The optical-mechanical analogy for stationary metrics in general relativity

American Journal of Physics

Volumn 66, Issue 9, 1998, Pages 779-790

  Alsing, Paul M ^a  

^a UNIVERSITY OF NEW MEXICO   (United States)

Author keywords

[No Author keywords available]

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EID: 0032335059     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18957     Document Type: Article

References (36)

1. See the excellent scientific/historical discussion of the optical-mechanical analogy in C. Lanczos, The Variational Principles of Mechanics (Dover, New York, 1986), 4th ed.; W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory (Saunders, Philadelphia, 1968); and J. Evans, K. K. Nandi, and A. Islam, Ref. 4, pp. 1411-1415, listed below.
2. See the excellent scientific/historical discussion of the optical-mechanical analogy in C. Lanczos, The Variational Principles of Mechanics (Dover, New York, 1986), 4th ed.; W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory (Saunders, Philadelphia, 1968); and J. Evans, K. K. Nandi, and A. Islam, Ref. 4, pp. 1411-1415, listed below.
3. See the excellent scientific/historical discussion of the optical-mechanical analogy in C. Lanczos, The Variational Principles of Mechanics (Dover, New York, 1986), 4th ed.; W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory (Saunders, Philadelphia, 1968); and J. Evans, K. K. Nandi, and A. Islam, Ref. 4, pp. 1411-1415, listed below.
4. J. Evans and M. Rosenquist, "F = ma optics," Am. J. Phys. 54 (10), 876-883 (1986); M. Rosenquist and J. Evans, "The classical limit of quantum mechanics from Fermat's principle and the de Broglie relation," ibid. 56 (10), 881-882 (1988); J. Evans, "Simple forms for equations of rays in gradient-index lenses," ibid. 58 (8), 773-778 (1990).
5. J. Evans and M. Rosenquist, "F = ma optics," Am. J. Phys. 54 (10), 876-883 (1986); M. Rosenquist and J. Evans, "The classical limit of quantum mechanics from Fermat's principle and the de Broglie relation," ibid. 56 (10), 881-882 (1988); J. Evans, "Simple forms for equations of rays in gradient-index lenses," ibid. 58 (8), 773-778 (1990).
6. J. Evans and M. Rosenquist, "F = ma optics," Am. J. Phys. 54 (10), 876-883 (1986); M. Rosenquist and J. Evans, "The classical limit of quantum mechanics from Fermat's principle and the de Broglie relation," ibid. 56 (10), 881-882 (1988); J. Evans, "Simple forms for equations of rays in gradient-index lenses," ibid. 58 (8), 773-778 (1990).
7. K. K. Nandi and A. Islam, "On the optical-mechanical analogy in general relativity," Am. J. Phys. 63 (3), 251-256 (1995).
8. J. Evans, K. K. Nandi, and A. Islam, "The optical-mechanical analogy in general relativity: New methods for the paths of light and the planets," Am. J. Phys. 64 (11), 1404-1415 (1996), and reference within.
9. J. Evans, K. K. Nandi, and A. Islam, "The optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons," Gen. Relativ. Gravit. 28 (4), 413-439 (1996).
10. 00(r) = h(r). We follow the latter, and reserve the symbol Ω for denoting constant angular velocity.
11. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, New York, 1975), 4th ed., pp. 234-237, 248-258.
12. See Ref. 7, Sec. 84, pp. 234-237.
13. Choosing the moment "halfway" between the departure and arrival of the light signal at B is the so-called Einstein convention. For an interesting discussion of other possible choices of this factor between the values of zero and unity and its relation to the question of "where, exactly does the aging occur?" in the twin paradox see T. A. Debs and M. L. G. Redhead, "The twin paradox and the conventionality of simultaneity," Am. J. Phys. 64 (4), 384-392 (1996).
14. See Ref. 7, Sec. 97, pp. 286-291.
15. β. 12 See C. Lanczos in Ref. 1, pp. 125-128, 132-134.
16. See Ref. 7, Sec. 88, pp. 250-253.
17. J. N. Islam, Rotating Fields in General Relativity (Cambridge U.P., Cambridge, 1985), p. 27.
18. -1.
19. R. Adler, M. Brazin, and M. Schiffer, Introduction to General Relativity (McGraw-Hill, New York, 1975), 2nd ed., Sec. 6.2, pp. 196-199.
20. 0 explicit throughout our formulas.
21. ij. See Ref. 7, p. 253.
22. E. Harris, "Analogy between general relativity and electromagnetism for slowly moving particles in weak gravitational fields," Am. J. Phys. 59 (5), 421-424 (1991).
