rela

tivity

Qu

an

tum

Classic physics

Modern physics

Einstein:The founder

of modern space-time

Time-space view

dynamics

Length contraction

Time

dilation

Relative nature of simultaneity

Prin

cip

le o

f P

rincip

le o

f re

lativ

ityre

lativ

ity

Lorentz transformatio

Lorentz transformatio

nn

Relativity

Einstain’s theory took us into a world far beyond that of ordinary experience,it led us to a deeper and more satisfying view of the nature of space and time

In modern long range navigation,the precise location and speed of moving craft are continuously monitored and updated.A system of navigation satellites called NAVSTAR permits locations and speeds anywhere on earth to be determined to within about 16m and 2cm/s.However if relativity effects were not taken into account,speeds could not be determined any closer than about 20cm/s,which is unacceptable for modern navigation systems.How can something as abstract as Einstain’s relativity be involved in something as practical as navigation?

A pair of twins ,A remains on earth,and B make the milk run to a nearby solar system in high speed,when B come back,who’s younger?

Take an account on a event in two different frame

1.Galileo transformation

Pro

Su

xxo

S

r

§1 Galileo relativity principle§1 Galileo relativity principle

utxx yy

zz tt

tt

zz

yy

tuxx

S

S

tzyxr ,,,

tzyxr ,,,

tzyxv ,,,

tzyxv ,,,

a

a

Pro

Su

xxo

S

r

Velocity and acceleration transformation

zz

yy

xx

vv

vv

uvv

zz

yy

xx

vv

vv

uvv

zz

yy

xx

aa

aadt

duaa

zz

yy

xx

aa

aatd

duaa

zz

yy

xx

aa

aa

aa

zz

yy

xx

aa

aa

aa

u

is constant

In two inertial frame aa

conclusion: 1.time interval is absolute

t=t t=t2.space separation is absolute

tuxx t u x t u x x

0t xx 3.the invariability of Newton’s law

ammaF

for ： so ：

amF FF

P957

S 2021012211 vmvmvmvm

S 2021012211 vmvmvmvm

For example ： conservation of momentum

2.Galileo relativity

3.the trouble in electromagnetic equation1)

01

2

2

22

2

t

E

cx

E

C :to which reference frame ?

Even though electromagnetism shares with mechanics concept such as energy and momentum.there appear to be a major difference between these two fundamental discipline.the laws of mechanics look the same in all inertial frames,but electromagnetism appears to violate the general law.According to Maxwell’s equation,electromagnetic waves propagate at speed C,with no restrictions on the state of the source of detector,this suggests the existence of an absolute frame for electromagnetism.

1.the relativity postulate:the laws of physics are the same for observers in all inertial reference frames.no frames is preferred

2.the speed postulate:the speed of light in vacuum has the same value c in all direction and in all inertial reference frames

Einstein relativity develop Newton’s theory

discussion

physicsrule

Mechanics rule

§2 the postulates

The light speed invariability is opposed to Galileo velocity transformation Difference in view

Newton

Time scale

Length scale

Mass measure

Has no relation

with frame

relativity Time,space,mass has relation with reference frame

C is

constant

transformation Inverse transformation

xc

tt

zz

yy

utxx

xc

tt

zz

yy

tuxx

2.lorentz transformation1.transformation formula

21

1

c

u

ttux ,,

1

tt

zz

yy

utxx

Galileo

transformation

cu <<

discussionHas relation with

No meaning.Maximum Speed is C

cu >

4.procedure to solve the problem

1)establish coordinate

2)determine the moving frame and rest frame

3)u is the positive speed of S’ to S

4)use formula to get relation among x, t, x’

t’ and solve problem

P

o

Su

xxo

S

Example:two persons a,b observe lighting pulses, from point of a, x1=6104m ， t1=2 10-4 s ； x2=12 104m ， t2=1 10-4 s ， from the point of b, two events happen at the same time （ 1 ） find the relative velocity of b to a （ 2 ） find the space separation of lighting pulses measured by b

x

c

vtt 221

1

2

2

1

xc

vt

t

Solution:

We get

0t

2

2

44

2

44

1

)1061012()102101(0

c

vc

v

2

cv

From lorentz transformation

vtxx

21

1

mtvx

x 4

21020.5

1

Example:a race track with length 100m,a sports man

Run from the origin point to end point with time interval 10s,a craft with velocity 0.8c fly along the direction of run way.from the point of craft man

find the space separation and time interval

Solution:x=100, t=10s,u=0.8c

mtux

x 9

2104

1

Negative sign means the sportsman run in the opposite direction

sx

cu

tt 6.16

1 2

2

§ 2 time and space of relativity§ 2 time and space of relativity1 、 relative nature of simultaneity

But in Einstain theory:

00 ttIn Newton’s theory

0

0

0

t

x

t

t

tv

xcv

c

2

2

21

?tso，

Relative nature of simultaneity 。

?0 tt

if

We have

Conclusion:in general,two events that appear simultaneous in one frame of reference don’t appear simultaneous in a second,unless the two events happen in the same place.

