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PH604 Special Relativity (8 lectures). 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. - PowerPoint PPT Presentation
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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
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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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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
35
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.
