1. Administrative Details: PHYS 344
Homework #1 Due in class next Wednesday, Sept 3rd Read Chapters 1 and 2 of Krane, Modern Physics Problems: Chapter 1: 2, 4, 6, 8, 10 Chapter 2: 3, 5, 7, 8, 10

2. Special Theory of Relativity 1
2.1 The Need for Ether
2.2 The Michelson-Morley Experiment
2.3 Einstein’s Postulates
2.4 The Lorentz Transformation
2.5 Time Dilation and Length Contraction
2.6 Addition of Velocities
2.7 Experimental Verification
2.8 Twin Paradox
2.9 Space-time
2.10 Doppler Effect
2.11 Relativistic Momentum
2.12 Relativistic Energy
2.13 Computations in Modern Physics
2.14 Electromagnetism and Relativity
Albert Michelson ( )
It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative and would, therefore, show that there is still a difficulty in the theory itself…
- Albert Michelson, 1907

3. Newtonian (Classical) Relativity
Newton’s laws of motion must be implemented with respect to (relative to) some reference frame. x z y
A reference frame is called an inertial frame if Newton’s laws are valid in that frame.
Such a frame is established when a body, not subjected to net external forces, moves in rectilinear motion at constant velocity.

4. Newtonian Principle of Relativity
If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system.
This is referred to as the Newtonian principle of relativity or Galilean invariance. x’ z’ y’ x z y

5. The Galilean Transformation
For a point P:
In one frame K: P = (x, y, z, t)
In another frame K’: P = (x’, y’, z’, t’) x’ z’ y’ K’ x z y K
P = (x, y, z, t)
P = (x’, y’, z’, t’)

6. Conditions of the Galilean Transformation
1. Parallel axes
2. K’ has a constant relative velocity (here in the x-direction) with respect to K.
3. Time (t) for all observers is a Fundamental invariant, i.e., it’s the same for all inertial observers.

7. The Inverse Relations Step 1. Replace -v with +v.
Step 2. Replace “primed” quantities with “unprimed” and “unprimed” with “primed.”

8. Swimming in a current (Galilean transformations)
Frame of reference K attached to the river bank
Frame of reference K-prime attached to the river (water)
The swimmer can swim at speed c in still water (relative to the water)
The river moves with velocity Vx relative to the ground (assume Vx is less than c)
Suppose the swimmer swims upstream a distance L and then returns downstream to the starting point.
Find the time necessary to make the round trip, and compare it with the time to swim across the river a distance L and return.

9. The swimmer always moves with velocity c relative to the water so
Calculate times
The swimmer always moves with velocity c relative to the water so
vx’ =-c for the upstream part
According to our transformations vx’ =vx-Vx
So that relative to the ground his velocity is: vx= vx’ + Vx = Vx-c
(the velocity relative to the ground is less than c on the upstream part and is negative since he is swimming in the negative x direction)
tup=L/(c-Vx)
On the downstream part, vx = Vx + c so the time downstream is:
tdown=L/(c+Vx) : the total time, up and back, is 𝑡= 2𝐿 𝐶 1 1− 𝑉 2 𝑐 2

10. Swimming across the river
In the frame of reference of the ground we need to have vx = 0, so v’x = -Vx
The speed relative to the water is always c, thus v’y = 𝑐 2 − 𝑉 2 and the round trip time is:
t=2tacross = 2𝐿 𝐶 − 𝑉 2 𝑐 2
Notice the difference in form between this result and the upstream-downstream result.

11. Maxwell’s Equations & Absolute Reference Systems
In Maxwell’s theory, the speed of light, in terms of the permeability and permittivity of free space, was given by: v
Remember e0 is constant from Coulomb’s law (electrostatics)and m0 is the constant from the Biot-Savart law (magnetostatics). Thus the velocity of light is a constant.
Aether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made.
The Michelson-Morley experiment was an attempt to show the existence of aether.

12. 2.1: The Need for Aether
The wave nature of light seemed to require a propagation medium. It was called the aether.
Aether had to have such a low density that the planets could move through it without loss of energy.
It had to have an elasticity to support the high velocity of light waves.
And somehow, it could not support longitudinal waves.
And (it goes without saying…) light waves in the aether obeyed the Galilean transformation for moving frames.

13. Constructive vs. destructive interference; Coherent vs
Constructive vs. destructive interference; Coherent vs. incoherent interference
Waves that combine in phase add up to relatively high intensity.
Constructive interference (coherent) =
Waves that combine 180° out of phase cancel out and yield zero intensity.
Destructive interference (coherent) =
Waves that combine with lots of different phases nearly cancel out and yield very low intensity. =
Incoherent addition

14. The Michelson Interferometer
Input
beam L2
Output
beam
Mirror
The Michelson Interferometer deliberately interferes two beams and so yields a sinusoidal output intensity vs. the difference in path lengths.
Beam-
splitter L1
Delay
Mirror
Fringes (in delay) I
“Bright fringe”
“Dark fringe”
DL = 2(L2 – L1) l

15. 2.2: Michelson-Morley experiment
Parallel and anti-parallel propagation
Michelson and Morley realized that the earth could not always be stationary with respect to the aether.
And light would have a different path length and phase shift depending on whether it propagated parallel and anti-parallel or perpendicular to the aether.
Beam-
splitter
Mirror
Mirror
Perpendicular propagation
Supposed velocity of earth through the aether

16. Michelson-Morley Experimental Prediction
The phase shift is w times this relative delay:
or:
The Earth’s orbital speed is: v = 3 × 104 m/s
and the interferometer size is: L = 1.2 m
So the time difference becomes: 8 × 10−17 s
which, for visible light, is a phase shift of: rad = periods
Although the time difference was a very small number, it was well within the experimental range of measurement for visible light in the Michelson interferometer.

