2. Course Outline...
References:
A. Beiser, Concept of Modern Physics (or Perspective of Modern
Physics), Tata-McGraw Hill, 2005
R. Resnick, Relativity, Wiley Eastern Pvt. Ltd., 2007
Special Relativity: Reference frames, inertial frames, Michelson-
Morley experiment (constancy of speed of light), Galilean relativity
and transformation ( concept of absolute time and length
interval), postulates of relativity, Lorentz transformation, length
contraction and time dilation, meson decay, twin paradox, space-
time, velocity addition, relativistic momentum, relativistic mass
and mass-energy relation, general relativity (introduction)
3. Classical and Modern Physics
Modern Physics-
• Relativity – Fast moving objects
• Quantum Mechanics – very
small
Classical Relativistic
Speed
10% c
Quantum Classical
Size
Atomic/molecular size
Classical Physics -
Larger, slow moving
• Newtonian Mechanics
• EM and Waves
• Thermodynamics
4. Aimed to answer some burning questions:
Could Maxwell’s equations for electricity and magnetism be
reconciled with the laws of mechanics?
Where was the ether?
Below 10% c, classical mechanics holds
(relativistic effects are minimal)
Above 10%, relativistic mechanics holds (more
general theory)
SPECIAL THEORY OF RELATIVITY
5. Reference frames
A frame of reference in physics, may refer to a coordinate
system or set of axes within which to measure the
position, orientation, and other properties of objects in it
Inertial frames
in which no accelerations are
observed in the absence of external
forces
that is not accelerating.
Newton’s laws hold in all inertial
reference frames
Non-Inertial frames
that is accelerating with respect
to an inertial reference frame
bodies have acceleration in the
absence of applied forces
y
x
z
O
K
6. Some related information:
Albert Einstein (1879–1955) was only two years old when Michelson reported
his first null measurement for the existence of the ether.
At the age of 16 Einstein began thinking about the form of Maxwell’s
equations in moving inertial systems.
In 1905, at the age of 26, he published his startling proposal about the
principle of relativity, which he believed to be fundamental.
7. Postulates of Special Relativity
The postulates of relativity as stated by Einstein1905)
1. Equivalence of Physical Laws
The laws of physics are the same in all inertial
frames of reference.
There is no absolute reference frame
of time and space
2. Constancy of the Speed of Light
The speed of light in a vacuum, c = 3.00 x 108 m/s, is the
same in all inertial frames of reference, independent of the
motion of the source or the receiver.
8. Searching for an absolute Reference System
Ether was proposed as an absolute reference system in which
the speed of light was constant and from which other
measurements could be made.
The Michelson-Morley experiment was an attempt to show the
existence of ether
Michelson-Morley Experiment
Experiment designed to measure small changes in the speed of light was
performed by A. Michelson(1852-1931, Nobel) and Edward W.
Morley(1838-1923)
“ether”: a hypothetical medium pervading the universe in which light
waves were supposed to occur.
“Motion of earth through the ether”
• Used an optical instrument called an interferometer that Michelson invented
• Device was to detect the presence of the ether
A. A. Michelson and E. W. Morley, American Journal of Science, 134, 333 (1887)
9. Experimental setup
Outcome of the experiment was negative
contradicting the “ether” hypothesis.Michelson’s Conclusion
Michelson noted that he should be able to detect a phase shift of light due to
the time difference between path lengths but found none.
He thus concluded that the hypothesis of the stationary ether must be
incorrect.
After several repeats and refinements with assistance from Edward Morley
(1893-1923), again a null result.
Thus, ether does not seem to exist!
11. Drawbacks:
1. Violates both of the postulate of special theory of relativity
[i] Same equations of physics in K and K, but the equations
of electricity and magnetism is entirely different.
[ii] c = c - v
Velocity component
zz
yy
xx
v
td
zd
v
v
td
yd
v
vv
td
xd
v
Go for different transformation
13. Matrix form
Inverse Lorentz Transformation:
22
2
22
/1
,,,
/1 cv
c
xv
t
tzzyy
cv
tvx
x
Measurements made in the moving frame K to their equivalents in K
H.W.: Check the validity of relativity postulates in Lorentz transformation ?
From a frame K(x, y, z, t) to a frame K (x, y, z, t) moving with
velocity v along the x-axis the space –time coordinates are transformed
as
14. Length Contraction
Length where observer is
moving relative to the
length being measured.
