relativistic kinematics -...
TRANSCRIPT
Gaitskell
PH0008
Quantum Mechanics and Special Relativity
Lecture 9 (Special Relativity)
Relativistic Kinematics
Relativistic Doppler Effect & Visualisation
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) • 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”)
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Homework / Office Hours
• Please pick up your HW #1-4 from outside my office B&H 516
• Special Office Hourso I will hold special office hours on Friday 1-3 pm
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SectionQuestion Section
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L09-Q1
•Two objects have velocity along x b1=0.5 and b2=-0.5measured in our frame? What is their apparentclosing velocity in our frame?
o(1) 0.0c
o(2) 0.5c
o(3) 0.8c
o(4) 1.0c
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L09-Q2
•Two objects have velocity b1=0.5 and b2 =-0.5measured in our frame S? What is their apparentclosing velocity viewed from object 1?
o(1) 0.0c
o(2) 0.5c
o(3) 0.8c
o(4) 1.0c
†
Let frame of object 1 be ¢ S moving at b1 in frame S
Velocity of 2 in ¢ S frame is
¢ b 2 =b2 - b( )1+ b2b( )
=(-0.5) - (+0.5)( )
1- (-0.5)(+0.5)( )
=-1( )
1+ 0.25( )= 0.8
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L09-Q3
•In the acoustic Doppler Effect - what is the fequencyshift dependent on?
o(1) Only the relative source-observer velocity
o(2) Velocity of source in medium
o(3) Velocity of observer in medium
o(4) Both (2) & (3)
†
¢ n = n1- ureceiver w1- usource w
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L09-Q4
•What colour is my tie, if am approaching you at b=1?o(1) Red
o(2) Infra-Red
o(3) Blue
o(4) None of above
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Question SpecRel L09-Q5
•How does a sphere, moving with high perpendicularvelocity, appear to us?
o(1) Contracted along direction of motion, but same heightas when stationary
o(2) Still spherical
o(3) Contracted vertically, but same width as when stationary
o(4) Need more information
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Doppler EffectRelativistic Doppler Effect
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Doppler Effect in Sound
• Acoustical Effecto (Reading).
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
†
Consider 1st and (n +1)th light pulses from source at x = 0 which are both observed at position ¢ x 1 = ¢ x 2 , (the observer is stationary in ¢ S ).(b is velocity of observer frame ¢ S measured in S)
In S frame, if we consider the propagation time of the light then the obs. evts #1 and # 2 are located at(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( )[ ]
= gcnt1- b( )
- bbcnt1- b( )
È
Î Í
˘
˚ ˙
†
x0
†
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
Sour
ce x
=0
Obs
erve
rx’
=con
st.
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
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
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Doppler Effect (4)
• Exampleso Red shift of galaxies (Hubble)
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
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Twin ParadoxTwin Paradox•Discuss
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
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Twin Paradox & Signal Exchange
• From Eartho 12 pulses are sent (including one at arrival)o Note only 2 arrive at ship during out bound leg
• From Ship (moving at 2/3c)o 12/1.34~8.9 pulses are sento Earth also sees a very uneven reception
pattern
x
ct
Light-Ray
†
Astronaut is moving at 2/3c on both legs
Time dilation will be¢ t = gt
g =1
1- 23( )2
~ 1.34
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
RelativisticRelativistic Visualisation Visualisation
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Apparent Rotation at Relativistic Speeds
• Apparent Rotation of Object due to finite propagation time of lighto J. Terrell, Physical Review 116, 1041 (1959).
†
To viewer the : -apparent length of perpendicular face is
=W0
cÊ
Ë Á
ˆ
¯ ˜ v
Apparent length of parallel face is
=L0
gNote this could be considered a rotation inrest frame of q since projected lengths would be
W0 sinq and L0 cosq , respectively
(cos2 q =1g 2 =1- b 2 =1- sin2 q)
†
L0
†
W0†
v
Side View
Assume oberving at perpendicular,large distance away
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
… and a sphere?
• How would a sphere appear?o R. Penrose, Proc. Camb. Phil. Soc. 55, 137 (1959).
o Always spherical…
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Tram
• Effectso Lorentz contraction of parallel length
• What is the g ?
o “Rotation” of trailing perpendicular edge• Time of flight effect
o Curve of verticals above and below centre of view• ditto
v~0.87c
• Images provided byo C.M. Savage and A.C. Searle,
Department of Physics andTheoretical Physics, AustralianNational University, ACT 0200,Australia
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Tram (2)
• Animation also showso Doppler shift of light
o Intensity effects
Tram moves fromright (when it ispartially comingtoward us) towardleft of scene (whenit is partiallyreceding)
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Relativistic Tram Shadow
• Shadow of tram (from light in upper portion of picture)
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Star Field
• Speed increases during movie, up to v~co The stars from sides and behind all pile up in narrow cone in forward direction
o Note also increase in intensity
PH0008 Gaitskell Class Spring2002 Rick Gaitskell
Next Lecture
• Mondayo Introduction to Relativistic Dynamics