56 lines
2.1 KiB
Plaintext
56 lines
2.1 KiB
Plaintext
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Generalized Doppler Effect
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C. P. Viazminsky Department of Physics University of Aleppo
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Aleppo-Syria Kayssarv@mail2world.com
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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.
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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
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(1)
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R = γr ( 1− β 2 sin 2 θ r + β cosθ )
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Now assume that the distance r in the moving frame corresponds to one wave-length,
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i.e. r = λ . With respect to the observer O the distance R corresponds to one wavelength λ′ .
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Generalized Doppler's formula
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Substituting in the last equation r = λ and R = λ′ , we obtain:
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(2)
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λ′ = γ ( 1− β 2 sin 2 θ + β cosθ )λ,
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with γ = 1/ 1− β 2 , which determines the wave length as measured by the
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stationary observer. Note that the radiating source here is at a position of azimuth
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angle θ , and that the polar axis is OX.
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Longitudinal Doppler's Formula
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Setting θ = 0 in the generalized formula (1) we obtain
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(3)
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λ′ = 1+ β λ
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1− β
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which is the red shift Doppler's formula, corresponding to the source and the observer
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receding from each other. For θ = π we obtain the blue shift Doppler's formula
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(4)
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λ′ = 1−β λ
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1+ β
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corresponding to the source and the observer approaching each other.
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Traverse Doppler's Effect
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Setting θ = π / 2 in (2) we find
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(5)
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λ′ = λ.
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Hence, and contrary to the relativistic prediction, there is no traverse Doppler's effect.
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References [1] C P Viazminsky, Generalized Lorentz transformations and Restrictions on Lorentz Transformation, Research Journal of Aleppo University, 48, 2007.
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