rev. 020307 lorentz tran s.: worke d ex...
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Gaitskell
PH0008Quantum Mechanics and Special Relativity
Lecture 5 (Special Relativity)
Rev. 020307
Lorentz Trans.: Worked Example Time Dilation, Lorentz Contractions - Rod and Single Clock
Use of Lorentz-Einstein Transformation
Prof Rick Gaitskell
Department of PhysicsBrown University
Main source at Brown Course Publisher
background material may also be available at http://gaitskell.brown.edu
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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 andTime 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 andConservation Laws
• (Review)• Ch3 Einstein & Lorentz Transforms
• Ch4 Realtivity: Measurement of Lengthand Time Inetrvals
• Homework #7 (M 3/11)o Start early - tough
(see web “Assignments”)
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SectionQuestion Section
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L04-Q1
•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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L04-Q2
•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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L04-Q3
•Which is the correct expression for g? (What is g?)o(1)
o(2)
o(3)
o(4)
†
g =1
1- b 2
†
g =1
1- b 2
†
g = 1- b 2
†
g 2 =1
1- b 2
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L04-Q4
•When we observe moving object …o(1) Time and length appear slower & shorter than propervalues?
o(2) Time and length appear faster & shorter than propervalues?
o(3) Time and length appear slower & longer than propervalues?
o(4) Time and length appear faster & longer than propervalues?
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
What is a photonWhat is a photon’’s views viewof the universe itof the universe itpasses through?passes through?
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Use of LorentzUse of LorentzTransformationTransformation
•to study rod and single clock events
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Lorentz Contraction - Formally
• Let’s rework the Lorentz Contraction example, more formally, usingLorentz Transformations
†
¢ x = g x - bct( ) x = g ¢ x + bc ¢ t ( )¢ y = y y = ¢ y ¢ z = z 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)
g =1
1- v 2 c 2=
11- b 2
Note the use of (ct) rather than t which accentuates the symmetry of the transforms
• Space and Time are mixingas move between frames
• v ≤ c
• Eqns are Linear• If Dx=Dt=0 then Dx’=Dt’=0
o Two events that take place atsame point in position and timein one frame will also becoincident (in space and timein another frame)
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Single Disk and Rod - using Lorentz Transformations
• Label Events in (space,time) in both frames (subscripts are event #)
†
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( ) ct = g c ¢ t + b ¢ x ( )b = v c g = 1- b 2( )
- 12
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
†
(x2,t2)
†
( ¢ x 2, ¢ t 2)
Event #1
Event #2
** Work Example on Board
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Single Disk and Rod (2)
• Let’s re-annotate variableso Remember S is rod frame, S’ is disk frame
o Define event #1 as “zero” in both frames• No less of generality
o And relable event #2 using proper subscript 0where appropriate
†
x = t = ¢ x = ¢ t = 0
Proper time in disk frame¢ t 2 = D ¢ t 0
Proper time in disk framex2 = Dx0
Also redesignatet2 = Dt
Disk isn't moving in ¢ S ¢ x 2 = ¢ x 1 = 0
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
†
(x2,t2)
†
( ¢ x 2, ¢ t 2)
Event #1
Event #2
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Single Disk and Rod (3)
• Use each Lorentz Transformation in turn
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
†
(x2,t2)
†
( ¢ x 2, ¢ t 2)
Event #1
Event #2
†
(1) ¢ x 2 = g x2 - bct2( ) fi gDx0 = gbcDtDx0 = vDt
