Special Relativity Galilean Transformations x,y,z,t x z z.

Special Relativity

Galilean Transformations

x,y,z,t

, , ,x y z t

x

z

z

x

y

z

Galilean Transformations

x x ut

y y

z z

t t

x x

y y

z z

v v u

v v

v v

, ,x x y y y ya a a a a a

4

30 m/sec 22 m/sec

200 m

An Example

t=25 s

در آب ساکن شنا کند، درنهری شنامی cشناگری که می تواند باسرعت مثال: است . فرض کنید شناگر مسافت مارا در خالف uکند که درآن سرعت جریان آب

جهت جریان شنا کند و سپس به نقطه شروع برگردد. زمان الزم برای این سفر رفت و در عرض رودخانه و بازگشت به Lبرگشت را محاسبه و ان را با زمان شنای مسافت

مکان اول مقایسه کنید.

u

O’

O

u-c

u+cu

O

O’

...ادامه:زمان رفت وبرگشت درطول رودخانه

:زمان رفت وبرگشت درعرض رودخانه

22 2

2

2 2 1

1

L L Lc Lt

uc u c u c u cc

2 2 2 2 2 2 2

2 2 2

2

2 2 12

1

x y y x

cross

v v c v c v c u

L Lt t

cc u uc

8

Michelson-Morley Experiment

Physics theories of the late 19th century postulated that, just as water waves must have a medium to move across (water), and audible sound waves require a medium to move through (such as air or water), so also light waves require a medium, the "luminiferous aether". Because light can travel through a vacuum, it was assumed that the vacuum must contain the medium of light. Because the speed of light is so great, designing an experiment to detect the presence and properties of this aether took considerable ingenuity.

9

Earth travels a tremendous distance in its orbit around the sun, at a speed of around 30 km/s or over 108,000 km per hour. The sun itself is travelling about the galactic centre at even greater speeds, and there are other motions at higher levels of the structure of the universe. Since the Earth is in motion, it was expected that the flow of aether across the Earth should produce a detectable "aether wind". Although it would be possible, in theory, for the Earth's motion to match that of the aether at one moment in time, it was not possible for the Earth to remain at rest with respect to the aether at all times, because of the variation in both the direction and the speed of the motion.


11

At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analysing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the aether.


12

Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether flow. The device he designed, later known as an interferometer, sent a single source of white light through a half-silvered mirror that was used to split it into two beams travelling at right angles to one another. After leaving the splitter, the beams travelled out to the ends of long arms where they were reflected back into the middle on small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive interference based on the spent time to transit the arms.


13

If the Earth is traveling through an ether medium, a beam reflecting back and forth parallel to the flow of ether would take longer than a beam reflecting perpendicular to the ether because the time gained from traveling downwind is less than that lost traveling upwind. The result would be a delay in one of the light beams that could be detected when the beams were recombined through interference. Any slight change in the spent time would then be observed as a shift in the positions of the interference fringes. If the aether were stationary relative to the sun, then the Earth’s motion would produce a fringe shift 4% the size of a single fringe.


14


15

The Principle of Relativity: The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.

The Principle of Invariant Light Speed: Light in vacuum propagates with the speed c (a fixed constant) in terms of any system of inertial coordinates, regardless of the state of motion of the light source.

Special Relativity

Classical Relativity

1,000,000 ms-1 1,000,000 ms-1

■ How fast is Spaceship A approaching Spaceship B?

■ Both Spaceships see the other approaching at 2,000,000 ms-1.

■ This is Classical Relativity.

Einstein’s Special Relativity

1,000,000 ms-1

0 ms-1

300,000,000 ms-1

Both spacemen measure the speed of the approaching ray of light. How fast do they measure the speed of light to be?

Special Relativity

Stationary man300,000,000 ms-1

Man travelling at 1,000,000 ms-1

301,000,000 ms-1?Wrong!

The Speed of Light is the same for all observers

Time Travel!

Time between ‘ticks’ = distance / speed of light

Light in the moving clock covers more distance… …but the speed of light is constant… …so the clock ticks slower!

Moving clocks run more slowly!V

20

Time Dilation!

22

Length Contraction! Spaceship Moving at the 10 % the Speed of Light

Spaceship Moving at the 86.5 % the Speed of Light

Spaceship Moving at the 99 % the Speed of Light

Spaceship Moving at the 99.9 % the Speed of Light

Length Contraction!

23

A B Cu

o

24

P

DF

particle

light

Einstein’s Velocity Addition

25

Classical Doppler Effect

o

s

v v

v v

1. Velocities are calculated with respect to the environment.

2. We have different situations concerning whether source is stationary and observers moves or vice versa or both move.

26

Relativistic Doppler Effect

o o

N Waves

-uThere is no distinction between inertial reference frames. So we are just concerned with the relative speeds of the observer and the source.