Einstein trainSS

S rest frame

In train

There’s a signal source

0tt M Give a signal

In the middle

M

S uA BM

Einstain’s train experiment

Event 1

Receive flashA

Event 2 Receive flashB

S MBMA

A B Receive the flash at same time

S Su

A BM

Two events happen in different time

M Give a signal BA Move with

S

Receive light early than

A

0tt M Give a signal

BS

discussion

S Su

A BM

1 ） simultaneous is absolutely only when two events happen in same place

tt

vx

cv

c

2

2

21

2) Relative nature of simultaneous is the result of constant c

3) When speed is far more less than C,the result in two inertial frame is same

tt

vx

cv

c

2

2

21

A君

B君

Example:two trains leave from two station A,B with space separation 1000km at the same time,a craft with u=9km/s along the direction ab,find

The interval from the point of spacemanSolution:x=106 t=0

7

2

2

101

xcu

tt

Negative means b train go first

1)proper time

a time interval between two events at the same space point in a frame is called a proper time in that frame

2. Time dilation

2 、 proper time is shortest time in all frames Take an account on a clock in S

0x t

2

2

2

1cu

xcu

tt

2

2

1cu

t

t

x 0

1 t

Time dilation,proper time is minimum time

In s,from lorentz transformation

3)physics reason of time dilation y′

x′

u

d

u t

dl

M′

A′C′ C′

In S ′， there’s a light source in A ′ M′is a reflected mirror

c

dtx

20

y′

x′

u

d

u t

dl

M′

A′C′ C′

In S ：

22

2

22

0

tu

dcc

lt

x

2

21

2

cu

cd

t

2

21

cu

tt

tt

Time dilation

(2)the proper time is the shortest one in all frames

discussion

（ 1 ） time dilation is the space-time effect of relativity,it has no relation with the structure of clock

（ 3 ） there’re a lot of experiment to proof the time dilation effect

example a rocket v=0.95c ， the time interval is 10min measured in rocket ， how long is it in earth frame ？

min01.32min95.01

10

1 22

'

tt

Solution:

Example:the lifelong time for is 2.5×10-8s while in rest,if its u=0.99c,the passing distance is 52m,is this ok ？

Solution:if with t′=2.5 ×10-8s times u ， we get 7.4m 。Take account of time dilation

2

2

1cu

tt

)(108.1

)99.0(1

105.2 7

2

8

s

＝

So s=uΔt=53m,it’s ok

3.length contraction S

uS

0l1 、 proper length

A length measured in the rest frame of the body is called proper length

2 、 proper length is the longest in all frames (length contraction)

S

S 120 xxl

12 xxl

012 ttt

Notes:in S we measure the rod length,we must measure end points in same time

2

2

1cu

tuxx

From lorenze transformation

2

2

0 1c

ull

2

2

0 1c

ull 11 0ll

Proper length is longest

discussiondiscussion

1)Relativity effect

2) In low speed Galileo transformation

4)length contraction is relativity effect,it’s different from what we say that the body become smaller.

3)proper length is the longest ,0ll

Example:a length of rocket measured in rocket frame is 15m, suppose v=0.95c,find the length in earth frame ？

2

0 1 ll

mml 6.495.0115 2

Solution:

cv2

3

Example:a 1m rod rest in O’x’y’ 。 the angle is 450 with x’ axis measured from s’ 。 find the length of the rod and angle with x in s 。 The related velocity is

'''''' sincos llll yx

solution:

z

y y'

S S'v

O O

z'

x

x'

ly'

lx''

l'

from lorentz transformation:

''' sinlll yy

'22'22 cos1 llll yx

2' 1cos llx

2

'

'2'

''

1cos1

sin

tg

l

l

l

ltg

x

y

,23

,1,45 '0' cvml

0

2

'

43.63,21

tg

tg

mll 79.0cos1 '22'

Conclusion:not only the rod have contraction,but also rotate in some angle.