Length where observer is
at rest relative to the
length being measured.
22
0 /1 cvLL
The length of an object is measured to be shorter when it is
moving relative to the observer than when it is at rest.
Faster means shorter
(a)
(b)
Observers from earth would see a
spaceship shorten in the length of travel
Only shortens in the direction of travel
15. Time Dilation
Actual difference of elapsed time between two events as measured
by observers either moving relative to each other or differently situated from
gravitational masses.
This effect arises neither from technical aspects of the
clocks nor from the fact that signals need time to propagate,
but from the nature of space-time itself.
22
0
/1 cv
t
t
Time where clock is moving
relative to the observer
Time where clock is at rest
relative to the observer.
Proper time
Clocks moving relative to an observer are
measured by that observer to run more
slowly, as compared to the clock at rest.
A moving clock ticks more slowly than a clock at rest
APPOLLO 11
16. Muon Decay
)693.0(
0
2/1tt
eNN
N: No. of muons at t
No :No. of muons at t=0
Half life: t1/2 =1.5 x 10-6 sec
Experimental verification
• Time Dilation and Muon Decay
Cosmic rays enter the upper atmosphere
and interact with particles in the upper
atmosphere creating mesons (pions),
decay into other particles called muons
Obey radioactive law:
17. A clock in a moving frame will be seen to be running slow, or "dilated"
according to the Lorentz transformation. The time will always be
shortest as measured in its rest frame. The time measured in the frame
in which the clock is at rest is called the "proper time".
Mean lifetime as measured in laboratory frame
0 = 2.2 s
v = 0.995 c
= 220 s = 100 0
= 22 s = 10 0
v = 0
v = 0.99995 c
The mean lifetime of a muon in its own reference frame, called the proper
life time, is 0 = 2.2 s. In a frame moving at velocity v with respect to that
proper frame, the lifetime is = 0 , where is the time dilation factor.
18. Velocity Addition
The light emitted from the K in the direction of its motion relative to another frame K
ought to have a speed of c + v as measured in K.
violets the postulate of relativity
Common sense is no more reliable as a guide in science than it is elsewhere
Suppose something is moving relative to both K and K . An observer in K measures its
three velocity components to be
dt
dx
Vx
dt
dy
Vy
dt
dz
Vz
td
xd
Vx
While to an observer
td
yd
Vy
td
zd
Vz
By differentiating the inverse Lorentz transformation equations for x, y, z and t, we have
22
/1 cv
tvdxd
dx
yddy zddz
22
2
/1 cv
c
xvd
td
dt
19. and so
td
xd
c
v
v
td
xd
c
xvd
td
tvdxd
dt
dx
Vx
22
1
Relativistic velocity transformation
2
1
c
Vv
vV
V
x
x
x
2
22
1
/1
c
Vv
cvV
V
x
z
z
2
22
1
/1
c
Vv
cvV
V
x
y
y
If Vx = c, if the light is emitted in the moving frame K in its direction of
motion relative to K, an observer in frame K will measure the speed:
c
vc
vcc
c
vc
vc
c
Vv
vV
V
x
x
x
)(
11 22
20. TWIN PARADOX
A longer life, but it will not seem longera thought experiment
The Set-up
Twins Mary and Frank at age 30 decide on two career paths: Mary decides to
become an astronaut and to leave on a trip 8 lightyears (ly) from the Earth at a
great speed and to return; Frank decides to reside on the Earth.
20 Yr20 Yr 50 Yr70 Yr
The Problem
Upon Mary’s return, Frank reasons that her clocks measuring her age must run
slow. As such, she will return younger. However, Mary claims that it is Frank
who is moving and consequently his clocks must run slow.
The Paradox
Who is younger upon Mary’s return?
21. However, this scenario can be resolved within the standard framework of
special relativity.
Accelerations are outside the realm of special
relativity and require general relativity.
The clear implication is that the travelling twin would indeed be younger,
but the scenario is complicated by the fact that the travelling twin must
be accelerated up to travelling speed, turned around, and decelerated
again upon return to Earth.
The Resolution
Frank’s clock is in an inertial system during the entire trip; however, Mary’s
clock is not. As long as Mary is traveling at constant speed away from Frank,
both of them can argue that the other twin is aging less rapidly
When Mary slows down to turn around, she leaves her original inertial system
and eventually returns in a completely different inertial system.
