Special Relativity

Mrio Pimenta

Udine, September 2009

Galileo Galilei 1564-1642

any observer doing experiments cannot determine whether he is at rest or moving at a steady speed

Salviatius :Shut yourself up with some friend in the main cabin below decks on some large ship,and have with you there some flies, butterflies, and other small flying animals. Havea large bowl of water with some fish in it; hang up a bottle that empties drop by dropinto a wide vessel beneath it. With the ship standing still, observe carefully how thelittle animals fly with equal speed to all sides of the cabin. The fish swimindifferently in all directions; the drops fall into the vessel beneath; and, in throwingsomething to your friend, you need throw it no more strongly in one direction thananother, the distances being equal; jumping with your feet together, you pass equalspaces in every direction. When you have observed all these things carefully (thoughdoubtless when the ship is standing still everything must happen in this way), havethe ship proceed with any speed you like, so long as the motion is uniform and notfluctuating this way and that. You will discover not the least change in all the effectsnamed, nor could you tell from any of them whether the ship was moving or standingstill.

Galileo GalileiDialogue Concerning the Two Chief World Systems,translated by Stillman Drake, University of California Press,1953, pp. 186 - 187 (Second Day).

There exists an absolute space, in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.

All inertial frames share a universal time.

The fundamental laws of physics are the same in all inertial frames

Isaac Newton 1642-1727

Not easy to get an inertial frame

aa

a = 0 !!!

no fictional forces

Frames in rotation

Wr

Centrifugalforce

Coriolis Force

Euler force

Coriolis force corrents

Low pressions (North)

Foucalt pendulum Pantheon, Paris

Really difficult

Solar system Milky way

Great attractor

A possible candidate ???CMB dipole

CMB rest frame

COBE, T/T ~ 10-3

COBE, T/T~10-5After dipole subtraction

Penzias and Wilson, 1965

The true inertial frame

The free-fall elevator !General relativity

Galileo Transformations

x = x V t

y = y

z = z

t = t

vx = vx V t

vy = vy

vz = vz

ax = ax

ay = az

az = az

Instantaneous forces

221

rmmGF =

no time scale!Crab SNR (1054)

any change in any object inthe Universe will be felt atonce in all points of theUniverse

The speed of light

Galileo

Fizeau ~ 1850

Roemer 1670

Michelson -Morley

1887 try to detect thepredicted fringe shift due to theEarth movement (30 km/s) inthe aether. Resolution 8 km/s.

Michelson

The laws of physics are the same for all observers in uniform motion relative to one another

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.

Albert Einstein 1879-1955

Simultaneity

Two events simultaneous for one observer may not be simultaneous for another observer

S

'SA B

S (bus)

Time dilatation

t = t / !!!22 /1( cv

S (Lab)

v t/2

Hard to test it

Muons at the Earth surface

Muons are produced on theupper layers of the atmospherebut they do arrive at the Earthsurface !!!

Length contraction

t = 2 l / c

t = (l + v t1) / c + (l - v t2) / c t = t1 + t2

l = l !!!22 /1( cv

Fotografia tirada em 1912 durante o Grande Prmio de Frana por Jacques-Henri Lartigues

End first lecture

Space-time

Hendrick Lorentz

Try to save the aether

Maxwell equations are not invariantunder the Galileo transformations !

Michelson and Morley experimentfailed to detector the aether wind !

Francis FitzGerald Joseph Larmor

FitzGerald introduced the hypothesis of the length contraction.

Larmor published the complete Lorentz transformation two years before Lorentz, and predicted time dilatation.

Lorentz published the Lorentz transformations in 1895, 1899 and 1904

222

22

/1/)/(

/1/)(

cvcxvtt

zzyy

cvtvxx

=

==

=

The Lorentz transformations

)(

)(

xzzyy

xx

===

= = v / c

= c t

= 1 / 21

The factor

A clock in S Praga astronomical clock

00

==

A

Ax

A

{{

B

Tcx

B

B

==

0

A

{{

0

0

=

=

A

Ax

Tc

Tcx

B

B

=

=

B

time dilatation !

)(

)(

x

xx+=

+=

A ruler in S

)(

)(

x

xx+=

+=

00

==

B

Ax

A

{{

B

0==

B

B Lx

A

{{

0

0

=

=

A

Ax

L

Lx

B

B

=

=

B

BAB TVLxx += )(

BL +=

)( LLL =

/LL =

Length contraction !

Space-time diagrams

B

ABA

L

L

S S

B

A

B

A

cTcT

Space-time interval

s 2= 2 - (2x+ 2y+ 2z) is a Lorentz invariant!

s2 > 0 time-like interval

s2 < 0 space-like interval

s2 = 0 light interval

present

past

future

light cone

if s2 > 0 : (s2) = proper time (0)

If s2 < o : (-s2)= rest length (L0)

Velocity transformations

)(

)(

xzzyy

xx

+=

=

=

+=

ddxcdtdxvx // ==

== dxdctdxdvx //cV /=

2/1)()(

cvVVv

xdddxd

cvx

xx +

+=

+

+=

)/1()( 2cvVv

xddzdcv

x

zz +

=

+

=

)/1()( 2cvVv

xddydcv

x

yy +

=

+

= !!!cvcv xx ==

!!!0 VvvV +=

Four vectors,

1000010000-00-

xyz

xyz

=

XX =

XXgs =

2

-10000-10000-100001=g

The twin paradox

One of the twin leaves on a spacejourney during which he travelsclose to the speed of light, while hissister remains on Earth. On his returnthe space traveller will find that hissister has aged more than himself!

