PH0008 Quantum Mechanics and Special Relativity Lecture 6 (Special Relativity) Use of Lorentz-Einstein Transformation.

Transcription

1 PH0008 Quantum Mechanics and Special Relativity Lecture 6 (Special Relativity) Minkowski Space Use of Lorentz-Einstein Transformation Minkowski Space 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 next 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 Example: Rod and Single Clock Time Dil., Lorentz Cont., Relativity of Simultaneity o Minkowski Space Lecture 6 (W 3/6) o Minkowski Space More Worked Example: Two Rods Time Dil., Lorentz Cont., Relativity of Simultaneity Lecture 7 (F 3/8) o Review with Further Worked Example 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 Question Section

5 Question SpecRel L06-Q1 Where will the Dow Jones (~10,000 today) be when you graduate? o(1) >+60% o(2) 40_60% o(3) 20_40% o(4) 0_20% o(5) -20_0% o(6) -40_-20% o(7)-60_-40% o(8) <-60%

6 Question SpecRel L06-Q2 Where will the Dow Jones (~10,000 today) be when Jan 99 you graduate? o(1) >+60% o(2) 40_60% o(3) 20_40% o(4) 0_20% o(5) -20_0% o(6) -40_-20% o(7)-60_-40% o(8) <-60% Oct ~ 1,000 10,000 Oct ,000

7 Question SpecRel L06-Q3 How do we view these events? (see demo) o(1) A and B simultaneous o(2) A before B o(3) B before A o(4) None of above

8 Can t t determine relative time of an event without specifying position

9 Use of Lorentz Transformation to study rod and single clock events

10 (x 1,t 1 ) ( x 1, t 1 ) Event #1 Single Disk and Rod (3) Use each Lorentz Transformation in turn ( ) fi gdx 0 = gbcdt (1) x 2 = g x 2 - bct 2 ( ) fi cd (2) c t 2 = g ct 2 - b x 2 Dx 0 = vdt ( ) ( ) t 0 = g cdt - bdx 0 = g cdt - bbcdt ( ) D t 0 = gdt 1- b 2 = 1 g Dt Event #2 (x 2,t 2 ) ( x 2, t 2 ) (3) x 2 = g x 2 + bc t 2 Dt = gd t 0 ( ) fi Dx 0 = gbcd (4) ct 2 = g c t 2 + b = gvd t 0 = gd x D x = 1 g Dx 0 ( ) fi cdt = gcd x 2 t 0 t 0

11 (x 1,t 1 ) ( x 1, t 1 ) Event #1 Event #2 Single Disk and Rod (4) (x 2,t 2 ) ( x 2, t 2 ) What do they mean (1) Dx 0 = vdt (2) Dt = gd t 0 (3) D x = 1 g Dx 0 (4) Dt = gd t 0 o (1) The velocity of disk is v in S rod frame The time interval between events in rod frame is simply L/v This must be the case o (2) Clock tick of disk when observed in rod frame is slower Moving clocks appear slower o (3) Apparent length of rod measured in disk frame is shorter Moving lengths appear shorter o (4) We already knew this

12 (x 1,t 1 ) ( x 1, t 1 ) Single Disk and Rod (5) Event #3 (x 3,t 3 ) ( x 3, t 3 ) v is velocity of frame S measured in S x = g( x - bct) x = g x + bc t y = y y = y z = z z = z c t = g ct - b x t + b x ( ) ( ) ct = g( c ) ( ) b = v c g = 1- b 2 Consider Event #3 o The right hand end of the rod when Event #1 occurs in rod frame S In rod frame S x 3 = Dx 0 = x 2 t 3 = t 1 = 0 In disk frame S t 3 =? t 1 = 0 No!!! - don t use common sense o Use Lorentz transforms (5) x 3 = g x 3 - bct 3 ( ) fi ( ) x 3 = g Dx 0 (6) c t 3 = g( ct 3 - b x 3 ) fi c t 3 = g( -bdx 0 ) t 3 = -g v c 2 Dx 0

