CHAPTER 2 Special Relativity “interpretation” Time Dilation

CHAPTER 2 Special Relativity “interpretation” Time Dilation

CHAPTER 2 Special Relativity “interpretation” Time Dilation

Dean, Jennifer, Features Reporter has reference to this Academic Journal, PHwiki organized this Journal 2.1 The Need as long as Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein’s Postulates 2.4 The Lorentz Trans as long as mation 2.5 Time Dilation in addition to Length Contraction 2.6 Muon observation on earth (experimental verification special relativity) 2.7 Twin Paradox (NOT a problem of special relativity) 2.8 Addition of Velocities 2.9 Doppler Effect Summary up to that point & Conclusions Michelson-Morley experiment 2.10 Relativistic Momentum 2.11 Relativistic Energy Pair Production in addition to Annihilation 2.13 Computations in Modern Physics 2.14 Electromagnetism in addition to Relativity Chapter 15, General Relativity CHAPTER 2 Special Relativity It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative in addition to would, there as long as e, show that there is still a difficulty in the theory itself – Albert Michelson, 1907 The “paradox” is only a conflict between reality in addition to your feeling of what reality “ought to be.” R. P. Feynman Space time diagrams are skipped Newtonian (Classical) Relativity / Invariance due to Galileo Galilei, 1564 – 1642 All of mechanics (i.e. Newton’s laws) is independent on the inertial reference frame in which it is happening. Not only do we not feel that the Earth is moving (around the sun while spinning), we also cannot prove this by mechanical experiments. So the Earth may well be moving around the sun despite the catholic church’s burning of Giordano Bruno (1548-1600) on the stake as long as that belief. Inertial Frames K in addition to K’ K is at rest in addition to K’ is moving with velocity Axes are parallel K in addition to K’ are said to be INERTIAL COORDINATE SYSTEMS (frames of reference)

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The Galilean Trans as long as mation For a point P In system K: P = (x, y, z, t) In system K’: P = (x’, y’, z’, t’) x K P K’ x’-axis x-axis Space in addition to time are separate, both are fundamental frames in which physical processes are happening Conditions of the Galilean Trans as long as mation Parallel axes K’ has a constant relative velocity in the x-direction with respect to K Time (t) as long as all observers is a Fundamental invariant, i.e., the same as long as all inertial observers The Inverse Relations Step 1. Replace with . Step 2. Replace “primed” quantities with “unprimed” in addition to “unprimed” with “primed.”

The Transition to Modern Relativity Although Newton’s laws of motion had the same as long as m under the Galilean trans as long as mation, Maxwell’s equations did not !!! So some corrections should be needed as long as their validity on Earth, some other corrections as long as their validity on Mars, is a constant according to Maxwell, speed of light supposed to be with respect to the medium in which light is traveling in In 1905, the 26 years young Albert Einstein proposed a fundamental connection between space in addition to time in addition to that Newton’s mechanics laws are only an approximation. 2.1: The Need as long as Ether The wave nature of light “requires” that there existed a propagation medium called the luminiferous ether or just ether. (Assuming that light is just like the other waves known to physics.) Ether had to have such a low density that the planets could move through it without loss of energy It also had to have an enormously high elasticity to support the high velocity of light waves No such material was known or seemed to exist Maxwell’s Equations In Maxwell’s theory the speed of light, in terms of the permeability in addition to permittivity of free space (a vacuum on earth), is given by So if Maxwell’s equations are correct ( in addition to experiments indicate that they are on earth), in addition to the speed of light is this constant on earth, planets, stars, that move with respect to the earth need to have different speeds of light

An Absolute Reference System Ether was proposed as an absolute reference system in which the speed of light was this constant in addition to from which other measurements could be made. The Michelson-Morley experiment was an attempt to show the Earth’s movement through the ether ( in addition to thereby it’s existence). Seemed to be unreasonable that the ether would be attached to the earth, which would then be a very special place in all of the universe. 2.2: The Michelson-Morley Experiment Albert Michelson (1852–1931) was the first U.S. citizen to receive the Nobel Prize as long as Physics (1907), “ as long as his optical precision instruments in addition to the spectroscopic in addition to metrological investigations carried out with their aid”. (interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions that classical mechanics predicted.) With which he didn’t get the anticipated result !!! Typical interferometer fringe pattern expected when the system is rotated by 90°

The Michelson Interferometer Whole apparatus is considered to move with respect to the ether light source is originally a selected range of rays from the sun, but can be any sort of light as the real aim is detecting motion with respect to ether rather then sun Near mono-chromaticity of light is required as long as sharp interference pattern 1. AC is parallel to the motion of the Earth inducing an “ether wind” 2. Light from source S is split by mirror A in addition to travels to mirrors C in addition to D in mutually perpendicular directions 3. After reflection the beams recombine at A slightly out of phase due to the “ether wind” as viewed by telescope E. The Michelson Interferometer 0

The Analysis Time t1 from A to C in addition to back: Time t2 from A to D in addition to back: So that the change in time is: · The Analysis (continued) Upon rotating the apparatus (we use (black) primes to mark the rotation), the optical path lengths 1 in addition to 2 are interchanged in order to produce a different change in time: (note the change in denominators) = Should be something small, but just measurable

