Further Logistical Consequences of Einstein’s Postulates

By: Everett Chu and Stephen D’Auria

Review • Lorentz Transformations

• Time Dilation

• Length Contraction

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1. Doppler Effect• Original non-relativistic effect discovered in

1803 by Christian Doppler

• Principle modified by Einstein to include light

http://fr.wikipedia.org/wiki/Christian_Doppler

Christian Doppler

http://antwrp.gsfc.nasa.gov/apod/ap951219.html

Albert Einstein

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

• Equations for the classical and relativistic Doppler effect

• http://physics.berea.edu/~king/Teaching/ModPhys/Relativity/animations.htm

Doppler Effect Cont.

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Relativistic Doppler effectSource/observer approaching

Source/observer receding

Nonrelativistic Doppler effect:

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2. Mass-Energy EquivalenceThis principle was discovered by Einstein using the following argument and his original two postulates

1. Einstein derived the Lorentz transformations, which describe how position and velocity can be related between two inertial reference frames

2. 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

Mass-Energy Equivalence Cont.

3. 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

)( BvEqdt

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ump Relativistic Momentum

Mass-Energy Equivalence Cont.

4. Using the work energy theorem, Einstein was then able to find a relativistic expression for the kinetic energy of an object

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Mass-Energy Equivalence Cont.

5. 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

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

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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

Review • Lorentz Transformations

• Time Dilation

• Length Contraction

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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

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

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Pole in the Barn cont.

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

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.

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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.

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?