Relativity 2Einstein’s PostulatesBasic DefinitionsRelativity of Simultaneity:Now that we are armed with these definitions, let’s explore the consequences of the constancy of the speed of light in all inertial frames.Time DilationLength ContractionThe Lorentz TransformationPHY2061 Enriched Physics 2 Lecture Notes Relativity 2Relativity 2Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete or even inaccurate. Some topics may be unclear. These notesare only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.Einstein’s PostulatesThe absence of any fringe shift in the Michelson-Morely experiment for any orientation of the interferometer and for any time of the year negated the ether hypothesisfor light propagation. Light waves are oscillations of the electromagnetic field, and no propagation medium is necessary, unlike sound waves. However, if Galilean transformations are correct, then Maxwell’s equations must be modified for every possible reference frame to account for different velocities for the speed of light. Einsteinassumed the opposite: that Maxwell’s equations are fundamentally correct, but that our intuitive Galilean transformation is not. This led to the following two postulates: 1. The laws of physics, including electromagnetism, are the same in all inertial frames. 2. Every observer measures the same value c for the speed of light (in vacuum) in allinertial frames.The second postulate is really a consequence of the first, because if Maxwell’s equations hold in all inertial frames, then the only possible value for the speed of light is c. These postulates embody Einstein’s Special Theory of Relativity, first published in 1905 in a paper titled On the Electrodynamics of Moving Bodies. Later he would incorporate gravity and acceleration in his General Theory of Relativity. As in Newtonian Relativity, there is no way to detect absolute motion. Only the relative velocities between two inertial reference frames matters.These seemingly simple postulates have extraordinary consequences. For example, when you turn on the headlights of a car, the light beam leaves the car at a relative velocity of c 3 0 108. m/s. However, someone standing on the sidewalk also measures the speed of the light beam as c independent of the velocity of the car! How can this be? As we shall see, our concepts of space and time must be modified. D. Acosta Page 1 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes Relativity 2Basic DefinitionsEvents are physical phenomena that occur independent of any reference frame. For example: a flash, explosion, return of a spaceship, or disintegration of a subatomic particle.Observers record events, both the time and spatial coordinates, in a particular reference frame. For example, Mission Control in Houston marking down the time and location of the splashdown of a space capsule. The reference frame in this case is the Earth.Simultaneous events occur when the light signals from two events reach an observer at the same timeRelativity of Simultaneity: Two events simultaneous in one inertial frame are not simultaneous in any other frame. This is a consequence of Einstein’s Postulates. Proper time is the time difference between two events occurring at the same position (Denoted by t0 or ).Rest frame is the inertial frame where two events are only separated by time. The frame in which the proper time is measuredProper length is the distance between two positions at rest, the length measured in therest frame. (Denoted by L0).Now that we are armed with these definitions, let’s explore the consequences of the constancy of the speed of light in all inertial frames.D. Acosta Page 2 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes Relativity 2Time DilationWe explore the rate of time in different inertial frames by considering a special kind of clock – a light clock – which is just one arm of an interferometer. Consider a light pulse bouncing vertically between two mirrors. We analyze the time it takes for the light pulse to complete a round trip both in the rest frame of the clock (labeled S’), and in an inertial frame where the clock is observed to move horizontally at a velocity v (labeled S).In the rest frame S’tLctLct tLc121 22= time up= time down=Now put the light clock on a spaceship, but measure the roundtrip time of the light pulse from the Earth frame S:ttttcL v t c tL c v ttLc vtLcv c v c122 2 2 2 22 2 2 2222 22 2 2 2224 4442 11 1 time up time downThe speed of light is still in this frame, so/ /// /chSo the time it takes the light pulse to make a roundtrip in the clock when it is moving by us is appears longer than when it is at rest. We say that time is dilated. It also doesn’t matter which frame is the Earth and which is the clock. Any object that moves by with a significant velocity appears to have a clock running slow. We summarize this effect in the following relation:D. Acosta Page 3 1/14/2019 mirror mirror L L c t / 2 v t / 2 tv c 1112 2/PHY2061 Enriched Physics 2 Lecture Notes Relativity 2Length ContractionNow consider using a light clock to measure the length of an interferometer arm. In particular, let’s measure the length along the direction of motion.In the rest frame S’Lc02Now put the light clock on a spaceship, but measure the roundtrip time of the light pulse from the Earth frame S:ttt t tL vt ct tLc vL vt ct tLc vt t tLcc vLc v cLctv ctv c121 21 1 12 2 21 22 2 2 22 22 22 2 11211 time out time back But, from time dilation///c hIn other words, the length of the interferometer arm appears contracted when it moves by us. This is known as the Lorentz-Fitzgerald contraction. It is closely related to time dilation. In fact, one implies the other, since we used time dilation to derive length contraction.Time dilation and length contraction are consequences of the assumption that all observers measure the same value for the speed of light. This means that time runs at different rates for different inertial frames. There is no absolute time, time only has a relative meaning. Likewise, length also has only a relative meaning. Everything dependson the relative velocity
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