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Classical Transformations & Galilean Relativity: Understanding Inertial Frames

Explore Galilean Transformations, Newton's Invariance Principle, Inertial Coordinate Systems & Classical Reference Frames in this overview. Learn about the Ether River, Michelson-Morley Experiments, and delve into the history of ether theories. Find insights on positioning, velocity, and acceleration in different reference frames. Dive into the world of special relativity with engaging visuals and references.

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Classical Transformations & Galilean Relativity: Understanding Inertial Frames

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  1. Chapter 02Special Relativity Version 110906, 110907, 110908, 110913 General Bibliography 1) Various wikipedia, as specified 2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated

  2. Outline • Galilean Transformations • Names & Reference Frames • The Ether River • Michelson-Morley Experiments • Einstein Postulates • Lorentz Transformations • Position • Velocity • Space-Time Diagrams • Relativistic Forces & Momentum • Relativistic Mass • Relativistic Energy

  3. CLASSICAL / GALILEAN / NEWTONIANTRANSFORMATIONS

  4. Galilean Transformations K’ frame moving with speed v K frame fixed v K & K’ coincided at t=0. Sketch shown at time t later. How do the position, velocity, acceleration, & time between the 2 frames compare?

  5. Galilean Transformations K’ frame moving with speed v K frame fixed v K & K’ coincided at t=0. Sketch shown at time t later. How do the position, velocity, acceleration, & time between the 2 frames compare? K  K’ x = x’ + vt K’  K x = x’ -vt

  6. 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 relativityor Galilean invariance.

  7. Inertial Frames K and K’ • K is at rest and K’ is moving with velocity • Axes are parallel • K and K’ are said to be INERTIAL COORDINATE SYSTEMS

  8. The Galilean Transformation For a point P • In system K: P = (x, y, z, t) • In system K’: P = (x’, y’, z’, t’) P x K K’ x’-axis x-axis

  9. Conditions of the Galilean Transformation • Parallel axes (for convenience) • K’ has a constant relative velocity in the x-direction with respect to K • Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers x’ = x – v t y’ = y z’ = z speed of frame NOT speed of object t’ = t

  10. Galilean Transformation Inverse Relations Step 1. Replace with . Step 2. Replace “primed” quantities with “unprimed” and “unprimed” with “primed.” x = x’ + v t y = y’ z = z’ speed of frame NOT speed of object t = t’

  11. General Galilean Transformations Position inertial reference frame Velocity frame K frame K’ Acceleration Newton’s Eqn of Motion the same at face-value in both reference frames

  12. Classical Reference Frames • Inertial Reference Frame • Non-accelerating • Newton’s Laws apply at face-value • Non-Inertial Reference Frame • Examples: • Rocket during acceleration phase • Rotating merry-go-round • Rotating Earth

  13. Youtube clips (part 1) • Galilean/Classical Relativity Part 1 – The Cassiopeia Project http://www.youtube.com/watch?v=6rl3Z9yCTn8 The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.http://www.cassiopeiaproject.com/To read more about the Theory of Special Relativity, you can start here:http://www.phys.unsw.edu.au/einsteinlight/http://www.einstein-online.info/en/elementary/index.htmlhttp://en.wikipedia.org/wiki/Special_relativity

  14. THE ETHER RIVERHISTORY OF ETHERMICHELSON-MORLEY EXPTS

  15. The Ether River D v C A Maximum speed of the boat is ‘c’ meters/sec

  16. The Ether River Time t1 from A to C and back: down river Time t2 from A to D and back: So that the difference in trip times is:

  17. Timeline of luminiferous aether(http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether) • 4th cent BC – Light propagates in air – Aristole • 1704 – Aetheral medium for light & heat – Newton • 1818 – aether – Fresnel wave theory • 1830 – problems emerge, explained by “aether drag”, Fresnel & Stokes • 1830 – ~1955 – mixed experimental conclusions Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg

  18. Timeline of luminiferous aether(http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether) • 1830 – ~1955 – mixed experimental conclusions • 1887 – 1st Michelson-Morley expt doesn’t find aether • 1889(1895) – Fitzgerald hypothesis (Lorentz) • 1902-1904 – Refined Michelson-Morley measurements • 1905 – Trouton-Rankine expt doesn’t support Fitz-Loentz hypothesis • 1958- nearly all measurements do not find evidence for aether Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg

  19. Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg Michelson-Morley Expt“the most famous failed experiment”

  20. D v C A Michelson-Morley: Ether River - Revisited v D A C Measure two orientations because don’t know direction of aether river

  21. Ether River - Revisited Orientation 1 down river Orientation 2 down river Difference in Orientations

  22. Michelson-Morley Measurements Apollo 11 Apollo 15 v=30 km/s c=3E8 m/s ~2002 accuracy ~1 mm http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_Experiment

  23. Crises with Reference Frame Xformations • Can’t find the Ether • Maxwell’s Equations not Galilean Invariant Speed of Light fixed by EM constants

  24. Fitzgerald-Lorentz Hypothesis1889 (1895){only a partial explanation} POSTULATE: the null results are due to changes in length in the direction of travel.

