1. 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
Books: “Special Relativity, a first encounter”, Domenico Giulini, Oxford “Introduction to the Relativity Principle”, G.Barton, Wiley many others in Section QC.6

2. Newtonian Mechanics and the Aether
Newtonian Mechanics and Newton’s law of Inertia
The relativity principle of Galileo and Newtonian
Questions with regard to Newtonian Mechanics
The “Aether” – does it exist?
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

3. 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?)
Newton discovered his laws in the 17th century. In 18th and 19th century, using Newton's laws, physicists in the 18th and 19th century were able to predict the motions of the planets, moons, comets, cannon balls, etc.
This law is actually not always correct! (surprised?) It depends on which frame you are using to describe the motion of the object.
One example is the motion of objects inside the space shuttle in orbit. They do not move relative to the space shuttle even though gravity is acting on them
Newton evaded this difficult by supposing all measurements to be made in ‘inertial’ co-ordinate systems, i.e., those that are in uniform motion relative to the ‘fixed stars’.
So when we talk about the law of inertia, we are assuming that a frame exists in which the law is correct. Such a frame is called an inertial frame
If one such inertial frame exists, then an infinite number of other inertial frames exist since any frame that is moving at a constant relative velocity to the first inertial frame is also an inertial frame.
--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.

4. 2. The relativity principle of Galileo and Newtonianz Y X O Z’ Y’ X’ O’ S’
Two inertial reference frames S and S’ moving with a constant velocity u relative to each other S u
( u is // to x and x’)
A moving object is described in
S: as (x,y,z,t) and in
S’: as (x’, y’, z’, t’)
N_M had a specific and clear ideas on space and time. Shown as in the figure,….
All measurements on space coordinate and time in two reference frames moving relatively to reach other are related by a simple “common sense” relationship between the coordinates,
………. Newtonian mechanics has been applied with singular success since the days of Newton. For most ‘daily’ applications it’s all that was needed.
Common sense shows the two measurements are related by:
Or in vector form:
v ’ = v - u
a’ = a
--This is the Galilean transformation. Note the universal time, t=t’
--They would assert that Mechanics only deals with relative motion and that ‘absolute’ motion can never be measured.

5. 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
We shall be concerned with case ii) in the 8 lectures in this course.
Modern experiment that shows the limitation of Newtonian mechanics:
Van de Graaf accelerator Experiment:
[American Journal of Physics, Volume 32, Issue 7, pp (1964). ]
About 100 years ago, it was realised that N-M could not accurately account for these
Phenomena: For very small scales and high velocities, an uniform “theory” (UNIFIED FIELD THEORY) has yet to be formulated!
We will start with a modern experi….. carried out in MIT.
This is the diagram of the experiment, Electrons are accelerated in Van de Graaf electrostatic generator and a linear accelerator shown as the box, they are beamed toward the target at point B with velocities V, which can be decided by the time for them to pass a distance D . Their K.E can be determined by measurement the rise in the target’s temperature as the consequence of the collision between the electrons and the target, therefore the K.E.s. ……..
Accelerator
Pulsed electrons beam
Measure the rise in temperature
Target B D
T  Kinetic Energy of electrons (between MeV)
The V. of electrons can be determined by: V = D / time
A relation between V2 vs K.E of the electrons can be plotted.

6. v2 Newtonian Mechanics: K.E. = ½ mv2 C2 O K.E.
Experiment
The velocities of the electrons appear to equal the speed of light in Vacum.
Irrespective of particle, there seems to be a unique limiting velocity beyond which nothing can go faster O
K.E.
--N-M prediction is valid at low energy (velocities).
--Experiment: Vmax  3108(ms-1)C
--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

7. 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:
How light propagates through a vacuum ?
--All other wave motions known, needed some form of ‘medium’
-- Wave velocity would be relative to the ‘medium’
In the late 19th centaury, Maxwell’s electromagnetic theory predicted that light should travel with a constant speed, where  and  are the permittivity and permeability constant in vacuum. This rises the question of how light propagates through a vacuum. All other wave motions known, needed some form of ‘medium’ to propagate in and the wave velocity would be relative to the ‘medium’. E.g., sound propagating in air. Perharps even a vacuum contains a very tenuous ‘medium’ --- the ‘ether’, then the constant velocity of light is relative to this absolute frame, and the speed of light in other ‘inertial’ systems would not be C.
if so, can we detect it?
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 the
speed 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).

