8. The special theory of relativity
Relativity is connected with the measurement of where and when events take
place, and how these events are measured in reference frames that are moving
relative to one another. It gives a new look at the notion of simultaneity.
The theory called „special” means that it holds only for the inertial reference
frames in which Newton’s laws are valid (the theory concerning the frames which
undergo gravitational acceleration is known as general theory of relativity) .
8.1. Einstein’s Postulates
In 1905 Albert Einstein introduced two postulates:
1. The Relativity Postulate: The laws of physics are the same for
observers in all inertial reference frames.
2. The Speed of Light Postulate: The speed of light in vacuum has the
same value c in all directions and in all inertial reference frames.
1
The speed of light, cont.
Many experiments have shown that the speed of light does not depend on the move
either a light source or an observer.
in the frame S
the source Z
is at rest
in the frame S’
the source Z is
moving but the
measurement of
light speed also
gives c
The speed of light has a limit. W. Bertozzi (1964)
accelerated electrons measuring their speed and kinetic
energy. Electrons reached the speed
0.999 999 999 95c but always less than the ulimate value
c = 299 792 458 m/s; the energy increased toward very
large values.
We essentially use the approximate value of the light speed
in vacuum as c = 3.0 · 108 m/s.
2
8.2. The Lorentz transformation equations
Consider two reference frames S and S’, where S’ is moving in respect to S with
constant velocity v along the x – axis.
We wish to find the transformation (relations between x,y,z and t in frame S in
relation to x’,y’,z’ and t’ in frame S’), for which the speed of light does not
depend on the move either the light source or the observer.
Suppose that at time instant t = t’ = 0 we emit
from the common origin of both frames the
light flash moving at speed c.
The wavefront of a light wave is a sphere of the radius
in frame S
r  ct
in frame S’
r   ct 
In S this sphere is described by
x 2  y 2  z 2  c 2t 2
(8.1a)
3
The Lorentz transformation, cont.
In S’:
x  2  y  2  z 2  c 2 t  2
(8.1b)
Applying to Eq.(8.1b) the clasical Galilean Transformation (GT) one obtains:
x 2  2xvt  v 2t 2  y 2  z 2  c 2t 2
(8.2)
Eq.(8.2) is in contradiction to (8.1a).
Conclusion:
GT is not valid if the second Einstein’s postulate is true (the speed of
light is the same in both frames).
We are looking for the transformation, which converts (8.1b) into (8.1a) and
reduces to GT for v / c  0 .
This transfotmation should be:
• simple for y’ and z’ to convert y’2 and z’2 without changes to y2 and z2.
• linear vs. x and t to obtain the spherical wavefront
• time should be also transformed to cause vanishing of extra terms
in Eq.(8.2)
4
The Lorentz transformation, cont.
In the first attempt we introduce the constant f
x   x  vt
y  y
z  z
t   t  fx






(8.3)
Substituting (8.3) into (8.1b) one obtains
x 2  2xvt  v 2t 2  y 2  z 2  c 2t 2  2c 2ftx  c 2f 2 x 2
(8.4)
The terms comprising xt should disappear
 2 xvt  2c 2 ftx  0  f  
v
c2
then
t  t 
v
x
c2
(8.5)
Substituting for f into (8.4) we have
 v2 
v2 
2
2
2 2
x 1  2   y  z  c t 1  2 
 c 
 c 
2
(8.6)
Eq.(8.6) transforms to (8.1a) if we reduce the unnecessary multiplication term
1  v
2
/ c2

5
The Lorentz transformation, cont.
The final shape of the transformation is then
x 
x  vt
v2
1 2
c
y  y
z  z
v
x
2
c
t 
v2
1 2
c
t


 (8.7)



