The optical-mechanical analogy in general relativity: New methods for the paths of light and of the planets

American Journal of Physics

Volumn 64, Issue 11, 1996, Pages 1404-1415

  Evans, James ^a   Nandi, Kamal K ^b   Islam, Anwarul ^c  

^a UNIVERSITY OF PUGET SOUND   (United States)

^b UNIVERSITY OF NORTH BENGAL   (India)

^c Department of Mathematics   (Bangladesh)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030485820     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18366     Document Type: Article

References (62)

1. J. Evans and M. Rosenquist, "'F = ma' optics," Am. J. Phys. 54, 876-883 (1986).
2. J. Evans, "Simple forms for equations of rays in gradient-index lenses," Am. J. Phys. 58, 773-778 (1990).
3. A. S. Eddington, Space, Time and Gravitation (Cambridge U. P., Cambridge, 1920), p. 109.
4. S. Plebanski, "Electromagnetic waves in gravitational fields," Phys. Rev. 118, 1396-1408 (1960).
5. F. de Felice, "On the gravitational field acting as an optical medium," Gen. Rel. Grav. 2, 347-357 (1971).
6. K. K. Nandi and A. Islam, "On the optical-mechanical analogy in general relativity," Am. J. Phys. 63, 251-256 (1995).
7. For more detail see James Evans, Kamal K. Nandi, and Anwarul Islam, "The optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons," Gen. Rel. Grav. 28, 413-439 (1996).
8. 0 for the speed of light.
9. The isotropic form of the Schwarzschild line element is mentioned in many standard texts, e.g., W. Misner, K. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco, 1973), p. 840. For a systematic technique for finding isotropic coordinates see R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity and Gravitation (McGraw-Hill, New York, 1965), pp. 174-177. Effective indices of refraction for a number of metrics are given in Ret. 7. Note in particular that any spherically symmetric static metric can be put into isotropic form.
10. The isotropic form of the Schwarzschild line element is mentioned in many standard texts, e.g., W. Misner, K. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Co., San Francisco, 1973), p. 840. For a systematic technique for finding isotropic coordinates see R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity and Gravitation (McGraw-Hill, New York, 1965), pp. 174-177. Effective indices of refraction for a number of metrics are given in Ret. 7. Note in particular that any spherically symmetric static metric can be put into isotropic form.
11. For passing between Hamilton's principle and Jacobi's form of Maupertuis's principle, see Cornelius Lanczos, The Variational Principles of Mechanics (Dover, New York, 1986), 4th ed., pp. 125-128 and 132-134.
12. We have obtained Eq. (19) directly from the geodesic condition. But it may also be considered a special case of the three-dimensional variational principle of Weyl and Levi-Civita. Hermann Weyl, "Zur Gravitations-theorie," Ann. Phys. 54, 117-145 (1917). T. Levi-Civita, "Statica einsteiniana," Atti della Reale Accademia Dei Lincei, Ser. 5, Rendiconti, Classe di scienze fisiche, matematiche e naturali 26, 458-470 (1917). To obtain Eq. (19) from the variational principle of Weyl and Levi-Civita, one goes into isotropic coordinates, introduces the index of refraction, and then invokes conservation of energy on the varied path.
13. We have obtained Eq. (19) directly from the geodesic condition. But it may also be considered a special case of the three-dimensional variational principle of Weyl and Levi-Civita. Hermann Weyl, "Zur Gravitations-theorie," Ann. Phys. 54, 117-145 (1917). T. Levi-Civita, "Statica einsteiniana," Atti della Reale Accademia Dei Lincei, Ser. 5, Rendiconti, Classe di scienze fisiche, matematiche e naturali 26, 458-470 (1917). To obtain Eq. (19) from the variational principle of Weyl and Levi-Civita, one goes into isotropic coordinates, introduces the index of refraction, and then invokes conservation of energy on the varied path.
14. That light obeys Fermat's principle in static metrics is, of course, a well-known fact. That is, the coordinate time of travel between two fixed points is invariant under infinitesimal deformations of the path. T. Levi-Civita, "La teoria di Einstein e il principio di Fermat," Nuovo Cimento 16, 105-114 (1918). For a good recent discussion see N. Straumann, General Relativity und Relativistic Astrophysics (Springer-Verlag, Berlin, 1984), pp. 99-100.
15. That light obeys Fermat's principle in static metrics is, of course, a well-known fact. That is, the coordinate time of travel between two fixed points is invariant under infinitesimal deformations of the path. T. Levi-Civita, "La teoria di Einstein e il principio di Fermat," Nuovo Cimento 16, 105-114 (1918). For a good recent discussion see N. Straumann, General Relativity und Relativistic Astrophysics (Springer-Verlag, Berlin, 1984), pp. 99-100.
