The document discusses Einstein's theory of special relativity. It provides background on Einstein's two postulates: 1) the laws of physics are the same in all inertial frames of reference, and 2) the speed of light in a vacuum is the same for all observers regardless of their motion. It describes how these postulates led Einstein to develop the Lorentz transformations, which show that time and space are relative between different frames of reference moving at a constant velocity with respect to each other.

1. Special
Theory
of Relativity

2. The dependence of various physical phenomena on
relative motion of the observer and the observed
objects, especially regarding the nature and behaviour
of light, space, time, and gravity is called relativity.
When we have two things and if we want to find out
the relation between their physical property
i.e.velocity,accleration then we need relation between
them that which is higher and which is lower.In
general way we reffered it to as a relativity.
The famous scientist Einstein has firstly found out the
theory of relativity and he has given very useful
theories in relativity.
Introduction to Relativity

3. What is Special Relativity?
In 1905, Albert Einstein determined that the laws
of physics are the same for all non-accelerating
observers, and that the speed of light in a vacuum
was independent of the motion of all observers.
This was the theory of special relativity.

4. FRAMES OF REFERENCE
A Reference Frame is the point of View, from which
we Observe an Object.
A Reference Frame is the Observer it self, as the
Velocity and acceleration are common in Both.
Co-ordinate system is known as FRAMES OF
REFERENCE
Two types:
1. Inertial Frames Of Reference.
2. non-inertial frame of reference.

6. FRAMES OF REFERENCE
 We have already come across idea of frames of
reference that move with constant velocity. In
such frames, Newton’s law’s (esp. N1) hold.
These are called inertial frames of reference.
 Suppose you are in an accelerating car looking at a
freely moving object (I.e., one with no forces
acting on it). You will see its velocity changing
because you are accelerating! In accelerating
frames of reference, N1 doesn’t hold – this is a
non-inertial frame of reference.

7. Conditions of the Galilean Transformation
Parallel axes (for convenience)
K’ has a constant relative velocity in the x-direction with
respect to K
Time (t) for all observers is a
Fundamental invariant,
i.e., the same for all inertial observers
speed of frame
NOT speed of object
x' x – v t
y' y
z' z



Galilean Transform

8. Galilean Transformation Inverse Relations
Step 1. Replace with .
Step 2. Replace “primed” quantities with
“unprimed” and “unprimed” with “primed.”
speed of frame
NOT speed of object
x x’ vt
y y’
z z’
t t’
 




9. General Galilean Transformations
'
'
'
tt
yy
vtxx



11
'
'
'
'
'
'



dt
dt
dt
dt
vv
dt
dy
dt
dy
vvvv
dt
dx
dt
dx
samethearetandt
yy
xx
yy
yy
xx
xx
aa
dt
dv
dt
dv
aa
dt
dv
dt
dv
samethearetandt
'
'
'0
'
'


inertial reference frame
  FamFam '

11
'
'
'
'
'
'



dt
dt
dt
dt
tt
dt
dy
dt
dy
vuuv
dt
dx
dt
dx
samethearetandt
yy
xx
frame K frame K’
Newton’s Eqn of Motion is same at
face-value in both reference frames
PositionVelocityAcceleration

10. Einstein’s postulates of special theory of
relativity
• The First Postulate of Special Relativity
The first postulate of special relativity states
that all the laws of nature are the same in all
uniformly moving frames of reference.

11. Einstein reasoned all motion is relative and all frames of
reference are arbitrary.
A spaceship, for example, cannot measure its speed
relative to empty space, but only relative to other objects.
Spaceman A considers himself at rest and sees
spacewoman B pass by, while spacewoman B considers
herself at rest and sees spaceman A pass by.
Spaceman A and spacewoman B will both observe only the
relative motion.
The First Postulate of Special Relativity

12. A person playing pool
on a smooth and fast-
moving ship does not
have to compensate
for the ship’s speed.
The laws of physics
are the same whether
the ship is moving
uniformly or at rest.
The First Postulate of Special Relativity

13. Einstein’s first postulate of special relativity
assumes our inability to detect a state of
uniform motion.
Many experiments can detect accelerated
motion, but none can, according to Einstein,
detect the state of uniform motion.
The First Postulate of Special Relativity

14. The second postulate of special relativity
states that the speed of light in empty space
will always have the same value regardless
of the motion of the source or the motion of
the observer.
The Second Postulate of Special Relativity

15. Einstein concluded that if an
observer could travel close to
the speed of light, he would
measure the light as moving
away at 300,000 km/s.
Einstein’s second postulate of
special relativity assumes that
the speed of light is constant.
The Second Postulate of Special Relativity

16. The speed of light is constant regardless of the
speed of the flashlight or observer.
The Second Postulate of Special Relativity
The speed of light in all reference frames is always the
same.
• Consider, for example, a spaceship departing from the
space station.
• A flash of light is emitted from the station at 300,000
km/s—a speed we’ll call c.

