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Relativity

Relativity. In the 19 th Century…. Ocean waves propagate through a medium called “water” Sound waves propagate through a medium called “air” Ergo: Light waves propagate through a medium Medium was called Ether Surrounded us and passed through us…. Michelson-Morely Experiment.

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Relativity

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  1. Relativity

  2. In the 19th Century… • Ocean waves propagate through a medium called “water” • Sound waves propagate through a medium called “air” • Ergo: • Light waves propagate through a medium • Medium was called Ether • Surrounded us and passed through us….

  3. Michelson-Morely Experiment • In 1887, Albert Michelson and Edward Morley decided to try detect the motion of the earth through the “ether”. • They measured the speed of light in the direction of the earths motion • They measured the speed of light in the against the earth’s motion

  4. What they suspected • Vlight += Vlight + Vearth • Vlight -= Vlight – Vearth

  5. What they found • Vlight += Vlight • Vlight -= Vlight • NO DIFFERENCE! • Uhoh!

  6. 1887-1905 • Many scientists, most famously Hendrik Lorentz, attempted to explain (or in Lorentz’s case gloss over) the Michelson-Morely Experiment • In 1905, this guy named Einstein published a paper called “On the Electrodynamics of Moving Bodies “

  7. Einstein “cuts through the ether” • Einstein said that the Michelson-Morely experiment could be explained under two postulates • The laws of physics are the same in every inertial frame of reference • The speed of light in vacuum is the same in all inertial frames of reference and is independent of the motion of the source

  8. What do these mean? • The first is self-evident: there are no special physics “rules” for one place compared to another • The second postulate is subtle. • What if a spaceship (traveling at 1000 km/s) fires a light beam ahead of it, what is the speed of the beam? • Answer: c not c+1000 km/s!

  9. More subtlety • A subtle implication here is that it is impossible for an inertial observer to travel at c • Consider: an observer on earth and an observer on a spaceship which is traveling away from earth at v=c • If the spaceship turns on a headlight, the earthbound-observer measures the beam to move at c • Thus, the earthbound observer never sees the beam of light move ahead: it is always at the same point in space as the ship. • However, the 2nd postulate says that the light must move away from the observer in the ship at c. • The only way out of this paradox is to not let the spaceship move at c

  10. Why does light have to travel at c? • First, the speed of light in vacuum is constant • The speed of light can change (always slower) in different materials. • Next, if we let c have different values than those allowed by electromagnetic equations, then we have violated the first postulate

  11. Is faster than light travel possible? • Yes, but right now you and I don’t qualify • We will discuss why later..

  12. Galilean Transforms Frame S’ y’ Frame S y x’ x S’ S u Frame S’ is moving away from Frame S at speed u

  13. Frame S’ y’ Frame S y x’ x S’ S u Galilean Transforms From Frame S’s point of view, x’=x-ut y’=y This is called a coordinate transformation (Galilean) Note that the velocity in the x’ direction in Frame S’ is

  14. Mathematically speaking • Previously, we used logic to solve the problem of the observer going at the speed of light • Now, let’s prove it mathematically • Recall : vx’=c, vx=c • So vx’=vx + u • Or c=c+u which is completely wrong! • Thus, we must make some changes

  15. Simultaneity Gedunken S’ S S’ S S’ S Two events which appear simultaneous in one inertial frame of reference may not appear coincident in another

  16. A simple experiment Consider the beam of flashlight reflecting from a mirror For a stationary observer, the time interval for the beam of light to reflect is d

  17. Now let the mirror and light move Consider the beam of flashlight reflecting from a mirror L d u Dt

  18. Mathematically L d u Dt A time interval in a frame (Dt0) measured by an observer at rest in this frame is always smaller than the measured value (Dt) of an observer moving at speed u relative to the frame

  19. Proper Time • The time interval wherein the clock is at rest relative to the observer is called the proper time (Dt0) • Or • proper time is the time measured between two events which happen in the same location

  20. Relativity of Length Consider the beam of flashlight reflecting from a mirror For a stationary observer, the time interval for the beam of light to reflect is L0

  21. Relativity of Length Consider the beam of flashlight traveling to the mirror L0

  22. Slowed Down L u Dt1 d = L + u Dt1 and d=c Dt1 c Dt1= L + u Dt1 And the interval, Dt2, for the return trip can be shown to be

  23. The total time is The length, L, in the frame where the length is moving is shorter than the length at rest (rest-length)

  24. Important Safety Note • There is no length contraction in a direction perpendicular to motion

  25. Coordinate Transformations Assume that Frame S is moving at speed, u, with respect to Frame S’ Use relativistic approach when u ≥ 10%c

  26. Velocity Transformation

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