the Earth twin, perhaps only 1 year (depending on the spacecraft’s speed) would pass for the traveler. Thus, when the traveler returns, the earthbound twin could expect to be 40 years old whereas the traveling twin would be only 21. ¢ t 0 = ¢ t 2 1 - v 2 c 2 = ( 100 yr ) 2 1 - ( 0.999 ) 2 = 4.5 yr. v = 0.999 c , 1 year = 3.0 * 10 8 m s * 3.16 * 10 7 s = 9.5 * 10 15 m ( 2.25 * 10 8 m s )( 600 s ) = 1.35 * 10 11 m. D 0 = v ¢ t 0 = 2.04 * 10 11 m. ( 0.75 c)( 15.1 min ) = ( 0.75 ) A 3.0 * 10 8 m s B ( 15.1 min * 60 s min ) = D = v ¢ t = ¢ t = ¢ t 0 3 1 - A v 2 c 2 B = 10.00 min 3 1 - (0.75) 2 = 15.1 min. ¢ t 0 ¢ t 0 ¢ t speed * time. EXAMPLE 26 ; 3 754 CHAPTER 26 The Special Theory of Relativity
SECTION 26 – 4 Time Dilation and the Twin Paradox 755 FIGURE 26–7 A visiting professor of physics uses the GPS on her smart phone to find a restaurant (red dot). Her location in the physics department is the blue dot. Traffic on some streets is also shown (green good, orange slow, red heavy traffic) which comes in part by tracking cell phone movements. = = = This is the viewpoint of the twin on the Earth. But what about the traveling twin? If all inertial reference frames are equally good, won’t the traveling twin make all the claims the Earth twin does, only in reverse? Can’t the astronaut twin claim that since the Earth is moving away at high speed, time passes more slowly on Earth and the twin on Earth will age less? This is the opposite of what the Earth twin predicts. They cannot both be right, for after all the spacecraft returns to Earth and a direct comparison of ages and clocks can be made. There is, however, no contradiction here. One of the viewpoints is indeed incorrect. The consequences of the special theory of relativity — in this case, time dilation — can be applied only by observers in an inertial reference frame. The Earth is such a frame (or nearly so), whereas the spacecraft is not. The spacecraft accelerates at the start and end of its trip and when it turns around at the far point of its journey. Part of the time, the astronaut twin may be in an inertial frame (and is justified in saying the Earth twin’s clocks run slow). But during the accelerations, the twin on the spacecraft is not in an inertial frame. So she cannot use special relativity to predict their relative ages when she returns to Earth. The Earth twin stays in the same inertial frame, and we can thus trust her predictions based on special relativity. Thus, there is no paradox. The prediction of the Earth twin that the traveling twin ages less is the correct one. Global Positioning System (GPS) Airplanes, cars, boats, and hikers use global positioning system ( GPS ) receivers to tell them quite accurately where they are at a given moment (Fig.26 – 7). There are more than 30 global positioning system satellites that send out precise time signals using atomic clocks. Your receiver compares the times received from at least four satellites, all of whose times are carefully synchronized to within 1 part in By comparing the time differences with the known satellite positions and the fixed speed of light, the receiver can determine how far it is from each satellite and thus where it is on the Earth. It can do this to an accuracy of a few meters, if
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