17. The speed of a light flash emitted by either the
spaceship or the space station is measured as c by
observers on the ship or the space station.
Everyone who measures the speed of light will get
the same value, c.
The Second Postulate of Special Relativity

18. 18
The Ether: Historical Perspective
Light is a wave.
Waves require a medium through which to
propagate.
Medium as called the “ether.” (from the Greek
aither, meaning upper air)
Maxwell’s equations assume that light obeys
the Newtonian-Galilean transformation.

19. The Ether: Since mechanical waves require a
medium to propagate, it was generally accepted that light
also require a medium. This medium, called the ether,
was assumed to pervade all mater and space in the
universe.

20. 20
The Michelson-Morley
Experiment
Experiment designed to measure small
changes in the speed of light was performed
by Albert A. Michelson (1852 – 1931, Nobel )
and Edward W. Morley (1838 – 1923).
Used an optical instrument called an
interferometer that Michelson invented.
Device was to detect the presence of the
ether.
Outcome of the experiment was negative,
thus contradicting the ether hypothesis.

21. Michelson-Morley
Experiment(1887)
Michelson developed a device called an inferometer.
Device sensitive enough to detect the ether.

22. Michelson-Morley
Experiment(1887)
Apparatus at rest wrt the ether.

23. Michelson-Morley
Experiment(1887)
Light from a source is split by a half silvered mirror (M)
The two rays move in mutually perpendicular directions

24. Michelson-Morley
Experiment(1887)
The rays are reflected by two mirrors (M1 and M2)
back to M where they recombine.
The combined rays are observed at T.

25. Michelson-Morley
Experiment(1887)
The path distance for each ray is the same (l1=l2).
Therefore no interference will be observed

26. Michelson-Morley
Experiment(1887)
Apparatus at moving through the ether.
u
ut

27. Michelson-Morley Experiment(1887)
First consider the time required for the parallel ray
Distance moved during the first part of the path is
|| ||
||
ct L ut
L
t
(c u)
 
 

(distance moved by
light to meet the mirror)
u
ut

28. Michelson-Morley
Experiment(1887)
(distance moved by light to meet the mirror))(
||
uc
L
t


|||| utLct 
Similarly the time for the return trip is
)(
||
uc
L
t


The total time
)()(
||
uc
L
uc
L
t




u
ut

29. Michelson-Morley
Experiment(1887)
The total time ||
2 2
2 2
( ) ( )
2
( )
2 /
1
L L
t
c u c u
Lc
c u
L c
u c
 
 




u
ut

30. Michelson-Morley Experiment(1887)
For the perpendicular ray
we can write,
ct
vt
2 2 2
2 2 2 2 2
2 2 2
2 2
( )
( )
ct L ut
L c t u t
c u t
L
t
c u

 


 
 
 


(initial leg of the
path)
The return path is the same as the
initial leg therefore the total time is
22
2
uc
L
t


u
ut

31. Michelson-Morley Experiment(1887)
ct
vt
2 2
2 2
2
2 /
1
L
t
c u
L c
t
u c




 

The time difference between t
two rays is, 1
212 2
|| 2 2
2 2
2 3
2
1 1
2
2
L u u
t t t
c c c
After a binomial expansi
L u Lu
t
c c c
on
 

    
          
     
   
u
ut

32. Michelson-Morley Experiment(1887)
The expected time difference is too small to be measured
directly!
Instead of measuring time, Michelson and Morley looked
for a fringe change.
as the mirror (M) was rotated there should be a shift in the
interference fringes.
Results of the Experiment
 A NULL RESULT
No time difference was found!
Hence no shift in the interference patterns
Conclusion from Michelson-Morley Experiment
the ether didn’t exist.