Lorentz transformation (LT) equations
valid at all physically possible speeds
For v / c  0 LT reduces to GT, what was expected.
v
 ,
c
Introducing the replacements:
the LT equations can be rewritten as:
x    x   ct 
y  y
z  z
 x 
t    t 

c 

(8.8)
(   1)
or in the inverse form:
1
1  2
,
(   1)
x    x    ct  
y  y
z  z
(8.9)
 x 

t   t  

c 

The inverse transformation (8.9) was obtained from (8.8) by reversing the sign 6of v.
8.3. Consequences of the Lorentz transformation
8.3.1. The relativity of length (length contraction)
We measure the length of a rod moving in the diection of its length.
Rod at rest in
a resting frame S
The observer in S measures
the coordinates of a rod end points
which are independend of time.
The length is: L0 = x2 – x1
The observer in S’ has to measure
the coordinates of a rod end points at the same
instant of time t’. So the inverse Lorentz
transformation is used:
x1   x1'   ct '

  x

  ct 
'
x2
2
Subtracting on both sides one obtains

'

x 2  x1   x '2  x1'   L  L0
or
L  L0 /   L0 because  >1
The length L of a moving rod decreased (length contraction).
The measurements in directions y or z give results independent of velocity.
7
Length contraction, cont.
The same result as previously we obtain if the rod rests in a moving frame S’.
L = x 2 – x1
The observer in S has to measure the
coordinates of a rod end points at the same
instant of time t. So we use the transformation:
L0  x'2  x1'
The observer in S’ measures
the coordinates of a rod end points
which are independend of time.
x '2   x 2  ct 

'
'
  x 2  x1   x 2  x1   L0   L
'
x1   x1  ct  

It is not important in which reference frame we place the rod, only whether it is
moving in relation to the observer or not (in diection of its length). The measurement
of length of a moving object gives lower value. The essential role plays here the
simultaneity. Two simultaneous events in S (Δt=0) separated by Δx, are separated
in S’ both in space and in time. From LT one obtains easily: x   x , c  t    x
8
8.3.2. The relativity of time (time dilation)
The problem concerns the dilation (expansion) of time measured by the
moving clocks. The time interval betwen two events occuring at the same
location measured by the clock placed in this location is called the proper time
interval. Measurements of the same time interval from any other inertial
reference frame are always greater.
The events occur in
point A, at rest in S’.
The clock is placed
at the same location,
i.e. is at rest in
relation to A(1,2).
The clock in S
moves in relation
to point A(1,2) in
which the events
occur.
We use the transformation for which Δt’ time interval occures for x’ = const, so
the inverse LT
 
 




t   t '  x
t   t  x  
c 

1


1
c


 

t 2    t 2'  x 
c 


t 2  t1   t 2'  t1'

      0
The proper time is the minimal time between events.
 0 - the proper time
9
Time dilation - examples
1. The lifetime of mesons π+
It is known that meson π+ is an unstable particle and decays giving a meson μ+
and a neutrino


0
    0n
The lifetime of π+ (the proper time, measured in a frame in which the meson
rests) is 2.5·10-8 s. What is the lifetime observed in a laboratory, if mesons are
moving with a speed of v = 0.9 c.
0
2.5  10 8 s
   0 

 5,7  10 8 s
2
1  0,81
1 
In this way the meson covers more than two times longer distance vs. that
obtained from non-relativistic calculations correct for speeds much less than c.
2. The paradox of twins
What will be the age of one of the twins sent in space after the birth with velocity
v = 0.9 c, when he comes back after 20 years according to the age of a twin in the
Earth.      1   2  20 1   0,9c  2  8,7 years For v = 0.5c τ0 = 17.3 years.
0



 c 
10
Time dilation – examples, cont.
3. Macroscopic Clocks. Super precision atomic clocks (large systems) flown in
airplanes (β ~7x10-7) enabled an experiment on a macroscopic scale. U. Maryland
carried out an experiment using an atomic clock flying over Chesapeake Bay
(round and round) and checked the time dilation within 1% of predictions.
If the clock on the U. Maryland flight registered 15.00000000000000 hours as the
flight duration, how much would a clock that stayed on earth (lab frame) have
measured for the duration? More or less? Does it matter whether airplane returns to
same place?
if   7 107   
1
1 
2
 1.000000000000245
t  t0  1.00000000000024515.00000000000000 hr 
 15.00000000000368 hr
t  t0  1 108 s!
11
8.4. The relativity of velocities
A particle has velocity u in the frame S.
What velocity is measured by the observer in S’,
which is moving with velocity v relative to S.
From the Lorentz transformation one gets
x    x   ct 
dx   dx   cdt 
  x
t    t 