23. See Ref. 7, pp. 254-255 as well as, e.g., H. Ohanian, Gravitation and Spacetime (Norton, New York, 1976), p. 311; J. Foster and J. D. Nightangale, A Short Course in General Relativity (Springer-Verlag, New York, 1994), 2nd ed., pp. 93-96; R. D'Inverno, Introducing Einstein's Relativity (Oxford U.P., New York, 1992), pp. 253-254; B. F. Schutz, A First Course in General Relativity (Cambridge U.P., New York, 1985), p. 298; T. A. Weber, "A note on rotating coordinates in relativity," Am. J. Phys. 65 (6), 486-487 (1997).
24. See Ref. 7, pp. 254-255 as well as, e.g., H. Ohanian, Gravitation and Spacetime (Norton, New York, 1976), p. 311; J. Foster and J. D. Nightangale, A Short Course in General Relativity (Springer-Verlag, New York, 1994), 2nd ed., pp. 93-96; R. D'Inverno, Introducing Einstein's Relativity (Oxford U.P., New York, 1992), pp. 253-254; B. F. Schutz, A First Course in General Relativity (Cambridge U.P., New York, 1985), p. 298; T. A. Weber, "A note on rotating coordinates in relativity," Am. J. Phys. 65 (6), 486-487 (1997).
25. See Ref. 7, pp. 254-255 as well as, e.g., H. Ohanian, Gravitation and Spacetime (Norton, New York, 1976), p. 311; J. Foster and J. D. Nightangale, A Short Course in General Relativity (Springer-Verlag, New York, 1994), 2nd ed., pp. 93-96; R. D'Inverno, Introducing Einstein's Relativity (Oxford U.P., New York, 1992), pp. 253-254; B. F. Schutz, A First Course in General Relativity (Cambridge U.P., New York, 1985), p. 298; T. A. Weber, "A note on rotating coordinates in relativity," Am. J. Phys. 65 (6), 486-487 (1997).
26. See Ref. 7, pp. 254-255 as well as, e.g., H. Ohanian, Gravitation and Spacetime (Norton, New York, 1976), p. 311; J. Foster and J. D. Nightangale, A Short Course in General Relativity (Springer-Verlag, New York, 1994), 2nd ed., pp. 93-96; R. D'Inverno, Introducing Einstein's Relativity (Oxford U.P., New York, 1992), pp. 253-254; B. F. Schutz, A First Course in General Relativity (Cambridge U.P., New York, 1985), p. 298; T. A. Weber, "A note on rotating coordinates in relativity," Am. J. Phys. 65 (6), 486-487 (1997).
27. See Ref. 7, pp. 254-255 as well as, e.g., H. Ohanian, Gravitation and Spacetime (Norton, New York, 1976), p. 311; J. Foster and J. D. Nightangale, A Short Course in General Relativity (Springer-Verlag, New York, 1994), 2nd ed., pp. 93-96; R. D'Inverno, Introducing Einstein's Relativity (Oxford U.P., New York, 1992), pp. 253-254; B. F. Schutz, A First Course in General Relativity (Cambridge U.P., New York, 1985), p. 298; T. A. Weber, "A note on rotating coordinates in relativity," Am. J. Phys. 65 (6), 486-487 (1997).
28. See Ref. 7, pp. 254-255 as well as, e.g., H. Ohanian, Gravitation and Spacetime (Norton, New York, 1976), p. 311; J. Foster and J. D. Nightangale, A Short Course in General Relativity (Springer-Verlag, New York, 1994), 2nd ed., pp. 93-96; R. D'Inverno, Introducing Einstein's Relativity (Oxford U.P., New York, 1992), pp. 253-254; B. F. Schutz, A First Course in General Relativity (Cambridge U.P., New York, 1985), p. 298; T. A. Weber, "A note on rotating coordinates in relativity," Am. J. Phys. 65 (6), 486-487 (1997).
29. A. Einstein, The Meaning of Relativity (Princeton U.P., Princeton, NJ, 1955), pp. 55-63.
30. R. P. Kerr, "Gravitational field of a spinning mass as an example of algebraically special metrics," Phys. Rev. Lett. 11, 237-238 (1963).
31. See Ref. 7, Sec. 104, pp. 321-330. See also Refs. 14 and 16, and H. Ohanian, R. D'Inverno, and B. F. Schutz in Ref. 20 and C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), Chap. 33, pp. 872-915 .
32. See Ref. 7, Sec. 104, pp. 321-330. See also Refs. 14 and 16, and H. Ohanian, R. D'Inverno, and B. F. Schutz in Ref. 20 and C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), Chap. 33, pp. 872-915 .
33. See Ref. 7, Sec. 104, pp. 321-330. See also Refs. 14 and 16, and H. Ohanian, R. D'Inverno, and B. F. Schutz in Ref. 20 and C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), Chap. 33, pp. 872-915 .
34. See Ref. 7, Sec. 104, pp. 321-330. See also Refs. 14 and 16, and H. Ohanian, R. D'Inverno, and B. F. Schutz in Ref. 20 and C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), Chap. 33, pp. 872-915 .
35. R. H. Boyer and R. W. Lindquist, "Maximal analytic extension of the Kerr metric," J. Math. Phys. 8, 265-281 (1967).
36. J. B. Marion, Classical Dynamics of Particles and Systems (Academic, New York, 1970), 2nd ed., Chap. 8, pp. 247-249.