Physics 3313 - Lecture 2 1/26/20091 3313 Andrew Brandt Monday January 26, 2009 Dr. Andrew Brandt...

3313 Andrew Brandt 1

Physics 3313 - Lecture 2

1/26/2009

Monday January 26, 2009Dr. Andrew Brandt

1. Special Relativity2. Galilean Transformations3. Time Dilation and Length Contraction


What do we mean by motion?

Throw a ball, fly in an airplane…

If you are watching the ground while you fly, you see

motion, but if you are watching the seat in front of you

you don’t.

Motion: Change of position relative to something else

Frame of reference is a part of description of motion

1905 Einstein Special Relativity (2005 World Year of Physics to celebrate) treats inertial frames of reference where Newton’s First law holds

(remember that ? )

Body at rest stays at rest, body in motion stays in motion with constant velocity if no external force)

What about if there is a force?

CH. 1 Relativity (Motion)

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Any frame moving with constant velocity relative to an inertial frame is

itself an inertial frame

This implies that there is no universal frame of motion—everything is relative

(especially in West Virginia)

Person in car moving at constant velocity relative to person on ground equivalent to person on ground moving at same speed next to car

(although he might get tired running 40 mph –backwards)

General Relativity 10 years later (still Einstein) deals with accelerated reference frames and gravity (harder math—graduate class)

Inertia/Relativity

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Postulates :

1) Laws of physics are the same in all inertial frames

2) The speed of light in free space is a constant in all inertial frames

Sounds fast! Is something misssing?

What if it was mirons/millenium. Need units in physics!!!

m/sec in this case; 186,000 miles/second

Special Relativity

83 10c x

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• Suppose an observer is in an inertial reference frame S, when an event occurs• What are its time and place in a frame S’ moving at constant velocity with

respect to frame S?• Is S’ an inertial frame?

Galilean Coordinate Transformation(A1)

'

'

'

'

x x vt

y y

z z

t t

• To get this relationship, suppose clocks started when origins of two frames coincide, then as S’ moves along x axis, measurements in S will be greater than S’ by vt (velocity x time)

• Can also consider the case where the event is fixed at the origin of the S’ frame (S’ frame represents the moving ojbect). After some time the value of the origin of the S’ frame is still zero, but the distance to the origin in the S frame is x=vt

• Generalized to any point x = x’+ vt or can rewrite:1/26/2009


• Suppose two events are separated by x and t

• Can divide two equations to obtain

• Take limit • But this does not work according to special relativity: shine a flashlight in

a car moving with velocity v; in S’ frame vx’=c so in S frame vx=v+c (Einstein says not Allowed!)

• SPACE and TIME not absolute: Depend on Relative motion

Galilean Coordinate Transformation

'

'

x x v t

t t

0

'

'limt

x xv

t t

''

'x

dx dxv v

dt dt

velocity addition

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'

'

x x

x x

v v v

v v v


Time Dilation

• Time Dilation: = moving clocks run slow• Measurements of time intervals affected by relative motion• Person on a spaceship measures time interval between two events as• On ground measure

• Proper time is the time interval between two events at the same place in an observers frame

• If that frame is moving with respect to an observer on ground , the observer

will see the events occurring at different places, so the duration appears

longer (hence time dilation)

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0t

0t t

0t


Light Clock

1/26/20090

0

2Lt

C

0L

At Rest: Moving with velocity v:


Light Clock Math• Pythagorean Theorem (an oldie, but a

goodie) :

simplify:

solve for t

take sqrt

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2 2 20( ) ( )

2 2

ct vtL

22 2 2

0( )4

tc v L

2 22 0 0

22 22

2

4 (2 )

( )(1 )

L Lt

vc vc

c

0

0

2 2

2 2

2

1 1

Ltct

v vc c

2

21v

c

2

2

11

1vc

v

c

0t t

Is >, < or =1 for v<c?

often use effect is reciprocal (spaceship guy sees zig-zag pattern of ground clock while his ticks normally )—who is observer!!