13 Single Disk and Rod (6) (x 1,t 1 ) ( x 1, t 1 ) In rod frame S x 3 = Dx 0 = x 2 t 3 = t 1 = 0 In disk frame S x 3 = gdx 0 c t 3 = -gbdx 0 Event #3 (x 3,t 3 ) ( x 3, t 3 ) Consider Event #3 o At right hand end of rod, an event simultaneous with Event #1 when in the rod frame, S In the disk frame S Event #3 o Occurs before t =0 (Event #1) i.e. before Event #1 o It is a distance >Dx 0 from Event #1 Not shortened, further away But remember it does not occur at same time as t 1 Let s introduce a 2nd disk separated by rigid bar to help visualise what is going on

14 Two Disks, a Rod, and an Excuse Me? (7) Viewed in rod frame Event #3 (x 1,t 1 ) Event #1 Viewed in (two) disk frame Event #3 Event #1 ( x 1, t 1 ) (x 3,t 3 ) ( x 3, t 3 ) Consider Event #3 o Event #1 & #3 simultaneous in rod frame In the disk frame S o Event # 3 occurs before Event #1 t 3 <0 In rod frame S x 3 = Dx 0 = x 2 t 3 = t 1 = 0 In disk frame S x 3 = gdx 0 c t 3 = -gbdx 0 o Event #3 is a distance >Dx 0 from Event #1 The disks are further apart than Dx 0 But remember it does not occur at same time as t 1

15 Space-Time Diagrams Help visualize consequences of Lorentz Transforms

16 Simple 1-D (space) world (Minkowski, 1908) Add time as 2nd dimension ct x

17 Time as extra dimension Simple 2-D (space) world (Video) ct y x

18 Space and Time Become Mixed Note that variable in S (x,t ) are formed from both (x,t) and vise versa ( ) x = g ( x + bc t ) x = g x - bct y = y z = z c t = g ct - b x ( ) ct = g( c t + b x ) b = v c,v is velocity of frame S measured in S) 1 g = 1- v 2 c = b 2 Note the use of (ct) rather than t which accentuates the symmetry of the transforms

19 Minkowski Path is described by unique locus in (x,t) ct x

20 Minkowski (Equally valid) Even though axis are not orthoganal, locus is still unique ct x

21 Minkowski: Trajectories Consider particles with different velocities in frame S ct Allowed trajectory Allowed (constant v) Light-Ray Prohibited trajectory x

22 Stationary point in frame S Minkowski: Stationary ct Path of x=constant (i.e. point stationary in S) x

23 Minkowski: 2nd frame Consider 2nd frame S of reference with constant velocity v ct Path of x =0 (i.e. point stationary in S ) ( ) x = g x - bct If x = 0 fi x = bct = vt x This could be Galilean?

24 Question SpecRel L06-Q4 The axis shown could be Galilean? o(1) Yes o(2) No

25 Minkowski: 2nd frame Consider 2nd frame S of reference with constant velocity v ct ct Path of x =0 (i.e. point stationary in S ) If x = 0 x = g x - bct ( ) fi x = bct = vt x Consider also t = 0 c t = g( ct - b x) fi x = 1 b ct Note symmetrical arrangement of x & ct x

26 Minkowski: -b Consider 2nd frame S of reference with constant velocity -v ct ct Path of x =0 (i.e. point stationary in S ) If x = 0 x = g x + bct ( ) fi x = -bct = -vt Consider also t = 0 c t = g( ct + b x) fi x = - 1 b ct x x

27 Minkowski: Light Path Light traces same velocity in either frame! ct ct Light-Ray x = ct x = c t x x

28 Video Video 00:30:30 -> 00:37:50

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

30 Lorentz Transformation Two frames S and S moving at relative velocity v ct ct Light-Ray ( ) x = g ( x + bc t ) x = g x - bct y = y z = z c t = g ct - b x ( ) ct = g( c t + b x ) b = v c,v is velocity of frame S measured in S) 1 g = 1- v 2 c = b 2 x Note the use of (ct) rather than t which accentuates the symmetry of the transforms x Video

31 Consider a stationary rod in S Stationary Rod in S ct ct x x

32 Consider a stationary rod in S Stationary Rod in S ct ct x x

33 Two Watches Discuss