Lecture 4 - Physics 325 Lecture 20 Relativistic Doppler...

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Physics 325 Lecture 20 Relativistic Doppler Effect We begin with an observer at 90 ˚ . Non-relativistically, as we just saw, this observer sees no frequency shift. However, relativity tells us that the clock on the moving train is different than the clock with the stationary observer, so time-dilation slows the period of emitted waves according to τ γτ = (20.1) The wavelength and frequency shift accordingly: f f λ γ τλ = == ′′ (20.2) This is a purely relativistic effect. Now the rest of the argument goes very much like the train example above. Let’s repeat it with a moving source emitting light waves v G S’ S The source is at rest in the S’ frame and is emitting light waves. An observer measures them from the frame S. S’ moves with velocity v G with respect to S. If S’ were at rest with respect to S, then S would see a wave train of length c t in time t . However, since S’ is approaching S at speed v , the wave train gets shorter because the end of it is emitted closer to S by a distance v t , therefore the length of the wave train observed by S is xc tv t ∆=∆−∆ If the wave train consists of n full waves, we can write ( ) nc v t = −∆ (20.3) and the frequency is () cc n f cv t (20.4)
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According to an observer moving with S’ there are also n full waves emitted, but they are emitted in a proper time . So the frequency of the emitted waves in S’ is t n f t ′ = (20.5) The time measured in the frame S is longer by a factor of γ tt γ ∆=∆ Using this in Equation (20.5) above, gives n t f ∆= and inserting this into Equation (20.4) we find () 1 1 1 f f f vc β + == (20.6) Equation (20.6) is valid when the source and receiver are approaching each other. If the source an receiver are receding from each other, then the v t term in Equation (20.3) changes sign and (20.6) becomes 1 1 f f = + (20.7) More generally, we can have the case where the observer is at angle θ with respect to the velocity of the moving source. As before the v t term in Equation (20.3) becomes v t cos , and carrying this through we find 2 1 1c o s f f = (20.8) which contains both (20.6) ( θ=0) and (20.7) ( θ=π ). Causality In relativity, event 1 that happens at ( x 1 ,t 1 ) can only cause event 2 that happens at ( x 2 ,t 2 ) if light leaving x 1 at t 1 can reach x 2 at or before t 2 . Mathematically this relationship is 21 x xc ≤− (20.9) or 22 2 0 ct x −∆ (20.10) Events that satisfy (20.10) are causally connected. It would be extremely strange, unphysical even, if two events that are causally connected in one frame are not causally connected in any other frame. So we had better check.
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Suppose the two events above occur in a frame S’ that is moving with respect to S at speed v (the usual story). In S’ we have a causal connection between the two events: 22 2 0 ct x ′′ −∆ () 11 , xt , v G S’ S Invoking the Lorentz transformations of Lecture 18 Equation (18.5) to connect S’ to S, we have ( ) x xc t t t x c
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This note was uploaded on 10/06/2011 for the course PHYS 225 taught by Professor Makins during the Spring '10 term at University of Illinois, Urbana Champaign.

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Lecture 4 - Physics 325 Lecture 20 Relativistic Doppler...

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