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Two Examples of Circular Motion for Introductory Courses in Relativity Stephanie Wortel and Shimon Malin∗ Department of Physics, Colgate University, Hamilton NY 13346 Mark D. Semon† Department of Physics, Bates College, 44 Campus Ave, Lewiston ME 04240 The circular twin paradox and Thomas Precession are presented in a way that makes both acces- sible to students in introductory relativity courses. Both are discussed by examining what happens during travel around a polygon and then in the limit as the polygon tends to a circle. Since rela- tivistic predictions based on these examples can be verified in experiments with macroscopic objects (e.g atomic clocks and the gyroscopes in Gravity Probe B), they are particularly convincing to introductory students. 1. I. INTRODUCTION when the twins first meet. The paradox is the same as in the linear case: since Twin A is moving relative to Twin Experimental confirmations of relativistic effects are B, and Twin B is moving relative to Twin A, each should especially important for students in introductory relativ- measure the other’s clock as running slow compared to ity courses. Each provides a particular situation that their own. However, what distinguishes the circular twin makes the abstract more concrete, and gives students an paradox from the standard twin paradox is that now both actual physical framework in which to understand what twins accelerate in the same manner. It is true that one is (and is not) happening. accelerates in the clockwise and the other in the coun- terclockwise direction but, since time dilation depends One of the most frequently discussed relativistic effects only on the magnitude of the velocity, this can’t matter. is the standard twin paradox. In its most common form Consequently, unlike what happens in the standard twin Twin A stays on Earth while Twin B travels at a constant paradox, the magnitude of acceleration measured on an velocity v to a neighboring star. Twin B then moves accelerometer can’t distinguish between the two travel- into a new inertial frame and travels with a velocity v ers. However, just as in the standard twin paradox, when back to Earth. The paradox is formulated by making the clocks meet and are face to face each cannot have run two predictions which can’t both be true. On the one slow relative to the other. There can be only one answer; hand, since Twin B is moving in the frame of Twin A, what will that answer be? Twin A should measure a clock held by Twin B to run 5 slow. On the other hand, since Twin A is moving in the Lightman, Press, Price and Teukolsky gave a short, formal solution to the circular twin paradox in 1975 and, frame of Twin B, Twin B should measure a clock held 6 by Twin A to run slow. The paradox, of course, is that in 2004, Cranor, Heider and Price considered the para- when the two twins meet their clocks are face to face and dox in more detail. One purpose of Lightman, et. al. was each cannot have run slow relative to the other. to “present an analysis that should be both mathemat- ically and physically intelligible to beginning relativity The generally acknowledged resolution to the paradox 7 is that the reference frames of Twin A and Twin B are students.” Their approach, however, involved a detailed arXiv:gr-qc/0703090v2 17 Apr 2007 not equivalent. Twin B had to move from one inertial analysis of four different ways in which clocks on a ro- frame to another while Twin A didn’t. Consequently, tating ring could be synchronized and a discussion of the since an accelerometer can distinguish between the two consequences of each synchronization method. The anal- twins, there is no conflict with the Postulate of Special ysis became quite complex, but the authors said the com- Relativity if the clock carried by Twin B runs slow rela- plexity was necessary because, in their opinion, “Special tive to the clock carried by Twin A and not vice versa. relativistic time dilation must be considered in the con- text of clock synchronization ...” and “Time dilation will Following their presentation of the twin paradox many only be observed from a reference frame in which the textbooks discuss its experiment confirmation in cy- 7 1 2 clocks are appropriately synchronized ...”