Derivation: How to calculate redshift using values of phase shift in frequency of wave equation?

Derived by Soumendra Nath Thakur. (ORCID: 0000-0003-1871-7803)

Summary

To calculate redshift using the values of phase shift in frequencies, follow the derivations. Here's a step-by-step explanation for each case:

Gravitational or Cosmic Redshift:

Start with the formula for 1° phase shift: Z = (λ₀/360) / λ₀, where λ₀ is the wavelength at the source.

Rearrange the equation to express λ₀ in terms of frequency: λ₀ = 1/f₀, where f₀ is the frequency at the source.

Substitute the expression for λ₀ into the redshift formula: Z = ((1/f₀)/360) / (1/f₀).

Simplify the equation to obtain the final formula for gravitational or cosmic redshift:

• Z = (1/360f₀) / (1/f₀) = 1/360.

Therefore, for gravitational or cosmic waves, the redshift (Z) can be calculated as 1/360 for each 1° phase shift.

Doppler Redshift:

Begin with the formula for 1° phase shift: Z = (λ(rest)/360 - λ(rest)) / λ(rest), where λ(rest) is the wavelength at rest.

Similarly to the previous derivation, express λ(rest) in terms of frequency: λ(rest) = 1/f(rest), where f(rest) is the frequency at rest.

Substitute the expression for λ(rest) into the redshift formula: Z = ((1/f(rest))/360 - (1/f(rest))) / (1/f(rest)).

Simplify the equation to obtain the final formula for Doppler redshift: 

• Z = (1/360f(rest) - 1/f(rest)) / (1/f(rest)).

Therefore, for Doppler redshift, the redshift (Z) can be calculated based on the given phase shift and the frequencies (or wavelengths) at rest.

In both cases, you'll need to know the frequency (or wavelength) at the source/rest and apply the appropriate formula to calculate the redshift. Additionally, keep in mind the specific velocities of the waves involved, whether it's the speed of sound (343 m/s) for acoustic waves or the speed of light (299,792,458 m/s) for electromagnetic waves.

Description:

The value of a redshift is denoted by the letter Z, corresponding to the fractional change in wavelength, positive for redshifts, negative for blueshifts, and by the wavelength ratio 1 + z, which is >1 for redshifts, <1 for blueshifts. And so, red-shift (z,>1) is the displacement of spectral lines towards longer wavelengths (Δλ+λ₀) > λ₀ i.e. the red end of the electromagnetic spectrum.

Where, velocity of the wave v = fλ, where acoustics waves speed 343 m/s and electromagnetic waves speed 299792458 m/s. 
                                   
For gravitational or cosmic waves, wavelength at the source is λ₀ and observed change in wavelength is Δλ.

The time interval T(deg) for 1° of phase is inversely proportional to the frequency (f). We get a wave corresponding to the time shift.

1° phase shift = T/360.

Since, T = 1/f,

1. Derivation of formula for gravitational and cosmic waves:

For 1° phase shift, T(1°) = T/360 = (1/f)/360 = ΔT. 

Since, λ = T = 1/f.

λ₀ = T₀ = 1/f₀. Where, T₀ is the period and f₀ frequency at the source.

Δλ = T₀/360 = (1/f₀)/360 for gravitational or cosmic waves.

Or, Δλ = λ₀/360 = (1/f₀)/360 

Since, Z = Δλ/λ₀ 

For 1° phase shift, Z = {(1/f₀)/360}/(1/f₀) or, (λ₀/360)/λ₀. formula for gravitational and cosmic redshift... (1).

2. Derivation of formula for Dopplar redshift:

For 1° phase shift, T(1°) = T/360 = (1/f)/360 = ΔT. 

Since, λ = T = 1/f.

λ(rest) = T(rest) = 1/f(rest). Where, T(rest) is the period and f(rest) frequency at the source.

λ(obs) = T(rest)/360 = {1/f(rest)}/360 for Doppler redshift.

Or, λ(obs) = {1/f(rest)}/360 = λ(rest)/360.

And since, Z = λ(obs)-λ(rest)}/λ(rest)

For 1° phase shift, Z = {1/f(rest)}/360 -1/f(rest)} / 1/f(rest) or λ(rest)/360 - λ(rest)}/λ(rest) formula for Dopplar redshift ... (2).