  25. EINSTEIN’s 1905 POSTULATES • All laws for physics have the same functional form in any inertial reference frame • Speed of Light (in vacuum) is same in any inertial reference frame.

  26. LORENTZ POSITION-TIME TRANSFORMATIONS

  27. Lorentz Transformations K’ K v K’ K x P x’ x’-axis x-axis

  28. Lorentz Transformations K’ K v KK’ x P x’ x’-axis x-axis

  29. K: 3km, 5us K’: 2.6km, -1.25us Example As observed from a large asteroid, an explosion occurs at x=3000, y=500, z=-500 and t=5 us. v P A spaceship approaches at a high speed v=0.6c . The reference frames coincided at t=0, t’=0 At what position does the spaceship observe the explosion to occur?

  30. K rear -5km, -10 us front +5km, +10 us ExampleThe reference frames coincide at x=0, x’=0 & t=0, t’=0 A spaceship has indicator lights which are flashed at the same time. At t’=0 the lights flash. The locations of the lights are x’rear=-4km & x’front=+4km. K’ v K x’-axis x-axis The spaceship is observed from the spacestation. The spaceship is observed to move at v=0.6c . At what position does the spacestation observe the lights to flash?

  31. 0m, 2.3 s 6.5E8, 3.2 s ExampleThe reference frames coincide at x=0, x’=0 & t=0, t’=0 As viewed from the Earth, a meteorite impacts the lunar surface at 3E8m and 2.5s . The impact is observed from 2 passing spaceships, one traveling to the right at 40% c and the other to the left at - 40% c. Where do the 2 spaceships observe the impact to occur ?

  32. Moving objects appear shorter Length Contraction(Lorentz-Fitzgerald) A meter stick, lying parallel to the x-axis, is moving with speed v v How long does the stick appear to be to a stationary observer who makes the observation of the length at t=0? xleft & tleft=0 xright & tright=0

  33. Moving clocks run slow Time Dihilation(distinct from the L-F) A clock, located at x’=0, makes ticks at t’1, t’2, … v What is the interval between ticks to a stationary observer, who observes the clock to move at speed v? x’1=0 & t’1 x’2=0 & t’2

  34. Distorted Pictures stationary moving to the right Our brain records photographs (frames in a movie) – light rays arriving at the same time.

  35. “Jump to Light Speed”

  36. Distorted Pictures

  37. Lorentz Transformation - DerivationLight propagates with speed c in all inertial reference frames K K’ Spherical wavefronts in K: Spherical wavefronts in K’: ct’ ct

  38. Derivation – see pages 30-31 • Let x’= (x – vt) so that x = (x’ + vt’) • By Einstein’s first postulate: • The wavefront along the x,x’- axis must satisfy:x = ct and x’ = ct’ • Thus ct’= (ct – vt) and ct = (ct’+ vt’) • Solving the first one above for t’and substituting into the second...

  39. Youtube clips (part 2) • Galilean/Classical Relativity Part 2 – The Cassiopeia Projecthttp://www.youtube.com/watch?v=WgsKlSnUO0k The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.http://www.cassiopeiaproject.com/To read more about the Theory of Special Relativity, you can start here:http://www.phys.unsw.edu.au/einsteinlight/http://www.einstein-online.info/en/elementary/index.htmlhttp://en.wikipedia.org/wiki/Special_relativity

  40. LORENTZ VELOCITY TRANSFORMATIONS

  41. Lorentz Velocity Transformationsee page 40

  42. Lorentz Velocity Transformationsee page 40

  43. Lorentz Velocity Transformationsee page 40 Note that because of the time transformation, the y- and z-components get messed up.

  44. A spaceship traveling at 60%c shoots a proton with a muzzle speed of 99%c at an asteroid. What is the velocity of the proton as viewed from a ‘stationary’ space station?

  45. MISC. LORENTZ TRANSFORMATIONEXAMPLES

  46. Cosmic Ray Muon Lifetime electron mo=9.1e-31 kg halflife = inf muon mo=207 * (mass e) halflife = 1.5e-6 sec http://www.windows2universe.org/physical_science/physics/ atom_particle/cosmic_rays.html http://landshape.org/enm/cosmic-ray-basics/

  47. Cosmic Rays Susan Bailey Nuclear News Jan 2000, pg 32

  48. Cosmic Ray references Cosmic Ray Muon Measurements http://www.youtube.com/watch?v=yjE5LHfqEQI http://www.ans.org/pubs/magazines/nn/docs/2000-1-3.pdf http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cosmic-rays.pdf http://hyperphysics.phy-astr.gsu.edu cosmic rays ashsd.afacwa.org/ radation

  49. Cosmic Ray Muon Lifetime muon mo=207 (9.1e-31 kg) halflife = 1.5e-6 sec Suppose muon traveling at 0.98c Q1. Classically, how far could the muon travel during a time 1.5e-6 sec ? Q2. What do we observe the lifetime to be ? Q3. How far do we observe the muon to travel during that time ? 2000 meters Q4. How high does the muon think the mountain is?

  50. Simultaneity • http://www.youtube.com/watch?v=wteiuxyqtoM • http://www.youtube.com/watch?v=KYWM2oZgi4E

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