ph604 special relativity (8 lectures)

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PH604 Special Relativity (8 lectures)

Books: “Special Relativity, a first encounter”, Domenico Giulini, Oxford “Introduction to the Relativity Principle”, G.Barton, Wiley + many others in Section QC.6

• Newtonian Mechanics and the Aether• Einstein’s special relativity and Lorentz transformation

and its consequences• Causality and the interval• Relativistic Mechanics• Optics and apparent effects

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Newtonian Mechanics and the Aether

1. Newtonian Mechanics and Newton’s law of Inertia

2. The relativity principle of Galileo and Newtonian

3. Questions with regard to Newtonian Mechanics

4. The “Aether” – does it exist?

5. Michelson – Morley Experiment

Books: “Special Relativity, a first encounter”, Domenico Giulini, Oxford “Introduction to the Relativity Principle”, G.Barton, Wiley + many others in Section QC.6

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1.Newtonian Mechanics and Newton’s law of Inertia

--Newton’s Law: m a = F: Predict the motions of the planets, moons, comets, cannon balls, etc

--This law is actually not always correct! (surprised?)

http://www.phys.vt.edu/~takeuchi/relativity/notes/section02.html

--Inertial Frame: A frame in which the Newton’s law is correct.

--Any frame that is moving at a constant relative velocity to the first inertial frame is also an inertial frame.

--The frames in which Newton’s law does NOT hold that are accelerating with respect to inertial frames and are called non-inertial frames.

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zz

YY

XXOO

Z’Z’

YY’’

X’X’OO’’

SSS’S’

uuTwo inertial reference frames S and S’ moving with a constant velocity u relative to each other

S: as (x,y,z,t) and in

Common sense shows the two measurements are related by:

tt

zz

yy

utxx

'

;'

)1.1(;'

;'

zz

yy

xx

vv

vv

uvv

'

'

'

)2.1(

Or in vector form:

v ’ = v - u v ’ = v - u

a’ = aa’ = a

( u is // to x and x’)

A moving object is described in

S’: as S’: as (x’, y’, z’, t’)(x’, y’, z’, t’)

--This is the Galilean transformation. Note the universal time, t=t’

2. The relativity principle of Galileo and Newtonian

--They would assert that Mechanics only deals with relative motion and that ‘absolute’ motion can never be measured.

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3. Questions with regard to Newtonian Mechanics

i) phenomena on a very small scale we need Quantum Mechanics;

ii) Phenomena where the speed of motion is near the speed of light “c”

we need relativity

Modern experiment that shows the limitation of Newtonian mechanics:

We shall be concerned with case ii) in the 8 lectures in this course.

T Kinetic Energy of electrons (between 0.5-15 MeV)

A relation between V2 vs K.E of the electrons can be plotted.

Van de Graaf accelerator Experiment: [American Journal of Physics, Volume 32, Issue 7, pp. 551-555 (1964). ]

Accelerator Pulsed electrons beam

Measure the rise in temperature

Target B

D

The V. of electrons can be determined by: V = D / time

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Newtonian Mechanics: K.E. = ½ mv2

Newtonian Mechanics

v2

K.E.

Experiment

--N-M prediction is valid at low energy (velocities).

--Experiment: Vmax 3108(ms-1)C

C2

O

--The Vmax of the electrons appear to equal the speed of light in Vacuum.

--Other ‘massless particles’ such as neutrinos appear only to move at C as well

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4. Speed of Light: existence of Aether ?

Maxwell’s electromagnetic theory predicted that light should travel with a constant speed in vacuum, irrespective of reference frames:

00

1

C

How light propagates through a vacuum ?

--All other wave motions known, needed some form of ‘medium’

-- Wave velocity would be relative to the ‘medium’

Suggestion:Perharps, even a vacuum contains a very tenuous ‘medium’ --- the ‘Aether’, then the constant velocity of light is relative to this absolute frame, and thespeed of light in other ‘inertial’ systems would not be C.

if so, can we detect it?

Direct measurement of the relative motion to aether is difficult, but If it existed in space, we should be able to measure the motion of the Earth relative to aether -- Michelson-Morley (1887).

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5. Michelson-Morley Experiment –Detect the Earth moving through the Aether??