Therefore, depending upon acoustic or, electromagnetic wave, we can calculate respective values of f₀ or f(rest), to obtain respective values of redshift (Z), using vales of respective phase shift in frequencies f₀ or f(rest) for gravitational and cosmic redshifts, or Doppler shift respectively, applying respective velocities of the waves, whether acoustics wave or electromagnetis wave.

Principles of Superheterodyne and Heterodyne:

The superheterodyne receiver is the most common configuration for radio communications. Its basic principle of operation is the translation of all received channels into an intermediate frequency (IF) band where weak input signals are amplified before being applied to a detecto

Heterodyne and superheterodyne principles:

A superheterodyne receiver combines amplification with frequency mixing and is the most popular architecture for a microwave receiver. Heterodyne means mixing two signals of different frequencies together, resulting in a "beat" frequency.

#superheterodyne #Heterodyne

Luminance signal, monochromatic signal, chrominance signal:

The luminance (luma) signal carries information about the brightness or brightness of the video scene and the chrominance signal carries color or chrominance information. Since the human eye's ability to perceive detail is most acute when viewing white light, light transmission carries the impression of fine detail. Luminance as different shades of light in grays while chroma are different hues of color. Colors have intensity while light has brightness. We see color in images because of light. In the absence of light, in total darkness, we do not see any colors.

The luminance signal is composed of a ratio of 30% red, 59% green and 11% blue from the color signal. This combined output becomes the luminance (brightness/monochromatic) signal. It is written as Y. RGB signal derived from camera or telesign by matrix or summation

A monochrome signal (Y signal) is commonly known as a black and white or grayscale signal. Black-and-white displays often use colored backlights such as green, blue, or orange.

A chrominance signal (chroma or C signal) is a signal used in video systems to convey image color information separately from the accompanying luma signal (Y signal)

The chrominance (chrome) signal in NTSC systems is an alternating current of precisely defined frequency (3.579545 ± 0.000010 MHz), which allows accurate recovery at the receiver even in the presence of strong noise or interference. The PAL (Phase Alternation Line) system is similar to the NTSC system in that the chrominance signal is simultaneously amplitude modulated to carry the color saturation (pastel-versus-vibrant) direction and phase modulated to carry the hue direction.

#Signal

Maximum speed of electromagnetic waves "c":

Abstract

The relationship between the speed of light, wavelength (λ), and frequency (f) is given by the equation v = λf, where v represents the speed of the electromagnetic wave. In a vacuum, the speed of any electromagnetic wave is equal to the speed of light, c. Therefore, electromagnetic waves can have various wavelengths and frequencies as long as their product, λf, equals c.

The maximum speed of electromagnetic waves, which is commonly referred to as the speed of light. The speed of light in a vacuum is approximately 3x10^8 meters per second (m/s). This speed is denoted by the symbol "c" and is a fundamental constant of nature.

The Planck length (ℓP) and Planck time (tP) are fundamental units in theoretical physics, derived from fundamental constants of nature. The Planck length is approximately 1.61626×10^−35 meters, and the Planck time is about 5.39×10^−44 seconds. The ratio of the Planck length to the Planck time (ℓP/tP) yields a value close to the speed of light, c:

• ℓP/tP ≈ c

This observation suggests a connection between the Planck scale and the speed of light, although our understanding of physics at the Planck scale is still speculative.

Maxwell's equations, developed in the 19th century, describe the behavior of electromagnetic waves and predict their propagation speed. The equation c = 1/√(ε₀μ₀) relates the speed of light to the electric constant (ε₀) and the magnetic constant (μ₀). The measured value of the speed of light, approximately 2.998x10^8 m/s, is in close agreement with this equation.

The Michelson-Morley experiment, conducted in 1887, aimed to detect the motion of the Earth through a hypothetical luminous ether medium. The experiment's results consistently showed that the speed of light was constant regardless of the Earth's motion, challenging the notion of an ether and leading to the development of Einstein's theory of special relativity.

In summary, the speed of light, which represents the maximum speed of electromagnetic waves, is approximately 3x10^8 m/s in a vacuum. It plays a fundamental role in physics and has been verified through various experiments and theoretical considerations.