c 

 dx 

dt     dt 

c 

Using the definition of u’ we have
u' 
dx   dx  cdt  u  c
u v




v
dx 
dt 

  dt 
 1 u 1 2 u
c
c
c 

the relativistic velocity
transformation
(8.10)
For v / c  0 one obtains from (8.10) the known Galilean velocity transformation:
u'  u  v
12
The relativity of velocities, cont.
Example
What is the speed of a photon in a reference frame at rest if it has velocity
c in a frame moving with velocity v relative to the resting frame. A photon
moves parallel to the x axis.
In this case u'  c
For the Galilean transformation we would obtain
u  v  c what is in contradiction to the Einstein’s second postulate.
From the relativistic velocity transformation (8.10) one obtains
u'  v
c v
c c  v 
u
v
1  2 u'
c

v
1 2 c
c

c v
c
in accordance with the relativistic theory.
The obtained result also indicates that it is impossible to find such a reference frame
in which a photon would be at rest. Even for v = -c, u = c.
13
8.5. Doppler effect for light
In the classical case of mechanical waves the frequency f’ detected by the observer
is equal
v  v0
(8.11)
f  f
v  vz
f – the proper frequency of the source
v – velocity of the wave in the medium
v0, vz – velocities of the observer and the source respectively
(the signs of velocities are positive if are directed
similarly to v).
In the case of light we expect that the change in frequency connected with the
Doppler effect will depend on the relative velocity v of source vs observer only.
There is no air (ether) for the relative move of light.
where
f  f
1 
1 
(8.12)
If source and detector move toward one another  →  
As is expected only the ratio of relative velocity of source and detector to the
velocity of light  = v/c is important.
14
Doppler effect for light, cont.
Eq.(8.12) for low speeds ( « 1) can be expanded in a series and approximated as
(8.13)
1
f   f (1     2 )  f (1   )
2
Taking into account that f = c/l, one can obtain from (8.13)
c
l'

c
l
(1   )
(8.14)
Introducing the Doppler wavelength shift l  l'l one obtains from (8.14)
v
l
l
c
(8.15)
Eq.(8.15) is used in astronomical observations to determine how fast the light
sources are moving and in which direction (toward or away from the observer).
The theory of the universe expansion was approved by observation of the so
called red shift (l’> l.
15
Doppler effect for light, cont.
The NAVSTAR Navigation System
v1
f03
f01
v2
vairplane
v3
f02
Given v1, v2, v3, f01, f02, f03,
and measured f1, f2, f3, can determine vairplane.
16
8.7. Relativistic dynamics
Relativistic momentum
The classical (nonrelativistic) momentum of a particle

(8.16)

p  m0 v
is not conserved in collisions of particles moving with high speeds. When we
define the momentum as
(8.17)


p  m0  v
it becomes invariant vs. Lorentz transformation. The relativistic momentum can
then be written as

where

p  mv  v
mv   m0
(8.18)
m – relativistic mass of a particle with rest mas m0
and velocity v
The dependence between mass and velocity was also
proved experimentally; in practice for
v / c  0,2
m  m0
17
Relativistic dynamics, cont.
Relativistic Energy
Elementary change in kinetic energy caused by the work of a net force F is


 


dp  dp 
dEk  dW  F  d r 
d r 
 v dt  d p  v  v  d  m v 
dt
dt




(8.19)
The total change in energy is obtained by integrating eq.(8.19) by parts as below
 udw  uw   wdu
In this way one obtains
v 
v
v
v
m0vdv
mc
Ek   v  d  m v   v  m v   m v  d v  mv 2  
 mv 2  0
2
2


