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

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

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 http://fr.wikipedia.org/wiki/Christian_Doppler http://antwrp.gsfc.nasa.gov/apod/ap951219.html

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

Doppler Effect Cont. Equations for the classical and relativistic Doppler effect http://physics.berea.edu/~king/Teaching/ModPhys/Relativity/animations.htm 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

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

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

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

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

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

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

Review Lorentz Transformations Time Dilation Length Contraction

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

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

Simultaneity Cont. Images concerning the train example are taken from Modern Physics 4th edition http://webphysics.davidson.edu/faculty/thg/physlets/html/247/illustration40_4.html

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

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

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

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

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

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?

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

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.

Twin Paradox cont.

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?

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

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

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

Questions?