Relativistic Kinematics -...

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