CHAPTER 2 Special Theory of Relativity Time Dilation Source in addition to Receiver Receding Momentum in addition to Energy (continued)

CHAPTER 2 Special Theory of Relativity Time Dilation Source in addition to Receiver Receding Momentum in addition to Energy (continued)

CHAPTER 2 Special Theory of Relativity Time Dilation Source in addition to Receiver Receding Momentum in addition to Energy (continued)

Berton, Justin, 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 Addition of Velocities 2.7 Experimental Verification 2.8 Twin Paradox 2.9 Spacetime 2.10 Doppler Effect 2.11 Relativistic Momentum 2.12 Relativistic Energy 2.13 Computations in Modern Physics 2.14 Electromagnetism in addition to Relativity CHAPTER 2 Special Theory of 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 Newtonian (Classical) Relativity Assumption It is assumed that Newton’s laws of motion must be measured with respect to (relative to) some reference frame. Inertial Reference Frame A reference frame is called an inertial frame if Newton laws are valid in that frame. Such a frame is established when a body, not subjected to net external as long as ces, is observed to move in rectilinear motion at constant velocity.

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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 uni as long as m velocity relative to the first system. This is referred to as the Newtonian principle of relativity or Galilean invariance. 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 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

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. In 1905, Albert Einstein proposed a fundamental connection between space in addition to time in addition to that Newton’s laws are only an approximation.

2.1: The Need as long as Ether The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether. 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 elasticity to support the high velocity of light waves Maxwell’s Equations In Maxwell’s theory the speed of light, in terms of the permeability in addition to permittivity of free space, was given by Thus the velocity of light between moving systems must be a constant. 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 existence of ether.

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), in addition to built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions. The Michelson Interferometer 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

Typical interferometer fringe pattern expected when the system is rotated by 90° 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: Assuming the Galilean Trans as long as mation The Analysis (continued) Upon rotating the apparatus, the optical path lengths 1 in addition to 2 are interchanged producing a different change in time: (note the change in denominators)

The Analysis (continued) in addition to upon a binomial expansion, assuming v/c 1, this reduces to Thus a time difference between rotations is given by: Results Using the Earth’s orbital speed as: V = 3 × 104 m/s together with 1 2 = 1.2 m 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. Michelson’s Conclusion Michelson noted that he should be able to detect a phase shift of light due to the time difference between path lengths but found none. He thus concluded that the hypothesis of the stationary ether must be incorrect. 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!

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. 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 Lorentz-FitzGerald Contraction Another hypothesis proposed independently by both H. A. Lorentz in addition to G. F. FitzGerald suggested that the length 1, in the direction of the motion was contracted by a factor of 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. 2.3: Einstein’s Postulates Albert Einstein (1879–1955) was only two years old when Michelson reported his first null measurement as long as the existence of the ether. At the age of 16 Einstein began thinking about the as long as m of Maxwell’s equations in moving inertial systems. In 1905, at the age of 26, he published his startling proposal about the principle of relativity, which he believed to be fundamental.

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Einstein’s Two Postulates With the belief that Maxwell’s equations must be valid in all inertial frames, Einstein proposes the following postulates: The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, in addition to no preferred inertial system exists. The constancy of the speed of light: Observers in all inertial systems measure the same value as long as the speed of light in a vacuum. Re-evaluation of Time In Newtonian physics we previously assumed that t = t’ Thus with “synchronized” clocks, events in K in addition to K’ can be considered simultaneous Einstein realized that each system must have its own observers with their own clocks in addition to meter sticks Thus events considered simultaneous in K may not be in K’ The Problem of Simultaneity Frank at rest is equidistant from events A in addition to B: A B 1 m +1 m 0 Frank “sees” both flashbulbs go off simultaneously.

The Problem of Simultaneity Mary, moving to the right with speed v, observes events A in addition to B in different order: 1 m 0 +1 m A B Mary “sees” event B, then A. We thus observe Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K’) moving with respect to the first frame. This suggests that each coordinate system has its own observers with “clocks” that are synchronized Synchronization of Clocks Step 1: Place observers with clocks throughout a given system. Step 2: In that system bring all the clocks together at one location. Step 3: Compare the clock readings. If all of the clocks agree, then the clocks are said to be synchronized.

15.5: Frame Dragging Josef Lense in addition to Hans Thirring proposed in 1918 that a rotating body’s gravitational as long as ce can literally drag spacetime around with it as the body rotates. This effect, sometimes called the Lense-Thirring effect, is referred to as frame dragging. All celestial bodies that rotate can modify the spacetime curvature, in addition to the larger the gravitational as long as ce, the greater the effect. Frame dragging was observed in 1997 by noticing fluctuating x rays from several black hole c in addition to idates. This indicated that the object was precessing from the spacetime dragging along with it. The LAGEOS system of satellites uses Earth-based lasers that reflect off the satellites. Researchers were able to detect that the plane of the satellites shifted 2 meters per year in the direction of the Earth’s rotation in agreement with the predictions of the theory. Global Positioning Systems (GPS) had to utilize relativistic corrections as long as the precise atomic clocks on the satellites.

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