Mary’s claim is no longer valid, because she does not remain in the same
inertial system. There is also no doubt as to who is in the inertial system. Frank
feels no acceleration during Mary’s entire trip, but Mary does.
22. 22
0
/1 cv
m
m
Mass
Mass observed by an
observer moving
relative to the mass
The mass of an object is measured to increase as its speed increases.
Mass measured when
object is at rest relative
to the observer-rest mass
Relativistic Momentum
Relativistic momentum: mup 22
/1
1
:
cv
where
Classical mechanics: Linear momentum: p = mv, v <<<<<<<<<<<<<<<<<<< c
Whether this formula is valid in relativistic inertial frames ?
In this form the conservation of momentum is valid in special relativity
Effect on mass:
23. As the speed gets closer and closer to c, the momentum increases without limit;
note that the speed must be close to the speed of light before
the difference between classical and relativistic momentum is noticeable:
24.
25. Relativistic Energy
Due to the new idea of relativistic mass, we must now redefine the
concepts of work and energy.
Therefore, we modify Newton’s second law to include our new
definition of linear momentum, and force becomes:
For simplicity, let the particle start from rest under the influence of
the force and calculate the kinetic energy K after the work is done.
For speeds u << c, we expand in a binomial series as follows:
Relativistic kinetic energy
26. Total Energy and Rest Energy
the rest energy
The total energy is denoted by E and is given by
Classic Kinetic
energy
27. • Even when a particle has no velocity and therefore no kinetic energy, it still
has energy by virtue of its mass.
• The laws of conservation of energy and conservation of mass must be
combined into one law: Law of conservation of mass-energy
• Mass and energy are equivalent
42222
2
0
222
cmcpE
EcpE
Relativistic energy and momentum: Massless Particles:
For a particle having no mass:
For a massless particle:
pcE
cu
Other Units
Rest energy of a particle:
Example: E0 (proton)
28. What about non-inertial frames of reference?
Einstein waited 10 years to publish his General Theory of
Relativity…
… you only have to wait until this class over!
General Theory of Relativity (GTR)(1916)
Describes the relationship between gravity and the geometrical
structure of space and time
Principles
General relativity is the extension of special relativity. It includes the
effects of accelerating objects and their mass on space-time.
As a result, the theory is an explanation of gravity
It is based on two concepts: (1) the principle of equivalence, which is an
extension of Einstein’s first postulate of special relativity and (2) the
curvature of space-time due to gravity.
29. When the elevator is moving at a constant speed, the light from the flashlight
travels in a straight line. When the elevator accelerates, the light bends.
The principle of equivalence then tells us that light should bend in a
gravitational field as well.
Illustration
Tests of GTR-II
Gravitational Redshift
Tests of GTR-I
Bending of Light
30. POINTS TO REMEMBER
All observers agree upon relative speed.
Distance depends on an observer’s motion. Proper length L0 is the distance between two points measured by
an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when
measuring the distance between two points that are stationary relative to the Earth.
Length contraction L is the shortening of the measured length of an object moving relative to the observer’s
frame:
Two events are defined to be simultaneous if an observer measures them as occurring at the same time.
They are not necessarily simultaneous to all observers—simultaneity is not absolute.
Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another
observer.
With classical velocity addition, velocities add like regular numbers in one-dimensional motion: u=v+u′, where v is the
velocity between two observers, u is the velocity of an object relative to one observer, and u′ is the velocity relative to
the other observer.
Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an
object moving at a relativistic speed:
The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum.
Relativistic momentum p is classical momentum multiplied by the relativistic factor γ.
At low velocities, relativistic momentum is equivalent to classical momentum.
oL
cvLL 22
0 /1
22
0
/1 cv
t
t
31. POINTS TO REMEMBER
Relativistic momentum approaches infinity as u approaches c. This implies that an object with mass
cannot reach the speed of light.
Relativistic momentum is conserved, just as classical momentum is conserved.
Relativistic energy is conserved as long as we define it to include the possibility of mass changing to
energy.
Total Energy is defined as: E=mc2
Rest energy is E0=mc2, meaning that mass is a form of energy. If energy is stored in an object, its mass
increases. Mass can be destroyed to release energy.
We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so
small for a large increase in energy.
