6593.relativity

Concept of Modern Physics A. Beiser

Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 1

 The concept of relativity had been
well known since the time of
Galileo.

It was used by Newton and Poincare
developed this idea.

Einstein said that he thought of the idea
whilst riding his bicycle.


Special Relativity

In 1905 the 26 year old Albert
Einstein described in his theory of
Special Relativity
“how measurements of time and
space are affected by the motion
between the observer and what is
being observed.”

The theory of special relativity
revolutionized the world of physics by
connecting space and time, matter and
energy, electricity and magnetism …


Introduction of the chapter
After completing the chapter you will be familiar with
the
1. Two basic postulates of the STR
2. Frame of reference, concept of ether and Michelson-
Morley Interferometer
3. Galilean transformation
4. Lorentz transformation
5. Time dilation
6. Length contraction
7. Velocity transformation
8. Relativistic momentum and energy

Postulates of Special Relativity

Einstein built the special theory of relativity on two postulates:

1. The Relativity Principle: The laws of motion are the same
in every inertial frame of reference.

2. Constancy of the speed of light: The speed of light
in a vacuum is the same independent of the speed
of the source or the observer.


Motion is always measured relative to a frame of
reference i.e. there is no absolute motion

Frame S Frame S’ V0 relative to frame S

V ’ = V - V0

Speed
measured In
frame S
Speed measured
In frame S’

What is an event ?

• When is it happening time t
• Where is it happening position (x,y,z)
• What reference frame

coordinate (t,x,y,z) measured with
respect to a particular observer at
(0,0,0,0) frame of reference


Measuring an event

 An event is something that happens, to which
an observer can assign three space coordinates
and one time coordinate
 A given event may be recorded by any number
of observers, each in a different reference
frame
 In general different observers will assign
different space-time coordinates for the same
event.

Inertial Reference Frames

 An Inertial Frame of Reference is one in which
the basic laws of physics apply- e.g., a train moving
at a constant velocity, in this, objects move “ normally”.

v

Objects obeying the Newton’s First Law.

Non-Inertial Frames
An accelerating or decelerating objects. If you are
sitting / walking on that, then during this period, you are in
non- inertial frame.

For example: If you are in a Ferris Wheel
you are always accelerating inwards
so non-inertial.


Relative Velocity:
What is the black car’s velocity relative to your
frame?

V= 70 km/hr V=50 km/hr

We know the answer intuitively (120 km/hr) ?


Same idea with velocity of light ?

C= 3 x 10^(8) m/s C= 3 x 10^(8) m/s

From the first example, we would expect the relative velocity
to be 2c = 6 x 10^(8) m/s.

This is in fact WRONG !!!!!

The Luminiferous Ether
The Ether was the basis of understanding, and the
term was used to describe a medium for the
propagation of light.

 It was hypothesized that the
Earth moves through this medium .

The Ether; Ether
Was transparent
Had zero density
Was everywhere
Was the substance which allowed light to propagate.

Airplane
wind

If the velocity of the wind is v1 relative to Earth
, and v2 is the velocity of airplane relative to the
wind, the speed of Airplane relative to the earth
is (a) v1+v2 in the same direction, (b) v2-v1 in the
opposite direction and (c) in the
Direction perpendicular to the wind.
Observer fixed on earth

v C c+ v

v c c-v

2 2 Determine the speed of light
c c v under these circumstances ?

In our case the ether wind is blowing through our
apparatus fixed to the Earth, determine the velocity
of light ?

If the Sun is assumed to be at rest in the ether, then the velocity
of the Ether wind would be equal to the orbital velocity of the earth
around the Sun. which has a magnitude of about 3 x 10^4 m/s
compared to c= 3 X 10^8 m/s.

The change in the speed of
light should be detectable !!!!


The Michelson-Morley Experiment
The Famous experiment designed to detect
small changes in the speed of light with motion
of an observer through the ether .

It was performed in 1887.
Albert A. Michelson
The negative results of the experiment not only
meant that the speed of light does not depend on
the direction of light propagated but also
contradicted the ether hypothesis.