The paradox arises because it can beargued that the sister is moving nearthe speed of light relative to herbrother and so the brother should begetting older instead.

The Lorentz transformations

)(

)(

xzzyy

xx

+=

=

=

+=

= v / c

= c t

= 1 / 21

Simultaneity planes

the simultaneity planes in two frames are not the same!

When the traveller twininverts his movementchanges his simultaneityplane!

The travel of the brother

End second lecture

E=mc2

It followed from the special theory of relativity that mass andenergy are both but different manifestations of the same thing -- asomewhat unfamiliar conception for the average mind. Furthermore,the equation E is equal to m c-squared, in which energy is put equalto mass, multiplied by the square of the velocity of light, showedthat very small amounts of mass may be converted into a very largeamount of energy and vice versa. The mass and energy were in factequivalent, according to the formula mentioned before. This wasdemonstrated by Cockcroft and Walton in 1932, experimentally.

momentum

pr must be conserved in all the references frames !!!B rest frame

CM frame

Relativistic mechanics

Classical mechanics

v of a body under a constant force

22 /1 cvvm

vmp

=

=r

vv

m - particle rest mass

EnergyKinetic energy

Low

2

2

22

2

21

)1...2

1(

)1(

vm

cvcm

cmEk

++

=

cEk

Rest energy

Total energy2cmE =

2cmE =

Rest mass is just a form of energy!

Fission and fusion

Binding energy

Fission the splitting of an heavy atom

Fusion the fusing of light atoms into heavy atom

mass Kinetic Energy

Particle Physics

Collides high energyparticles and observeswhat comes out !

An example: the LEP @CERN

Kinetic Energy mass

DELPHI- the first WW event

Energy-momentum

1000010000-00-=cE /

xpyp

zp

cE /xp

ypzp

cE /pr

is a four vector !p =

Lorentz transformation

Useful formulae

p2 = (E/c)2-| |2 = m c2pr

prE2 = | |2 c2 +m2 c4

)//( cEpr=)/( 2cmE=

two body 1 2

3

4

ECM2 =(p1+p2)2

(p1+p2) = (p3+p4)

if m

u (p1 p4)2 s + t + u = m12 + m22 + m32 + m42

Mandelstam Variables

Particle Physics examples(I)

0 mass

Particle Physics examples(II)

Fix target vs collider

Particle Physics examples(III)0 decay

m = 0

The light is emitted by the lamp and is absorbed in the wall.

Does the wagon moves?

a) Yes - the light carries Energy and momentum !

b) NO there is no external Force !

The photon -

m = 0

E = p c

E = h = h c/

Compton effect

Energy- momentum conservation

hc/ i + mec2 = hc/f + ((mec2)2+(pec)2)

h/i = h/f cos() + pecos()

0 = h/f sin() + pe sin()

f i = h/(mec) (1 cos())

Doppler effectobserver at rest, source in movement

signal

source

x1 x0 x

t

T0

TS

observer rest frame

Doppler effect

observer at rest, source in movement

signal

source

x1 x0 x

t

T0

TS

observer rest frame

SSSS

S

S

TvVv

vTVT

vxxTT

==

= 100

v signal velocity

VS source velocity

Classical Mechanics

SS TT = fVv

vfS

=

Relativistic Mechanics

SS TT = fVvvf

S

=)(

fVcVcf

S

S +

=cvif =

signal

source

x1 x0 x

t

T0

TS

observer rest frame

Doppler effect

source at rest, observer in movement

source at rest, observer in movement

signal

source

x1 x0 x

t

T0

TS

observer rest frame

Classical Mechanics

vVv

TT

R

SS

+=

=

0

fv

Vvf += 0

Doppler effect

SR

R

R

SS

RS

Tv

VvvTVT

vxxTT

00

100

==

=

vR signal velocity (obs)

V0 observer velocityv signal velocity (medi)

Relativistic Mechanics

20

0

1c

vVvVv

TT

R

SS

+

+=

=

fVcVcf

S

S +

=

fvVvf +=

0

cvif =

The GPS System

Cosmological redshift

The galaxies are notmoving, is the spaceitself that is inexpansion!

End third lecture

The speed of light has been an elusive thing through out history. In 1670, OleRoemer stumbled upon a measurement that was accurate to 10% in the negativedirection of the now accepted value. Roemer was an astronomer studying Jupiter,specifically its moon Io. He noticed that as the year went by, the time at whichJupiter eclipsed the moon seemed to grow as the two planets moved apart. Hereasoned that this time delay was due to the extra time it took light to travel extradistance. This experiment was reproduced using a smaller change in time, 30 days,and greater precision in the measurements of distance between the planets, obtainedfrom pre-existing sources. This will hopefully yield a more accurate value for c. Anadditional value for c was obtained through a Fitzeau wheel. Fitzeau, in the mid1800s, used a rotating toothed wheel and a light, concentrated through lenses,measured the speed of light to about 10% of the accepted value in the positivedirection. In a Fitzeau wheel, the light is directed through the teeth, 360 in this case,of a quickly rotating wheel, 10,000 RPM, reflected off a plain mirror back to thewheel, at which point it passes through a different tooth. The time it takes for thewheel to rotate from one tooth to the next is equal to the round trip time of thelight. The round trip distance and the time were then used to calculate an additionalvalue for c. The experiment is ongoing; results are pending. The predicted result ofthe Jupiter calculation is close to 2.9*10^8, and the predicted result of the Fitzeauwheel calculation 3.1*10^8.

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