(2) c ¢ t 2 = g ct2 - b x2( ) fi cD ¢ t 0 = g cDt - bDx0( )= g cDt - bbcDt( )
D ¢ t 0 = gDt 1- b 2( )=
1g
Dt
Dt = gD ¢ t 0
(3) x2 = g ¢ x 2 + bc ¢ t 2( ) fi Dx0 = gbcD ¢ t 0= gvD ¢ t 0= gD ¢ x
D ¢ x =1g
Dx0
(4) ct2 = g c ¢ t 2 + b ¢ x 2( ) fi cDt = gcD ¢ t 0
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Single Disk and Rod (4)
• What do they meano (1) The velocity of disk is v in S rod frame
• The time interval between events in rod frame issimply L/v
• This must be the case…
o (2) Clock tick of disk when observed in rod frameis slower
• Moving clocks appear slower
o (3) Apparent length of rod measured in diskframe is shorter
• Moving lengths appear shorter
o (4) We already knew this…
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
†
(x2,t2)
†
( ¢ x 2, ¢ t 2)
Event #1
Event #2
†
(1) Dx0 = vDt(2) Dt = gD ¢ t 0(3) D ¢ x =
1g
Dx0
(4) Dt = gD ¢ t 0
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Matrix Form of Equations
• We can rewrite Lorentz Transformationsin a matrix form
o Don’t worry if this is new to you... just enjoythe simplicity of the representation
†
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( ) ct = g c ¢ t + b ¢ x ( )b = v c g = 1- b 2( )
- 12
†
¢ x c ¢ t
Ê
Ë Á
ˆ
¯ ˜ =
g -gb
-gb g
Ê
Ë Á
ˆ
¯ ˜
xct
Ê
Ë Á
ˆ
¯ ˜
¢ X = R XandR-1 ¢ X = R-1R X = X
R-1 =1
det(R)g gb
gb g
Ê
Ë Á
ˆ
¯ ˜
(see box on calculating inverse matrix)but, det(R) = g 2 1- b 2( ) =1
X = R-1 ¢ X =g gb
gb g
Ê
Ë Á
ˆ
¯ ˜ ¢ X
fixct
Ê
Ë Á
ˆ
¯ ˜ =
g gb
gb g
Ê
Ë Á
ˆ
¯ ˜
¢ x c ¢ t
Ê
Ë Á
ˆ
¯ ˜
†
Note on Inverse of 2 ¥ 2 matrix :
R =a bc d
Ê
Ë Á
ˆ
¯ ˜ R-1 =
1det R( )
d -b-c a
Ê
Ë Á
ˆ
¯ ˜
where deta bc d
Ê
Ë Á
ˆ
¯ ˜ = ad - bc
Check R R-1 =1
det R( )a bc d
Ê
Ë Á
ˆ
¯ ˜
d -b-c a
Ê
Ë Á
ˆ
¯ ˜ =
1ad - bc
ad - bc -ab + bacd - dc -cb + ad
Ê
Ë Á
ˆ
¯ ˜
=1 00 1
Ê
Ë Á
ˆ
¯ ˜ = I
I is the identity matrix, such that XI ≡ X
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
So how do photons viewSo how do photons viewthe universe?the universe?
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Next LectureNext Lecture
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Single Disk and Rod (5)
• Consider Event #3o The right hand end of the rod when Event #1occurs in rod frame S
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
Event #3
†
In rod frame Sx3 = Dx0 = x2 t3 = t1 = 0
In disk frame ¢ S
¢ t 3 =?
¢ t 1 = 0†
(x3,t3)
†
( ¢ x 3, ¢ t 3)
• No !!! - don’t use “common” senseo Use Lorentz transforms
†
(5) ¢ x 3 = g x3 - bct3( ) fi ¢ x 3 = g Dx0( )
(6) c ¢ t 3 = g ct3 - b x3( ) fi c ¢ t 3 = g -bDx0( )
¢ t 3 = -gvc 2 Dx0
†
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( ) ct = g c ¢ t + b ¢ x ( )b = v c g = 1- b 2( )
- 12
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Single Disk and Rod (6)
• Consider Event #3o At right hand end of rod, an eventsimultaneous with Event #1 when in the rodframe, S
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
Event #3
†
In rod frame Sx3 = Dx0 = x2t3 = t1 = 0
In disk frame ¢ S ¢ x 3 = gDx0
c ¢ t 3 = -gbDx0
†
(x3,t3)
†
( ¢ x 3, ¢ t 3)• In the disk frame S’ Event #3
o Occurs before t’=0 (Event #1)• i.e. before Event #1
o It is a distance >Dx0 from Event #1• Not “shortened”, further away
• But remember it does not occur at sametime as t1‘
Let’s introduce a 2nd diskseparated by rigid bar to helpvisualise what is going on
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Two Disks , a Rod, and an “Excuse Me?” (7)
• Consider Event #3o Event #1 & #3 simultaneous in rod frame
†
(x1,t1)
†
( ¢ x 1, ¢ t 1)
Event #3
†
In rod frame Sx3 = Dx0 = x2t3 = t1 = 0
In disk frame ¢ S ¢ x 3 = gDx0
c ¢ t 3 = -gbDx0†
(x3,t3)
†
( ¢ x 3, ¢ t 3)
• In the disk frame S’…o Event # 3 occurs before Event #1
• t3‘<0
o Event #3 is a distance >Dx0 from Event #1• The disks are further apart than Dx0
• But remember it does not occur at sametime as t1‘
Event #1
Viewed in rod frame
Viewed in (two) disk frame
Event #3
Event #1