The constancy of light speed, i.e., that velocity, V = c, in a specific reference frame, was first discovered by Hertz. So, using this in the relation V = λν, we get
c = λν (1)
Now, Applying relativity principle to Eq.(1) means, that Eq.(1) is valid for all observers, i.e.,
c = λν (a) and c = λ’ν’ (b) (2)
Since, special relativity theory's (SRT's) assumption of the invariance of light speed is based completely on the concept of time dilation. If time is not delayed as presented by SRT, then light speed invariance cannot be maintained. According to Eqs. (2), one may conclude that the light speed postulate is an inevitable consequence of the relativity principle. Now assume that particle q in frame S, as an ion (source), emits a light wave(photon) that moves at an angle θ with the positive x — axis. Light is received by the observer in frame S' at angle θ' relative to the x’ — axis.
The momentum of the photon in frame S has components;
Pₓ = P Cos(θ), Pᵧ = P Sin(θ), and Pᴢ = 0 (3)
Now, invoking the well known relationship;
E’² - c²P’² = E² - c²P² = m₀²c⁴ (4)
and using the fact that for a photon, the rest mass mo = 0, we obtain from Eqn (4);
E² - c²P² = 0 ⟹ E = Pc (5)
Now, inserting Pₓ = (E/c)Cos(θ) in E' = γ(E — vPₓ), which I derived but cannot explain here. Then we have
E’ = γE[1 – (v/c)Cos(θ)] (6)
But for a photon we know E = hν and E' = hν’
Hence, Eqn(6) becomes;
ν’ = γν[1 – (v/c)Cos(θ)] (7a)
But the connect between frequencies based on reference frame S and S' is given by;
ν = γν’[1 – (v/c)Cos(θ)] (7b)
Now, If we consider frame S to beco-moving with the source and receding/approaching from the observer, Eq (7), becomes;
a). For θ = θ' = 0°, i.e., source is moving away from observer;
ν’ = γν₀[1 – v/c] = ν₀√[(1-v/c)/(1+v/c)] = ν₀√[(1-β)/(1+β)]
b). For θ = θ' = 180°, i.e., source is moving towards observer;
ν’ = γν₀[1 + v/c] = ν₀√[(1+v/c)/(1-v/c)] = ν₀√[(1+β)/(1-β)]
where, γ = 1/√(1-v²/c²), is the Lorentz factor.
ν’ = ν(observed) and ν₀ = ν(source)
Hence, the proof!
The relativistic Doppler effect is the change in frequency (and wavelength, λ) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special theory of relativity (STR). The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry. Astronomers know of three sources of redshift/blueshift: Doppler shifts; gravitational redshifts (due to light exiting a gravitational field); and cosmological expansion (where space itself stretches). This post is only for Doppler's Effect!
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