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Exploring the transition from special to general relativity
1.H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity (Dover, New York, 1952), pp. 99–108 and 111–118.
2.J. Stachel, “The rigidly rotating disk as the ‘missing link’ in the history of general relativity,” in Einstein from B to Z, edited by J. Stachel (Plenum, New York, 1980), pp. 245–261.
3.A. Einstein, Relativity: The Special and the General Theory (Three Rivers, New York, 1961), pp. 75–95.
4.A. Einstein, The Meaning of Relativity (Princeton U. P., Princeton, NJ, 1956), pp. 55–61.
5.A. Einstein and L. Infeld, The Evolution of Physics (Simon and Schuster, New York, 1966), pp. 209–238.
6.In this argument Einstein implicitly assumes the material disk is not affected by rotation which, as discussed in the following, many authors have pointed out is incorrect.
7.Reference 1, pp. 115–117.
8.Reference 2, pp. 251–252.
9.E. T. Whittaker, From Euclid to Eddington (Cambridge U. P., Cambridge, 1979), pp. 108–111.
10.A. S. Eddington, The Mathematical Theory of Relativity (Cambridge U. P., Cambridge, 1923), pp. 112–113.
12.Apparently it didn’t occur to Ehrenfest (nor to anyone but Einstein) that one resolution of the paradox is that the geometry on the disk is non-Euclidean. Interestingly, Pauli (Ref. 13) stated that the conclusion to be drawn from Ehrenfest’s argument is that it is impossible to set a material disk into rotation and have it remain rigid according to the definition of rigidity given by Born (Ref. 14).
13.W. Pauli, Theory of Relativity (Pergamon, New York, 1958), pp. 131–132.
16.See Ref. 17, Chap. 15 for an annotated historical survey.
17.G. Rizzi and M. L. Ruggier, Relativity in Rotating Frames (Kluwer Acameric, Boston, 2004).
18.J. R. Taylor, C. D. Zafiratos, and M. A. Dubson, Modern Physics for Scientists and Engineers, 2nd ed. (Pearson/Prentice Hall, Upper Saddle River, NJ, 2004), pp. 72–79.
19.S. T. Thornton and A. Rex, Modern Physics for Scientists and Engineers, 2nd ed. (Saunders, Philadelphia, 2005), pp. 507–509.
20.B. Ryden, Introduction to Cosmology (Addison-Wesley, Reading, MA, 2002), pp. 27–30.
21.S. Weinberg, Gravitation and Cosmology: Applications of the General Theory of Relativity (Wiley, Hoboken, NJ, 1972).
22.S. Weinberg, Cosmology (Oxford U. P., Oxford, 2008).
23.R. M. Wald, General Relativity (U. of Chicago, Chicago, 1984).
24.W. Rindler, Relativity: Special, General and Cosmological, 2nd ed. (Oxford U. P., Oxford, 2006), pp. 15–27.
25.The length of each side of the -gon can be measured by the relativistic traveler using any of the standard methods. For example, two observers in the traveler’s frame can measure the location of each end of a side of the -gon at the same time, or a single observer in the traveler’s frame can measure the time necessary for both ends of the -gon to move by and then multiply this time by the speed at which the side of the -gon is moving, or an observer in the traveling frame can simply use the Lorentz transformation. Students will have studied these methods earlier in the course as part of the derivation and discussion of length contraction. In the Appendix we use the Lorentz transformation to derive the length of one side of the -gon as measured in the traveler’s frame.
26.S. Wortel, S. Malin, and M. D. Semon, “Two examples of circular motion for introductory courses in special relativity,” Am. J. Phys. 75, 1123–1133 (2007).
27.When considering the rotating material disk, Einstein’s conclusions about the length of its circumference and the rate of a clock traveling on the circumference don’t satisfy this consistency check because he concluded that the rotating observer measures the circumference as and the time to complete one rotation as . The ratio of these two quantities is not equal to the speed measured by the inertial observer.
28.We are only considering points on the axis of the traveling frame whose speed relative to an inertial observer is .
29.Translated variously as “remarkable,” “illustrious,” “eminent,” or “distinguished” theorem. However, as John Rhodes points out, since “,” a more literal translation is “A theorem that sticks out from the herd.”
31.Note that Einstein was mistaken when he concluded that the spatial curvature of a frame traveling on a circular path is hyperbolic, that is, when he concluded that and .
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