The relativistic work-energy theorem is Wnet=E−E0=mc2−mc2=(−1)mc2.
Relativistic ally, Wnet=KErel , where KErel is the relativistic kinetic energy.
Relativistic kinetic energy is KErel=(−1)mc2, At low velocities, relativistic kinetic energy reduces to classical
kinetic energy.
No object with mass can attain the speed of light because an infinite amount of work and an infinite
amount of energy input is required to accelerate a mass to the speed of light.
The equation E2=(pc)2+(mc2)2 relates the relativistic total energy E and the relativistic momentum p. At
extremely high velocities, the rest energy mc2 becomes negligible, and E=pc
32. TUTORIAL-II
1. The dinosaur’s had a length of about 25 m. Calculate the dinosaur’s length if it was running
at:
a) ½ lightspeed
b) ¾ lightspeed
c) 95% lightspeed
2. What is the lifetime of a muon travelling at 0.60 c (1.8 X 108 m/s) if its rest lifetime is 2.2
s?
22
0
/1 cv
t
t
s
cc
s
t 6
22
6
108.2
/)6.0(1
102.2
3. How long will a 100 year trip (as observed from earth) seem to the astronaut who is
travelling at 0.99 c?
22
0 /1 cvtt 4.5 year
Ans.
Ans.
a) 21.7 m
b) 15.5 n
c) 7.8 m
Ans.
4. A particle travels at 1.90×108 m/s and lives 2.10×10−8 s when at rest relative to an observer.
How long does the particle live as viewed in the laboratory?
Ans:
s
s
s
s
cv
t
t 8
28
28
8
22
0
1071.2
100.3
109.1
1
1010.2
/1
33. TUTORIAL-II
5. What is the speed of the second stage of the rocket shown with
respect to the earth?
u = v + u’
1 + vu’/c2
= 0.60c + 0.60c
1 + [(0.60c)(0.60c)/c2 ]
u = 0.88 c
(classical addition would give you 1.20c,
over the speed of light)
6. Suppose a car travelling at 0.60c turns on its headlights. What is the speed of the light
travelling out from the car?
u = v + u’
1 + vu’/c2
u = 0.60c + c = 1.60c
1 + [(0.60c)(c)/c2 ] 1.60
u = c
7. Now the car is travelling at c and turns
on its headlights.
u = v + u’
1 + vu’/c2
u = c + c = 2c
1 + [(c)(c)/c2 ] 2
u = c
Ans.
Ans.
34. 8. Calculate the mass of an electron moving at 0.98 c in an accelerator for cancer therapy.
)5(1058.4
/)98.0(1
1011.9
0
30
22
31
mkg
cc
kg
m
Ans.
9. How much energy would be released if a p0 meson (mo=2.4 X 10-28 kg) decays at rest.
Ans.
E = mc2
E = moc2 (particle is at rest)
E = (2.4 X 10-28 kg)(3.0 X 108 m/s)2
E = 2.16 X 10-11 J
10. An electron is moving at 0.999c in the CERN accelerator.
a. Calculate the rest energy
b. Calculate the relativistic momentum
c. Calculate the relativistic energy
d. Calculate the Kinetic energy
TUTORIAL-II
Do this.
35. ASSIGNMENT-III
1. The length of a spaceship is measured to be exactly half of its proper length. What is the speed of
the spaceship relative to the observer’s frame ?
2. (a) If the average life time of a muon() is 2.3 x 10-6 sec, what average distance would it travel in
a vacuum before dying as measured in reference frames in which its velocity is 0.0 c, 0.60c, 0.90c
and 0.99c ? (b) Compare each of these distances with the distance the muon sees itself travelling
through.
3. Suppose that a particle moves parallel to the x-x axis, that v = 25,000 mi/hr and Vx = 25,000
mi/hr. What percent error is made in using the Galilean rather than the Lorentz equation to
calculate Vx ? The speed of light is 6.7 x 108 mi/hr.
4. A painting is 1.00 m tall and 1.50 m wide. What are its dimensions inside a spaceship moving at
0.90 c?
5. Calculate the mass of an electron moving at 4.00 107 m/s in the CRT of a television tube.
6. A p0 meson (mo=2.4 X 10-28 kg) travels at 0.80 c.
a. Calculate the new mass
b. Calculate the relativistic momentum
c. Calculate the energy of the particle
Due date: 21-10-2013
