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Page 1: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

RelativityRelativity

Chapter 14Chapter 14

Page 2: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

2

本章目录

14-0 Basic Requirements

14-1 Galileo’s Transformations; Space-time Concepts in Newtonian mechanics

14-2 Michelson-Morley Experiment

14-3 Fundamental Principles of Special Relativity; Lorentz Transformations

14-4 Space-time Concepts in Special Relativity

14-6 Relativistic Energy and Momentum

*14-5 Doppler’s Effect of Light

*14-7 Introduction to General Relativity

Page 3: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

3

1 Understand Galileo transformations and space-time concepts in Newtonian mechanics

2Understand Michelson-Morley experiment

3Understand basic principles of SR, Lorentz

transformation formulae

14-0 Basic Requirements

Page 4: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

4

4Understand the relativity of simultaneity,

length contraction and time delay, Grasp

space-time concepts in SR

5 Grasp the relations of mass, momentum

and velocity, and that of mass and energy, in

SR

14-0 Basic Requirements

Page 5: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

5

x'x

y 'y

v

o 'o

z 'z

'ss

*)',','(

),,(

zyx

zyxP

x'xtv

z 'z

'yy

1. Galileo Transformation and Relativity

Principle in Classical MechanicsSuppose S’ frame moves along X-axis w.r.t. S frame with uniform velocity v, we observe a event in two frames

When t = t’ = 0, O and O’ coincident

Page 6: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

6

x'x

y 'y

v

o 'o

z 'z

'ss

*)',','(

),,(

zyx

zyxP

x'xtv

z 'z

'yy

then at any later time t = t’, the transformations of coordinates are:

tt

zz

yy

vtxx

'

'

'

'

Coordinatetransformations

Page 7: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

7

v xx uu

yy uu

zz uu

velocity transformation

x'x

y 'y

v

o 'o

z 'z

'ss

*)',','(

),,(

zyx

zyxP

x'xtv

z 'z

'yy

Page 8: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

8

zz aa yy aa

xx aa

acceleration transformation

aa amF

amF

Newtonian laws of motion take the same form in two inertial reference frames moving uniformly with each other

x'x

y 'y

v

o 'o

z 'z

'ss

*)',','(

),,(

zyx

zyxP

x'xtv

z 'z

'yy

Page 9: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

9

2. Space-time Concepts in Classical Mechanics

Absolute Space: Space has nothing to do

with motion, the measurement of space is

absolutely invariant and has nothing to do

with inertial frame

Absolute Time: Time goes by

homogeneously , it has nothing to do with

the motion of matter, all inertial frames

have a consistent time

Page 10: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

10

Newtonian Absolute Space-time View

Relativity Principle of Newtonian Mechanics

Note

Relativity principle of Newtonian mechanics

loses its validity in high speed limit and

microscopic area.

Page 11: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

11

Studies on EM phenomena show that

Maxwell’s group of equations obeyed by EM

phenomena violate Galileo transformation

Light speed in vacuum

?v cc

x'x

y 'yv

o 'oz 'z

'ssc

Page 12: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

12

Calculate the time durations that light signals reach the observer. The signals are sent right before and after the ball is thrown.

Beforec

dc

dt 1

21 tt v

c

dt2

Aftervc

v

E.g.

Page 13: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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13

As a result, (from Galileo transformation), the observer sees first the ball’s image after throwing, then the image before throwing

cd

c

dt 1

21 tt v

c

dt2vc

v

Page 14: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

14

When supernovae explodes, its out-layer materials fly away in all directions. We now study the light propagates during this explosive process.

l = 5000 light yearcvc

km/s1500vmatter fly-away speed

A

B

Page 15: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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15

v

c

ltA

Time of light from point A hits the earth

Time of light from point B hits the earth c

ltB

l = 5000 light-yearcvc

km/s1500vmatter fly-away speed

A

B

Page 16: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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16

l = 5000 light-yearcvc

km/s1500vmatter fly-away speed

A

B

Theoretical calculations show that the strong

light emissions in supernovae explosion last

about , while the real

duration is about 22 months, explain this.

yrttt AB 25

Page 17: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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17

Michelson-Morley Experiment

The Michelson–Morley experiment was

performed in 1887 by Albert Michelson

and Edward Morley at what is now Case

Western Reserve University in

Cleveland, Ohio, U.S.A. Its results are

generally considered to be the first strong

evidence against the theory of a ether and

in favor of special relativity.

Page 18: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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18

Testing the invariance of the speed of light on the motion of the observer: the reconstructd setup of the first experiment by Albert Michelson in Potsdam, performed in 1881, and a modern high-precision, laser-based set-up that keeps the mirror distances constant to less than a proton radius and constantly rotates the whole experiment around a vertical axis.