Light is now understood to be a phenomenon
that requires no medium for its propagation.
Edward W. Morley

Light source

Mirror
Compensating plate
Semi-silvered plate

Telescope
Actual Experimental set-up


The Michelson interferometer produces interference fringes
by splitting a beam of monochromatic light so that one beam
strikes a fixed mirror and the other a movable mirror. When
the reflected beams are brought back together, an
interference pattern results.


M1
C
V Ether wind

Arm two
L
Light velocity -c

Source B
A Arm one
M2
L

A = Semi silvered plate
B= Mirror total
C = Mirror total
Observer


Michelson Interferometer

Precise distance measurements can be made with the
Michelson interferometer by moving the mirror and
counting the interference fringes which move by a
reference point. The distance d associated with m
fringes is


Michelson-Morley Interferometer has two arms of equal
length L.
First, the beam traveling parallel to the direction of the
ether wind
Velocity of light beam moves to the right, with respect to
the Earth is = c - v
Velocity of light beam moves to the left, with respect to the
Earth is = c + v
The total time of travel for the round –trip along the
horizontal path is

2 1
L L 2L v
t1 1 2
c v c v c c


Now, the light beam traveling perpendicular to the wind, so in
this case the speed of the beam relative to the
1
Earth is 2 2 2
c v
Total time of travel for the round-trip is
2 12
2L 2L v
t2 1
2 2 12 c c2
c v
Thus the time difference between the right beam traveling
horizontally and the beam traveling vertically is

2 1 2 12
2L v v
t t1 t2 1 2 1 2
c c c

2
Lv
After simplification , t t1 t 2 3
c Because

After rotating the interferometer through 90 degree v2
1.
The path difference corresponding to this time difference is c2

2 Lv 2
d c2 t
c2
The corresponding fringe shift is equal to this path difference
divided by the wavelength of light, lembda, because a change
in path of 1 wavelength corresponds to a shift of 1 fringe,

2 Lv 2
Shift
c2

Change in path of the order of lambda = one fringe shift
If change is one then fringe shift is equal to 1 over lambda
If change is d in path then change in shift would be equal to d
into one upon lambda.

The speed of the Earth about the Sun, gives a path difference of

2(11)(3 10 4 m / s ) 2 7
d 2.2 10 m
(3 10 8 m / s ) 2
7
d 2.2 10 m
Shif t 7
0.40
5.0 10 m

Conclusion:
1. No detection of fringe shift in the pattern
2. No motion of Earth with respect to Ether.
3. The speed of light is same for all observers

“ Most famous negative result in the History of Physics”.


Questions/ doubts
1) How Michelson able detect change in the speed of light?
2) Concept of aether ?
3) Concept of frame of reference ?
4) How shifting of fringes decides whether the speed of light is same or not ?
5) In theory of relativity fourth coordinate Is of time , why we take it so?
because to define position of particles only distance from 3 co-ordinates
is needed?
6) How we come to know light is electromagnetic wave through
Moreley Experiment?
7) In Morley experiment is source of light is moveable or mirror is moveable?
8) The basic points how Morely had thought before doing experiment ?
9) Is theory of relativity and special theory of relativity same?
9) Conclusion about the persences of aether from M-M expt.?


d
11) Why there is change in velocity of light when shift = 1 ?


12) Why do hypothetical concept of ether was needed?
13) Mathematics of M-M Expt?
14) When Interferometer is rotated through 90 degree what happens then?
15) Derivation of M-M expt. ?


What Time Dilation means…………?
• A moving clock ticks more slowly than a clock
at rest

When two events are occurs at the same location in an inertial reference
frame , the time interval between them, measured in that frame, is called
the proper time interval or the “ proper time”

Measurements of the same time interval from any other inertial reference
frame are always greater.

The amount by which a measured time interval is greater than the
corresponding proper time interval is called time dilation.


v

Time Dilation

One clock is at rest in a laboratory on the ground and
the other is in a spacecraft that moves at the speed v
relative to the ground. An observer in the laboratory
watches both clocks; Does he/she find that they tick
at the same rate ?


moving clocks run slow. This means that if two
events occur at the same place, such as the ticks of
a clock, a moving observer will measure the time
between the events to be longer. The relation
between a time measured by a stationary observer
t0 to the time t measured by an observer moving
with velocity v is:

The gamma factor is common in relativity, and we
will use it often. It is always greater than unity. If the
velocity were greater than c, it would be undefined.

