Generalized Doppler Effect
C. P. Viazminsky Department of Physics University of Aleppo
Aleppo-Syria Kayssarv@mail2world.com

Let S ≡ OXYZ and s ≡ oxyz be inertial frames in standard configuration, and assume that s translates parallel to OX with a constant velocity u (u>0). Let b be a source of light that is stationary in s, and hence moving with a constant velocity u relative to S. suppose that the source b is radiating a monochromatic light of wavelength λ . This will be received by o as monochromatic light of the same wavelength. Let (R,θ ,ϕ) and ( (r,θ ,ϕ) be the spherical coordinates of the source b in S and s respectively.
The moment at which light first reaches the contiguous observers o and O corresponds to r=ct. Setting r=ct in the generalized Lorentz transformations [1] yields

(1)

R = γr ( 1− β 2 sin 2 θ r + β cosθ )

Now assume that the distance r in the moving frame corresponds to one wave-length,
i.e. r = λ . With respect to the observer O the distance R corresponds to one wavelength λ′ .

Generalized Doppler's formula
Substituting in the last equation r = λ and R = λ′ , we obtain:

(2)

λ′ = γ ( 1− β 2 sin 2 θ + β cosθ )λ,

with γ = 1/ 1− β 2 , which determines the wave length as measured by the
stationary observer. Note that the radiating source here is at a position of azimuth
angle θ , and that the polar axis is OX.

Longitudinal Doppler's Formula
Setting θ = 0 in the generalized formula (1) we obtain

(3)

λ′ = 1+ β λ

1− β

which is the red shift Doppler's formula, corresponding to the source and the observer
receding from each other. For θ = π we obtain the blue shift Doppler's formula

(4)

λ′ = 1−β λ

1+ β

corresponding to the source and the observer approaching each other.

Traverse Doppler's Effect
Setting θ = π / 2 in (2) we find
(5)

λ′ = λ.

Hence, and contrary to the relativistic prediction, there is no traverse Doppler's effect.

References [1] C P Viazminsky, Generalized Lorentz transformations and Restrictions on Lorentz Transformation, Research Journal of Aleppo University, 48, 2007.