Relativistic Kinematics

Transcription

1 PH0008 Quantum Mechanics and Special Relativity Lecture 8 (Special Relativity) Relativistic Kinematics Velocities in Relativistic Frames & Doppler Effect Prof Department of Physics Brown University Main source at Brown Course Publisher background material may also be available at Gaitskell

2 Section: Special Relativity Week 4 Homework (due for M 3/11) Please hand in now Reading (Prepare for 3/11) o SpecRel (also by French) Ch5 RelativisticKinematics Lecture 8 (M 3/11) o Relativistic Kinematics Velocities Doppler Effect Reading (Prepare for 3/18) o SpecRel (also by French) Ch6 Relativistic Dynamics: Collisions and Conservation Laws (Review) Ch3 Einstein & Lorentz Transforms Ch4 Realtivity: Measurement of Length and Time Inetrvals Ch5 RelativisticKinematics Lecture 6 (W 3/13) o General Relativity Guest Lecture from Prof Ian Dell Antonio Lecture 7 (F 3/15) Doppler Effect Reanalysis of Twin Paradox with signal exchange Introdution to Relativistic Dynamics Homework #8 (M 3/18) o Start early! (see web Assignments )

3 Homework / Office Hours Homework - please hand in Please pick up your HW #1-3 from outside my office B&H 516 I will not be available on Tuesday or Thursday this week o I will hold special office hours on Friday 1-3 pm o

4 Question Section

5 Question SpecRel L08-Q1 New problem: Clock coming directly towards us at near light speed? o(1) The clock appears to be running slow o(2) The clock appears to be running fast o(3) Not enough information to judge above

6 Twin Paradox Discuss

7 Twin Paradox The phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. First Law o Body continues at rest, or in uniform motion During acceleration and deceleration this frame is not inertial o We will return to this problem at end of Relativistic Kinematics Section Einstein, quoted in Physics, Structure and Meaning, p288 Leon Cooper

8 Review Space-Time Intervals

9 Minkowski: Interval Separation of two events in Space-Time ct ct Light-Ray x x Consider "invariant" [ x] 2 - [ ct] 2 = g ( x + bc t ) È [ x ] x bc t = g 2 Í Î Í - c t Define Ds 2 = cdt [ ] 2 - [ g( c t + b x )] 2 = g 2 1- b 2 = x [ ] + [ bc t ] 2... [ ] 2 - [ 2 x bc t ] - [ b x ] 2 [ x ] 2 - [ c t ] 2 [( )] [ ] 2 - [ c t ] 2 [ ] 2 - [ Dx] 2 If events are simultaneous (but spatially separated) in one frame then Ds 2 < 0 "Space - like" and events cannot be causally connected If events occur in same place in one frame (separated only by time) then Ds 2 > 0 "Time - like" and events can be causally connected Ds 2 = 0 "Light - like" Events are on light - cone

10 Relativistic Kinematics

11 Relativistic Treatment of Velocities Start with new definitions o (Board) New beta notation Derivatives w.r.t. cdt Look at how velocity will transform o Consider derivatives of variable w.r.t. time x = g ( ) x = g x - bct x + bc t y = y y = y c t = g ct - b x t + b x ct = g( c ) x = g x + bc t dx = g Ê d x + b ˆ Á Ë = g b x + b y = y dy = d y = b y ct = g c t + b x c dt = g Ê + b d x ˆ Á Ë = g 1+ b b x

12 Relativistic Treatment of Velocities (2) Use previous expressions to get o b x and b y o By symmetry we can also quickly calculate b x and b y x = g x + bc t dx = g Ê d x + b ˆ Á Ë = g b x + b y = y dy = d y = b y ct = g c t + b x c dt = g Ê + b d x ˆ Á Ë = g 1+ b b x b x = dx c dt = dx = g ( b x + b) g 1+ b b x c dt = ( b x + b) ( 1+ b ) b y = dy c dt = dy b = y g 1+ b b x c dt b x = b y g ( 1+ b ) b x = d x = b x - b 1- bb x b y = d y = b y g 1+ bb x b x