Show Me The Derivation?

For our derivation, we will consider two
measurements. One taken by a rider with the
clock and the other measurement for the
clock will be made by a stationary observer
(referred to as the stationary for the mover).


A light pulse clock at rest
On the ground as seen by
An observer on the ground.
mirror
The light travels the total
Distance 2L at speed c, therefore
Recording device The time for entire trip is

' 2L
t
c
Lo
Meter stick

Light pulse

mirror

Photosensitive surface


Clock is moving with v velocity in spacecraft. What observer notice who is in the
Rest with respect to spacecraft (seen from the ground). The time interval between
ticks is t. v

0

t/2
t

B
ct
2
Lo
A C
D

vt
2


Observer in same inertial frame and notice the events

to = 2L/c


Now, observer in another frame and seen events
from the ground …………


The light ray travels the path AB and mirror AD with velocity v.
The distance AB= ct/2 and AD=vt/2

2 2
ct 2 vt
L0
2 2
2 L0 / c t0
t
1 v2 c2 1 v2 c2

To= time interval onclock at rest relative to an observer= proper time
T= time interval on clock in motion relative to an observer
v-= speed of relative motion
C= speed of light.


Example 1:
If you were to board a craft and travel at 0.2 c, how long would 1 hour be?


Example 2:
If you were to board a craft and travel at 0.8 c, how long would 1 hour be?


Example 3:
If you were to board a craft and travel c, how long would 1 hour be?


Example 4:
If you were to board a craft and travel at 300 m/s, how long would 1 hour be?


Question:

As we watch, a spaceship passes us in time t. The crew of the
spaceship measures the passage time and finds it to be t'.
Which of the following statements is true?

A) t is the proper time for the passage and it is smaller t'
B) t is the proper time for the passage and it is greater than t'
C) t' is the proper time for the passage and it is smaller than t
D) t' is the proper time for the passage and it is greater than t
E) None of the above statements are true.
Ans. C


Question
Spaceship A, traveling past us at 0.7c, sends a message
capsule to spaceship B, which is in front of A and is
traveling in the same direction as A at 0.8c relative to us.
The capsule travels at 0.95c relative to us. A clock that
measures the proper time between the sending and
receiving of the capsule travels:

A) in the same direction as the spaceships at 0.7c relative to us
B) in the opposite direction from the spaceships at 0.7c relative to us
C) in the same direction as the spaceships at 0.8c relative to us
D) in the same direction as the spaceships at 0.95c relative to us
E) in the opposite direction from the spaceships at 0.95c relative to us

Ans. D


Problem
You wish to make a round trip from Earth in a spaceship, traveling at
constant speed in a straight line for 6 months and then returning at the
same constant speed. You wish further, on your return, to find Earth as it
will be 1000 years in the future.

(a)How fast must you travel?

(b) Does it matter whether you travel in a straight line on your journey?
If, for example, you traveled in a circle for 1 year, would it still find 1000
years had elapsed by Earth clock when you returned?


Question

A millionairess was told in 1992 that she had exactly 15
years to live. However, if she travels away from the Earth
at 0.8 c and then returns at the same speed, the last New
Year's day the doctors expect her to celebrate is:

A) 2001
B) 2003
C) 2007
D) 2010
E) 2017

Ans. E

The Time Dilation equation for Relativity is:

What is t, to v, and c?


The relativity of length and Length Contraction

The Relativity of Length

The length L0 of an object measured in the rest frame of the
object is its proper length or rest length. Measurements of the
length from any reference frame that is in relative motion
parallel to that length are always less that the proper length

2
v
L L0 1 2
c

Speed of Spaceship Observed Length Observed Height

At rest 200 ft 40 ft

10 % the speed of light 199 ft 40 ft

86.5 % the speed of light 100 ft 40 ft

99 % the speed of light 28 ft 40 ft

99.99 % the speed of light 3 ft 40 ft


Question
A measurement of the length of an object that is moving
relative to the laboratory consists of noting the coordinates
of the front and back:

A) at different times according to clocks at rest in the
laboratory
B) at the same time according to clocks that move with
the object
C) at the same time according to clocks at rest in the
laboratory
D) at the same time according to clocks at rest with
respect to the fixed stars
E) none of the above
Ans. C


Question

A certain automobile is 6 m long if at rest. If it is
measured to be 4/5 as long, its speed is:

A) 0.1c
B) 0.3c
C) 0.6c
D) 0.8c
E) > 0.95c
Ans. C


Problem

A cubical box is 0.50 m on a side.

