Doppler Effect ¾The Doppler effect is the shift in frequency and wavelength of waves that results from a moving source and/or moving observer relative to the wave medium ¾Consider a point source of traveling waves (a siren) at some frequency f0 We have λ=V/f0 where V is the wave velocity and V depends on the characteristics of the medium alone 1 Doppler Effect ¾ A source moving towards an observer with velocity vs The observer sees wave fronts that are “squeezed” in the direction of vs but the wave velocity remains unchanged v V − v λ′ = λ − s = s f0 f0 V ⎛ V ⎞ f ⎜ ⎟ 0 f = = f0 ⎜ ⎟ = λ′ V − v vs ⎝ s ⎠ 1− ¾ A source moving away from an observer V f f = 0 v 1+ s V 2 Doppler Effect 3 Doppler Effect Optical spectrum Optical spectrum of sun of cluster of distant galaxies 4 Doppler Effect ¾ An observer moving towards a source with velocity vo Here an observer will see the same wavelength λ but with a higher wave velocity (e.g. wading into an ocean) v′ = V + | vo | v′ V + | vo | | vo | ⎛ | vo |⎞ f = = = f0 + = f0 ⎜1+ ⎟ λ λ λ ⎝ V ⎠ ¾ An observer moving away from a source ⎛ | v |⎞ f = f ⎜1− o ⎟ 0 v ⎝ ⎠ 5 Doppler Effect ¾In the case of light waves Classical, moving source f f = 0 V 1m Classical, moving observer c ⎛ V ⎞ f = ⎜1± ⎟ f0 ⎝ c ⎠ ¾Makes sense, the frequency is higher if observer and source approach each other and lower if they are receding 6 Doppler Effect ¾But This is a classical calculation There is an asymmetry between moving source and moving observer Moving source f ⎛ V V 2 ⎞ f = 0 ≈ ⎜1± ± ± ...⎟ f V ⎜ 2 ⎟ 0 1 ⎝ c c ⎠ m c Moving observer ⎛ V ⎞ f = ⎜1± ⎟ f0 ⎝ c ⎠ 7 Doppler Effect ¾The relativistic Doppler effect calculation differs from the classical one because of time dilation ¾Consider a source of frequency f0 fixed in S’ As usual, S’ is moving with velocity V relative to S N = f0Δt’ waves are emitted in Δt’ Δt=γΔt’ 8 Doppler Effect ¾In S, with the source in S’ moving towards the observer in S During Δt, the first wave moves cΔt and the source moves VΔt The wavelength in S is cΔt −VΔt c λ = and f = N λ cN f = cΔt −VΔt Now Δt N = f0Δt′ = f0 γ 9 Doppler Effect ¾Still in S, we find the relativistic Doppler effect for the source approaching cf 1− β 2 1+ β f = 0 = f c()1− β 1− β 0 ¾For the source receding, we have 1− β f = f 1+ β 0 10 Doppler Effect ¾For completeness, consider the source of frequency f0 be fixed in S and let Δt=λ/c ¾In S’, let the observer see the source receding λ cΔt cΔt′ = VΔt′ + = VΔt′ + γ γ V 2 V 1− 2 cΔt 1+ Δt′ = c = c Δt c −V V 1− c 1− β f ′ = f0 1+ β 11 Doppler Effect ¾In S’, if the observer sees the source approaching we would find 1+ β f ′ = f 1− β 0 12 Doppler Effect ¾ In summary For source and/or observer approaching 1+ β f = f 1− β 0 For source and/or observer receding 1− β f = f 1+ β 0 ¾ In special relativity, the two situations (source moving or observer moving) give the same result ¾ There is also a transverse Doppler effect that does not occur classically f f = 0 γ 13 Doppler Effect ¾An astronaut tries to get out of a traffic violation for running a red light (λ=670nm) by arguing the Doppler effect made the light appear green (λ=540nm) . ¾What was the velocity of the astronaut? 14 Spacetime Diagrams 15 Spacetime Diagrams 16 Spacetime Diagrams 17 Spacetime Diagrams 18 Spacetime Diagrams ¾Barn and pole paradox – Barn frame ct Back of pole Front of pole Pole inside barn x Front Back of barn of barn 19 Spacetime Diagrams ¾Barn and pole paradox – Pole frame ct Back of barn Front of barn Front of pole leaves Back of pole enters Back of barn front of barn x Back Front of pole of pole 20 Spacetime Diagrams ct ct’ x’ x 21 Spacetime Diagrams 22.