chap14 relativity physics relativity chapter 14. chap14 relativity physics 2 本章目录 14-0 basic...
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Chap14 Relativity
Physics
RelativityRelativity
Chapter 14Chapter 14
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
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
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
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
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
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
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
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
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.
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
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.
Chap14 Relativity
Physics
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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
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
Chap14 Relativity
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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
Chap14 Relativity
Physics
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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
Chap14 Relativity
Physics
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.
Chap14 Relativity
Physics
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.
Chap14 Relativity
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“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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
22
Plots based on data from the Michelson-Morley experiment
Chap14 Relativity
Physics
23
Chap14 Relativity
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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.
Chap14 Relativity
Physics
25
Albert Einstein (1879-1955)
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
29
)tx(tx
x vv
21
yy
zz
)xc
t(x
ct
t22
2
1
vv
(1) Lorentz Transformation
cv211
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
31
(2) Lorentz velocity transformation
x
xx
uc
uu
21vv
x
zz
uc
uu
21v
x
yy
uc
uu
21v
Chap14 Relativity
Physics
32
Inverse
x
yy
uc
uu
21v
x
zz
uc
uu
21v
x
xx
uc
uu
21vv
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
34
1. Relativity of simultaneity
Event 1: back wall detector receives light signal
Event 2: front wall detector receives light signal
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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?
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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'
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
48
3. Time Dilation
Chap14 Relativity
Physics
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:
Chap14 Relativity
Physics
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
'Δ(Δ2c
xtt
v
'ΔΔ 12 tttt
0x
21
tt
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
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:
Chap14 Relativity
Physics
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.
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!
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
Chap14 Relativity
Physics
star field at rest star field at 0.99c
lattice at rest lattice at 0.99c
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
58
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
Chap14 Relativity
Physics
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
Chap14 Relativity
Physics
60
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
Chap14 Relativity
Physics
61
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
Chap14 Relativity
Physics
62
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
Chap14 Relativity
Physics
63
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
Chap14 Relativity
Physics
64
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
Chap14 Relativity
Physics
65
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
Chap14 Relativity
Physics
66
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.
Chap14 Relativity
Physics
67
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.
Chap14 Relativity
Physics
68
The Mass-Energy relation in special relativity provides the theoretical foundation for the opening of the age of atomic energy
Chap14 Relativity
Physics
69
The ‘Big European Bubble Chamber’ and an example of tracks of relativistic particles it produced, with the momentum values deduced from the photograph (@ CERN).
Chap14 Relativity
Physics
70
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
Chap14 Relativity
Physics
71
Explosion of A-bom
Chap14 Relativity
Physics
72
(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
Chap14 Relativity
Physics
73
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
Chap14 Relativity
Physics
74
Photon c,m v00
mccEp
Wave-particle duality
/hp
hE
Chap14 Relativity
Physics
75
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
Chap14 Relativity
Physics
76
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.
Chap14 Relativity
Physics
77
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
Chap14 Relativity
Physics
78
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