17. Michelson-Morley Experiment: Results
The Michelson interferometer should’ve revealed a fringe shift as it was rotated with respect to the aether velocity. MM expected 0.4 periods of shift and could resolve periods. They saw none!
Their apparatus
Pictures from (G. Joos, Lehrbuch der Theoretischen Physik, Akademische Verlags., Leipzig, 1930)
Expt explanation from
Photograph of lab from
After the development of Maxwell's theory of electromagnetism, several experiments were performed to prove the existence of ether and its motion relative to the Earth. The most famous and successful was the one now known as the Michelson-Morley experiment, performed by Albert Michelson ( ) and Edward Morley ( ) in 1887.
Michelson and Morley built a Michelson interferometer, which essentially consists of a light source, a half-silvered glass plate, two mirrors, and a telescope. The mirrors are placed at right angles to each other and at equal distance from the glass plate, which is obliquely oriented at an angle of 45° relative to the two mirrors. In the original device, the mirrors were mounted on a rigid base that rotates freely on a basin filled with liquid mercury in order to reduce friction.
Prevailing theories held that ether formed an absolute reference frame with respect to which the rest of the universe was stationary. It would therefore follow that it should appear to be moving from the perspective of an observer on the sun-orbiting Earth. As a result, light would sometimes travel in the same direction of the ether, and others times in the opposite direction. Thus, the idea was to measure the speed of light in different directions in order to measure speed of the ether relative to Earth, thus establishing its existence.
Michelson and Morley were able to measure the speed of light by looking for interference fringes between the light which had passed through the two perpendicular arms of their apparatus. These would occur since the light would travel faster along an arm if oriented in the "same" direction as the ether was moving, and slower if oriented in the opposite direction. Since the two arms were perpendicular, the only way that light would travel at the same speed in both arms and therefore arrive simultaneous at the telescope would be if the instrument were motionless with respect to the ether. If not, the crests and troughs of the light waves in the two arms would arrive and interfere slightly out of synchronization, producing a diminution of intensity. (Of course, the same effect would be achieved if the arms of the interferometer were not of the same length, but these could be adjusted accurately by looking for the intensity peak as one arm was moved. Changing the orientation of the instrument should then show fringes.)
Although Michelson and Morley were expecting measuring different speeds of light in each direction, they found no discernible fringes indicating a different speed in any orientation or at any position of the Earth in its annual orbit around the Sun.
In 1895, Lorentz concluded that the "null" result obtained by Michelson and Morley was caused by a effect of contraction made by the ether on their apparatus and introduced the length contraction equation
where L is the contracted length, is the rest length, v is the velocity of the frame of reference, and c is the speed of light. Although the main interpretation of Lorentz for this equation was rejected later, the equation is still correct and was the first of a sequence of new equations developed by Poincaré, Lorentz, and others, resulting in a new branch of physics ultimately brought to fruition by Albert Einstein in special relativity. Einstein's idea of space-time contraction replaced Lorentz's interpretation of the contraction equation, and once and for all relegated ether to the history books.
Interference fringes showed no change as the interferometer was rotated.
Michelson and Morley's results from A. A. Michelson, Studies in Optics

18. Michelson’s Conclusion
In several repeats and refinements with assistance from Edward Morley, he always saw a null result.
He concluded that the hypothesis of the stationary aether must be incorrect.
Thus, aether seems not to exist!
Albert Michelson ( )
Edward Morley ( )

19. 2.3: Einstein’s Postulates
Albert Einstein was only two years old when Michelson and Morley reported their results.
At age 16 Einstein began thinking about Maxwell’s equations in moving inertial systems.
In 1905, at the age of 26, he published his startling proposal: the Principle of Relativity.
It nicely resolved the Michelson and Morley experiment (although this wasn’t his intention and he maintained that in 1905 he wasn’t aware of MM’s work…)
Albert Einstein ( )
It involved a fundamental new connection between space and time and that Newton’s laws are only an approximation.

20. Einstein’s Two Postulates
With the belief that Maxwell’s equations must be valid in all inertial frames, Einstein proposed the following postulates:
The principle of relativity: All the laws of physics (not just the laws of motion) are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.
The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

21. Re-evaluation of Time!
In Newtonian physics, we previously assumed that t’ = t.
With synchronized clocks, events in K and K’ can be considered simultaneous.
Einstein realized that each system must have its own observers with their own synchronized clocks and meter sticks.
Events considered simultaneous in K may not be in K’.
Also, time may pass more slowly in some systems than in others.