(a) What are the dimensions of the box as measured by an
observer moving with a speed of 0.88c parallel to one of the
edges of the box?

(b) What is the volume of the box as measured by this
observer?


2
v
L L0 1 2
c

L= 0.5 (1-(0.88)**2)1/2
=0.24m
The observed dimension are= 0.24*0.5*0.5=0.059m**3


An Important general question:

If we know the co-ordinate x, y, z and time t of an
event, as measured in a frame S, How can we find
the coordinates x’, y ’ , z ’ and t’ of the same event as
measured in a second frame S?


Galilean Transformation


To convert velocity components measured in the
S frame to their equivalents in the S’ frame according
to the Galilean Transformation, we simply differentiate
x’, y’, and z’ with respect to time:
' Galilean transformation
' dx
v x vx v violate both of the
dt ' postulates of special
dy ' relativity.
v 'y vy 1). If we measure the speed
dt ' of light in the x-direction in
the S-system to be c,
' dz ' however, in the S’ system it
vz vz
dt ' will be
c’= c-v


Lorentz’s
Transformation

The primed frame moves with velocity v in the x
direction with respect to the fixed reference frame. The
reference frames coincide at t=t'=0. The point x' is
moving with the primed frame.

The reverse transformation is:

Much of the literature of relativity uses the
symbols β and γ as defined here to simplify
the writing of relativistic relationships


Relativistic Velocity Transformation

No two objects can have a relative velocity greater than c! But what if I observe
a spacecraft traveling at 0.8c and it fires a projectile which it observes to be
moving at 0.7c with respect to it!? Velocities must transform according to the
Lorentz transformation, and that leads to a very non-intuitive result called
Einstein velocity addition.


Just taking the differentials of these quantities leads to the velocity
transformation. Taking the differentials of the Lorentz
transformation expressions for x' and t' above gives

Putting this in the notation introduced in the illustration above:


The reverse transformation is obtained by just solving for u
in the above expression. Doing that gives

Applying this transformation to the spacecraft traveling at 0.8c which fires a
projectile which it observes to be moving at 0.7c with respect to it, we obtain a
velocity of 1.5c/1.56 = 0.96c rather than the 1.5c which seems to be the
common sense answer.


When ux
and v are both much smaller than c (the non
relativistic case) , the denominator of equation
approaches unity and so u’x = ux –v. This
corresponds to the Galilean velocity
transformation, In the other Extreme, when ux =c
;
the equation becomes U’x = c ,
From this result, we see that an object moving with
a speed c relative to an observer in S also has a
speed c relative to an observer in S’- independent
of the relative motion of the S and S’.


Question:
Imagine a motorcycle rider moving with a speed of
0.800c past a stationary observer, as shown in figure
below, If the rider tosses a ball in the forward direction
with a speed of 0.700c with respect to himself, what is
the speed of the ball as seen by the stationary observer?

0.700c

0.800c

In this situation, the velocity of the motorcycle with respect to
the stationary observer is v=0.800c. The velocity of the ball
in the frame of reference of the motorcyclist is ux’=0.700c.
Therefore, the velocity, ux, of the ball relative to the stationary
observer is

'
u x v
ux '
uxv
1
c2
0.700c 0.800c
0.9615c
1 0.700c 0.800c 1 2
c


The length of any object in a moving frame
will appear foreshortened in the direction
of motion, or contracted. The amount of
contraction can be calculated from the
Lorentz transformation. The length is
maximum in the frame in which the object
is at rest.


