Announcements. Today s class. The Lorentz transformation. Lorentz transformation (Relativistic version of Galileo transformation)

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1 Announements Reading for Monda:. -.5 HW 3 is posted. Due net Wed. noon. The Frida was a TYPO! IT I DUE WEDNEDAY! Toda s lass Lorent transformation Doppler shift First Midterm is on the 6 th. Will oer relatiit (Pratie questions will be put on CULearn. Homeworks returned and solutions up on CULearn. Final date and time: Ma 9, 4:30 p.m. 7:00 p.m. Lorent transformation (Relatiisti ersion of Galileo transformation) ' ' ' (,,,t) (',',',t') ' The Lorent transformation 0 A stik is at rest in. Its endpoints are the eents (,t ) = (0,0) and (,0) in. is moing to the right with respet to frame. Eent left end of stik passes origin of. Its oordinates are (0,0) in and (0,0) in. Lorent transformation An obserer at rest in frame sees a stik fling past her with eloit : 0 As iewed from, the stik s length is /g. Time t passes. Aording to, where is the right end of the stik? (Assume the left end of the stik was at the origin of at time t=0.) a) = gt b) = t + /g ) = -t + /g d) = t /g e) something else Origin of (left end of stik) Length of the stik The Lorent transformation = t + /g. This relates the spatial oordinates of an eent in one frame to its oordinates in the other. Algebra 0 = g(-t) = t + /g

2 ' (',',',t') ' ' ' If is moing with speed in the positie diretion relatie to, then the oordinates of the same eent in the two frames are related b: Transformations Galilean transformation (lassial) t t t Lorent transformation (relatiisti) g ( t) ee homework #3 t g ( t Note: This assumes (0,0,0,0) is the same eent in both frames. ) A note of aution: The wa the Lorent and Galileo transformations are presented here (and in the tetbook) assumes the following: An obserer in would like to epress an eent (,,,t) (in his frame ) with the oordinates of the frame ', i.e. he wants to find the orresponding eent (',',',t') in '. The frame ' is moing along the -aes of the frame with the eloit (measured relatie to ) and we assume that the origins of both frames oerlap at the time t=0. Note: Veloit has a sign! ' ' ' (,,,t) (',',',t') ' o we are alread nearl done with the transformation laws We now an onert: loations: et. time: t t But we still hae to figure out: eloities: u u Veloit transformation Remember that the Galileo eloit transformation u = u doesn t work for light? (Beause!) Let s fi it! Veloit transformation (D) A An objet moes from eent A=(,t ) to eent B =(,t ). As measured b, its speed is u with: Δ = - t Δt = t - t As measured b, its speed is u with: Δ = - t Δt = t - t Δ B Veloit transformation (D) u, u, where Δ= t t -, Δ = - Use Lorent: = g( -t ) g ( t) u t = g( -t ) t g ( t ( / ) ). t u u u / Galilean result New in speial relatiit (Note: this is again due to time-dilation and length-ontration)

3 Does this work for light? For light: u. We epet to get u = u u u / / / / ( / ) Great! It works!! How about for er small? (Classial limit: << ) u u u / For << we know that u/ 0. Therefore: u u u u u / 0 Classial limit: Galileo eloit transformation! Veloit transformation in 3D (,,,t) ' ' ' u (',',',t') In a more general ase we want to transform a eloit u (measured in frame ) to u in frame. Note that u an point in an arbitrar diretion, but still points along the -aes. ' Veloit transformation (3D) The eloit u=(u, u, u ) measured in is gien b: u = Δ / Δt, u = Δ / Δt, u = Δ / Δt, where Δ= - To find the orresponding eloit omponents u, u, u in the frame, whih is moing along the -aes in with the eloit, we use again the Lorent transformation: =γ( -t ), and so on t =γ(t - / ), and so on We find: g ( t) and: t g ( t ( / ) ) (just as before) ' ' Now: = Δ / Δt et... Veloit transformation (3D) (aka. Veloit-Addition formula ) u u / u g u / ome eamples u g u / Q 3

4 Relatiisti transformations = 0.5 g ( t) u t u g ( t ) u / u u u u / u / uppose a spaeraft traels at speed =0.5 relatie to the Earth. It launhes a missile at speed 0.5 relatie to the spaeraft in its diretion of motion. How fast is the missile moing relatie to Earth? (NOTE: Remember whih oordinates are the primed ones! And: Does our answer make sense?) a) 0.8 b) 0.5 ) d) 0.5 e) 0 u =0.5 The Doppler Effet We are all familiar with the aousti Doppler effet. It relies on the speed of sound in air. For light: No ether! We epet some differenes. (Also must inlude time dilation!) t: proper time between soure = t t = γ t (time dilation) =? t = γ t (time dilation) =? t t was emitted from here was emitted from here was emitted from here was emitted from here t = γ t (time dilation) =? t = γ t (time dilation) =? t was emitted from here was emitted from here was emitted from here t τ (τ - t ) was emitted from here = τ - (τ - t )- t = t - t = t ( - ) = γ t ( - ) 4

5 = τ - (τ - t )- t = t - t = t ( - ) = γ t ( - ) The obsered frequen (from the trak) is: with ' g ( ) t g ( ) t Note that the frequen of the soure in its rest-frame is: f soure t t Therefore: f obs fsoure g ( ) The optial Doppler effet f soure f soure with If soure is approahing f soure, if soure and obserer are approahing eah other If soure is reeding f soure, if soure and obserer are reeding from eah other Questions? 5