. clotrons (e.g. Bailey et. al., Hay, et.al. ) or with atomic clocks flown in airplanes (e.g. Hafele and Keating,3 Alley, In this paper we show that the circular twin paradox et. al.4). However, neither experimental arrangement in- can be resolved more simply by using the results of the linear twin paradox and that, contrary to what is claimed volves clocks moving along linear trajectories. Rather, 6 they more closely resemble what is called the “circular by Cranor et. al., it is not necessary to examine clock twin paradox,” which is formulated as follows: Suppose synchronization in the reference frame of either twin. Our Twin A and Twin B are each on different rings and that approach is to first consider what happens when the twins the rings are rotating in opposite directions about a com- travel on concentric polygons and then to take the limit mon axis through their centers. One ring can either be as the polygons tend to a circle. just above the other or right next to it. Assume each The polygon method also can be used to discuss twin is holding a clock and that both clocks read zero Thomas Precession8 which, as far as we know, has never 2 been included in an introductory relativity class. Using elementary arguments involving rotations and boosts, we show that the coordinate axes of an inertial frame trav- eling around a polygon are rotated when they return to their starting point by an angle of 2π(γ − 1) relative to their initial orientation. Surprisingly, the direction of the (a) Beginning of Trip as seen in Earth Frame. rotation depends upon whether the polygon is traversed in the clockwise or counterclockwise direction. In fact, Thomas Precession is one of the few effects in special rel- ativity that depends upon the direction of the velocity (b) Turn Around as seen from Earth. rather than only upon its magnitude. (Other examples occur in the addition of non-collinear velocities.9 10). As noted by Uhlenbeck, “even the cognoscenti of the relativ- ity theory (Einstein included!) were quite surprised.”11 The first experimental confirmation of Thomas Preces- sion occurred just after the introduction of spin. In fact, (c) End of Trip as seen from Earth. Thomas Precession played an important role in estab- lishing the concept of spin (and the validity of quantum theory) since without it quantum mechanics is unable FIG. 1: The “Triplet Paradox.” to correctly predict the fine-structure in the spectrum of hydrogen, hydrogen-like atoms, alkali atoms, etc. Perhaps less well-known is that another experiment to measure Thomas Precession is currently underway. The Stanford-NASA satellite Gravity Probe B (GP-B), launched in 2004, contains a gyroscope predicted to pre- its resolution. In Section IV we discuss predictions based cess, in part, due to Thomas Precession. Consequently on the circular twin paradox and their confirmation in Thomas Precession is especially interesting to students experiments with elementary particles in cyclotrons and because it involves the confirmation of a prediction of macroscopic atomic clocks flown in airplanes. In Sec- special relativity with a macroscopic object, and because tion V we discuss how acceleration can be treated in the the results of the gyroscope experiment in GP-B should various twin paradoxes. In Section VI we derive Thomas be announced during 2007 and so are part of a contem- Precession and illustrate how it affects a gyroscope orbit- poraneous physics research experiment that introductory ing the Earth. In Section VII we discuss the anticipated students can understand. Also, as we mention in our dis- confirmation of Thomas Precession with gyroscopes on cussion, the technology involved in the gyroscope experi- the Stanford-NASA satellite Gravity Probe B. We then ment is quite impressive and provides excellent topics for use the gyroscope analogy to discuss spin and the exper- student projects and papers. imental confirmation of Thomas precession in the spec- After discussing the major predictions of relativity – trum of hydrogen. In Section VIII we summarize our time dilation, length contraction, etc. – most texts dis- results and discuss where they fit into an introductory cuss their experimental confirmation with muons, pions, relativity course. and other elementary particles. However, many stu- dents in introductory relativity courses are unfamiliar with these elementary particles and thus come to think of One final note: the main body of this paper is writ- relativity as an interesting philosophical theory that has ten so that it can be understood by students in relativ- little to do with their everyday life. Many are surprised ity courses for nonmajors, such as those using the texts to learn about relativistic effects on macroscopic objects by Mermin12 or Baierlein.13 The more quantitative parts such as atomic clocks and gyroscopes, and studying ex- have been removed from each section and placed in one perimental confirmations made with these objects causes of the three appendices. In this sense the paper has two many students to take the concepts of relativity more tracks, one designed for a more conceptually based course seriously and to develop a new understanding of their and the other for courses in which students have worked importance. Consequently, not only are the circular twin with Lorentz transformations and the relativistic form paradox and Thomas Precession interesting from a con- of Newton’s Second Law; i.e., for courses such as the ceptual point of view, they also are especially effective modern physics part of a standard calculus-based intro- at convincing students of the validity of relativity theory ductory course or any of the higher level courses in the and the importance of its predictions.

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