Example:in 6000m altitude ， a meson fly to earth with v=0.998c 。 Suppose meson ‘s life span in its rest frame is 210-6 s ， from relativity ， 1)can meson arrive in earth from earth frame ？ 2) from mesonFrame?

s

cv

tt 6

2

0 106.31

1

solution ： 1) proper time for meson is t0=210-6 s 。 Because of time dilation ， the life span from earth

The distance for to travel m6000m9460 tvL

It can go straight through the earth

2 ） from frame

mc

c

c

ull 360)

998.0(160001 2

2

2

0

The distance for to travel in frame

mcvl 599102998.0 6

It can go straight through the earth

4 、 comparison of two time-space view

space,time is absolutely,there’r no relation among time,space and the motion of body

Classic view

4 、 light speed is c,which is utmost speed of motion body

Relativity view

1 、 there are relation among time space and the motion of body

2 、 every inertial frame has its own time scale,and found that the clock in other frame go slow

3 、 every inertial frame has its own space scale,and found the ruler in other frame become shorter.

§ 5 the dynamic of relativity§ 5 the dynamic of relativity 1 、 momentum and dynamic equation

vmp

m

But the utmost of v is c

2

2

0

1cv

mm

dt

pdF

Must change with speed

0

dtmF

v

mF

dtdv

vmp A:It can be

proofed

notes ：

m m 0（ 1 ） if ， v c

（ 2 ） if ， cv m

（ 3 ） if ， v c 00m

B: dynamic equation

vmp

d

dtv F

m

v c( )0

2 21

v cif amF

0

(4) v>c ， m is negative,no meaning

2 、 mass and energy

1 、 kinetic energy

2

0

2 cmmcEK

If v<<c:

2

0

2

2

2

0k 21 E cmc

c

vm

2

02

1 vm

2

2

0

1cv

mm

2

2

2

2 21

11

1cv

cv

p986

三、相对论能量 质能关系

2

0

2 cmmcEK

Rest energyKinetic energy Total energy

Mass and energy formula

56

2 、 energy

Notes:

（ 1 ） a body in rest,it still has substantial

energy 。

（ 2 ） mass is not the measure of inertial,but also the energy

（ 3 ） change in mass implies change in energy

（ 4 ） to isolated system,the total energy keeps constant

In 1955,atomic age has arrive!

application

Example:there’re mass loss because of radiation energy of the sun

S

ESEr

sJt

E/1029.4 26

skgtc

E

t

m/104.5 9

2

stm

m/105.8 14

3 、 the relation between momentum and energy

In relativity p mvvm

v c

0

2 21

E mcm c

v c

2 0

2

2 21From above

2 2 2 2 220

2( ) ( )mc m c m v c

2 2 2 20E E p c

E0E

pc

0E

kE

Basic formula in dynamics

2 2 2 20E E p c

2

0

2 cmmcEK

2

2

0

1cv

mm

Example:a particle move with v=0.80c 。 Find the total energy,kinetic energy and momentum

MeV

MeV

cv

cmmcE

1563

8.01(

938

)/1(

2/12

2/122

2

02

）

Kinetic energy

MeVMeVMeVcmEEk 62593815632

0

Solution: E0=m0c2=938MeV

momentum

119

1

2/12

827

2/122

0

..1068.6

..8.01(

1038.01067.1

)/1(

smkg

smkg

cv

cmmvp

）

Alternative way for momentum

MeVMeVcmEcp 12509381563)( 2222

0

2

cMeVp /1250

Example: m0 of electron is known （ 1 ） find the E0 ；（ 2 ） find the work for the particle moving from rest to 0.60c

2 ） the work for accelerating particle

J10025.160.01

10199.8

1

13

2

14

2

22

cmmcE o

J 10 05 . 214

k E

evcmE 52

0 1012.50

solution ：（ 1 ） rest energy of electronic

0EEE

k

Example: two rest particle with rest mass m0 each collide head on with v to become a combined particle 。 Find the rest mass and speed of the combined particle 。

2

202

1

2

cmMc

MVvmvm －

2

0

1

2

mM

Solution from the conservation law of momentum and energy

Notes:the rest mass of combined particle is larger than 2m0 ， the difference

2

2)

1(

20

221

02

02

02

02

2

0

2

c

KE

cmcm

cm

mmM

－－

The difference of rest mass comes from kinetic energy

V=0

(1)

(2)

We’v