Description:

According to our current understanding of physics, the speed of electromagnetic waves, including light, is about 3x10^8 meters per second in a vacuum. This value is usually called the "speed of light" and is denoted by the symbol "c".

v = λf. The speed of any electromagnetic wave in free space is the speed of light c = 3x10^8 m/s. Electromagnetic waves can have any wavelength λ or frequency f as long as λf = c.

Generally speaking, we say that light travels in waves and that all electromagnetic radiation travels at the same speed, which is about 3x10^8 meters per second through a vacuum. This is what we call "the speed of light"; nothing can travel faster than the speed of light in a gravitational field

It is worth noting that the Planck time tP is the time required for light to travel a distance of 1 Planck length = 1.62×10^-35 m in a vacuum, which is a time interval of about 5.39×10^−44 second, and the smallest possible time interval that can be measured.

The Planck length and Planck time are fundamental units in the field of theoretical physics, and they are indeed related to the speed of light in a vacuum

The Planck length, denoted "ℓP" is about 1.61626×10^−35 meters, and the Planck time, denoted "tP" is about 5.39×10^−44 seconds. These values are derived from fundamental constants of nature, such as the gravitational constant, the speed of light, and the reduced Planck constant.

The speed of light in a vacuum, denoted as "c", is about 3x10^8 meters per second. Interestingly, if you divide the Planck length by the Planck time (ℓP/tP), you get a value close to the speed of light:

• ℓP/tP ≈ c.

This observation suggests a fundamental connection between the Planck scale and the speed of light.

According to Max Planck, the speed of electromagnetic waves or light is equal to one Planck length per Planck period; The limit of a photon's travel.

• Planck length = 1.61626×10^−35 m.

• Planck time = 5.39×10^−44 seconds.

• Therefore, c = 3x10^8 m/s

Maxwell's equations, developed in the 19th century, describe the behavior of electromagnetic waves and predict their propagation speed. The equation, c = 1/√(ε₀μ₀), relates the speed of light to the electric constant (ε₀) and the magnetic constant (μ₀). The value of c was measured to be about 2.998 x 10^8 meters per second, which is very close to the currently accepted value.

The Michelson-Morley experiment, conducted in 1887, aimed to detect the motion of the Earth through hypothetical luminous ether, a medium believed to be responsible for the propagation of light waves. However, experimental results consistently show that the speed of light is constant regardless of the direction of the Earth's motion. 

This experiment played an important role in the development of Albert Einstein's theory of special relativity, which introduced the concept of a universal speed limit, the speed of light

However, the electromagnetic fields Maxwell was calculating were a medium for waves, such as waves across the surface of a pond. And the equations show that these waves travel at constant speed. Doing the sums, the speed was about 300000 km s^-1, otherwise known as the speed of light.

c = 1/ (e0m0) ^1/2 = 2.998x10^8m/s. Light is an electromagnetic wave. Maxwell realized this around 1864, when the equation c = 1/ (e0m0) ^1/2 = 2.998x10^8m/s was discovered, since the speed of light was accurately measured. By then, and its agreement with c is unlikely to be coincidental.

Michaelson and Edward W. Morley, in 1887, conducted an experiment known as the Michelson-Morley Experiment to prove that the speed of light was always the same.

What is a phase shift, and how does it relate to the frequency of the wave?

[Author ORCID: 0000-0003-1871-7803]

A phase shift refers to the displacement of a wave form in time. A complete wave cycle, also known as a period (T), corresponds to a phase shift of 360 degrees or 2π radians.

When representing a complete wave cycle in degrees (°), it can be denoted as T(deg). In this notation, T(deg) represents the angular measure of one complete cycle of the waveform in degrees.

In terms of frequency (f), which represents the number of wave cycles per unit of time, there is an inverse relationship between the period and the frequency. The period (T) is the reciprocal of the frequency (f), and the relationship can be expressed as: 

• T = 1 /f

If we express the period in degrees, T(deg), the relationship still holds

• T(deg) = 360° / f

In this case, T(deg) represents the angular measure of one complete cycle of the waveform in degrees, and it is inversely proportional to the frequency (f). Phase shifts can occur under the effects of relative velocities of observers and gravitational potential differences. 

These effects can introduce changes in the perception of time and the behavior of clocks, which may manifest as phase shifts in oscillatory systems and cause errors in time between relative clock oscillations under the effects of both relative velocities an. gravitational potential differences.

#PhaseShift #Frequency #Wave #Oscillation