Page 19: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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19

“ether”reference frame is absolute rest frame

G M1 G

vv

c

l

c

lt1

2

2

cltcΔv

Suppose ether reference frame is , and the lab frame is s

sMichelson-Morley Experiment

l 12 GMGM

vs G

M1

M2

T

Page 20: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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20

l 12 GMGM

vs G

M1

M2

T(Observe from S’ frame

G M2

c 22 vc

v-

M2 G

cv-

22 vc

G M2 G

2221

2

cc

lt

v

Page 21: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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21

2

2

22

cl

ΔN

v

m/s103,nm500,m10 4 vl

4.0N Observable accuracy: 01.0N

Exp. Result: , the relative motion of the earth to“ether”is un-observed!

Conclusion: ether-as the absolute reference frame-does NOT exist

0N

Page 22: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

22

Plots based on data from the Michelson-Morley experiment

Page 23: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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23

Page 24: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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24

The same experiment were repeated many

times in different seasons, directions, as

well as with higher precision lasers in

recent years, no relative motion are

observed.In a lecture given at the Royal Institute, Nineteenth Century Clouds over the dynamic theory of heat and light, April 1900, Lord Kelvin said that the physics knowledge is almost complete. Two small ``clouds" remain over the horizon.

Page 25: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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25

Albert Einstein (1879-1955)

Page 26: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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26

Light propagates through empty space with

a definite speed independent of the speed of

the source or of the observer

(1) The Principle of Relativity

The laws of physics have the same form in

all inertial reference systems.

(2) The Principle of Invariant Light Speed

1. Basic Principles of SR

Page 27: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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27

Keys: Relativity and Invariance

Galileo transformation and special relativity’s principles conflict

Special relativity’s principles agree well with experiments!

The two principles are the foundation of SR

Page 28: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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28

2. Lorentz Transformation

Let coincident

at ; the space

time coordinate of

event P is shown on

right figure

0ttoo ,

z 'z

'y

x'x

yv

o 'o

'ss * )',',','( tzyx),,,( tzyxP

Page 29: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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29

)tx(tx

x vv

21

yy

zz

)xc

t(x

ct

t22

2

1

vv

(1) Lorentz Transformation

cv211

Page 30: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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30

)tx(x v yy

zz

)xc

t(t2

v

)tx(x vyy

zz

)xc

t(t 2

v

Inverse

Note When ,

Lorentz transformation goes

back to Galileo transformation

cv 1 cv

Page 31: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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31

(2) Lorentz velocity transformation

x

xx

uc

uu

21vv

x

zz

uc

uu

21v

x

yy

uc

uu

21v

Page 32: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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32

Inverse

x

yy

uc

uu

21v

x

zz

uc

uu

21v

x

xx

uc

uu

21vv

Page 33: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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33

Discussions:

c

cc

cux

21vv

Invariance of light speed

If one observer sends a light signal along

x direction in S frame, the other observer

in S′ frame sees:

Light propagates through empty space with

a definite speed independent of the speed of

the source or of the observer

Page 34: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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34

1. Relativity of simultaneity

Event 1: back wall detector receives light signal

Event 2: front wall detector receives light signal

Page 35: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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35

v

'x

'y

'o

1 212

36

9

12

36

9

S Frame (Earth frame)

Event 2 ),,,( 2222 tzyx

),,,( 1111 tzyxEvent 1

Suppose: in S frame two events occur at x1 、 x2, with a time interval . What is the time interval of the two event occurring in S′ frame?

12Δ ttt

1Δ ttt 2

Page 36: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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36

'x

'y

'o

1 2

x

y

o

v

12

36

9

12

36

9

12

36

9

)',',','( 1111 tzyx

frame(moving frame)

)',',','( 2222 tzyx

S'

2

2

1

ΔΔΔ

xc

tt

vQuestion: Do two events,

occurring simultaneously

in one inertial frame,

simultaneously occur in

another frame?

Page 37: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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37

Discussions:2

2

1

ΔΔ'Δ

xc

tt

v

-- NO simultaneity

-- NO simultaneity2

Simultaneity in place but not in time

0Δ 0Δ tx

1

Simultaneity in time but not in place

0Δ0Δ tx

Page 38: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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38

xc

ut

2-- Simultaneity

-- Simultaneity

-- NO Simultaneity

Discussions (cont’d): 2

2

1

ΔΔΔ

xc

tt

v

3

Simultaneous in place & time

0Δ 0Δ tx

4

Not simultaneous

0Δ0Δ tx

Page 39: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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39

Conclusion: Simultaneity is relative

The simultaneity is relative: only two

events occurring at the same time and

the same location in one inertial frame,

is simultaneous in another inertial frame

Page 40: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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40

The measurement of length is closely related to the concept of simultaneity

2. Length Contraction

x

y

oz

s1'x 2'x0l

'y

'x

v

'o'z

's

1x 2x

A rod along axis at

rest w.r.t. frame,

simultaneously measure

the coordinates of

both ends in frame

21, xx

xO S

S

Page 41: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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41

