B.tech sem i engineering physics u iii chapter 1-the special theory of relativity

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.


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 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.


5

The speed of light is constant regardless of the

speed of the flashlight or observer.


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.


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


Apparatus at rest wrt the ether.


Light from a source is split by a half silvered mirror (M)

The two rays move in mutually perpendicular directions


The rays are reflected by two mirrors (M1 and M2)

back to M where they recombine.

The combined rays are observed at T.


The path distance for each ray is the same (l1=l2).

Therefore no interference will be observed


Apparatus at moving through the ether.

u

ut

8


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


(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


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


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


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

http://postimg.org/image/b3ca819jx

B.tech sem i engineering physics u iii chapter 1-the special theory of relativity

Education

inertial frames of reference

theory of special relativity

speed relative

relative motion

idea of frames

frames of referencewe

speed of light

constant relative velocity