PH0008 Quantum Mechanics and Special Relativity Lecture 7 (Special Relativity) Symmetry in Transformations Twin Paradox & Couple of Watches

Transcription

1 PH0008 Quantum Mechanics and Special Relativity Lecture 7 (Special Relativity) Reciprocity Symmetry in Transformations Twin Parado & Couple of Watches 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 3 Homework (none due for M 3/4) (see Assignments on web pages) [Please start on net homework) Reading (Prepare for 2/4) o SpecRel (also by French) Ch3 Einstein & Lorentz Transforms Ch4 Realtivity: Measurement of Length and Time Inetrvals Lecture 5 (M 3/4) o Lorentz Transformation Worked Eample: Rod and Single Clock Time Dil., Lorentz Cont., Relativity of Simultaneity o Minkowski Space Lecture 6 (W 3/6) o Minkowski Space More Worked Eample: Two Rods Time Dil., Lorentz Cont., Relativity of Simultaneity Lecture 7 (F 3/8) o Reciprocity o Intervals Reading (Prepare for 3/11) o SpecRel (also by French) Ch5 RelativisticKinematics Ch6 Relativistic Dynamics: Collisions and Conservation Laws (Review) Ch3 Einstein & Lorentz Transforms Ch4 Realtivity: Measurement of Length and Time Inetrvals Homework #7 (M 3/11) o Start early - tough (see web Assignments )

3 Homework I have moved last question to week after o See web site Please pick up your HW #1-3 from outside my office B&H 516

4 Typo s In SpecRel L05 Matri Eample o Couple of serious typo s so please download new version, if you have stored old one.

5 Question Section

6 Question SpecRel L07-Q1 How is S frame moving viewed in S frame? o(1) S has +ve velocity and b=tan q o(2) S has +ve velocity and g=tan q o(3) S has -ve velocity and b=tan q ct q ct o(4) S has -ve velocity and g=tan q q o(5) None of above

7 Question SpecRel L07-Q2 Which graph best reflects a Galilean (rather than Lorentz Transform)? (3) ct ct (1) ct ct (2) ct ct (4) ct ct

8 Space-Time Diagrams Help visualize consequences of Lorentz Transforms

9 Minkowski: Calibrating Aes Calibrating aes o If we define =1, where is =1? ct ct Light-Ray Consider "invariant" [ ] 2 - [ ct] 2 = g ( + bc t ) È [ ] bc t = g 2 Í Î Í - c t [ ] 2 - [ g( c t + b )] 2 = g 2 1- b 2 = [ ] + [ bc t ] 2... [ ] 2 - [ 2 bc t ] - [ b ] 2 [( )([ ] 2 - [ c t ] 2 )] [ ] 2 - [ c t ] 2 =1 =1 Draw hyperbola [ ] 2 - [ ct] 2 =1 Since [ ] 2 - [ ct] 2 = [ ] 2 - [ c t ] 2 =1 So point where it intersects - ais [ ] 2 =1 c t = 0 fi This is true generally for any S

10 Minkowski: Calibrating Aes Calibrating aes o If we define =1, where is =1? ct ct A O =1 =1 Light-Ray Draw hyperbola [ ] 2 - [ ct] 2 =1 Since [ ] 2 - [ ct] 2 = [ ] 2 - [ c t ] 2 =1 So point where it intersects - ais [ ] 2 =1 c t = 0 fi This is true generally for any S Find the length from origin to point A where the hyperbola instersects ais [ A ] 2 - ct A 2 A 1- b 2 [ ] 2 =1 and ct A = b A 2 = g 2 ( ) =1 fi A ( O A ) 2 = [ A ] 2 + [ ct A ] 2 ( = g 2 + g 2 b 2 = 1+ b 2 ) ( 1- b 2 ) O A = 1+ b 2 1- b 2

11 Steiger & Stewart: The Two Rods Help visualize consequences of Lorentz Transforms

12 Consider a stationary rod in S Stationary Rod length l 0 in S ct ct

13 Consider a stationary rod in S Stationary Rod also length l 0 in S ct ct Note scale for S is slightly larger by O A = 1+ b 2 1- b 2 so length l 0 is drawn slightly longer than rod in S

14 Event #1 Event #1 Left hand ends of rods coincide ct ct ( 1,c t 1 ) ( 1,ct 1 )

15 Event #2 Event #2 Right hand ends of rods coincide ct ct ( 2,ct 2 ) ( 2,c t 2 )

16 Time Order And Event #1 occurs before Event #2, right?

17 Event #1 ct ct 1 1

18 Event #2 ct ct 2 2

19 Order of Events is Opposite in Frames ct ct

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

21 Material For Net Lecture

22 Twin Parado An eplanation?

23 Twin Parado 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 Einstein, quoted in Physics, Structure and Meaning, p288 Leon Cooper

24 Intervals

25 Minkowski: Interval Calibrating aes o If we define =1, where is =1? ct ct Light-Ray Consider "invariant" [ ] 2 - [ ct] 2 = g ( + bc t ) È [ ] bc t = g 2 Í Î Í - c t Define Ds 2 = cdt [ ] 2 - [ g( c t + b )] 2 = g 2 1- b 2 = [ ] + [ bc t ] 2... [ ] 2 - [ 2 bc t ] - [ b ] 2 [ ] 2 - [ c t ] 2 [( )( )] [ ] 2 - [ c t ] 2 [ ] 2 - [ D] 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

26 Two Watches Class Discussion

27 To Follow Relativistic Kinematics