8. 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
In 1887 Michelson and Morley built an interferometer which to use interferometer to measure the movement of the Earth through the Eather.
In this interferometer, a beam of light is shot at a beam splitter. The beamsplitter splits the light into two parts, which each bounce off of their own mirror. The light is then recombined by the beam splitter and goes to a detector. Light is a wave, so when the path lengths are equal, the waves add to produce a bright spot at the detector. When the path length changes by only half a wavelength (which for visible light comes out to be about 250nm) the waves cancel each other to produce a dark spot. Even though this instrument can be a few meters big, it can detect changes in distance of hundreds of nanometers

9. Interferometer, stationary in the Aether
Interferometer Moving Through the Aether
Interferometer, stationary in the Aether
Here is what they thought the interferometer would look like if it was standing still in the eather. Think of the sphere as a photon or a wavefront traveling at the speed of light when you first turn on the light source. Each arrow is an equal distance from the previous one and since the speed of light is constant, one arrow turns on for every clock tick.
If the interferometer is moving through the aether, the light that is going in the direction of motion takes longer to get to the detector. They thought that by rotating the whole interferometer, one arm would go from being in the direction of motion to being perpendicular to it and the light traveling down this arm would at first have a longer distance to travel and then a shorter distance. They should have been able to detect this change by seeing bright and dark fringes pass by their detector

10. The time for light to travel along l2 arm (cross stream) v
Aether wind speed l2 l1
The time for light to travel along l1 arm and back: (downstream)
V t
C t
travel along l2 arm and back:
Let’s look at in detail how it works when interferometer is moving in aether (aether is moving relative to the earth) see the diagram: two arm length l1 and l2, assuming the aether is drift with a speed of v toward left relative to the earth (apparatus)
So we have:………..
To magnify the time difference between the two paths, in the actual experiment the light was reflected backwards and forwards several times, like a several lap race.  Michelson calculated that an aether windspeed of only one or two miles a second would have observable effects in this experiment, so if the aether windspeed was comparable to the earth’s speed in orbit around the sun, it would be easy to see.  In fact, nothing was observed.
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:

11. 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:

12. --But they could detect no shift at all (at any time of year!)
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.
After the negative result of the M-M experiment, a serious re-examination of the concepts of space and time were requested. Various ‘explanations’ were proposed. Einstein didn’t like the idea of an ether and a specific universal reference frame. He felt that all frames should be regarded equally. He proposed a theory of relativity (1905) based on two postulates:

13. Einstein’s special relativity and Lorentz transformation and its consequences
Events and space-time in Relativity
3. Proper time and the invariant interval
4. Lorentz transformation
5. Consequences of the Lorentz transformation
6. Velocity transformation

14. 1. Principle of Relativity by Einstein (1905)
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
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. A B V
1) Means that the formula for a basic law of nature (such as the conservaton of Momentum ) should look the same in all references frames moving with constant speed relative to each other, Hence….
2) We see that, if the velocity of light is always measured to have the same value, we will have trouble to understand Galileo’s idea on a universal time!
In the figure: A car is moving with a velocity V.
Light source midway in the car detectors at A and B, releases light pulse
To an observer in this car, the light pulse reaches A and B at the same time irrespective of the speed of the car. To an observer to whom the car is moving with velocity V relative to: ………
Hence Two events which are simultaneous in one frame are not simultaneous in another.
Simultaneit…..entity.
This is not compatible with the Galileo’s transformation. We need a different …… A B V
Observer to whom the car is moving with relative V: the light pulse reaches A before B
Observer in the car: the light pulse reaches A and B at the same time
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

15. 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 r
constant
accelerated
decelerated
Time (ct)
Before we can figure out what kind of transformation will fit with Einstein’s postulation we have to specify how to describe the state of motion of an object in relativity, i.e., when and where the object is ?
The concept of an Event is of key importance in relativity.

16. 3 Proper time and the invariant interval
In 3-dimensional EUCLIDIAN space:
In coordinate system O:
In coordinate system O’:
In 3-dimensional space, two different observers can set up different coordinate systems, so they will not in general
assign the same coordinates to a pair of points P1 and P2. However they will agree on the distance between them.
some kind of `length' in 3+1-dimensional spacetime that is frame-independent, or the same for all observers.
There is such a quantity, and it is called: ……. Why? Because speed of light is invariant but time passed is not.
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
∆t is the difference in time between the events
∆r is the difference between the places of occurrence of the events.

17. 3.2 Events, INTERVAL AND THE METRIC
Before we can figure out what kind of transformation will fit with Einstein’s postulation we have to specify how to describe the state of motion of an object in relativity, i.e., when and where the object is ?
The concept of an Event is of key importance in relativity.
A metric specifies the interval between two events

18. 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:
, the interval is said to be “timelike”
--there always is such a frame since positive interval means:
If the interval between two events is less than zero…..
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
, the interval is said to be “spacelike”
--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).
--There is no such frame because necessarily it would have to move faster than the speed of light.

19. ,the interval is said to be “light-like” or null.
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
,the interval is said to be “light-like” or null.
Of course the interval (s)2 can also be exactly equal to zero. This is the case
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.

20. 4. A transformation formula – Lorentz Transformation
4.1 The formula fits into the Einstein’s two postulates
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,
Einstein said that all of the consequences of special relativity can be found from examination of the Lorentz transformations.
where  is a constant which can depend on u and c but not on the coordinates. (based on Postulate 1)
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.