13 Relativistic Treatment of Velocities (3) Consider o b x =1 Tests of this extreme case o Pions decay in flight o Accelerators If b x =1 b x = ( b x + b) ( 1+ b b x ) ( 1+ b) = 1+ b =1

14 Relativistic Treatment of Velocities (4) In low velocity limit o b x <<1 and b <<1 o Denominator becomes ~1 Both denominator and g are second order in velocities o Becaomes simple addition of velocities Galilean b b x = x + b 1+ b b x ª b x + b b b y = y g 1+ b b x ª b y b x = b - b x 1- bb x ª b x - b b y = b y g 1+ bb x ª b y

15 Doppler Effect in Sound Acoustical Effect o (Board).

16 Relativistic Doppler Effect Source in S frame, Observer in S frame ct t = nt t = 0 x 0 ct (x 2,t 2 ) (x 1,t 1 ) x 1 = x 2 (n+1) Pulse 1st Pulse x x b is velocity of observer frame S measured in S (1) x 1 = ct 1 = x 0 + bct 1 (2) x 2 = c( t 2 - nt) = x 0 + bct 2 Therefore, subtracting (2) - (1) above c( t 2 - t 1 ) - cnt = bc( t 2 - t 1 ) = cnt ( 1- b) = cnt ( 1- b) c t 2 - t 1 x 2 - x 1 = bcnt ( 1- b) In observer frame S using Loretz Trans. c ( t 2 - t 1 ) = g[ c( t 2 - t 1 ) - b( x 2 - x 1 )] È cnt = g ( 1- b) - b bcnt Í Î ( 1- b) The time interval covers n periods, and the apparent period t in S is t = t - t 2 1 n È t = gí Î 1- b = gt 1- b = g( 1+ b)t - b bt ( 1- b) [ ] 1- b 2

17 Relativistic Doppler Effect (2) Source in S frame, Observer in S frame, moving away from source with velocity b o The frequency the observer sees is lower than that of the source o This answer depends only on relative velocity of source and observer, unlike acoustic effect Remember Acoustical Doppler Effect : - Stationary source, receeding receiver n = ( 1- b)n Receeding source, stationary receiver 1 n = ( 1+ b) n where b is the velocity of moving object divided by wave velocity in medium The time interval covers n periods, and the apparent period t in S is t = g( 1+ b)t Ê = 1+ b Á Ë 1- b ˆ Ê = 1+ b ˆ 2 Á t Ë 1- b Or in terms of frequencies n Ê n = 1- b ˆ Á Ë 1+ b The time interval covers n periods, and the apparent period t in S is t = t 2 - t 1 n È t = gí Î 1- b = gt 1- b = g( 1+ b)t 1 2 n 1 2 t - b bt ( 1- b) [ ] 1- b 2

18 Relativistic Doppler Effect (3) Source in S frame, Observer in S frame, moving away from source with velocity b o The frequency the observer sees is lower than that of the source: RED SHIFTED If source and observer approach one another then sign of b is reversed o The frequency is increased: BLUE SHIFTED o (Frequency of blue light is higher than red light) The frequency of a clock approaching us directly will appear to be higher, not (s)lower o This in contrast to viewing clock from side o We must be clear about situation we are studying! Receeding at b Ê n = 1- b ˆ Á Ë 1+ b 1 2 n Approaching at b Ê n = 1+ b ˆ Á Ë 1- b 1 2 n

19 Relativistic Doppler Effect (4) Examples o Red shift of galaxies (Hubble)

20 Transverse Doppler Effect Relativistic Doppler Effect (5) o Classically when velocity of object is perpendicular to sight line there is no Doppler Effect o However, relativistically there is still time dilation to consider Perpendicular at velocity b, observer S t = gt n = 1 g n

21 Next Lecture Wednesday o Guest Lecture: General Relativity, Prof Ian Dell Antonio Friday o Doppler Effect o Reanalysis of Twin Paradox with signal exchange o Introduction to Relativistic Dynamics

22 Material For Next Lecture

23 ct ct Light-Ray x x

24 (Board) Discuss Symmetry of Problem o Basic Lorentz Relations under exchage of DT <-> -DT and b <-> -b