The Analysis (continued) in addition to upon a binomial expansion, assuming v/c 1, this reduces to Possible 1st part of first Homework is the derivation of the classical analysis as long as this experiment, justifying all the steps from t1 = from slide 16 onwards, show all of your intermediate steps in addition to end in the result below, also convince yourself in addition to the teaching assistant that the expressions as long as t1, t2, t1’ in addition to t2’ are all correct But was measured to be zero !!!!!! Results Using the Earth’s orbital speed as: V = 3 × 104 m/s together with 1 2 = 1.2 m (the longer the better as long as experiment) So that the time difference becomes t’ t v2(1 + 2)/c3 = 8 × 1017 s Although a very small number, it was within the experimental range of measurement as long as light waves. But was measured to be zero !!!!!! “interpretation” R. P. Feynman: The first principle is that you must not fool yourself, in addition to you are the easiest person to fool. 1887

Michelson’s in addition to almost all other’s Conclusions Michelson should have been able to detect a phase shift of light interference fringes due to the time difference between path lengths but found none. (Speed of Earth in orbit 30 km/s would be sufficiently fast as long as these kinds of measurements were classical physics applicable) After several repeats in addition to refinements with assistance from Edward Morley (1893-1923), again a null result. Thus, ether does not seem to exist – we then have a problem, there needs to be something wrong with Maxwell’s equations, the wave theory of light seems to wrong, but all of wave optics depends on it Possible Explanations Many explanations were proposed but the most popular was the ether drag hypothesis. This hypothesis suggested that the Earth somehow “dragged” the ether along as it rotates on its axis in addition to revolves about the sun. Earth would then be the only place in the universe where Maxwell’s equations would be valid without further modifications (correction factors) This was contradicted by stellar abberation wherein telescopes had to be tilted to observe starlight due to the Earth’s motion. If ether was dragged along, this tilting would not exist. The FitzGerald Contraction Another hypothesis proposed independently by both G. F. FitzGerald (just an assumption) in addition to H. A. Lorentz (as part of his trans as long as mations) suggested that the length 1, in the direction of the motion was contracted by a factor of On turning by 90 degrees, it will be 2 that gets contracted thus making the path lengths equal to account as long as the zero phase shift. This, however, was an ad hoc assumption that could not be experimentally tested. But it could be derived from Lorentz (Vogt) trans as long as mation in which Maxwell’s equations are invariant,1895, but no deeper underst in addition to ing of what these relations mean at that time as long as modern relativity

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The Michelson Interferometer Time t2 from A to D in addition to back: So that the change in time is: · So that Time t1 from A to C in addition to back: l2 can be considered to be “set equal” to l1 by nature of fringe shift measurement, but there is also an additional (Fitzgerald) contraction – marked by two green primes – in direction of motion of the apparatus with respect to light source ‘’ ‘’ ‘’ ‘’ ‘’ = 0, a stable setting of the interferometer Upon rotating the apparatus (we use black primes to mark the rotation), the optical path lengths 1 in addition to 2 are interchanged producing a different change in time: (note the change in denominators) ‘ ‘ ‘’ = 0, the same stable setting of the interferometer, so no change on turning apparatus by 90 degrees can be observed in addition to indeed it is not ‘ But now ‘ = 0 – 0 = 0

What if we have two observers, one stationary at the telescope, one far away looking with another telescope while moving with a constant speed in a straight line parallel (or anti-parallel) to the direction of the light that goes into the apparatus in addition to eventually produce the interference pattern Will they obtain different results No, just the spacing of the fringes will be ever so slightly reduced (or exp in addition to ed depending on the direction), but the interference pattern will not change there is nothing as long as the experiment to measure according to Einstein, because there is no electrodynamics experiment that could detect if earth was moving with respect to the sun or ether, it would not matter how fast the apparatus is moving with respect to the source of light (or the sun) length contraction would just compensate as long as that, so conceptually that velocity can also be set zero in addition to we also get an analytical result that agrees with the experimental outcome !!! The Analysis as long as v = 0 as it does not have any effect on the result of physical experiments Time t1 from A to C in addition to back: Time t2 from A to D in addition to back: So that the change in time is: · = 0 Regardless of the actual lengths l1 in addition to l2, we won’t see a change in the interference pattern on turning the apparatus by 90 degrees V = 0

The Analysis as long as c +- v = c as experiment was done with light, its consequences are the same as v = 0 Time t1 from A to C in addition to back: Time t2 from A to D in addition to back: So that the change in time is: · = 0 Regardless of the actual lengths l1 in addition to l2, we won’t see a change in the interference pattern on turning the apparatus by 90 degrees Going back to Michelson–Morley experiment, it is clear that it could not measure any “ether wind” as it was done with light, see next 2 slides y x x x x x x The Analysis as long as c +- v = c (continued) Upon rotating the apparatus by +- 90° = 0 – 0 = 0, we observe no change in interference pattern as we in addition to the apparatus can be considered at rest with respect to the non-existing ether, so no length contraction in addition to time dilation (delay) as long as us, there is no preferred inertial frames of reference, as long as trans as long as mations between frames, we need to use the Lorentz trans as long as mations = 0

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