Relativistic Kinematics - Brown...
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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
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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”)
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
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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 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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Twin ParadoxTwin Paradox•Discuss
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Review Space-TimeReview Space-TimeIntervalsIntervals
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic KinematicsRelativistic Kinematics
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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( )
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Treatment of Velocities (3)
• 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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Treatment of Velocities (4)
• 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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Doppler Effect in Sound
• Acoustical Effecto (Board).
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
†
The time interval covers n periods, andthe apparent period ¢ t in ¢ S is
¢ 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
†
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
†
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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Doppler Effect (3)
• 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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Doppler Effect (4)
• Exampleso Red shift of galaxies (Hubble)
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Doppler Effect (5)
• 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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
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
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Material For Next LectureMaterial For Next Lecture
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
x’
ct’
x
ctLight-Ray
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PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Discuss Symmetry of Problem
• (Board)o Basic Lorentz Relations under exchage of DT <-> -DT and b <-> -b