causing wavelength compression. Calculating the ratio of "ΔE" to "E" allows us to determine the extent of blueshift.

Redshift (z) as a Function of Phase Shift T(deg):

The relationship between redshift (z) and phase shift T(deg) (z = 360 * T(deg) * ΔE/h) is discussed, highlighting the role of energy changes (ΔE) and the Planck constant (h). When phase shift or energy change increases, it can lead to a corresponding increase in redshift.

Blueshift (z) as a Function of Phase Shift T(deg):

The relationship between blueshift (z) and phase shift T(deg) (z = -Δf * 360 * T(deg)) is explored concerning frequency changes (-Δf). When phase shift or frequency change increases, it can lead to a corresponding increase in blueshift.

Phase Shift T(deg) as a Function of Redshift (z):

The relationship between phase shift T(deg) and redshift (z) T(deg) = h / (-360 * z * E) is examined, emphasizing the role of energy (E) and the Planck constant (h). When redshift increases, phase shift decreases, and vice versa.

Phase Shift T(deg) as a Function of Blueshift (z):

The relationship between phase shift T(deg) and blueshift (z) T(deg) = h / (-360 * z * E) is explored, highlighting the role of energy (E) and the Planck constant (h). As blueshift increases, phase shift decreases, and vice versa.

Conclusion:

In conclusion, the equations governing redshift, blueshift, and phase shift in electromagnetic waves are essential tools in astrophysics, cosmology, and telecommunications. Understanding these equations enhances our comprehension of wave behavior and its implications across diverse scientific disciplines.

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.

Can light speed changes and photon turns into electron-anti electron pair? Questioned.

Answered:

Dear Suvankar Majumder , 

As the photon leaves a gravitational potential well, it does not change its speed but changes its wavelength (λ) and frequency (f), resulting in energy (ΔE) expenditure or, it's infinitesimal wavelength (Δλ) and infinitesimal frequency (Δf) changes, specifying the equation v = λf. Photon energy is defined by Planck's energy-frequency equation. E = hf.

The ratio of Planck length and the Planck time (lp/tp), set the upper speed limit without requiring other constants such as the proportionality constant or universal gravitational constant, the reduced Planck constant, and the speed of light in vacuum. Photons crossing the gravitational potential well can change energy, called gravitational redshift. Photons are massless, so they always travel at speed of light

1. Therefore, according to the above statement, the speed of photon does not vary with time in free space.

A photon is an elementary particle. Elementary particles are either elementary fermions or elementary bosons. A photon is a gauge boson, the carrier of the electromagnetic force. Pair production often refers specifically to a photon producing an electron-positron pair near a nucleus. To produce a pair the photon's contributing energy must be above a threshold of the total rest mass energy of the two particles produced.

2. Therefore, photons can actually become electron and positron pairs, subejct to above said conditions.

Best regards,

Soumendra Nath Thakur