--In 1887 Michelson and Morley built an interferometer

To measure the movement of the Earth through the Aether.

beam splitter

Light source

Mirror 1

Mirror 2

Detector

Even though this instrument can be a few meters in size, it can detect changes in distance of hundreds of nanometers

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Interferometer, stationary in the Aether

Interferometer Moving Through the Aether

10

v

Aether wind speedl2

l1

The time for light to travel along l1 arm and back: (downstream)

)1(

2

2

21

111

c

vc

lvc

l

vc

lt

22

222 )( lvct

2

22

1

22

c

vc

ltt

The time for light to travel along l2 arm (cross stream)

V t

C t

travel along l2

arm and back:

If the light has frequency of f, the number of fringes that corresponds with differences, t1-t2 of the light travel in the two arms is:

2

2

2

2

21

21

1

2

1

2)(

cv

c

l

cv

c

lfttf

11

2

22

2

2

121

1

2

1

2)''(

cv

c

l

cv

c

lfttf

2

221

2

221

2

2

2

221'

2'121

)(

1

1

1

1)(2)()(

c

vll

c

v

c

llf

cv

cvc

llfttfttf

Since the test was to see if any fringes moved as the whole apparatus was turned through 90o.Then the roles of l1 and l2 would be exchanged, and the new number of fringes would be

So the observed number of fringe shift on rotation through 90o should be:

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Michelson & Morley made apparatus long enough to detect 1/3 of a fringe, with =500nm, so that l1 + l2 =17m, Nfringe = 108v2 /(3c2)

--But they could detect no shift at all (at any time of year!)

--The only possible conclusion from this series of very difficult experiments was that the whole concept of an all-pervading aether was wrong from the start.

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1. Einstein’s special relativity 2. Events and space-time in Relativity3. Proper time and the invariant interval4. Lorentz transformation5. Consequences of the Lorentz transformation6. Velocity transformation

Einstein’s special relativity and Lorentz transformation and its consequences

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1. Principle of Relativity by Einstein (1905)

2) The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light. http://en.wikipedia.org/wiki/Theory_of_relativity

It is based on the following two postulates:1) The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity),

- need a transformation of coordinates which preserves the laws of physics

A B

V

Observer in the car: the light pulse reaches A and B at the same time

A B

V

Observer to whom the car is moving with relative V: the light pulse reaches A before B

Simultaneity breaks down time cannot be regarded as a universal entity

- need a different transformation from Galileo’s but will converge to it for V<<C

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2. Events and space-time in Relativity

When and where is the object under our interest An Event in Relativity.

--An event is a point defined by (t, x, y, z), which describes the precise location of a “happening” which occurs at a precise point in space and at a precise time.

--“Space-time” is often depicted as a “Minkowski diagram”.

Space rSpace r

constantconstant

acceleratedaccelerated

decelerateddeceleratedTime (ct)Time (ct)

3 Proper time and the invariant interval

In 3-dimensional EUCLIDIAN space: P1

P2

In coordinate system O: ),,(),,,( 22221111 zyxPzyxP

In coordinate system O’: )',','(),',','( 22221111 zyxPzyxP

2222222 ''' zyxzyxr

In relativity, we would like to find a similar quantity for pairs of events, that is frame-independent, or the same for all observers, that is

invariant interval

2222222 )()()()()()()( zyxtcrtcs

∆t is the difference in time between the events

∆r is the difference between the places of occurrence of the events.

3.1 invariant interval

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3.2 Events, INTERVAL AND THE METRIC

A metric specifies the interval between two events

3.3 Proper time (length) the invariant interval

The proper time between two events is the time experienced by an observer in whose frame the events take place at the same point.

According to the definition of the interval between two events:

rc || --there always is such a frame since positive interval means:

0)()()() 222 rtcsIfii

0)()()() 222 rtcsIfi

2)( sc

so a frame moving at vector v = (∆r) /(∆t), in which the events take place at the same point, is moving at a speed < c

--It is still invariant even though there is no frame in which both events take place at the same point. (or (c∆t)2 < 0).

, the interval is said to be “timelike”

, the interval is said to be “spacelike”

--There is no such frame because necessarily it would have to move faster than the speed of light.

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19

0)()()() 222 rtcsifiii ,the interval is said to be “light-like” or null.

22 )()(.,. rtcei This is the case in which

Or, in which the two events lie on the worldline of a photon.

Because the speed of light is the same in all frames……. …. an interval equal to zero in one frame must equal zero in all frames.

The three cases have different causal properties, which will be discussed later.

Sometimes the proper distance is defined to be the distance separating two events in the frame in which they occur at the same time. It only makes sense if the interval is negative, and it is related to the interval by 2S

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We assume that relative transformation equation for x is the same as the Galileo Trans. except for a constant multiplier on the right side, i.e,

)2()''(

)1()('

utxx

utxx

where is a constant which can depend on u and c but not on the coordinates. (based on Postulate 1)

4. A transformation formula – Lorentz Transformation

To find the factor ?

By tracing the propagation of a light wave front in two different reference frames, one of which is moving with a velocity of V along x-axis w.r.t. the other.

