Relativistic Kinematics - Brown...

Gaitskell

PH0008Quantum Mechanics and Special Relativity

Lecture 8 (Special Relativity)

Relativistic Kinematics

Velocities in Relativistic Frames& Doppler Effect

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 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

• 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

• 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

• Homework #8 (M 3/18)o Start early!

(see web “Assignments”)


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 weeko I will hold special office hours on Friday 1-3 pm

o


Question SectionQuestion Section


Question SpecRel L08-Q1

•New problem: Clock coming directly towards us atnear 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


Twin ParadoxTwin Paradox•Discuss


Twin Paradox

The phenomena of electrodynamics as well as ofmechanics possess no properties corresponding tothe idea of absolute rest. They suggest rather that… the same laws of electrodynamics and optics willbe valid for all frames of reference for which theequations of mechanics hold good.

Einstein, quoted in Physics, Structure and Meaning, p288 Leon Cooper

• First Lawo Body continues at rest, or in uniform motion …

• During acceleration and deceleration this frame is not inertialo We will return to this problem at end of Relativistic Kinematics Section


Review Space-TimeReview Space-TimeIntervalsIntervals


Minkowski: Interval

• Separation of two events in Space-Time

x

ct

x’

ct’

Light-Ray†

Consider "invariant"x[ ]2

- ct[ ]2= g ¢ x + bc ¢ t ( )[ ]2

- g c ¢ t + b ¢ x ( )[ ]2

= g 2 ¢ x [ ]2+ 2 ¢ x bc ¢ t [ ] + bc ¢ t [ ]2 ...- c ¢ t [ ]2

- 2 ¢ x bc ¢ t [ ] - b ¢ x [ ]2

È

Î Í Í

˘

˚ ˙ ˙

= g 2 1- b 2( ) ¢ x [ ]2- c ¢ t [ ]2( )[ ]

= ¢ x [ ]2- c ¢ t [ ]2

†

DefineDs2 = cDt[ ]2

- Dx[ ]2

If events are simultaneous (but spatiallyseparated) in one frame then

Ds2 < 0 "Space - like"and events cannot be causally connected

If events occur in same place in one frame(separated only by time) then

Ds2 > 0 "Time - like"and events can be causally connected

Ds2 = 0 "Light - like"Events are on light - cone


Relativistic KinematicsRelativistic Kinematics


Relativistic Treatment of Velocities

• Start with new definitionso (Board)

• New beta notation

• Derivatives w.r.t. cdt

• Look at how velocity will transformo Consider derivatives of variable w.r.t. time†

¢ x = g x - bct( ) x = g ¢ x + bc ¢ t ( )¢ y = y y = ¢ y

c ¢ t = g ct - b x( ) ct = g c ¢ t + b ¢ x ( )

†

x = g ¢ x + bc ¢ t ( )dx

c d ¢ t = g

d ¢ x c d ¢ t

+ bc d ¢ t c d ¢ t

Ê

Ë Á

ˆ

¯ ˜

= g ¢ b x + b( )

†

ct = g c ¢ t + b ¢ x ( )c dtc d ¢ t

= gc d ¢ t c d ¢ t

+ bd ¢ x c d ¢ t

Ê

Ë Á

ˆ

¯ ˜

= g 1+ b ¢ b x( )†

y = ¢ y dy

c d ¢ t =

d ¢ y c d ¢ t

= ¢ b y


Relativistic Treatment of Velocities (2)

• Use previous expressions to geto bx and by

o By symmetry we can also quickly calculatebx‘ and by‘

†

x = g ¢ x + bc ¢ t ( )dx

c d ¢ t = g

d ¢ x c d ¢ t

+ bc d ¢ t c d ¢ t

Ê

Ë Á

ˆ

¯ ˜

= g ¢ b x + b( )

†

ct = g c ¢ t + b ¢ x ( )c dtc d ¢ t

= gc d ¢ t c d ¢ t

+ bd ¢ x

c d ¢ t Ê

Ë Á

ˆ

¯ ˜

= g 1+ b ¢ b x( )†

y = ¢ y dy

c d ¢ t =

d ¢ y c d ¢ t

= ¢ b y

†

bx =dx

c dt=

dxc d ¢ t

c d ¢ t c dt

=g ¢ b x + b( )g 1+ b ¢ b x( )

=¢ b x + b( )

1+ b ¢ b x( )

†

by =dy

c dt=

dyc d ¢ t

c d ¢ t c dt

=¢ b y

g 1+ b ¢ b x( )=

¢ b y g

1+ b ¢ b x( )