16. 0. This is only a matter of convention. We call A the optical action because dA is proportional to c(r) dl, and thus is analogous to the action u(r) dl of classical mechanics.
17. 2 should be regarded as coming from the optical situations, in which it is perfectly apt. For particle dynamics, whether classical or relativistic, the term is perhaps not a perfect choice.
18. Other metrics amenable to the same reatment include the de Sitter metric, the Reissner-Nordström metric, the Bertotti-Robinson metric and the Halilsoy metric. These are all discussed in Ref. 7.
19. For an example of a radar-echo delay calculation which requires the use of Eq. (26), see Ref. 7.
20. p is not the canonical momentum used above, but rather the optical analog to the classical Newtonian momentum.
21. The same result is obtained by standard methods in Misner, Thorne and Wheeler (Ref. 9), p. 671.
22. 0).
23. For a detailed solution of Eq. (74), see, for example, Jerry B. Marion, Classical Dynamics of Particles and Systems (Academic, New York, 1970), 2nd ed., pp. 266-270.
24. Fermat made his principle of least time known through correspondence, starting around 1662. He wrote two short pieces for this purpose, which were never published in a regular way during his lifetime. They first appeared in print in the published correspondence of Descartes. These two pieces can be found in the standard edition of Fermat's works: Oeuvres de Fermat, edited by Paul Tannery and Charles Henry (Gautier-Villars et Fils, Paris, 1891-1912). See "Analysis ad refractiones," in Vol. 1, pp. 170-172, with French translation in Vol. 3, pp. 149-151; and "Synthesis ad refractiones," Vol. 1, pp. 173-179, with French translation in Vol. 3, pp. 151-156. For an English translation of the first piece, see William Francis Magie, A Source Book in Physics (McGraw-Hill, New York, 1935), pp. 278-280.
25. Fermat made his principle of least time known through correspondence, starting around 1662. He wrote two short pieces for this purpose, which were never published in a regular way during his lifetime. They first appeared in print in the published correspondence of Descartes. These two pieces can be found in the standard edition of Fermat's works: Oeuvres de Fermat, edited by Paul Tannery and Charles Henry (Gautier-Villars et Fils, Paris, 1891-1912). See "Analysis ad refractiones," in Vol. 1, pp. 170-172, with French translation in Vol. 3, pp. 149-151; and "Synthesis ad refractiones," Vol. 1, pp. 173-179, with French translation in Vol. 3, pp. 151-156. For an English translation of the first piece, see William Francis Magie, A Source Book in Physics (McGraw-Hill, New York, 1935), pp. 278-280.
26. Fermat made his principle of least time known through correspondence, starting around 1662. He wrote two short pieces for this purpose, which were never published in a regular way during his lifetime. They first appeared in print in the published correspondence of Descartes. These two pieces can be found in the standard edition of Fermat's works: Oeuvres de Fermat, edited by Paul Tannery and Charles Henry (Gautier-Villars et Fils, Paris, 1891-1912). See "Analysis ad refractiones," in Vol. 1, pp. 170-172, with French translation in Vol. 3, pp. 149-151; and "Synthesis ad refractiones," Vol. 1, pp. 173-179, with French translation in Vol. 3, pp. 151-156. For an English translation of the first piece, see William Francis Magie, A Source Book in Physics (McGraw-Hill, New York, 1935), pp. 278-280.
27. Fermat made his principle of least time known through correspondence, starting around 1662. He wrote two short pieces for this purpose, which were never published in a regular way during his lifetime. They first appeared in print in the published correspondence of Descartes. These two pieces can be found in the standard edition of Fermat's works: Oeuvres de Fermat, edited by Paul Tannery and Charles Henry (Gautier-Villars et Fils, Paris, 1891-1912). See "Analysis ad refractiones," in Vol. 1, pp. 170-172, with French translation in Vol. 3, pp. 149-151; and "Synthesis ad refractiones," Vol. 1, pp. 173-179, with French translation in Vol. 3, pp. 151-156. For an English translation of the first piece, see William Francis Magie, A Source Book in Physics (McGraw-Hill, New York, 1935), pp. 278-280.