0
0
0
0
v 
1  
c
m c2
 mv 2  0
2

v

0




 v2 
d  2 
2
2
 c   mv 2  m0c  2 1  v
2
c2
v2
1 2
c
v

2v
dv
c2

v2
1 2
c
2 v

0

 mv 2  m0c 2 1   2  1 
0
 v2



2
2
2
m0c 2
2
2 
2
2  1 
2
c
 m0c
 1    m0c  m0c
 m0c 
 m0c 2
2
2
2
 1 

1 
1 




18
Relativistic energy, cont.
and thus we have
Ek  mc 2  m0c 2  m  m0 c 2
(8.20)
It is interesting to prove if eq.(8.20) valid for a relativistic particle transforms into
the known classical expression for kinetic energy of a particle moving with v  c .
First we rewrite Eq.(8.20) as
2 1 / 2




v
Ek  m0c 2 1  2   1
 c 

(8.20a)
The expression
 v2 
1  2 
 c 
1 / 2
 1  x 

where
can be expanded into a series:
1  x 
 1  x 
v2
x 2
c
,  1
2
1
   1...  n  1 n
   1x 2  ... 
x  ...
2!
n!
For x  1 only the term to the first power is relevant, since the other terms are
much smaller, hence
(8.21)
1  x   1  x
19
Relativistic energy, cont.
Therefore, for the condition

E k  m0 c  1   2

2



1
2
v 2 / c 2  1
one obtains from (8.21)

1 2   m0 c 2  2 m0v 2
2 
 1  m0 c 1     1 

2
2
2





(8.22)
what is a known classical expression for the kinetic energy.
From Eq.(8.20) it follows that the increase in kinetic energy
of a particle is connected with the inrease of its mass.
The total energy E of a particle is a sum of its kinetic energy Ek
and a rest energy m0c 2
(8.23)
E  mc 2  Ek  m0c 2
Equation
Comparison of relativistic and
E  mc 2
(8.24)
classical expressions for the
states that a mass and an energy are equivalent. It is
kinetic energy of an electron
one of the most important consequences of the theory of with experimental data (x)
special relativity. From (8.24) it follows that the change m in a mass is
equivalent to the change E in energy:
(8.25)
E  m c 2
and also the energy can be converted into a mass:
E
m  2
(8.26)
c
20
Relativistic energy, cont.
The conversions of mass into energy (and vice versa) are clearly seen in nuclear
reactions.
We consider the reaction:
a particle a collides with a nucleus X producing another nucleus Y which
emits a particle b
X+a  Y+b
In reactions of this type the inertial mass (or the total energy) is conserved
m01 
Ek 3
E k1
Ek 2
Ek 4

m


m


m

02
03
04
c2
c2
c2
c2
m01  m02  ( m03  m04 ) 
where:
m01…m04 - rest masses
Ek1…Ek4 - kinetic energies
E k 3  E k 4   E k1  E k 2 
m01  m02  ( m03  m04 ) 
c2
Q
c2
Q – the energy of a reaction
if Q > 0 the energy is released (exothermic reaction)
if Q < 0 the energy is absorbed (endothermic reaction)
21
Nuclear reaction - example
What is the energy released in the following nuclear reaction
7
3
Li  11H  2 24 He 
Q
c2
Q
 mLi  mH  2mHe  0,0186 u  0
c2
Q  m  c 2  0,0186 u  931,5 MeV  17,33 MeV
u – mass unit
1u is equivalent to 931.5 MeV
1 MeV = 106 eV = 1.6 ·10-13J
For one mole we obtain the energy NA times higher (NA – Avogadro’s number)
Qmole  17,33  1,6  1013 J  6,02  1023  1012 J / mole
For comparison the chemical reaction of burning hydrogen in oxygen gives the
energy:
H2 
1
O2  H 2O
2
Q  3 105 J / mole
For chemical reactions the variation of a mass is then negligible and the rest mass
is conserved:
m01  m02  m03
22