4.1 The formula fits into the Einstein’s two postulates

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Assume a light pulse that starts at the origins of S and S’ at t =t’=0

zz

YY

XXOO

Z’Z’

YY’’

X’X’O’O’

After a time interval the front of the wave moves

Z’Z’

YY’’

X’X’O’O’

zz

YY

XXOO

uu(X, t) in S

(X’, t’) in S’

and

It is recorded as:

By Einstein’s postulates 2: x = ct

x’=ct’

SSS’S’

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Substituting ct for x and ct’ for x’ in eqs. (1) and (2)

)'3()('

)3(c

uc

t

t

)2()''(

)1()('

utxx

utxx

)3()()(' tucutctct

)4(')()''( tucutctct

)'4()(

')4(

uc

c

t

t

22

2

222

22

/1

1

1

1

cuc

uuc

c

Let (3’) = (4’)Let (3’) = (4’)u < c so u < c so is is always > 1always > 1

When u << cWhen u << c

~ 1 ~ 1

If u~c, If u~c,

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)2()''(

)1()('

utxx

utxx

The relativistic transformation for x and and x’ is The relativistic transformation for x and and x’ is

If u << c If u << c

~1 ~1 utxx

utxx

'

;'

Lorentz transf.Lorentz transf. Galileo transf.Galileo transf.

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The transformation between t and t’ can be derived:

)1()()('' tucutxctx

)(2c

uctt ][

2c

uxt )(' utct

ct

Divide c into Eq.(1)

For the wave front of light, x=ct, x’=ct’

Divide c into Eq(2) )2(')( tucctx

]'

'[2c

uxt )''( utct

ct

)

''(

2c

uctt

The complete relativistic transformation (L.T.) is

)5(]'

'[,','),''(2c

uxttzzyyutxx

)6(][',','),('2c

uxttzzyyutxx

4. 2 The interval of two events under Lorentz transformation.

For two events, (t1, x1,y1,z1) and (t2,x2,y2,z2), we define:

(T, X, Y, Z) = (t1-t2,x1-x2,y1-y2,z1-z2)

)('

)('

cTc

vXX

Xc

vcTcT

then Lorentz transformation becomes

]2

2['

222

22

222222'2

vXTTvX

vXTXc

vTcXTc

22' )()(.,. SSei The interval of two events is an invariant under Lorentz Transformation.For short: the interval is a Lorentz scalar.

2222'2'2

2

22222 .,.),1)((

XTcXTc

eic

vXTc

25

26

4.3 Lorentz transfermation in 4-dimensional formula

The L-T could be formally defined as a genernal linear transformation that leaves all intervals between any pair of events unaltered.

Introduce 4-D vector

1

1

1

1

z

y

x

ct

x

)xx()xx( 2121122 gS

1111 ,,,ct x zyx

Here we have introduced:

1000

0100

0010

0001

g

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L-T can be expressed as

1000

0100

00

00

c

vc

v

z

y

x

ct

'

'

'

'

z

y

x

ct

LXX'

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5. Consequences of the Lorentz transformation

5.1 Time dilation

ZZ

YY

XXOO

Z’Z’

YY’’

X’X’OO’’

The flash starts at time t’1, and goes off at time t’2,

A light source in S’ a flash of light lasting t’ seconds

t’ = t’2 – t’1

Also: x’ = x’2 – x’1 = 0

as measured by a clock in S’

How does an observer in S views the light on and off events?

]'

'[21

11c

uxtt

]'

'[22

22c

uxtt

Light on time:

Light off time:

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Light duration time measured by a clock in S:

2

2

2

2

2'1

'2

'1

'2

12

1

'

1

/)()(

c

u

t

c

u

cuxxttttt

x’ = 0x’ = 0

'tt

we have

2

2

1c

u

tt restmoving

--S’ is the rest frame w.r.t the strobeflash -- t’ is called trest(the proper time)

tmoving > trest, this effect is called time dilation on a moving clock, or moving clock runs slower, i.e., Observer in S see’s the clock in S’ runs slower because the clock in S’ is moving w.r.t to him. trest is also called the proper time (the shortest) of the two events.

--S is the moving frame w.r.t the strobe light-- t is called tmoving

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Experimental demonstration of time dilation effect:

Cosmic Ray evidence for ‘time dilation’

http://www.jlab.org/~cecire/cosmic.jpg

s62/1 102

kmm 10600)103(102 86

Mesons are formed at heights > 10 km in atmosphere. Observations found that most of them manage to survive down to sea level –despite their half-life being only

Even moving in C, half should have decayed in a distance of:

But: as they move so fast their clocks (proper time) run slower due to “ time dilation”.

