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Chapter 13 the Theories of Special and General Relativity Special Ron Ferril SBCC Physics 101 Chapter 13 2017Jul23A Page 1 of 14 Chapter 13 The Theories of Special and General Relativity Special Relativity The Theory of Special Relativity, often called the Special Theory of Relativity or just “special relativity” for a shorter name, is a replacement of Galilean relativity and is necessary for describing dynamics involving high speeds. Galilean relativity is very useful as long as the speeds of bodies are fairly small. For example, the speeds of a supersonic jet, a rifle bullet, and a rocket (such as the Apollo vehicle that went to and from the Moon at about 25,000 miles per hour) are all handled well by Galilean relativity. However, special relativity is required in explanations of the dynamics at the higher speeds of high-energy subatomic particles in cosmic rays and in large particle accelerators. Both Galilean relativity and special relativity involve “frames of reference” which are also called “reference frames.” A frame of reference is the location of an observer of physical processes. For a more mathematical view, a frame of reference can be viewed as a coordinate system for specifying the locations of bodies or physical events from the viewpoint of the observer. An observer may specify positions of bodies and events as distances relative to his or her own position. The positions can also be specified by a combination of distance from the observer and angles from the direction the observer is facing. The distances and angles can be called “coordinates.” An “inertial frame of reference” or “inertial reference frame” is a frame of reference that does not accelerate. If an observer is accelerated, you can expect the observer would experience inertial forces. Thus, an observer does not experience inertial forces in an inertial frame of reference. The observer can consider himself or herself to be at rest in his or her own inertial frame. For example, suppose two people on a moving train are throwing a ball back and forth in a game of “catch” and this game is observed by people on the train and people on the ground outside the train. Each observer of the game can specify the position of the ball relative to himself or herself at any time. © Copyright 2015, 2016 and 2017 by Ron Ferril. All rights reserved. Ron Ferril SBCC Physics 101 Chapter 13 2017Jul23A Page 2 of 14 Thus, a man throwing the ball forward, in the direction the train is moving, to be caught by a woman can specify the position of the ball as the distance from him. The woman can specify the position of the ball by the distance of the ball from her. An observer on the ground can specify the position of the ball by the distance of the ball past a reference point on the ground. Since we can expect the speeds (magnitudes of velocities) involved to be much smaller than a rifle bullet, we should be able to use either Galilean relativity (which was introduced in a previous chapter) or special relativity which is being introduced in this chapter. The Bohr correspondence principle assures us that the newer Theory of Special Relativity should give either the same or almost the same results as Galilean relativity for such speeds. The Bohr correspondence principle allows small differences in the results because of tiny corrections special relativity may introduce. To make our example of the game of catch on a train easier to visualize, let us assign some numerical values to the example. Suppose the train is moving as 40 miles per hour relative to the ground, and the man throws the ball forward, in the direction the train is moving, at a speed of 30 miles per hour relative to himself. The man does not need to notice any movement of the train. By Galilean relativity, an observer on the ground sees the train moving at 40 miles per hour and the ball moving at 70 miles per hour (40+30=70) relative to this observer on the ground. Special relativity presents these speed values as very good approximations and the differences in results between the two types of relativity are so small that only the most sensitive laboratory devices (devices called “interferometers”) can detect them. To make the differences introduced by special relativity more plain, we can consider a similar example involving a high speed. We will still have a train moving at 40 miles per hour, but suppose the man and woman on the moving train of the previous example replace their ball with light. The man can shine a flashlight forward at the woman, and instruments can measure the speed of the light. Earlier chapters told us that the speed of light is maximum in vacuum. Thus, let us assume the light shines forward through a vacuum. This is surely a high enough speed that we will need to use special relativity. The man and woman on the train can measure the speed and confirm that it travels at the speed of light in vacuum and can denote this speed by c. However, observers on the ground can also measure the speed of the light. By Galilean relativity, we would expect the speed of the light to be the sum of the speed of the train and the speed measured by the man and woman. If the speed of the train is v then we speak of light relative to observers on the ground would be expected to be v+c. However, a major claim of special relativity is that observers on the ground would measure the same speed c for the light in vacuum as the man and woman who are both on the train measure. © Copyright 2015, 2016 and 2017 by Ron Ferril. All rights reserved. Ron Ferril SBCC Physics 101 Chapter 13 2017Jul23A Page 3 of 14 Similar experiments were actually done with light in partial vacuums inside pipes and the motion of the train replaced with the motion of a point on the Earth's surface as the Earth rotated and revolved around the Sun. One such experiment, the Michelson-Morley experiment, used a “Michelson interferometer” in an attempt to detect a medium in which light was thought to travel. One reason for suspecting the existence of such a medium is that all other waves known at the time traveled in media. Thus, it was thought that light may travel in a special type of media that existed even in vacuum. The experiment did not detect any such medium. However, this “null” result of the experiment caused much thought about how light travels. Several questions were answered by new fundamental physical laws introduced in special relativity. When physicists develop a new fundamental theory of physics, they find there are several ways they can choose and express the basic postulates—the fundamental laws—of the theory. When Einstein was developing the Theory of Special Relativity, he chose the following two postulates as being the foundation for his new theory. In these postulates, the term “uniformly moving” means moving without acceleration. The Relativity Principle: All laws of nature are the same in all uniformly moving frames of reference. Law of Invariance of the Speed of Light in Vacuum: The speed of light in vacuum remains the same constant for all frames of reference regardless of the motions of the source of the light or observers. Galilean relativity and special relativity can both assert that the same physical laws apply in all uniformly moving frames, but Galilean relativity is incompatible with the invariance of the speed of light in vacuum. The different results between Galilean and special relativity become tiny when all the speeds involved are much smaller than c, but the differences are significant when speeds closer to c are involved. The consequences of the invariance of the speed of light in vacuum are severe. The concept of simultaneous events (events that occur at the same time) becomes complicated because two events can be simultaneous when viewed by observers in one frame of reference, but not simultaneous when viewed from another frame of reference which is moving relative to the first frame of reference. Also, measurements of lengths, distances and time may be different for different observers. © Copyright 2015, 2016 and 2017 by Ron Ferril. All rights reserved. Ron Ferril SBCC Physics 101 Chapter 13 2017Jul23A Page 4 of 14 Consider motion in one direction to avoid confusion from multiple directions. Suppose observers in two frames of reference measure the speed of a body or wave. Let the measured speed of the body for the first frame be vb1 and the speed be vb2 relative to the second frame. Let the speed of the second frame relative to the first frame be v21 and in the same direction as the velocity of the body or wave. Let the speed of the first frame to the second be v12. Both Galilean and special relativity assure us that v21 = -v21. Galilean relativity has vb2=vb1-v21 and vb1=vb2+v21. However, special relativity has 2 vb2 = [vb1 - v21] / [1 - vb1v21 /c ] 2 vb1 = [vb2 + v21] / [1 + vb2v21 /c ] 2 2 When vb1 is much smaller than c, 1+vb1v21/c and 1-vb2v21/c are both approximately 1 and the equations from special relativity give approximately the same values for vb1 and vb2 as the equations from Galilean relativity. (Thus, the Bohr Correspondence Principle is demonstrated.) Time Dilation and Length Contraction Measurements of length and time can be different for different frames of reference if the frames move relative to each other. Consider again the example of the train.
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