33. The Lorentz Transformation
We are now ready to derive the correct transformation
equations between two inertial frames in Special
Relativity, which modify the Galilean Transformation.
We consider two inertial frames S and S’, which have a
relative velocity v between them along the x-axis.
x
y
z
S
x'
y'
z'
S'
v

34. Now suppose that there is a single flash at the origin of S and S’ at
time , when the two inertial frames happen to coincide. The
outgoing light wave will be spherical in shape moving outward
with a velocity c in both S and S’ by Einstein’s Second Postulate.
We expect that the orthogonal coordinates will not be affected by
the horizontal velocity:
But the x coordinates will be affected. We assume it will be a
linear transformation:
But in Relativity the transformation equations should have the
same form (the laws of physics must be the same). Only the
relative velocity matters. So
x y z c t
x y z c t
2 2 2 2 2
2 2 2 2 2
  
      
 
 
y y
z z
  
   
x k x vt
x k x vt
a f
a f
 k k

35. Consider the outgoing light wave along the x-axis
(y = z = 0).
Now plug these into the transformation equations:
Plug these two equations into the light wave equation:
  

x ct
x ct
in frame S'
in frame S
     
     
1 / &
1 /
x k x vt k ct vt kct v c
x k x vt k ct vt kct v c
      
         
ct x kct v c
ct x kct v c
t kt v c
t kt v c
    
   
  
  
1
1
1
1
/
/
/
/
a f
a f
a f
a f

36. Plug t’ into the equation for t:
So the modified transformation equations for the
spatial coordinates are:
Now what about time?
t k t v c v c
k v c
k
v c
  
 
 


2
2 2 2
2 2
1 1
1 1
1
1
/ /
/
/
a fa f
c h

  
 
 
x x vt
y y
z z
 a f
  
   
   
x x vt
x x vt
x x vt vt


  
a f
a f
a f
inverse transformation
Plug one into the other:

37. Solve for t’:
So the correct transformation (and inverse transformation)
equations are:
 
 
 
2 2
2 2
2 2
2
2 2
2 2 2 2
2 2 2 2
2
1
1 / 1
1 /
/
1
/
/
x x vt vt
x vt vt
v c
x vt vt
v c
xv c vt vt
t xv c vt
v
t t vx c
  
  
 
  
 


  
  
 
 

  
   
  
      
   
   
      
x x vt x x vt
y y y y
z z z z
t t vx c t t vx c
 
 
a f a f
c h c h/ /2 2
The Lorentz
Transformation

38. Application of Lorentz Transformation
 Time Dilation
We explore the rate of time in different inertial frames by
considering a special kind of clock – a light clock – which is
just one arm of an interferometer. Consider a light pulse
bouncing vertically between two mirrors. We analyze the
time it takes for the light pulse to complete a round trip
both in the rest frame of the clock (labeled S’), and in an
inertial frame where the clock is observed to move
horizontally at a velocity v (labeled S).
In the rest frame S’
t
L
c
t
L
c
t t
L
c
1
2
1 2
2
 
 
   
= time up
= time down
=
mirror
mirror
L

39. Now put the light clock on a spaceship, but measure
the roundtrip time of the light pulse from the Earth
frame S:t
t
t
t
c
L v t c t
L c v t
t
L
c v
t
L
c v c v c
1
2
2 2 2 2 2
2 2 2 2
2
2
2 2
2 2 2 2
2
2
4 4
4
4
2 1
1 1
 
 
 
 






time up
time down
The speed of light is still in this frame, so
/ /
/
/ /
c h

L
c t/ 2
v t / 2

40. So the time it takes the light pulse to make a
roundtrip in the clock when it is moving by us is
appears longer than when it is at rest. We say
that time is dilated. It also doesn’t matter which
frame is the Earth and which is the clock. Any
object that moves by with a significant velocity
appears to have a clock running slow. We
summarize this effect in the following relation:
2 2
1 2
, 1,
1 /
L
t
cv c
      


41. Length Contraction
Now consider using a light clock to measure the
length of an interferometer arm. In particular, let’s
measure the length along the direction of motion.
In the rest frame S’:
Now put the light clock on a spaceship, but measure
the roundtrip time of the light pulse from the Earth
frame S:
L
c
0
2


1 2
1 2
1 1 1
2 2 2
time out, time backt t
t t t
L
L vt ct t
c v
L
L vt ct t
c v
 
 
   

   

A A’ C C’
vt1L

42. In other words, the length of the interferometer arm
appears contracted when it moves by us. This is
known as the Lorentz-Fitzgerald contraction. It is
closely related to time dilation. In fact, one implies the
other, since we used time dilation to derive length
contraction.
 
1 2 2 2 2 2
2 2
2 2
0
2 2
2 2 1
1 /
1 /
2
But, from time dilation
1 /
1
1
1 /
Lc L
t t t
c v c v c
ct
L v c
t
v c
L
L
v c



   
 
 


  


43. Engineering physics By Dr. M N Avadhnulu, S
Chand publication
ENGINEERING PHYSICS
ABHIJIT NAYAK
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