1. Further Logistical Consequences of Einstein’s Postulates
By: Everett Chu and Stephen D’Auria

2. Review Lorentz Transformations Time Dilation Length Contraction
All of these equations are important in the following concepts and are very useful

3. 1. Doppler Effect
Original non-relativistic effect discovered in 1803 by Christian Doppler
Principle modified by Einstein to include light
Christian Doppler
Albert Einstein
Original principle used to explain anomalies in sound
Ex: ambulance driving towards you and such

4. Doppler Effect Cont.
In both the nonrelativistic and relativistic cases, the Doppler effect predicts shifts in the frequency of a wave based on the speed of the observer and wave source
In both cases, there are predictions for both the “redshift” and “blueshift” of the waves, which depends on the directions of the motion of the wave source and the observer

5. Doppler Effect Cont.
Equations for the classical and relativistic Doppler effect
Relativistic Doppler effect
Source/observer approaching
Nonrelativistic Doppler effect:
Source/observer receding
Talk about difference between relativistic and classical shifts, which is due to addition of time dilation factor to the nonrelativistic equations
Nonrelativistic
The v refers to the speed of the wave
The Vo is the speed of the observer
The Vs is the speed of the source
The signs are determined by the direction of motion for the different objects
As the source approaches, the frequency must increase
Relativistic
Utilizes beta, which is v/c
Takes into account time dilation and other relativistic ideas to give the correct picture for the doppler shift

6. 2. Mass-Energy Equivalence
This principle was discovered by Einstein using the following argument and his original two postulates
Einstein derived the Lorentz transformations, which describe how position and velocity can be related between two inertial reference frames
He then utilized Maxwell’s equations and the Lorentz transformations to show how electric and magnetic fields as well as energy can be transformed from one frame to another
He also showed that these same transformations can be derived from Planck’s energy equation (E=hf) and the Doppler shift
Einsteins two initial postulates
1. the laws of physics are the same in all inertial reverence frames
2. the speed of light in a vacuum is constant no matter the speed of the source
He derived the lorentz transformations from his basic postulates, and then found the equations for time dilation and length contraction
Write the transformations for the electric and magnetic fields, as well as for energy

7. Mass-Energy Equivalence Cont.
Einstein then utilized the Lorentz force equation for two reference frames and the transformations for E and B between these frames to develop a relativistic expression for momentum
He used the noted previous equations as well as the first equations listed to derive the relativistic momentum
The mass used in the momentum expression is known as the rest mass, and is the value for the mass of an object in its rest frame
Relativistic Momentum

8. Mass-Energy Equivalence Cont.
Using the work energy theorem, Einstein was then able to find a relativistic expression for the kinetic energy of an object
Work-energy theorem is that work equals change in kinetic energy
This proof also used the equation that the work equals the integral of the force dotted with the net displacement, which can be written as the velocity times the chance in time
This, coupled with the previous equations for the force and relativistic momentum, allow for the above derivation for the relativistic kinetic energy

9. Mass-Energy Equivalence Cont.
Using the previous information, Einstein was then able to show that energy and mass are in fact equivalent
The conclusion of this proof yielded E=mc2
In his proof, Einstein used an object which emitted two light pulses as it traveled
He compared these pulse energies in two reference frames using the following proof
Using the previous described equations, he was then able to derive the famous e=mc2 equation

10. Invariant Mass
Similar to the previous idea, except it is a way to equate energy (and therefore mass) between reference frames
The magnitude of this quantity is given as the rest energy of the object, which is defined by E=mc2
The equation can be derived from the relativistic equations for the energy and the momentum
The reason for the p times the speed c is to aid calculations

11. Invariant Mass Cont.
From this statement, it can be shown that the rest energy, and therefore the mass, will be the same in all inertial reference frames
From this equation, there is the possibility for the mass=0
In this case, it is still possible to calculate the energy of massless particles (like photons, gluons, etc.): E=pc
This says that all massless particles move at the speed of light

12. Review
Lorentz Transformations
Time Dilation
Length Contraction

13. 3. Spacetime Diagrams Simple diagrams which illustrate movement
The x axis represents motion in one dimension while the y axis represents time

14. 4. Simultaneity
Einstein represented this idea in a simple thought experiment
In this proof, Einstein uses the idea of a train moving at a constant velocity next to a platform
Lightning would then hit point A and B simultaneous to a observer on the platform
Images concerning the train example are taken from Modern Physics 4th edition

15. Simultaneity Cont.
Images concerning the train example are taken from Modern Physics 4th edition

16. Simultaneity Cont.
“He (the observer) is hasting towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light coming from B earlier than he will see that emitted from A….Events which are simultaneous with reference to the embankments are not simultaneous with respect to the train, and vice versa.”
Albert Einstein
*NB* previous diagram shows observer on train hasting towards light from A and riding ahead of light from B
Therefore he will see light from A earlier than B

17. Coexistence
From the problem presented by simultaneity, events that seem simultaneous may not be coexistent and vise versa
Therefore, coexistence is not a result of simultaneity and vise versa

18. 5. Paradoxes There are two main relativistic paradoxes
Pole in the Barn Paradox
The Twin paradox
These paradoxes illustrate questions dealing with the principles of length contraction and time dilation

19. Pole in the Barn
This paradox illustrates several questions presented by the length contraction concept
The setup: a man with a 10m pole runs at 0.866c toward a barn 5m long. He enters the barn through the front door and exits out the back. A farmer is standing at the side observing

20. Pole in the Barn cont.
Diagram of the pole in the barn paradox

21. Pole in the Barn cont.
The paradox: The farmer will see the man and the 10m pole contract to 5m. Therefore, the whole pole will fit inside the 5m barn. But, the man will see the 5m barn contract to 2.5m. Therefore, the front of the pole will exit the barn before the end of the pole is inside the barn. Who is right?

22. Pole in the Barn cont.
The so called “paradox” is a result of different frames of reference
The events are not simultaneous, so the pole will seem to fit into the barn for the farmer but will not for the runner
The simultaneous events in one frame are not simultaneous in another, and therefore there is no paradox in this example

23. Twin Paradox
This paradox illustrates several questions presented by the time dilation concept
The set up: Homer and Ulysses are identical twins. Ulysses travels at 0.8c to a distant star and returns to Earth while Homer remains at home.

24. Twin Paradox cont.

25. Twin Paradox cont.
The paradox: The round trip takes Ulysses 6 years. When he returned, he found that Homer has aged 10 years.
However, if motion is relative, we can consider Ulysses as being at rest and Homer as moving away. In this case, Homer will have aged 3.6 years while Ulysses aged 6 years.
Who is right?

26. Twin Paradox cont.
This anomaly is a result of time dilation. Because Ulysses was in an accelerating frame of reference, he actually did age less than Homer.
This effect is a real and measurable event: two experiments have shown them to be real (using airplanes and a rocket)
Both tests showed a real time shifts based on motion (moving clocks ticked slower than the “stationary” one on earth)
For the jet clocks, the time differences were on the order of nanoseconds, and the rocket clock had similar results

27. Twin Paradox cont. Another paradox to consider:
If both Ulysses and Homer were in different spaceships and they flew past each other (near the speed of light), they will each see the other as younger than himself.
They would both appear to be younger than the other. However, you could never actually compare both of them as one would have to enter into an accelerating reference frame to see the other

28. Conclusion Doppler effect Mass energy equivalence Space time diagrams
Simultaneity and coexistence
Length contraction and time dilation
Examples of the paradoxes

29. Questions?