28. Fermat made his principle of least time known through correspondence, starting around 1662. He wrote two short pieces for this purpose, which were never published in a regular way during his lifetime. They first appeared in print in the published correspondence of Descartes. These two pieces can be found in the standard edition of Fermat's works: Oeuvres de Fermat, edited by Paul Tannery and Charles Henry (Gautier-Villars et Fils, Paris, 1891-1912). See "Analysis ad refractiones," in Vol. 1, pp. 170-172, with French translation in Vol. 3, pp. 149-151; and "Synthesis ad refractiones," Vol. 1, pp. 173-179, with French translation in Vol. 3, pp. 151-156. For an English translation of the first piece, see William Francis Magie, A Source Book in Physics (McGraw-Hill, New York, 1935), pp. 278-280.
29. Fermat made his principle of least time known through correspondence, starting around 1662. He wrote two short pieces for this purpose, which were never published in a regular way during his lifetime. They first appeared in print in the published correspondence of Descartes. These two pieces can be found in the standard edition of Fermat's works: Oeuvres de Fermat, edited by Paul Tannery and Charles Henry (Gautier-Villars et Fils, Paris, 1891-1912). See "Analysis ad refractiones," in Vol. 1, pp. 170-172, with French translation in Vol. 3, pp. 149-151; and "Synthesis ad refractiones," Vol. 1, pp. 173-179, with French translation in Vol. 3, pp. 151-156. For an English translation of the first piece, see William Francis Magie, A Source Book in Physics (McGraw-Hill, New York, 1935), pp. 278-280.
30. Descartes had derived the sine law from a particle model of light. The component of the velocity parallel to the interface was assumed to remain constant when light crossed from one medium to another. Since the ray lies closer to the normal in water than in air, this implies that it travels faster in the water. René Descartes, La dioptrique (published as a supplement to Discours de la méthode (Leiden, 1637). Oeuvres de Descartes, edited by Charles Adam and Paul Tannery (J. Vrin, Paris, 1874-1882), 2nd ed., Vol. 6., pp. 93-105. Discourse on Method, Optics, Geometry, and Meteorology, translated by Paul J. Olscamp (Bobbs-Merrill, Indianapolis, 1965).
31. Descartes had derived the sine law from a particle model of light. The component of the velocity parallel to the interface was assumed to remain constant when light crossed from one medium to another. Since the ray lies closer to the normal in water than in air, this implies that it travels faster in the water. René Descartes, La dioptrique (published as a supplement to Discours de la méthode (Leiden, 1637). Oeuvres de Descartes, edited by Charles Adam and Paul Tannery (J. Vrin, Paris, 1874-1882), 2nd ed., Vol. 6., pp. 93-105. Discourse on Method, Optics, Geometry, and Meteorology, translated by Paul J. Olscamp (Bobbs-Merrill, Indianapolis, 1965).
32. Descartes had derived the sine law from a particle model of light. The component of the velocity parallel to the interface was assumed to remain constant when light crossed from one medium to another. Since the ray lies closer to the normal in water than in air, this implies that it travels faster in the water. René Descartes, La dioptrique (published as a supplement to Discours de la méthode (Leiden, 1637). Oeuvres de Descartes, edited by Charles Adam and Paul Tannery (J. Vrin, Paris, 1874-1882), 2nd ed., Vol. 6., pp. 93-105. Discourse on Method, Optics, Geometry, and Meteorology, translated by Paul J. Olscamp (Bobbs-Merrill, Indianapolis, 1965).
33. On Fermat versus the Cartesians, see René Dugas, A History of Mechanics, translated by J. R. Maddox (Dover, New York, 1988), pp. 258-259.
34. Christiaan Huygens, Traité de la lumière (Leiden, 1690). Oeuvres complète de Christiaan Huygens (Martinus Nijhoff, La Haye, 1888-1950), Vol. 19, pp. 451-537. English translation: Treatise on Light...by Christiaan Huygens, translated by Silvanus P. Thompson (Dover, New York, 1962; first published in 1912). For Huygens's treatment of Fermat's principle, see Oeuvres, Vol. 19, pp. 489-490; Thompson, pp. 42-45.
35. Christiaan Huygens, Traité de la lumière (Leiden, 1690). Oeuvres complète de Christiaan Huygens (Martinus Nijhoff, La Haye, 1888-1950), Vol. 19, pp. 451-537. English translation: Treatise on Light...by Christiaan Huygens, translated by Silvanus P. Thompson (Dover, New York, 1962; first published in 1912). For Huygens's treatment of Fermat's principle, see Oeuvres, Vol. 19, pp. 489-490; Thompson, pp. 42-45.
36. Christiaan Huygens, Traité de la lumière (Leiden, 1690). Oeuvres complète de Christiaan Huygens (Martinus Nijhoff, La Haye, 1888-1950), Vol. 19, pp. 451-537. English translation: Treatise on Light...by Christiaan Huygens, translated by Silvanus P. Thompson (Dover, New York, 1962; first published in 1912). For Huygens's treatment of Fermat's principle, see Oeuvres, Vol. 19, pp. 489-490; Thompson, pp. 42-45.