Lorentz Transform• General relationship between P=x,y,z,t in frame S and P’=x’,y’,z’,t’ in

frame S’ (derived in Appendix); these assume measurements made in S frame and want to transfer to S’ frame

• or

• and

• note for time dilation derivation we did special case of x=0 so we got

t’=t and no extra term• Suppose v<<c then =1 and Lorentz transform reduces to Galilean one• Inverse transformation

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'

2

21

x vtx

vc

( )x vt ( )x ct

'

'

y y

z z

2'

2

21

vxtctvc

2

( )vx

tc

( )x

tc

v v '( )x x vt '2

( )vx

t tc


Time Dilation Example

• Muons are essentially heavy electrons (~200 times heavier)• Muons are typically generated in collisions of cosmic rays in upper

atmosphere and, unlike electrons, decay ( sec)• For a muon incident on Earth with v=0.998c, an observer on Earth would

see what lifetime of the muon?• 2.2 sec?

• t=35 sec • Moving clocks run slow so when an outside observer measures, they see a

longer time than the muon itself sees.

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0 2.2t

2

2

116

1vc


Length Contraction• Proper length (length of

object in its own frame:

• Length of object in observer’s frame:

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2 1L x x

2 1 2 1( ) ( ) ( )x vt x vt x x

' '0 2 1L x x

'2x

'1x

' '0 2 1L x x

0L L0 /L L >1 so the length is shorter in the direction of motion


More about Muons

• 1/cm2/minute at Earth’s surface (so for a person with 600 cm2 that would be 600/60=10 muons/sec passing through!)

• They are typically produced in atmosphere about 6 km above surface of Earth and frequently have velocities that are a substantial fraction of speed of light, v=.998 c for example

• How do they reach the Earth?• We see the muon moving so t=35 sec not 2.2 sec , so they can travel 16 time

further, or about 10 km, so they easily reach the ground• But riding on a muon, the trip takes only 2.2 sec, so how do they reach the

ground???• Muon-rider sees the ground moving, so the length contracts and is only

• At 1000 km/sec, it would take 5 seconds to cross U.S. , pretty fast, but does it give length contraction? (for v=0.9c, the length is reduced by 44%)

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8 60 2.994 10 2.2 10 sec 0.66

sec

mvt x x x km

0 6 /16 0.38L

km

0.999994L L


Twin Paradox• Famous relativistic example: Carla and Darla 20 years old• Darla travels to a planet 20 light-years away at 0.8c• To Carla, Darla’s life (clock) is slower by

• Darla’s heart beats 3 times for every 5 of Carla• Round trip takes 50 years (according to Carla)• So she knows she is 70 and thinks Darla should have aged only 30 years

and should be 50 years old• But … according to SR, it looks to Darla like Carla is the moving one!• So who is 70, and who is 50, or what?• Darla changed inertial frames in accelerating from 0 to 0.8c, decelerating

to stop, etc., so time dilation only applies to Carla’s observations since she stayed in inertial frame (so shes right—and old)

• Verified by airline trip around world (not the 20 years part!)1/26/2009

2 2 2

2 2

.81 1 0.6v c

c c

2025

.8

lightyearsyears

speedoflight


Velocity Addition (Appendix)

• ~Worth reading 1.6 on connection of electricity and magnetism; electric charge Q is relativistically invariant (same in all frames).

• Galilean Velocity addition where

and • From inverse Lorentz transform and• So

• Thus ; Note this relativistic velocity addition for

gives Observer in S frame measures c too!

Strange but true!

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'x xv v v

x

dxv

dt

''

'x

dxv

dt

' '( )dx dx vdt ' '2

( )v

dt dt dxc

' ' '

'' '

2

( )

( )x

dx dx vdt dtv

vdt dtdt dxc

'xv c

2

2

2

( )

( )1

x

c v c c vv c

v c c vc

'

'

'

2 '1

dxv

dtv dxc dt

'

'

21

xx

x

v vv

vv

c


Velocity Addition Example

• Lance is riding his bike at 0.8c relative to observer. He throws a ball at 0.7c in the direction of his motion. What speed does the observer see?

• What if he threw it just a bit harder? • Doesn’t help—asymptotically approach c, can’t exceed (it’s not just a

postulate it’s the law)

2

2

.7 .80.962

.7 .81

x

c cv c

cc

1/26/2009

'

'

21

xx

x

v vv

vv

c

Physics 3313 - Lecture 2 1/26/20091 3313 Andrew Brandt Monday January 26, 2009 Dr. Andrew Brandt...

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