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Medical Physics
Chapter 5
Introduction to Relativity
A special and important theory not only for objects at a high speed but also for
revealing the sources of energies
§5.1 Introduction to relativity
At the end of 19th century, physicists were satisfied with the Newtonian
mechanics as it could explains almost all the phenomena observed from the earth and
universe, such as fluids, wave motion and sound. For light, Maxwell unified the
theory of electricity and magnetism, founding the theory of electromagnetic field. At
the same time, Maxwell’s theory predicted a constant which should be the speed of
electromagnetic field traveling in free space. Everything was fine except for asking
which inertia frame of reference is the right one for the speed of light. A lot of
experiments were made in order to find out the “special inertia frame of reference” at
that time and no positive results were obtained.
The above problem is one of so-called two patches of clouds in the physics sky.
Another patch of cloud is the blackbody radiation which could not be explained either
by the well-known “perfect” classical theory of statistics and thermodynamics.
It is worth pointing out that the clearance of the two patches of clouds lead to the
foundation of two modern physics theories, one is special relativity and the other is
quantum theory which will be introduced later in our lectures.
Generally speaking, relativity contains two parts, one is called special relativity
and the other is called general relativity. The former describes the phenomena of
objects moving at a very high velocity and the latter explains the behavior of objects
moving closely to a strong gravitational field.
§5.2 Galileo transformation
•
•
The absolute outlook of space-time of classical mechanics and the classical
relative principle
Galileo transformation
 x   x  ut
 y  y


 z  z
 t   t
A frame of reference S  moves with velocity u relative to a frame S shown in Fig 5.1.
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Chapter 5. Relativity
y´
y
ut
x´
x
•P
y
y´
x
O
O´
x
´
Speed and acceleration of the particles can be obtained by the differentiation to the
equation of motion.
vx  vx  u

 vy  v y
 v  v
z
 z
 ax  a x
dv

 a ay  a y
dt
 a  a
z
 z
This explains that Newton’s mechanical laws are identical in all inertia reference
frames.



F  ma  ma
§5.3 Lorentz transformation
5.3.1 Tow postulates:
1. The relativity principle: All the laws of physics have the same form in al
inertial reference frames
2. Constancy of the speed of light: Light propagates through empty space with a
definite speed c independent of the speed of the source or observer
5.3.2 Lorentz transformation
x  ut

  ( x  ut )
 x 
2
2
1

u
c

 y  y
 z  z

u

t 2 x
u
c
t  
  (t  2 x)

c
u2

1 2
c

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where

1
1
u2
c2
Time and space are no longer independent. The reversal relations between the two
frames of reference are given below:
 x   ( x   ut )
 y  y

 z  z

u
t   (t   2 x )
c

The speed of an object after Lorentz transformation can be found by differentiation
with respect to time. Note that
dx 
dx
vx 
vx 
dt 
dt
The following relations are obtained:
v u
vx  x
,
u
1  2 vx
c
vy
u2
vy 
1 2 ,
u
c
1  2 vx
c
vz
u2
vz 
1 2
u
c
1  2 vx
c
Example 5-1. A spaceship moving away from earth with a speed 0.9c fires a missile
in the same direction as its motion, with a speed of 0.9c relative to the spaceship.
What is the missile’s speed relative to earth?
Solution: Let the earth’s frame of reference be S, the spaceship’s S´. Then vx´=0.9c
and u = 0.9c
The non-relativistic velocity addition formula would give a velocity relative to the
earth of 1.8c. The correct relativistic result can be obtained from:
v u
v  u
0.9c  0.9c
vx  x
 vx  x

 0.994c
2
u
u
1

(
0
.
9
c
)(
0
.
9
c
)
/
c
1  2 vx
1  2 vx
c
c
When u is less than c, a body moving with a speed less than c in one frame of
reference also has a speed less than c in every other frame of reference. This is one
reason of thinking that no material body may travel with speed greater than that of
light., relative to any frame of reference.
§5.4 Special relativistic conception of space-time
5.4.1 Time dilation (moving clocks are measured going slowly).
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Chapter 5. Relativity
Time dilation indicates that intervals of time are not absolute but are relative to
the motion of the observers. If two identical clocks are synchronized and placed side
by side in an inertial frame of reference they will read the same time as long as they
both remain side by side. However, if one of the clocks has a velocity relative to the
other, which remains beside a stationary observer, the traveling clock will show, to
that observer, that less time has elapsed than the stationary clock.
If two events ((x1, t1 ), (x2, t2 )) happen in S  in the same place but at different
time. What is the difference between the time intervals in the two relative moving
frames S  and S. It is easy to find that


  t 2  t1   t 2  t1   2 x 2  x1 
c


u
   t 2  t1    0   0
Where 0 is called proper time (固有时) which is less than the time interval observed
by the stationary observer. This is so called “the moving clock runs slower”.
5.4.2 Lorentz contraction (moving sticks become shorter)
As before, a frame of reference S  moves with velocity u relative to a frame S.
L0  x2  x1
x   x  ut 
L0  x2  x1   x2  x1    L
u2
L
 L0 1  2  L0 

c
L0
Moving ruler becomes shorter!
y
y
´
u
t
x´1
x´2
L0
O
x O
´
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5.4.3 The relativity of simultaneity
There two events ((x1, t1 ), (x2, t2 )) happen in S at the same time but in
different places. Using the formula
u
t   (t   2 x)
c
Therefore,
u


