Relativistic events and frames of reference are universal, follow universal applicability:

The Special Theory of Relativity introduces the relativistic frames of reference by introducing an additional frame of reference, mathematically, for presenting relativistic events within the universal frame of reference. 

A co-ordinate geometrical digram of relativistic frames of reference can be presented within the universal frame of reference under the influence of gravity.

As for example, when we present the digit 1, it's magnitude is considered as the difference between 1 and 0 so that  (1-0) = 1, necessarily we do not mention 0 or the difference between them to present the magnitude of 1. 

Similarly, when introducing an additional frame of reference, that would physically mean a mathematical or geometrical introduction of another frame of reference, within the universal frame of reference even when not defined. 

Therefore, relativistic events are relative universal events, and they occur within universal frame of reference, and therefore follow universal applicability. 

The Lorentz transformation is no exception in terms of universal applicability.

Author: Soumendra Nath Thakur, ORCID: 0000-0003-1871-7803

A photon passes near a massive object simultaniously gains and loses momentum:

The momentum of a photon is given by the equation p = E/c, where p represents momentum, E represents energy, and c represents the speed of light. Since the energy of a photon is directly related to its frequency or inversely related to its wavelength, any change in the photon's energy will result in a corresponding change in its momentum.

When a photon passes near a massive object, it simultaniously gains and loses energy (and thus momentum) due to gravitational interactions. This exchange of momentum causes the photon's path to be curved or deflected in the presence of a gravitational field.

Despite the exchange of momentum, a photon continues to travel at the speed of light (c) and covers the same distance (d) relative to its constant speed. The curvature of the photon's path is a result of the gravitational interaction and exchange of momentum.

Redshift due to motion, when phase shift T(deg) of the observed frequency and the source frequency f(rest) are know.:

Redshift due to motion - an alternative formula. 

A 1° phase shift in the wave oscillation is due to motion, we can calculate the redshift using the formula.

z = {1 - 360 * f(rest)} / {360 * f(rest)}.

Where, T(deg) represents the phase shift in degrees and f(rest) represents the source frequency

Frequency and phase in sin:

The equation y(x) = sin(f * x + p). What does f represent and what does p represent?

In the equation y(x) = sin(f * x + p), the variables f and p represent parameters that affect the behavior of the sine function.

f: The variable f represents the frequency of the sine wave. It determines how many oscillations occur over a given interval of x values. The frequency f controls the rate at which the sine function completes a full period, which is 2π radians or 360 degrees. Higher values of f result in more oscillations within the same interval, while lower values lead to fewer oscillations.

p: The variable p represents the phase shift of the sine wave. It indicates the horizontal displacement or translation of the wave along the x-axis. Specifically, it determines the initial position of the wave at x = 0. The phase shift p is measured in radians or degrees and can move the wave to the left or right. Positive values of p shift the wave to the left, while negative values shift it to the right.

By adjusting the values of f and p, you can modify the frequency and position of the sine wave, respectively, in the y(x) equation.

Equations of f frequency; T time period; v velocity; λ wavelength; λ₀ source wavelength; Δλ changed wavelength; T(deg) phase shift in degree; z gravitational or cosmic redshift:

Author ORCID: 0000-0003-1871-7803

Given below are the simplified expressions based on the equation and the relationship between the variables. Where, f is the frequency; T time period; v velocity; λ is the wavelength; λ₀ is the source wavelength; Δλ observed change in wavelength; T(deg) phase shift in degree; z gravitational/cosmic redshift. When,  f = 1/T = v/λ = v/λ₀; T(360) => T(deg) = z * 360; For 1° phase shift, T(deg)  = T/360 = (1/f)/360 = z * 360; z = Δλ/λ₀. 

• f is the frequency.

• We are given f = 1/T = v/λ = v/λ₀.
  

• Therefore, frequency f is equal to the reciprocal of the time period T, the velocity v divided by the wavelength λ, and also v divided by the source wavelength λ₀

• The specific velocities of the waves involved (343 m/s) for acoustic waves or (299,792,458 m/s) for electromagnetic waves.

• T time period.

• We are given a complete time period T in 360°, T(360) => T(deg) = z * 360.

• This equation represents the phase shift in degrees T(deg) being equal to the gravitational or cosmic redshift z multiplied by 360.

• For a complete time period T, the phase shift in degrees T(deg) is equal to z * 360, where z represents the gravitational or cosmic redshift.

• v velocity

• We are given v/λ = f = 1/T = v/λ₀.
  

• From these equations, we can see that v is equal to the product of the frequency f and the wavelength λ, and it is also equal to the product of the frequency f and the source wavelength λ₀.

• The specific velocities for acoustic waves (343 m/s) and electromagnetic waves (299,792,458 m/s) indicate the speed at which the waves propagate.

• λ is the wavelength

• We are given v/λ = f = 1/T =  v/λ₀.

• This equation indicates that the wavelength λ is equal to the velocity v divided by the frequency f, and it is also equal to the source wavelength λ₀.

• The wavelength (λ) is related to the velocity (v) and frequency (f) through the equation λ = v/f. Using the specific velocities provided, the wavelength can be calculated by λ = v/f.

• λ₀ is the source wavelength

• We are given v/λ₀ = v/λ = f = 1/T.

• Therefore, the source wavelength λ₀ is equal to the velocity v divided by the frequency f, and it is also equal to the wavelength λ.

• The source wavelength (λ₀) can be obtained by dividing the velocity (v) by the frequency (f), using the specific velocities given. Thus, λ₀ = v/f.

• Δλ observed change in wavelength

• We are given z = Δλ/λ₀.
  

• This equation represents the gravitational or cosmic redshift z being equal to the change in wavelength Δλ divided by the source wavelength λ₀

• T(deg) phase shift in degree

• We are given T(deg) = T/360 = (1/f)/360 = z * 360.

• This equation states that the phase shift in degrees T(deg) is equal to the time period T divided by 360, which is also equal to the reciprocal of the frequency f divided by 360, and it is equal to the gravitational or cosmic redshift z multiplied by 360. For a complete time period T, the phase shift in degrees T(deg) is equal to z * 360, where z represents the gravitational or cosmic redshift

• z gravitational/cosmic redshift.

• We are given z = Δλ/λ₀.

• This equation indicates that the gravitational or cosmic redshift z is equal to the change in wavelength Δλ divided by the source wavelength λ₀.

These are the simplified expressions based on the given equations and the relationships among the variables.

Reference : Relativistic effects on phaseshift in frequencies invalidate time dilation II 

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