O:

:

27

Lorenz Transformations

Three Specifications:1.They have to be linear as space and time are homogenous. 2.They have to be consistent with principles of relativity.3.In the low relative speeds (compared to the speed of light) they have to turn into Galilean transformations.

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2

2

2

1 /

1 /

x utx

u c

y y

z z

ut x

ctu c

2

2

2

1 /

1 /

x utx

u c

y y

z z

ut x

ctu c

29

Length Contraction!1 1 2 2

1 22 2

2 2

2 1 2 12 1 2 2

2 2

2 12 1 2

2

2

2

,

1 1

( )

1 1

1

1

x ut x utx x

u uc cx x u t t

x xu uc c

x xx x

uc

uL L

c

1 2t t

u

30


xx

x2

V uV

u1 V

c

2

uc

uc

2

y

y

x

V 1

V1 V

2

uc

uc

2

z

z

x

V 1

V1 V

xx

x2

V uV

u1 V

c

yy

x2

V uV

u1 V

c

zz

x2

V uV

u1 V

c

31

Special Relativity: Synchronizing Clocks

32


Let’s look carefully at the clock-synchronizing operation as seen from the ground. In fact, an observer on the ground would say the clocks are not synchronized by this operation! The basic reason is that he would see the flash of light from the middle of the train traveling at c relative to the ground in each direction, but he would also observe the back of the train coming at v to meet the flash, whereas the front is moving at v away from the bulb, so the light flash must go further to catch up.

33


34


Remember, this is the time difference between the starting of the train’s back clock and its front clock as measured by an observer on the ground with clocks on the ground. However, to this observer the clocks on the train appear to tick more slowly, by the factor , so that although the ground observer measures the time interval between the starting of the clock at the back of the train and the clock at the front as seconds, he also sees the slow running clock at the front actually reading seconds at the instant he sees the back clock to start.

Twin Paradox

The total time of the journey calculated by Gosper

12 light years20 years

0.6c

12 light years20 years

0.6c

L=12 Light years

Gosper Amelia V=0.6c2 2

2 2

0.60.6

1 11.25

(0.6 )1 1

v c

c c

v cc c

Going

Coming back

12years9.6 years

1.25

12years9.6 years

1.25

From Amelia’s frame of reference

L=12 Light years

V=0.6c2 2

2 2

0.60.6

1 11.25

(0.6 )1 1

v c

c c

v cc c

Twin Paradox

Going

Coming back

Gosper Amelia

12 light years 116 years

1.25 0.6

c

12 light years 116 years

1.25 0.6

c

L=12 Light years

V=0.6c2 2

2 2

0.60.6

1 11.25

(0.6 )1 1

v c

c c

v cc c

Twin Paradox

AmeliaGosper

Going

Coming back

The total time of the journey calculated by Amelia

39

Relativistic Dynamics

x

y

v

vx

yv

v

( ) 0

0

0

( ) 0

xi

yi

xf

yf

p mv mv

p

p

p mv mv

40


U=-v

2

2

2

1

vvc

x

y U=-v

x

y

v

v

2

22

vv

c

2

22

vv

c

2

21

vv

c

2

21

vv

c

2 2

2 2

2 2

2 2

2 2( ) (0)1 1

0

2

1 ( 1 ) 0

xi

yi

xf

yf

v mvp m m

v vc c

p

p mv mv mv

v vp mv m v

c c

xf xip p So concludes that momentum is not conserved.

41


If we wish to keep conservation of momentum consistent with the first relativity principle, we have to define momentum in a way that:

1.The definition is consistent with the first relativity principle, that is, if momentum is conserved in one inertial frame of reference, it has to be the same for all other observers in every inertial frame of reference. 2.In the low speeds it has to turn into the non-relativistic formulation, that is: P=mV

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U=-v

2

2

2

1

vvc

x

y U=-v

x

y

v

v

2

22

vv

c

2

22

vv

c

2

21

vv

c

2

21

vv

c

Kinetic energy is not conserved in frame of reference.

43


If we wish to keep conservation of kinetic energy consistent with the first relativity principle, we have to define momentum in a way that:

1.The definition is consistent with the first relativity principle, that is, if kinetic energy is conserved in one inertial frame of reference, it has to be the same for all other observers in every inertial frame of reference. 2.In the low speeds it has to turn into the non-relativistic formulation, that is:3.There should be no restriction increasing kinetic energy infinitely, however, increasing kinetic energy infinitely in classical mechanics requires increasing velocity infinitely which is prohibited in the second principle of special relativity.

Special Relativity Galilean Transformations x,y,z,t x z z.

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