Physics 2D Lecture Slides Lecture 5: Jan 12th 2004

Transcription

1 The Final Exam is on Mar 18 th, Time and Loation TBA NOT on Monday Mar 15 th as previosly annoned in the Handot et!! Pl. make a note of this hange!! This date hange is also posted in the ANNOUCEMENT setion of lass web page Pl. make it a point to reglarly hek the annonements setion of lass web page Physis D Letre Slides Letre 5: Jan 1th 004 Vivek Sharma UCSD Physis 1

2 Relativisti Doppler Shift 1+(v/) f obs = f 1-(v/) sore Doppler Shift & Eletromagneti Spetrm RED BLUE

3 Fingerprint of Elements: Emission & Absorption Spetra Example : The Atomi Energy levels of Hydrogen Doppler Shift in Spetral Lines and Motion of Stellar Objets Laboratory Spetrm, lines at rest wavelengths Lines Redshifted, Objet moving away from me Larger Redshift, objet moving away even faster Lines bleshifted, Objet moving towards me Larger bleshift, objet approahing me faster 3

4 Seeing Distant Galaxies Throgh Hbble Telesope Throgh enter of a massive galaxy lsters Abell 1689 Edwin Hbble, Mont Palomar & Expanding Universe Hale 100 inh Telesope, Mont Palomar Edwin Hbble 190 4

5 Spetral lines are shifted from Laboratory (at rest) Speimen Galaxies at different loations in Universe moving away at different veloities Hbble s Measrement of Reessional Veloity of Galaxies Reesional Veloity V distane; V = H d Farther things are, faster they go H = 75 km/s/mp (3.08x1016 m) Play the movie bakwards! Or Universe is abot 10 Billion Years old 5

6 Cosmologial Redshift & Disovery of the Expanding Universe: [ Spae itself is Expanding ] New Rles of Coordinate Transformation Needed The Galilean/Newtonian rles of transformation old not handles frames of refs or objets traveling fast V C (like v = 0.1 or 0.8 or 1.0) Einstein s postlates led to Destrtion of onept of simltaneity ( t t ) Moving loks rn slower Moving rods shrink Lets formalize this in terms of general rles of oordinate transformation : Lorentz Transformation Reall the Galilean transformation rles x = (x-vt) t = t These rles that work ok for ferraris now mst be modified for roket ships with v 6

7 Disovering The Corret Transformation Rle x' = x vt gess x' = G( x vt) x = x' + vt' gess x = G( x' + vt' ) Need to figre ot the fntional form of G!0 G mst be dimensionless G does not depend on x,y,z,t Bt G depends on v/ G mst be symmetri in veloity v As v/ 0, G 1 Gessing The Lorentz Transformation Do a Thoght Experiment : Wath Roket Moving along x axis Roket in S (x,y,z,t ) frame moving with veloity v w.r.t observer on frame S (x,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 travels as a spherial wave, origin is at O,O Speed of light is for both observers: Postlate of SR Examine a point P (at distane r from O and r from O ) on the Spherial Wavefront 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 7

8 Disovering Lorentz Transfromation for (x,y,z,t) Motion is along x-x axis, so y, z nhanged y =y, z = z Examine points x or x where spherial wave rosses the horizontal axes: x = r, x =r x = t = G( x' + vt') x' = t' = G( x - vt), G t' = ( x- vt) x = t = G( t ' + vt ') = v ( vt) + vt = G [ v ] or G = 1 1 ( v / ) = γ t G t t x' = γ ( x vt) x' = γ ( x vt), x= γ ( x' + vt') x = γγ ( ( x vt) + vt ') x γ x + γ vt = γ vt ' x γ x γ vt x x t' = + = γ + t γv γv γv γ v v x 1 1 v t' = γ t+ 1, sine 1 = v γ γ x v t ' = γ vx t + [1 1 γ = t v Lorentz Transformation Between Ref Frames x' = γ ( x vt) y' = y y = y ' z' = z z = z ' t' Lorentz Transformation = γ t vx Inverse Lorentz Transformation x= γ ( x' + vt') t = γ t ' + vx ' As v 0, Galilean Transformation is reovered, as per reqirement Notie : SPACE and TIME Coordinates mixed p!!! 8

9 Not jst Spae, Not jst Time New Word, new onept! SPACETIME Lorentz Transform for Pair of Events S S rler x x 1 x 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 t did S measre? two events or at same plae in S frame => x= 0 t = γ t ( t = proper time) Length Contration: Rler measred in S between x 1 & x : What x did S measre? two ends measred at same time in S frame => t = 0 x = γ ( x + 0 ) => x = x / γ ( x = proper length) 9

10 Lorentz Veloity Transformation Rle S S and S are measring ant s speed along x, y, z axes S v In S' frame, ' ' ' 1 x' = = ' ' ' t t1 dt ' v dx = γ ( dx vdt), dt ' = γ ( dt dx) dx vdt x' =, divide by dt' v dt dx x v x' = vx 1 For v <<, = v (Galilean Trans. Restored) x' x x dx x Veloity Transformation Perpendilar to S-S motion v dy ' = dy, dt ' = γ ( dt dx) ' dy ' dy y = = dy ' v γ ( dt dx) divide by dt on RHS ' y y = v γ (1 ) x There is a hange in veloity in the diretion to S-S' motion! Similarly Z omponent of Ant' s veloity transforms as ' z z = γ v (1 ) x 10

11 Inverse Lorentz Veloity Transformation Inverse Veloity Transform: x' + v x = vx' 1+ ' y y = v ' γ (1+ ) x ' z z = v ' γ ( 1+ ) x As sal, replae v -v Does Lorentz Transform work? Two rokets travel in opposite diretions An observer on earth (S) measres speeds = 0.75 And 0.85 for A & B respetively What does A measre as B s speed? Plae an imaginary S frame on Roket A v = 0.75 relative to Earth Observer S Consistent with Speial Theory of Relativity 11

12 Example of Inverse veloity Transform Biker moves with speed = 0.8 past stationary observer Throws a ball forward with speed = 0.7 What does stationary observer see as veloity of ball? Speed of ball relative to stationary observer X? Plae S frame on biker Biker sees ball speed X =0.7 Hollywood Yarns! 1

13 Terminator : Can yo be seen to be born before yor mother? A frame of Ref where seqene of events is REVERSED?!! S S I take off from SD 1 1 ' ' 1 1 I arrive in SF ' ' ' ' v x t' t t1 γ t = = Reversing seqene of even ts t ' < 0 I Cant be seen to arrive in SF before I take off from SD S S 1 1 ' ' 1 1 v x γ For what vale of v an t ' < 0 ' ' t' = t t1 = t ' ' v x v x v t' < 0 t < 1 < = t v > v> : Not allowed 13

14 A A B B C C D D 14