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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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