If v=0.999c, =22.37

kmkm 104.13)103(102 86

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The Twin Paradox

http://www.phys.vt.edu/~takeuchi/relativity/notes/section15.html

Right after their 20th birthday, L blasts off in a rocket ship for a space trip, travelling at a speed 0.99c to a nearby star at 30 light year away, then come back with same speed, while M stay on Earth..

-- In the view of M: The journey will take time T = 2*30*c*year/0.99c = 60 year, so L will return when M is 20+60=80 yr. How much will L aged over the same period? L was travelling at a high speed and L’s clock, including her internal biological clock, were running slowly compared to M’s, therefore when L reunite with M, L will have aged by T’ = T/ = 60/7 < 9 yr. So L is younger than M.

Conclusion: In one frame of reference, L is younger while from the other frame of reference, M is younger. This is the paradox.

-- In the view of L: M was travelling away at a high speed and M’s clock, including her internal biological clock, were running slowly compared to L’s, therefore when M reunite with L, M is younger than L.

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222 )( xtcS M: x=0 the proper time for her is tDR

L: moves quickly, so (x’) 0, so her proper time out to event T and back again will be much smaller by factor of than tDR.

x

ct

M

L

T

D

R

Worldlines in the (M’s or Earth’s frame)

The invariant between two events D and R:

Let's draw this now in L's frame:

A problem: just what frame do we choose?

Frame S’ that is L's rest frame on her way out to the space?

ORFrame S’ that is L's rest frame on her way back?

L changes frames at event T

This breaks the symmetry and resolves the paradox: M travels from event D to event R in a single frame with no changes, while L changes frames. L's worldline is crooked (non-inertial) while M's is straight (inertial)!

Therefore: M’s point of view is right, L will be younger than M

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Causality and prohibition of motion faster than light.

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5.2 Length contraction

O

x

S

y

zx

O’

S’y’

z’

x’

u

- A rod lies in x’ axis in S’, at rest relative to S’ - A rod lies in x’ axis in S’, at rest relative to S’

-its two ends measured as x’-its two ends measured as x’11 and x’ and x’22

The length of the rod in S’ isThe length of the rod in S’ is

'1

'2' xxL

What is the length of the rod measured in S?

Because the rod is moving relative to S, we should measure the x-coordinates x1 and x2 of the ends of the rod at the same time, i.e., t=t2-t1=0, L = x2 –x1

Using Eq 6Using Eq 6 )(' utxx )(' 11 utxx

)(' 22 utxx 2

2

2

2

1212'

1

'

2

11

)()('

c

u

L

c

u

ttuxxxxL

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Call L’ as the Lrest, since the rod is at rest to S’

L the Lmoving since it moves with velocity u relative to S,

-The length or the distance is measured differently by two observers in relative motion

22 /1 cuLL restmoving which shows the effect of length contraction on a moving rod.

- One observer will measure a shorter length when the object is moving relative to him/her

-The longest length is measured when the rod is at rest relative to the observer---proper length

-Only lengths or distances parallel to the direction of the relative motion are affected by length contraction

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6. Velocity addition

A) Velocity Transformation

How velocities are transformed from one Ref. frame to another by differentiating L.T. equations?

--In order to avoid confusion, we now use for the speed of the reference frame S’ w.r.t. S in x direction.

Suppose a particle has a velocity in S’ '

''

dt

dxu x

in S:

dt

dxux

Differentiate Eq. (5)

]'

'[),''(2c

vdxdtdtvdtdxdx

37L5L5

]'

'[

)''(

2c

vdxdt

vdtdx

dt

dxux

'

'1

'

'

2 dt

dx

c

v

vdt

dx

2

'

1

'

c

vu

vu

x

x

]'

'[

'

2c

vdxdt

dy

dt

dyu y

]'

'[

'

2c

vdxdt

dz

dt

dzu z

)1(

'

'

'

'

2 dt

dx

c

vdt

dy

)1(

'

2

'

c

vu

u

x

y

)1(

'

2

'

c

vu

u

x

z

)1(

'

'

'

'

2 dt

dx

c

vdt

dz

38

)1(

'

)7(

)1(

'

1

'

2

'

2

'

2

'

c

vu

uu

c

vu

uu

c

vu

vuu

x

zz

x

yy

x

xx

)1(

'

)8(

)1(

'

1

'

2

2

2

cvu

uu

cvu

uu

cvu

vuu

x

zz

x

yy

x

xx

The inverse velocity transformation equation is

The velocity transformation equation from S’ to S is

From Eq(7) and (8) we have:

i). When v and ux, ux’ << C, , the L.T G.Tuuxx = u = uxx’ +v’ +v

ii). When ux‘=C, ux=C, and when ux=Cux’=C

L.T. includes the constancy of the speed of light, as well as G.T. for the low speed world.