37. Christiaan Huygens, Traité de la lumière (Leiden, 1690). Oeuvres complète de Christiaan Huygens (Martinus Nijhoff, La Haye, 1888-1950), Vol. 19, pp. 451-537. English translation: Treatise on Light...by Christiaan Huygens, translated by Silvanus P. Thompson (Dover, New York, 1962; first published in 1912). For Huygens's treatment of Fermat's principle, see Oeuvres, Vol. 19, pp. 489-490; Thompson, pp. 42-45.
38. Christiaan Huygens, Traité de la lumière (Leiden, 1690). Oeuvres complète de Christiaan Huygens (Martinus Nijhoff, La Haye, 1888-1950), Vol. 19, pp. 451-537. English translation: Treatise on Light...by Christiaan Huygens, translated by Silvanus P. Thompson (Dover, New York, 1962; first published in 1912). For Huygens's treatment of Fermat's principle, see Oeuvres, Vol. 19, pp. 489-490; Thompson, pp. 42-45.
39. Pierre-Louis Moreau de Maupertuis, Oeuvres (Georg Olms, Hildesheim and New York, 1965-1974), 4 Vols. (This is a reprint based on the editions of Lyon, 1768 and Berlin, 1758.) Maupertuis announced his principle of least action in a paper read in 1744 to the Paris Academy of Sciences, and titled, "The Agreement between Different Laws of Nature, Which Had until Now Appeared Incompatible," (Oeuvres, Vol. 4, pp. 3-28). This first paper addressed the refraction of light. A second paper, "Research on the Laws of Motion," read to the Royal Academy of Sciences of Berlin in 1746, treated the collisions of elastic and inelastic bodies by the method of least action (Oeuvres, Vol. 4, pp. 31-42). It is one of the many ironies of this story that Maupertuis applied least action first to light, and only later to mechanics: the principle was applied first to the domain in which it is invalid.
40. Pierre-Louis Moreau de Maupertuis, Oeuvres (Georg Olms, Hildesheim and New York, 1965-1974), 4 Vols. (This is a reprint based on the editions of Lyon, 1768 and Berlin, 1758.) Maupertuis announced his principle of least action in a paper read in 1744 to the Paris Academy of Sciences, and titled, "The Agreement between Different Laws of Nature, Which Had until Now Appeared Incompatible," (Oeuvres, Vol. 4, pp. 3-28). This first paper addressed the refraction of light. A second paper, "Research on the Laws of Motion," read to the Royal Academy of Sciences of Berlin in 1746, treated the collisions of elastic and inelastic bodies by the method of least action (Oeuvres, Vol. 4, pp. 31-42). It is one of the many ironies of this story that Maupertuis applied least action first to light, and only later to mechanics: the principle was applied first to the domain in which it is invalid.
41. Pierre-Louis Moreau de Maupertuis, Oeuvres (Georg Olms, Hildesheim and New York, 1965-1974), 4 Vols. (This is a reprint based on the editions of Lyon, 1768 and Berlin, 1758.) Maupertuis announced his principle of least action in a paper read in 1744 to the Paris Academy of Sciences, and titled, "The Agreement between Different Laws of Nature, Which Had until Now Appeared Incompatible," (Oeuvres, Vol. 4, pp. 3-28). This first paper addressed the refraction of light. A second paper, "Research on the Laws of Motion," read to the Royal Academy of Sciences of Berlin in 1746, treated the collisions of elastic and inelastic bodies by the method of least action (Oeuvres, Vol. 4, pp. 31-42). It is one of the many ironies of this story that Maupertuis applied least action first to light, and only later to mechanics: the principle was applied first to the domain in which it is invalid.
42. Fermat, Oeuvres (Ref. 21), Vol. 4, p. 15.
43. For an account of Euler's and Lagrange's evelopment of Maupertuis's principle, see Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Mechanics (Dover, New York, 1979).
44. Pierre-Simon de Laplace, Oeuvres complète de Laplace, (Gauthier-Villars, Paris, 1878-1904), 13 vols., Vol. 12, pp. 267-298.
45. Hamilton's work is most easily consulted in The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U.P., Cambridge, 1931-1967), 3 vols.; Vol. 1, Geometrical Optics, edited by A. W. Conway and J. L. Synge; Vol. 2, Dynamics, edited by A. W. Conway and A. J. McConnell; Vol. 3, Algebra, edited by H. Halberstam and R. E. Ingram.
