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Class 2: the Special Theory of Relativity

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Class 2: the Special Theory of Relativity Class 2: The special theory of relativity Einstein’s postulates Although some of results of special relativity were known earlier, it was Einstein in 1905 that provided the logical basis for the theory by making two postulates: 1. The laws of physics are the same in all inertial reference frames. 2. The speed of light (in vacuum) is independent of the motion of the light source and has value c. A corollary of the second postulate is that all observers measure the same value for the speed of light. To see why, consider a light source S and two observers A and B. The light source and observer A are stationary and observer B is moving away from S at speed u, as shown in figure (a). The speed of light measured by observer A is c. By postulate 1, the situation in figure (b) is equivalent to that in figure (a). By postulate 2, observer B also measures the speed of light to be c. (a) (b) A u A u B u B Events, observers and simultaneity A physical event is something that happens, like a spark, or the collision of two particles, or an explosion. Events are not tied to any individual reference frames. Events are measured by observers who do 00:00 00:00 00:00 belong to particular inertial reference frames. To be specific, we will consider an observer to be a 1 m set of recording clocks arrayed throughout an inertial reference frame. It is also useful to consider the inertial reference frame as consisting 00:00 00:00 00:00 of a set of identical measuring rods (e.g. meter sticks) arranged to form a cubic lattice, with a recording clock at every intersection. A two dimensional analog is shown at right. The clocks are all identical and synchronized. 00:00 00:00 00:00 Synchronization can be done by selecting one clock as the reference clock. It is useful to take the origin of coordinates to be at the position 1 of the reference clock. A pulse of light is sent out from the reference clock when it reads zero. Since we know by construction the distance of any clock from the reference clock, we know how long it takes the light pulse to reach a particular clock, When the light pulse reaches a clock, we set the clock to the time it takes the pulse to reach it. Note that this procedure works to synchronize clocks in any reference frame but does not synchronize clocks in reference frames that move with respect to each other. Once the clocks are synchronized we can record the position and time of event by using the nearest clock. Two events in a particular reference frame are simultaneous if they occur at the same time, i.e. when we look at what is recorded on the clocks, the clocks nearest the events measure the same time. An important result of special relativity is the relativity of simultaneity , which states that two separated events which are simultaneous in one reference frame will not be simultaneous in any other reference frame moving relative to the first . It follows that events that are simultaneous in all reference frames cannot be spatially separated, i.e. the events must occur at the same location. Einstein gave an example of the relativity of simultaneity. A train is moving past a station platform at speed u. As seen from an observer on the platform, the ends of the train are simultaneously struck by lightning. In particular consider an observer C (i.e. a recording clock) on the platform located midway between the ends of the train when the lightning strikes. Since the light travel times from the two events (the lightning strikes) to C are the same, C records them as simultaneous. Now consider an observer C’ located in the train midway between the ends of the train. As seen from the observer C on the platform, C’ moves forward in the time it takes the flash from the front of the train to reach C’. So C’ records the flash from the front of the train before recoding the flash from the back of the train, i.e. the two events are not simultaneous in the frame moving with the train. v A’ A’ A A’ A A C’ C’ C C’ C C B’ B’ B B’ B B (a) (b) (c) The figure above illustrates the sequence of events. In (a) the lightning strikes the ends of the train simultaneously as seen by the observer on the platform. In (b) the light pulse from the lightning strike at A reaches C’, again as seen from the observer on the platform. At a later time, the light pulse from the lightning strike at B reaches C’, also as seen from the observer on the platform. Hence the clock at C’ records different times for the light pulses to travel from A’ and B’ and the observer in the train deduces that the lightning strikes were not simultaneous. 2 The relativity of simultaneity also implies that clocks which are synchronized in one reference frame are not synchronized (as measured by an observer) in any other reference frame moving relative to the first . The Lorentz transformation The Lorentz transformation relates the space-time coordinates of one inertial frame to those of another inertial frame moving relative to the first. Lorentz and others derived the transformation based on invariance of Maxwell’s equations and the need to resolve the paradoxical results of the Michelson – Morley experiment. This was done before Einstein developed special relativity Consider two inertial frames with relative velocity u. To make the analysis simple, take the x- axes of each coordinate system parallel to u, with the two y-axes parallel to each other. Also assume that the coordinate origins coincide at time zero. We want to derive the relation between the space-time coordinates of an event as observed in the two frames. Let the coordinates in frame S be ( x, y, z, t) and in frame S’ be ( x’, y’, z’, t’). The classical Galilean coordinate transformation is x′=− xut, y ′′′ = y , z = zt , = t , (1.1) with inverse transformation xxut=+′′, yy = ′ , zz = ′ , tt = ′ . (1.2) We have seen that this transformation is inconsistent with Einstein’s postulates of relativity. We assume that the correct equation for x′ is of form x′ =γ ( x − ut ), (1.3) where γ depends on only u and c. The inverse transformation should look the same except for the sign of u: x=γ ( x′ + ut ′ ), (1.4) Eliminating x′ from these last two equations gives 1−γ 2 x t′ =γ t + . (1.5) γ 2 u Now consider a pulse of light emitted from the origin at time t= t ′ = 0. Because of Einstein’s second postulate, the wave front is described in the two frames by x2+ y 2 + z 2 = ct 22 , (1.6) in frame S, and by x′2+ y ′′ 2 + z 2 = ct 22 ′ , (1.7) in frame S’. Using equations (1.3) and (1.5) to eliminate x′ and t′ from the last equation, we obtain 3 22 2 22 2 22222 222 2 2x 2 1−γ x γγγx−2 utx + vtyz ++= γ ct + 21 c() − γ tc + 2 . (1.8) uγ u This has to be consistent with equation (1.6). Since x and t are independent, this is possible only if the terms proportional to xt cancel. This requires c2 γ 2 = . (1.9) c2− u 2 Using this result in the terms in equation (1.8) that are proportional to x2 and t2 gives equation (1.6), as required. The factor γ is called the Lorentz factor and is usually written as 1 γ = . (1.10) u2 1− c2 A common notation is to use β = u c , so that 1 1 γ=, β = 1 − 2 . (1.11) 1− β 2 γ We see that γ increases from 1 to infinity as u increases from 0 to c. The final form for the Lorentz transform, once equation (1.9) is used in equation (1.5), is ux x′=−γ() xut,,, yyzzt ′′′ = = =− γ t , (1.12) c2 with inverse transformation ux ′ x=+γ() xut′′,,, yyzzt = ′ = ′ =+ γ t ′ . (1.13) c2 Note that from the last equation in equations (1.12), we see that separated clocks synchronized in frame S are not synchronized in frame S’. 4 .
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