b.tech sem i engineering physics u iii chapter 1-the special theory of relativity
TRANSCRIPT
SPECIAL
THEORY
OF RELATIVITY
Course: B.Tech
Subject: Engineering Physics
Unit: III
Chapter: 1
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
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.
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.
1
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.
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
x ' x – v t
y ' y
z ' z
speed of frame
NOT speed of object
Galilean Transform
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’
GENERAL GALILEAN TRANSFORMATIONS
'
'
'
tt
yy
vtxx
11'
''
''
'
dt
dt
dt
dt
vvdt
dy
dt
dy
vvvvdt
dx
dt
dx
samethearetandt
yy
xx
FamFam '
yy
yy
xxxx
aadt
dv
dt
dv
aadt
dv
dt
dv
samethearetandt
''
'0'
'
inertial reference frame
11'
''
''
'
dt
dt
dt
dt
ttdt
dy
dt
dy
vuuvdt
dx
dt
dx
samethearetandt
yy
xx
frame K frame K’
Newton’s Eqn of Motion is same at
face-value in both reference frames
Positio
nV
elo
city
Accele
ration
2
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.
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
3
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
4
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
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
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
5
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.
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
6
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.
18
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.
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.
20
MICHELSON-MORLEY EXPERIMENT(1887)
Michelson developed a device called an inferometer.
Device sensitive enough to detect the ether.
7
MICHELSON-MORLEY EXPERIMENT(1887)
Apparatus at rest wrt the ether.
MICHELSON-MORLEY EXPERIMENT(1887)
Light from a source is split by a half silvered mirror (M)
The two rays move in mutually perpendicular directions
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.
MICHELSON-MORLEY EXPERIMENT(1887)
The path distance for each ray is the same (l1=l2).
Therefore no interference will be observed
MICHELSON-MORLEY EXPERIMENT(1887)
Apparatus at moving through the ether.
u
ut
8
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
Lt
(c u)
(distance moved by
light to meet the mirror)
u
ut
9
MICHELSON-MORLEY EXPERIMENT(1887)
(distance moved by light to meet the mirror))(||
uc
Lt
|||| utLct
Similarly the time for the return trip is )(
||uc
Lt
The total time
)()(||
uc
L
uc
Lt
u
ut
MICHELSON-MORLEY EXPERIMENT(1887)
The total time ||
2 2
2 2
( ) ( )
2
( )
2 /
1
L Lt
c u c u
Lc
c u
L c
u c
u
ut
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
Lt
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
Lt
u
ut
10
MICHELSON-MORLEY EXPERIMENT(1887)
121
2 2
|| 2 2
2 2
2 3
21 1
2
2
L u ut t t
c c c
After a binomial expansi
L u Lut
c c c
on
ct
vt
2 2
2 2
2
2 /
1
Lt
c u
L ct
u c
The time difference
between the
two rays is,
u
ut
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.
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
11
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 fa f
k k
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 fa f
a fa f
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
kv c
2
2 2 2
2 2
1 1
1 1
1
1
/ /
/
/
a fa fc h
x x vt
y y
z z
a f
x x vt
x x vt
x x vt vt
a fa f
a f
inverse transformation
Plug one into the other:
Solve for t’:
So the correct transformation (and inverse transformation)
equations are:
2 2
2 2
2 22
2 2
2 2 2 2
2 2 2 2
2
1
1 / 1
1 /
/
1/
/
x x vt vt
x vt vt
v cx vt vt
v c
xv c vt vt
t xv c vtv
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
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
tL
c
t tL
c
1
2
1 2
2
= time up
= time down
=
mirror
mirror
L 12
Now put the light clock on a spaceship, but measure the
roundtrip time of the light pulse from the Earth frame S:
tt
tt
c
L v t c t
L c v t
tL
c v
tL
c v c v c
1
2
2 2 2 2 2
2 2 2 2
22
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
13
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 /
Lt
cv c
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:
Lc
02
1 2
1 2
1 1 1
2 2 2
time out, time back
t t
t t t
LL vt ct t
c v
LL vt ct t
c v
A A’ C C’
vt1 L
14
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 dilation1 /
1 1
1 /
Lc Lt t t
c v c v c
ctL v c
tv c
LL
v c
IMAGE REFERENCES LINKS
1. http://postimg.org/image/3wfn9a4yl
2. http://postimg.org/image/b3ca819jx
3. http://postimg.org/image/iklhn8z31
4. http://postimg.org/image/9tegszfrx
5. http://postimg.org/image/v9fs6x57h
6. http://postimg.org/image/xulcuful9
7. http://postimg.org/image/bdjsrdo65
8. http://postimg.org/image/obbo0z731
9. http://s6.postimg.org/kb9klumxp/New_Picture_26.png
10. http://s6.postimg.org/h0g5ouqot/New_Picture_27.png
11. http://s6.postimg.org/uet8kvxct/New_Picture_28.png
12. http://s6.postimg.org/5a287gxwd/New_Picture_29.png
13. http://s6.postimg.org/pk4ljq5ct/New_Picture_30.png
14. http://s6.postimg.org/k9zmsfl3x/New_Picture_31.png