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.
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