120 xxl then the rod’s Proper Length is

Proper length: the length measured when

they are at rest

Question: how

long is the rod

when measured

in S frame?x

y

oz

s1'x 2'x0l

'y

'x

v

'o'z

's

1x 2x

Page 42: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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42

2

2

12120

1

)()(

c

txtxxxl

v

vv

Suppose in S frame at time t, simultaneously

measured coordinates of both ends are x1 、 x2 ,

then in S frame the length is l= x2 - x1 , l and l0

satisfies:

2

2

2

2

12

11c

l

c

xx

vv

Page 43: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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43

Conclusion: Length is relative

Discussions: 2

2

0 1c

llv

1 Length contraction l<l0

V

2 If measure in frame (at rest in

frame) ,one also has length contraction

S S

The length of the object is contracted along The length of the object is contracted along

its moving directionits moving direction

Page 44: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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44

E.g-1 Suppose a rocket linearly moves w.r.t the

Earth with speed , if in rocket frame

one measures the length of the rocket 15 m,

then in the earth frame, what is the length of

the rocket one measures?

c95.0v

s'

s

Rocket Frame

Earth Frame

m150 lv

x'x

y 'y

o 'o

Page 45: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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45

s'

s

Rocket Frame

Earth Frame

m150 lv

x'x

y 'y

o 'o

Solu: Proper length ml 15'0l

ml 68.495.01151' 22 l

Moving length

Page 46: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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46

E.g-2 An 1m-long rod is placed at rest in the

plane , observer in S frame finds that

the angle between the rod and the axis is

. What are the corresponding length of the

rod and angle with Ox axis? Suppose S’

frame’s speed w.r.t S frame.23cv

45

''' yxO

'' xO

'

v

x'x

y 'y

o 'o''xl

''yl m1'l

Solu: in frame

,45' S'

Page 47: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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47

m2/2'' '' yx ll 23cv

m79.022 yx lll

43.63arctan x

y

l

l

m2/2' ' yy llin S frame

421 22 l'//cl'l xx v

'

v

x'x

y 'y

o 'o''xl

''yl

Page 48: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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48

3. Time Dilation

Page 49: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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49

'y

x'x

y v

o 'o

s'sd

B

12

36

9

)','( 1txemit light signal

)','( 2txreceive light signal

cdttt 2Δ 12 time duration

In S’ frame, two events occur at the same location B:

Page 50: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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50

x

y

o

sd

12

36

9

12

36

9

1x 2x

12

36

9

in S frame observe: ),(),,( 2211 txtx

)(211 c

xtt

v

)(222 c

xtt

v

'Δ(Δ2c

xtt

v

'ΔΔ 12 tttt

0x

21

tt

Page 51: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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51

x

y

o

sd

12

36

9

12

36

9

1x 2x

12

36

9

'y

x'x

y v

o 'o

s'sd

B

12

36

9

21

tt

Proper time:time

duration of two events

occurred at the same

location

Time dilation: moving

clock runs slower

0Δ'ΔΔ ttt

Page 52: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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52

3. When , cv tt ΔΔ

1. Time dilation is a relativistic effect

2. Time’s eclapse is not absolute,

movements change the process of time

(e.g., radioactivity decay, life etc.)

Note:

Page 53: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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53

Flying through three straight and vertical columns with 0.9 times the speed of light (0.9 c) as visualized by a person: on the left with the original colours; in the middle including the Doppler effect; and on the right including brightness effects, thus showing what an observer would actually see.

Page 54: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

March 2, 2002Natalia KuznetsovaSaturday Morning Physics

54

Relativistic aberrationSpeed Limit

cHere we are on a remote (desert) highway, where the speed limit is the speed of light

Now we are moving at about 3/4 the speed of light (0.75c). Note the relativistic aberration!

Page 55: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

Now we turn on Doppler shifting, so that the desert and the sky are blueshifted ahead

Now we turn on the "headlight" effect. Light is concentrated in the motion direction, which seems brighter, while everything around appears dimmer

Page 56: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

Physics

star field at rest star field at 0.99c

lattice at rest lattice at 0.99c

Page 57: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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57

Spacetime View in Special Relativity

(1) The spatial relation of two events is relative from different inertia frames, and also the temperal relation, it only makes sense when space-time is tied-up

(2)Space-Time is not independent with each other , they are in unity

(3)Light speed c is the bridge connected different space-time transformations

Page 58: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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E.g-3. Suppose a rocket moves linearly with respect to the earth with velocity ,it takes astronaut in the rocket 10min to record his observations on a nebulae, how much time does it take an observer on the earth to record ?