†

¢ b x =d ¢ x

c d ¢ t =

bx - b( )1- bbx( )

†

¢ b y =d ¢ y

c d ¢ t =

by g

1+ bbx( )



• Considero bx‘=1

• Tests of this extreme caseo Pions decay in flight

o Accelerators

†

If ¢ b x =1

bx =¢ b x + b( )

1+ b ¢ b x( )

=1+ b( )1+ b( )

=1



• In low velocity limito bx <<1 and b <<1

o Denominator becomes ~1• Both denominator and g are second order in

velocities

o Becaomes simple addition of velocities• Galilean

†

bx =¢ b x + b( )

1+ b ¢ b x( )ª ¢ b x + b

†

by =¢ b y g

1+ b ¢ b x( )ª ¢ b y

†

¢ b x =bx - b( )1- bbx( )

ª bx - b

†

¢ b y =by g

1+ bbx( )ª by


Doppler Effect in Sound

• Acoustical Effecto (Board).


Relativistic Doppler Effect

• Source in S frame, Observer in S’ frame

x’

ct’

x

ct

1st Pulse

(n+1) Pulse

†

(x1,t1)

†

t = nt†

(x2,t2)

†

¢ x 1 = ¢ x 2

†

t = 0

†

b is velocity of observer frame ¢ S measured in S(1) x1 = ct1 = x0 + bct1(2) x2 = c t2 - nt( ) = x0 + bct2

Therefore, subtracting (2) - (1) abovec t2 - t1( ) - cnt = bc t2 - t1( )

c t2 - t1( ) =cnt

1- b( )=

cnt1- b( )

x2 - x1 =bcnt1- b( )

In observer frame ¢ S using Loretz Trans.c ¢ t 2 - ¢ t 1( ) = g c t2 - t1( ) - b x2 - x1( )[ ]

= gcnt

1- b( )- b

bcnt1- b( )

È

Î Í

˘

˚ ˙

†

¢ x 0

†

The time interval covers n periods, andthe apparent period ¢ t in ¢ S is

¢ t =t2 - t1

n

= gt

1- b( )- b

bt1- b( )

È

Î Í

˘

˚ ˙

=gt

1- b( )1- b 2[ ]

= g 1+ b( )t


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 thanthat of the source

o This answer depends only on relative velocity ofsource and observer, unlike acoustic effect

†


¢ t = g 1+ b( )t

=1+ b( )2

1- b 2( )Ê

Ë Á Á

ˆ

¯ ˜ ˜

12

t

=1+ b1- b

Ê

Ë Á

ˆ

¯ ˜

12t

Or in terms of frequencies n

¢ n =1- b1+ b

Ê

Ë Á

ˆ

¯ ˜

12n

†


¢ t =t2 - t1

n

= gt

1- b( )- b

bt1- b( )

È

Î Í

˘

˚ ˙

=gt

1- b( )1- b 2[ ]

= g 1+ b( )t

†

Remember Acoustical Doppler Effect : -Stationary source, receeding receiver

¢ n = 1- b( )nReceeding source, stationary receiver

¢ n =1

1+ b( )n

where b is the velocity of moving objectdivided by wave velocity in medium



• Source in S frame, Observer in S’ frame,moving away from source with velocity b

o The frequency the observer sees is lower than thatof the source: RED SHIFTED

• If source and observer approach one anotherthen 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 usdirectly 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

¢ n =1- b1+ b

Ê

Ë Á

ˆ

¯ ˜

12n

†

Approaching at b

¢ n =1+ b1- b

Ê

Ë Á

ˆ

¯ ˜

12n



• Exampleso Red shift of galaxies (Hubble)



• Transverse Doppler Effecto Classically when velocity of object is perpendicular to sight linethere is no Doppler Effect

o However, relativistically there is still time dilation to consider

†

Perpendicular at velocity b, observer ¢ S ¢ t = gt

¢ n =1g

n


Next Lecture

• Wednesdayo Guest Lecture: General Relativity, Prof Ian Dell’Antonio

• Fridayo Doppler Effect

o Reanalysis of Twin Paradox with signal exchange

o Introduction to Relativistic Dynamics


Material For Next LectureMaterial For Next Lecture


x’

ct’

x

ctLight-Ray


Discuss Symmetry of Problem

• (Board)o Basic Lorentz Relations under exchage of DT <-> -DT and b <-> -b

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