t 2  t1   t 2  t1   2 x2  x1 
c


u
   2 x2  x1   0
c
The two events happen simultaneously in S  but not in S frame.
5.4.4 Causality and signal speed
In S frame, two events P(xp, tp) and Q(xQ, tQ) has causality.
t  tQ  t P  0
The propagating speed in S is
vs 
xQ  xP
tQ  t P

x
t
In S´ frame,
u
u x 



t     t  2 x   t 1  2  
c


 c t 
u


 t 1  2  vs 
 c

As vs < c, t´> 0. Therefore, causality is unchanged.
§5.5 Relativistic Mechanics
In relativity, the concepts of mechanics are facing redefinition. However, this
redefinition has to be satisfied with the correspondence principle, that is, when v << c,
the redefined physical quantities have to be their classical corresponding physical
quantities. On the other hand, the conservational laws are kept valid as much as
possible.
5.5.1 Momentum and mass-speed relation
In order to satisfy the conservation of momentum and to hold the correctness of
Lorentz transformation, the momentum can still have the save form:


p  mv
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Chapter 5. Relativity
But mass m must be a function of velocity and is given by
m
m0
1 u 2 / c2
This is a very important mass-velocity equation in relativity. It reveals the relation
between the mass and velocity for an object in motion. When the velocity of the
object approaches to the light speed, its mass will approach infinity and for an object
with infinity mass, it is impossible to accelerate it. This also explains that the light
speed is the upper speed limit for all objects.


p  m0v
When u<<c, the relativistic momentum is very closely equal to the classical result.
5.5.2 Force and kinetic energy
1. Force
Newtonian mechanics:

 dp d


dv
F
 mv   m
 ma
dt dt
dt
Relativistic mechanics:

 dp d


dv
F
 mv   m
 ma
dt dt
dt

d
 m0 v 
dt
2. Mass-energy relation
E  mc2 
m0c 2
1 u c
2
2
 m0c 2  E0
This relation not only reveals the relation between mass and energy but also unifies
the mutual independent two conservational laws of mass and energy
3. Energy-momentum relation:
E 2  p 2 c 2  m02 c 4
This relation predicts the possibility of the existence of zero-mass particles and it can
be shown that these zero- mass particles are moving with the speed of light. (p = mc)
§5.6 Introduction to General relativity
1.
In 1915, Einstein proposed the equivalent principle based on the assumption
that the inertia mass and gravitational mass are equivalent and founded the general
theory of relativity. Einstein extended his earlier work to include accelerated system,
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Medical Physics
which led to his analysis of gravitation. He interpreted the universe in terms of a
four-dimensional space-time continuum in which the presence of mass curves space in
such a way that the gravitational field is created. This explains that the mass curves
space or the presence of gravitational field will also curve space.
5.6.1 The two hypotheses of general relativity
1. Equivalent principle:
For all physical processes, the reference frame with uniform acceleration is
equivalent to the local region of gravitational field and the inertial force is equivalent
to the local region of gravitation.
2. General relativity principle:
Physics law has the same form in all reference frames, no matter inertial or
non-inertial.
5.6.2 The characteristics of space-time in gravitational field
1. Light is curved in gravitational field.
2. Space bends
The three-dimensional space used in Newton’s mechanics and the
four-dimensional space used in special relativity are Euclidian space. Light traveling
linearly in such a space can be considered as the result of the characteristics of the
level-straight space.
In gravitational field, light rays are curved. This is determined by the
characteristics of space-time in gravitational field which curves the four-dimensional
space.
3. Time dilation effect in gravitational field
Curved space
A
B
Straight space
t0 
t 
AB
c
AB
c
where AB is the arc path length, t  t0 is called time dilation caused by gravitational
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Chapter 5. Relativity
filed.
4. Gravitational collapse and black hole
Gravitational collapse is the phenomenon in the process of star evolution while
black hole is a strange star with infinite density. High density stars can be divided into
three kinds: white dwarf star (108/m3), Neutron star (107/m3) and black hole.
5. Gravitation wave
An accelerating electrical charge will emit electromagnetic wave. Einstein proposed
that the accelerating body in gravitational field will excite gravitational wave.
Problems
1. The reference frame S moves at a constant speed u with respect to reference
frame S along x-axis. Write the relations of coordinates and speed between S and
S frame in non-relativistic case, i. e. Galilean transformation equations, and in
relativistic case, i. e. Lorentz transformation equations；
2. What are the two basic postulates of special relativity?
3. David holds a watch and moves to the north by train; William holds another
watch but moves to the south by air. According to the observations by David and
William, whose watch seems slower? (hint, give their results respectively)
4. If David and William hold a meter stick respectively, what are their observations?
(hint: Answer should be like this: David (William) thinks that William’s (David’s)
meter stick becomes shorter or longer?).
5. A spaceship moving away from earth with a speed 0.9c fires (发射) a missile (导
弹) in the same direction as its motion, with a speed 0.9c relative to the spaceship.
What’s the missiles speed relative to the earth.
(hint: You should take the earth as frame S and the spaceship as S, therefore you
v  u
may probably use the formula v 
.)
1  uv  / c 2
6. An electron moves so rapidly in a linear acceleration that its relativistic mass is
5.0252 times its rest mess. Calculate the speed and relativistic kinetic energy of
this moving electron (hint: kinetic energy = mc2 – m0c2).
7. What are the characteristics of time-space in gravitational field in the opinion of general
relativity?
8. The Sun radiates energy equally in all directions. At the position of the Earth
(r=1.501011m), the Sun’s radiation is 1.4kw/m2.How much mass does the Sun
lose per day because of the radiation?
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