Physics 2D Lecture Slides Lecture : Jan 11th 200. First Quiz This Friday!

Transcription

1 Physis D Letre Slides Letre : Jan 11th 00 Viek Sharma UCSD Physis First Qiz This Friday! Bring a Ble Book, allator; hek battery Make sre yo remember the ode nmber for this ose gien to yo (reord it some plae safe!) No heat Sheet please, I will gie yo eqations and onstants that I think yo need When yo ome for the qiz, pl. opy seats in the front first. Pl. obsere one seat distane in the bak rows (there is plenty of spae) Aademi Honesty is for yo to obsere and for me to enfore: Be a good itizen, in this orse and foreer! 1

2 Time Dilation Eample: Relatiisti Doppler Shift Light : eloity f λ, f1/t A sore of light S at rest Obserer S approhes S with eloity S measres f or λ, f λ Epet f > f sine more wae rests are being rossed by Obserer S de to its approah diretion than if it were at rest w.r.t sore S Relatiisti Doppler Shift!T-T, now se f /! " f ; bt T (-)T sbstitting for T, se f 1/T T 1- (/) # T " f 1- (/) 1- (/) Eamine two sessie waefronts emitted by S at loation 1 and In S frame, T time between two waefronts In time T, the Sore moes by T w.r.t 1 Meanwhile Light Sore moes a distane T Distane between sessie waefront λ T T " f 1+(/) 1-(/) f better remembered as: 1+(/) f obs 1-(/) f sore f obs Freqeny measred by obserer approhing light sore

3 Relatiisti Doppler Shift 1+(/) f obs f 1-(/) sore Doppler Shift & Eletromagneti Spetrm RED BLUE 3

4 Fingerprint of Elements: Emission & Absorption Spetra Eample : The Atomi Energy leels of Hydrogen Doppler Shift in Spetral Lines and Motion of Stellar Objets Laboratory Spetrm, lines at rest waelengths Lines Redshifted, Objet moing away from me Larger Redshift, objet moing away een faster Lines bleshifted, Objet moing towards me Larger bleshift, objet approahing me faster 4

5 Seeing Distant Galaies Throgh Hbble Telesope Throgh enter of a massie galay lsters Abell 1689 Edwin Hbble, Mont Palomar & Epanding Unierse Hale 100 inh Telesope, Mont Palomar Edwin Hbble 190 5

6 Spetral lines are shifted from Laboratory (at rest) Speimen Galaies at different loations in Unierse moing away at different eloities Hbble s Measrement of Reessional Veloity of Galaies Reesional Veloity V distane; V H d Farther things are, faster they go H 75 km/s/mp ( m) Play the moie bakwards! Or Unierse is abot 10 Billion Years old 6

7 Cosmologial Redshift & Disoery of the Epanding Unierse: [ Spae itself is Epanding ] New Rles of Coordinate Transformation Needed The Galilean/Newtonian rles of transformation old not handles frames of refs or objets traeling fast V C (like 0.1 or 0.8 or 1.0) Einstein s postlates led to Destrtion of onept of simltaneity ( Δt Δt ) Moing loks rn slower Moing rods shrink Lets formalize this in terms of general rles of oordinate transformation : Lorentz Transformation Reall the Galilean transformation rles (-t) t t These rles that work ok for ferraris now mst be modified for roket ships with 7

8 Disoering The Corret Transformation Rle! t gess " G(! t) + t gess " G( + t ) Need to figre ot the fntional form of G!0 G mst be dimensionless G does not depend on,y,z,t Bt G depends on / G mst be symmetri in eloity As / 0, G 1 Gessing The Lorentz Transformation Do a Thoght Eperiment : Wath Roket Moing along ais Roket in S (,y,z,t ) frame moing with eloity w.r.t obserer on frame S (,y,z,t) Flashblb monted on roket emits plse of light at the instant origins of S,S oinide That instant orresponds to t t 0. Light traels as a spherial wae, origin is at O,O Speed of light is for both obserers: Postlate of SR Eamine a point P (at distane r from O and r from O ) on the Spherial Waefront The distane to point P from O : r t The distane to point P from O : r t Clearly t and t mst be different t t 8

9 Disoering Lorentz Transfromation for (,y,z,t) Motion is along - ais, so y, z nhanged y y, z z Eamine points or where spherial wae rosses the horizontal aes: r, r t G( + t ) t G( - t), G " t ( - t) # t G( t + t ) $ % # t G ( t & t) + t & t ( ) * " G [ & ] 1 or G! 1 & ( / ) #! ( & t)! ( # t),! ( + t ) $ "! (! ( # t) + t ) #! +! t! t %!! t & % & $ t # +! t!!! ( # +! ( ) * ) * % + 1,& + 1, +, $ t! t + - # 1, sine # 1 #.( ) /! 0* /! 0 / 0 % & % & +, +, " t! t + [1 # -/. # 1(! t # -. 0 ( ) *( ) / 0* Lorentz Transformation Between Ref Frames! ( $ t) y y y y z z z z t Lorentz Transformation " #! % t $ & t ( Inerse Lorentz Transformation! ( + t ) " #! $ t + % & As 0, Galilean Transformation is reoered, as per reqirement Notie : SPACE and TIME Coordinates mied p!!! 9

10 Not jst Spae, Not jst Time New Word, new onept! SPACETIME Lorentz Transform for Pair of Eents S S rler 1 X Can nderstand Simltaneity, Length ontration & Time dilation formlae from this Time dilation: Blb in S frame trned on at t 1 & off at t : What Δt did S measre? two eents or at same plae in S frame > Δ 0 Δt γ Δt (in this eample Δt proper time) Length Contration: Rler measred in S between 1 & : What Δ did S measre? two ends measred at same time in S frame > Δt 0 Δ γ (Δ + 0 ) > Δ Δ / γ ( in this eample Δ proper length) 10

11 Lorentz Veloity Transformation Rle S S and S are measring ant s speed along, y, z aes S In S frame, 1 t " t1 dt d! ( d " dt), dt! ( dt " d) d " dt, diide by dt dt " d " 1" For <<, " (Galilean Trans. Restored) " d Veloity Transformation Perpendilar to S-S motion dy dy, dt! ( dt # d) dy dy y dy! ( dt # d) diide by dt on RHS y y! (1 # ) There is a hange in eloity in the diretion " to S-S motion! Similarly Z omponent of Ant s eloity transforms as z z! (1 " ) 11

12 1 Inerse Lorentz Veloity Transformation Inerse Veloity Transform: (1 ) 1 1 ( ) y y z z!! As sal, replae -