46. Hamilton's work is most easily consulted in The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U.P., Cambridge, 1931-1967), 3 vols.; Vol. 1, Geometrical Optics, edited by A. W. Conway and J. L. Synge; Vol. 2, Dynamics, edited by A. W. Conway and A. J. McConnell; Vol. 3, Algebra, edited by H. Halberstam and R. E. Ingram.
47. Hamilton's work is most easily consulted in The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U.P., Cambridge, 1931-1967), 3 vols.; Vol. 1, Geometrical Optics, edited by A. W. Conway and J. L. Synge; Vol. 2, Dynamics, edited by A. W. Conway and A. J. McConnell; Vol. 3, Algebra, edited by H. Halberstam and R. E. Ingram.
48. Hamilton's work is most easily consulted in The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U.P., Cambridge, 1931-1967), 3 vols.; Vol. 1, Geometrical Optics, edited by A. W. Conway and J. L. Synge; Vol. 2, Dynamics, edited by A. W. Conway and A. J. McConnell; Vol. 3, Algebra, edited by H. Halberstam and R. E. Ingram.
49. Hamilton, Ref. 29, Vol. 1, p. 14.
50. Hamilton, Ref. 29, Vol. 1, pp. 168-169.
51. See, for example, Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1964), 2nd ed., p. 112; or Miles V. Klein, Optics (Wiley, New York, 1970), p. 28.
52. See, for example, Max Born and Emil Wolf, Principles of Optics (Pergamon, Oxford, 1964), 2nd ed., p. 112; or Miles V. Klein, Optics (Wiley, New York, 1970), p. 28.
53. x.
54. Reprinted in Hamilton, Ref. 29, Vol. 1, pp. 311-332. For an excellent summary of Hamilton's further development of his dynamical theory (leading to the canonical equations of motion and the Hamilton-Jacobi equation) see Thomas L. Hankins, Sir William Rowan Hamilton (Johns Hopkins U.P., Baltimore and London, 1980), pp. 181-198. Hankins also provides a good summary of Hamilton's work in geometrical optics (pp. 61-81).
55. Reprinted in Hamilton, Ref. 29, Vol. 1, pp. 311-332. For an excellent summary of Hamilton's further development of his dynamical theory (leading to the canonical equations of motion and the Hamilton-Jacobi equation) see Thomas L. Hankins, Sir William Rowan Hamilton (Johns Hopkins U.P., Baltimore and London, 1980), pp. 181-198. Hankins also provides a good summary of Hamilton's work in geometrical optics (pp. 61-81).
56. Cf. editors' introduction to Hamilton, Ref. 29, Vol. 1, p. xxi.
57. Hankins, Ref. 34, p. 86.
58. Hankins, Ref. 34, p. 87.
59. Louis de Brogue, Recherches sur la théorie des quanta (Masson, Paris, 1963). (A reprint of the thesis of 1924). Here, we have expressed the de Broglie relation nonrelativistically. de Broglie's relation does, of course, hold relativistically: We need only replace mv by the relativistic momentum p.
60. E. Schrödinger, ∂Quantisierung als Eigenwertproblem,∂ Ann. Phys. 79, 361-376 (1926). This paper is available in English translation in J. F. Shearer and W. M. Deans (translators), Collected Papers on Wave Mechanics by E. Schrödinger (Blackie and Son, Glasgow, 1928). For an account of Schrödinger's procedure, see Ref. 27, pp. 118-119, or W. Moore, Schrödinger: Life and Thought (Cambridge U.P., Cambridge, 1989), pp. 200-207.
61. E. Schrödinger, ∂Quantisierung als Eigenwertproblem,∂ Ann. Phys. 79, 361-376 (1926). This paper is available in English translation in J. F. Shearer and W. M. Deans (translators), Collected Papers on Wave Mechanics by E. Schrödinger (Blackie and Son, Glasgow, 1928). For an account of Schrödinger's procedure, see Ref. 27, pp. 118-119, or W. Moore, Schrödinger: Life and Thought (Cambridge U.P., Cambridge, 1989), pp. 200-207.
62. E. Schrödinger, ∂Quantisierung als Eigenwertproblem,∂ Ann. Phys. 79, 361-376 (1926). This paper is available in English translation in J. F. Shearer and W. M. Deans (translators), Collected Papers on Wave Mechanics by E. Schrödinger (Blackie and Son, Glasgow, 1928). For an account of Schrödinger's procedure, see Ref. 27, pp. 118-119, or W. Moore, Schrödinger: Life and Thought (Cambridge U.P., Cambridge, 1989), pp. 200-207.