c.950v

min01.32min95.01

10

1

'ΔΔ

22

t

t

The moving clock runs slower

min10'Δ tSolu: Let the rocket be frame, the earth be S frame,

S

Page 59: Chap14 Relativity Physics Relativity Chapter 14. Chap14 Relativity Physics 2 本章目录 14-0 Basic Requirements 14-1 Galileo’s Transformations; Space-time Concepts

Chap14 Relativity

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59

(1)Relativistic momentum follows

Lorentz transformation

vvv

mmm

p

020

1

when cv vv

0mmp

1. Momentum and speed

(2)Relativistic mass2

0

1

mm

Rest mass : m0

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Relativistic mass

2

0

1

mm

mass

is a function of

velocity

)(vm

m 0m

1

2

3

4

0.2 0.4 1.00 0.6 0.8

v c

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Mass of the object which is in rest w.r.t

inertial reference frame

0mRest mass:

Conclusion: Mass is relativistic

, , which is a const.,

such that Newtonian mechanics still

holds

when cv 0mm

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2. Basic Equations of SR

)m

(tt

pF

2

0

1d

d

d

d

v

tmFmmc

d

d0

vv

when

Newton’s 2nd Lawi.e., amF

0

0i

iF

, is invariant

i

ii

ii

mp

2

0

1 v

Conservation law of relativistic momentum

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3. Mass and Energy

Kinetic energy theorem

pxx

x pxt

pxFE

000k ddd

dd vSuppose

2

0

1

vmp pp)p( ddd vvv

v

vv

vv0 22

0

2

0k d

11 c

mmE

2

integral

20

20

20

k 11

cmcmm

E

22

22cv

cv

v

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0mm 20

20

20

k 11

cmcmm

E

22

22cv

cv

v

)(cmcmmcE 11

12

20

20

2k

Relativistic Kinetic Energy

20k 2

1vv mEc 当 时,

Rest Energy 200 cmE

Energy of an object in rest

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2mcE Relativistic Mass-Energy Relation:

This points out that the mass and energy of an object is closely related

Total Energy

Relativistic energy and mass conservations are united in one law

22 mccmEE 0K

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Experimental values (black dots) for the electronvelocity as function of their momentum p and as function of their kinetic energy T. The predictions of Galilean physics (blue) and the predictions of special relativity (red) are also shown.

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Physical Meaning:

2mcE 2)( cmE

Energy and (inertia) mass are equivalent,

and hence the changes of energy results in

the corresponding changes of the mass.

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The Mass-Energy relation in special relativity provides the theoretical foundation for the opening of the age of atomic energy

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The ‘Big European Bubble Chamber’ and an example of tracks of relativistic particles it produced, with the momentum values deduced from the photograph (@ CERN).

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4. Applications of Mass-Energy Relation in Nuclear Fission and Fusion

n2SrXenU 10

9538

13954

10

23592

u22.0mMass Defect

Atomic mass units kg 1066.1u1 27

Energy released MeV2002 cmE

(1)Nuclei Fission

1-gram Uranium-235 releases J105.8 10

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Explosion of A-bom

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(2) Light Nuclear Fusion HeHH 42

21

21

J1087.3)( 122 cmEQEnergy release

kg103.4u026.0 29mMass defect

Light nuclear fusion Conditions: ,

the kinetic energy of hence reaches

such that Column repulsive force between

nucleus can be overcome

K108TH2

1 keV10

Deuterium

Helium

kg103437.3)H( 27210

m

kg106425.6)He( 27420

m

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5. Momentum and Energy Relations

22

202

1 c

cmmcE

v

22

0

1 c

mmp

v

vv

222220

22 cm)cm()mc( v

E

200 cmE

pc

2220

2 cpEE

Ultra-relativistic limit pcE,EE 0

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Photon c,m v00

mccEp

Wave-particle duality

/hp

hE

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Eg-1. Suppose a proton’s velocity is

Calculate its total energy, kinetic energy and

momentum.

c80.0v

Solu: Proton’s rest energy

MeV15631 22

202

c

cmmcE

vMeV6252

0k cmEE

119 smkg1068.6

22 c1

mmp

v

vv

0

MeV 938200 cmE

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MeV1250)( 220

2 cmEcp

cp MeV1250

We can also work out momentum in the

following way

c80.0vEg-1. Suppose a proton’s velocity is

Calculate its total energy, kinetic energy and

momentum.

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nHeHH 10

42

31

21

Solu: for this light nuclear fusion reaction

E.g-2. A Tritium and a Deuterium

can fuse into a Helium , and generates

a neutron , How much energy does this

fusion reaction release?

H)(21H)(3

1

He42

n10

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MeV628.1875)H(21

20 cm

MeV944.2808)H(31

20 cm

MeV409.3727)He(42

20 cm

MeV573.939)n(10

20 cm

During the process of Deuterium and

Tritium fusing to Helium , the rest

energy decreases MeV17.59E