Questions/ Problems/ Doubts:

1). I know the derivation but I have problem in basic concept, I did not
understand the application of Galilean and Lorentz transformation?
2.) I know the concept, but I face problem when I try to solve the numerical
problem?
3.) No proper notes/material of subject?
4.) Problem in length contraction, when observer is moving or object is
moving?
5.) Why Galilean transformation do not obey laws of physics?
6.) Transformation equation?
7.) When we move away from a building it becomes smaller and smaller, is it
the case of length contraction or some other physical process?
8.) If object is in moving frame of reference, is there any change of
dimension?
9.) Why there is need of transformation, what we get from it?
10.) How a large building contract, when seen in the glass/mirror of moving
vehicle?

Questions/doubts:
11. WHY THERE IS CHANGE IN VELOCITY OF LIGHT
WHEN FRINGE SHIFT = 1 ?
12. When an observer and object of length L is moving
with velocity v1 and v2 respectively, what will be the
length contraction, v2> v1 ?
13. Not able to understand why the length seems less
while moving……..Length contraction?
14. How length contraction takes place in MUON’S
DECAY ?


Questions/ doubts……..417
 We consider earth an inertial frame of reference, and
there are hardly any motions comparable to speed of
light. Why do we refer to the relativity then ?

 does time, length and velocity all relativistic
quantities?


Questions/ doubts
 How does time change in S’ as per Lorentz
transformation & Physically why is this change of time?
 Explain the use of telescope and compensating plate
in the setup of experiment?
 How does the height of the object changes when
observer is along x-axis and object is moving along y-
axis ?


Questions/ doubts/ problems
 How to implement these formulas in typical
questions?
 Galilean & Lorentz transformation with numericals?
 Tried to understand the topic ‘ Length contraction’
using text book but was unable to understand, also
tried numerical problems on the topic ‘ time dilation’
and was unable to solve them too…......!!!!!
 Explain length contraction of a thin road, when
observer moves perpendicular to the rod ?


 How much Do I know?

What Do I need to learn?

What I have Learned ?


Relativistic mass
 Mass m-0 of an object measured when it was at rest and
mass m measured when it was in motion with velocity
v,
Relativistic Mass Increase

6

m0
m 5

4
2
1 v
Mass

3

2
c 2

1

0
0 0.2 0.4 0.6 0.8 1
Speed( c = 1)


Q. The total energy of a proton is three times its rest energy.
(a) Find the proton’s rest energy in electron volts.
2
mpc
27 8 2
(1.67 10 kg)(3.0 10 m / s )
10 19
(1.50 10 J )(1eV / 1.6 10 J)
938MeV


Q. With what speed is the proton moving?
E mc 2
mpc2
E 3m p c 2
u2
1
c2
1
3
1 u 2 c2
solv ing f or u giv es
u2 1
1
c2 9
or
8
u c 2.83 108 m / s
3

Relativistic Energy
 How Does the Total Energy of a Particle Depend on
Speed?
 We have a formula for the total energy E = K.E. + rest
energy,
2
2 m0c
E mc
2 2
1 v /c
so we can see how total energy varies with speed


How Does the Total Energy of a Particle Depend on Momentum?
It turns out to be useful to have a formula for E in terms of p.
Now
m0 c 4
2
E2 m2c 4
1 v2 / c2

m 2 c 4 (1 v 2 / c 2 ) m0 c 4
2

m2c 4 m2v 2c 2 m0 c 4
2

m2c 4 E2 m0 c 4
2
m2c 2v 2
2 4 2 2
hence using p = mv we find E mc
0 c p

If p is very small, this gives p2
E m0 c 2
2m0
the usual classical formula.
If p is very large, so c2p2 >> m02c4, the approximate formula is E = cp


The High Kinetic Energy Limit: Rest Mass Becomes
Unimportant!
Notice that this high energy limit is just the energy-momentum
relationship Maxwell found to be true for light, for all p. This could only
be true for all p if m02c4 = 0, that is, m0 = 0.
Light is in fact composed of “photons”—particles having zero “rest
mass”, as we shall discuss later. The “rest mass” of a photon is
meaningless, since they’re never at rest—the energy of a photon

m0c 2
E mc 2
1 v2 / c2
is of the form 0/0, since m0 = 0 and v = c, so “m” can still be nonzero.
That is to say, the mass of a photon is really all K.E. mass.
For very fast electrons, such as those produced in high energy
accelerators, the additional K.E. mass can be thousands of times the rest
mass. For these particles, we can neglect the rest mass and take E = cp.


6593.relativity

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