21. Assume a light pulse that starts at the origins of S and S’ at t =t’=0
After a time interval the front of the wave moves
It is recorded as: Z’ Y’ X’ O’ Z’ Y’ X’ O’ z Y X O S’ z Y X O
(X, t) in S u S
and
(X’, t’) in S’
x = ct
By Einstein’s postulates 2:
x’=ct’

22. Substituting ct for x and ct’ for x’ in eqs. (1) and (2)
u < c so  is always > 1
Let (3’) = (4’)
If u~c,  
When u << c  ~ 1

23. The relativistic transformation for x and and x’ is
If u << c  ~1
Lorentz transf.
Galileo transf.

24. The transformation between t and t’ can be derived:
For the wave front of light, x=ct, x’=ct’
Divide c into Eq.(1)
Divide c into Eq(2)
With equation (5) and (6), we can investigate how events recorded in one reference frame are recorded in another reference frame moving with velocity v relative to the first. We next discuss some characteristics of Lorentz transformation.
These transformations, and hence special relativity, lead to different physical predictions than Newtonian mechanics when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter-intuitive
Essential concept: Spacetime.
Since space affects time, cannot think of 3-D space and time separately, but only as a single 4-dimensional manifold.
The complete relativistic transformation (L.T.) is

25. 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)
then Lorentz transformation becomes
First, how about the interval of two events under Lorentz transformation
The interval of two events is an invariant under Lorentz Transformation.
For short: the interval is a Lorentz scalar.

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
Here we have introduced:

27. L-T can be expressed as

28. O’ O X’ X Z’ Z 5. Consequences of the Lorentz transformation
5.1 Time dilation
A light source in S’ a flash of light lasting t’ seconds Z Y X O Z’ Y’ X’ O’
The flash starts at time t’1, and goes off at time t’2,
as measured by a clock in S’
t’ = t’2 – t’1
A strobe light located at rest in S’, produces a flash of light lasting t’ seconds. We take the flash on and off as
two events.
Also: x’ = x’2 – x’1 = 0
How does an observer in S views the light on and off events?
Light on time:
Light off time:

29. x’ = 0 Light duration time measured by a clock in S:
--S is the moving frame w.r.t the strobe light
-- t is called tmoving
--S’ is the rest frame w.r.t the
strobeflash
-- t’ is called trest(the proper time)
The clock in S’ is at rest relative to the strobe light, so the frame S’ is called the clock’s rest frame and t’ is called trest, or the proper time.
Similarly the clock in the frame S is moving relative to the strobe light, t is called tmoving
we have
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.

30. Experimental demonstration of time dilation effect:
Cosmic Ray evidence for ‘time dilation’
 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
Muons are sub-atomic particles generated when cosmic rays strike the
upper levels of our atmosphere. They have a half lifetime of about 2
microseconds (µs) meaning that every 2 µs, their population will
reduce by half.
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

31. The Twin Paradox
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.
There are two twins M and L who were born at exactly the same moment
(a biological impossibility).
There are numerous ways of trying to resolve
the paradox. Here we shall briefly mention one of the most common and
simple one.
We know that we can only do physics in inertial frames of reference.
However, L is not in such a frame, the principal of inertia is violated
for L when she blasts off from Earth, again when she fires her thrusters
to turn around, and once again, when she lands on Earth.
If you were riding with L, you would “feel” these non-inertial force when
you are pushed into your seat or lifted from it when M’s rocket is firing.
-- 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.
Conclusion: In one frame of reference, L is younger while from the other frame of reference, M is younger. This is the paradox.

32. Worldlines in the (M’s or Earth’s frame)
The invariant between two events D and R: x ct M L T D R
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.
Let's draw this now in L's frame:
We can explain this paradox in worldlines of L and M. The worldlines of L
and M are plotted in the rest frame of the Earth (frame S), with L's
departure marked as event D, L's turnaround at the distant
star as T and her return home as R.
……… We cannot choose both because they are different frames: L changes
frames at event T.
The time dilation effect is reciprocal: as observed from the point of view of any two clocks which are in motion with respect to each other, it will be the other party's clock that is time dilated.
A problem: just what frame do we choose?
Frame S’ that is L's rest frame on her way out to the space?
Frame S’ that is L's rest frame on her way back? OR
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

33. Causality and prohibition of motion faster than light.

34. What is the length of the rod measured in S?
5.2 Length contraction
- A rod lies in x’ axis in S’, at rest relative to S’ O x S y zx O’ S’ y’ z’ x’ u
-its two ends measured as x’1 and x’2
The length of the rod in S’ is
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 6

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,
which shows the effect of length contraction on a moving rod.
-The length or the distance is measured differently by two observers in relative motion
- One observer will measure a shorter length when the object is moving relative to him/her
Note: the ladder paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage
-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

36. 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’
in S:
Differentiate Eq. (5)

37. L5

38. The velocity transformation equation from S’ to S is
The inverse velocity transformation equation is
From Eq(7) and (8) we have:
i). When v and ux, ux’ << C,
ux = ux’ +v
, the L.T G.T
ii). When ux‘=C,
ux=C,
and when ux=Cux’=C
L.T. includes the constancy of the speed of light, as well as G.T. for the low speed world.