Rev. 020307 Lorentz Tran s.: Worke d Ex...

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

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”)


Question SectionQuestion Section


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



•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



•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



•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?


What is a photonWhat is a photon’’s views viewof the universe itof the universe itpasses through?passes through?


Use of LorentzUse of LorentzTransformationTransformation

•to study rod and single clock events


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)


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


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



• 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



• 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


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

†



- 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


So how do photons viewSo how do photons viewthe universe?the universe?


Next LectureNext Lecture



• 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

†



- 12



• 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


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

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