Advertisement

Astrophysics and Space Science

, 306:205 | Cite as

The Gravitational “Plane Waves” of Liu & Zhou and the Nonexistence of Dynamic Solutions for Einstein’s Equation

  • C. Y. Lo
Original Article

Abstract

Although both the electromagnetic wave and the gravitational wave can be produced approximately from Maxwell-type equations, there are subtle differences in their respective exact equations. Since gravitational wave carries energy-momentum, the exact field equation of a gravitational wave must have a nonzero source term along its path, whereas a field equation for an electromagnetic wave does not. This explains that there is no weak wave solution of Einstein equation. Historically, neither Einstein & Rosen nor the Physical Review was aware that the nonexistence of gravitational wave solutions is due to a violation of the principle of causality. It is pointed out that the criterion of Liu & Zhou on plane-waves is valid since the principle of causality requires the existence of weak limits. However, due to the influence of the popular but unverified assumption of the existence of dynamic solutions, they made careless errors in their calculations and incorrectly concluded that their plane-waves have weak limits. It is shown that “plane-waves” of Liu & Zhou, is actually unbounded in amplitude, and have no weak limit. Therefore, Liu & Zhou provide additional evidence in supporting the nonexistence of dynamic solutions.

Keywords

Einstein’s equivalence principle Einstein-Minkowski condition Euclidean-like structure Dynamic solution Gravitational radiation Principle of causality Relativistic causality Plane-wave 

References

  1. Au, C., Fang, L.Z., To, F.T.: In: Jantzen, R.T., Mac Keiser, G. (ed.), Ruffini, R. (ser. ed.) Proceedings of the 7th Marcel Grossmann Meeting On Gen. Relat., Stanford, 1994, World Scientific, Singapore, p. 289 (1996)Google Scholar
  2. Bondi, H., Pirani, F.A.E., Robinson, I.: Proc. R. Soc. Lond. A 251, 519–533 (1959)Google Scholar
  3. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Princeton University Press (1993)Google Scholar
  4. Damour, T., Taylor, J.H.: Astrophys. J. 366, 501–511 (1991)CrossRefADSGoogle Scholar
  5. Damour, T., Taylor, J.H.: Phys. Rev. D 45(6), 1840–1868 (1992)CrossRefADSGoogle Scholar
  6. Eddington, A.S.: The mathematical theory of relativity (1923). Chelsea, New York (1975)Google Scholar
  7. Ehlers, J., Kundt, W.: Exact solutions of gravitational field equations. In: Witten, L. (ed.) Gravitation: an introduction to current research, Wiley, New York, London, p. 86. (1962)Google Scholar
  8. Einstein, A., Infeld, L., Hoffmann, B.: Ann. Math. 39(1), 65–100 (1938)CrossRefMathSciNetGoogle Scholar
  9. Einstein, A., Lorentz, H.A., Minkowski, H., Weyl, H.: The principle of relativity. Dover, N.Y. (1923)Google Scholar
  10. Einstein, A., Rosen, N.: J. Franklin Inst. 223, 43 (1937)CrossRefGoogle Scholar
  11. Einstein, A.: Sitzungsberi, Preuss, Acad. Wis. 1, 154 (1918)Google Scholar
  12. Einstein, A.: The Born Einstein Letters: Friendship, Politics, and Physics in Uncertain Times, p. 122. MacMillan, New York (2005)Google Scholar
  13. Einstein, A.: The meaning of relativity. Princeton University Press (1954)Google Scholar
  14. Fang, L.Z., Ruffini, R.: The fundamentals of relativity and astrophysics. Technology Press, Shanghai (1981)Google Scholar
  15. Feynman, R.P.: The Feynman Lectures on Gravitation. Addison-Wesley, New York (1996)Google Scholar
  16. Fock, V.A.: Rev. Mod. Phys. 29, 325 (1957)CrossRefADSMathSciNetzbMATHGoogle Scholar
  17. Fock, V.A.: The theory of space time and gravitation, translated by N. Kemmer. Pergamon Press (1964)Google Scholar
  18. Gullstrand, A.: Ark. Mat. Astr. Fys. 16(8) (1921);Google Scholar
  19. Gullstrand, A.: Ark. Mat. Astr. Fys. 17(3) (1922)Google Scholar
  20. Hawking, S.: A brief history of time. Bantam Books, New York (1988)Google Scholar
  21. Hogarth, J.E.: Particles, fields, and rigid bodies in the formulation of relativity theories. Ph. D. Thesis 1953, Dept. of Math., Royal Holloway College, University of London, p. 6 (1953)Google Scholar
  22. Hu, N., Zhang, D.-H., Ding, H.G.: Acta Phys. Sinica 30(8), 1003–1010 (1981)Google Scholar
  23. Infeld, L.: Quest: An Autobiography. Chelsea, New York (1980)Google Scholar
  24. Isaacson, R.A.: Phys. Rev. 116, 1263–1280 (1968)CrossRefADSGoogle Scholar
  25. Kennefick, D.: Einstein versus the physical review. Physics Today (September, 2005)Google Scholar
  26. Kramer, D., Stephani, H., Herlt, E., MacCallum, M.: 1In: Schmutzer, E. (ed.) Exact Solutions of Einstein's Field Equations, pp. 233–236. Cambridge University Press, Cambridge (1980)Google Scholar
  27. Lerner, E.J.: The big bang never happened, Vintage, New York p. 331, (1992)Google Scholar
  28. Liu, H.Y., Zhou, P.-Y.: Scientia Sincia (Series A) XXVIII(6), 628–637 (1985)Google Scholar
  29. Liu, L.: General Relativity, High Education Press, Shanghai, China p. 42, (1987)Google Scholar
  30. Lo, C.Y.: Astrophys. J. 455, 421–428 (1995)CrossRefADSGoogle Scholar
  31. Lo, C.Y.: Astrophys. J. 477, 700–704 (March 10, 1997a)Google Scholar
  32. Lo, C.Y.: Phys. Essays 10(3), 424–436 (September, 1997b)Google Scholar
  33. Lo, C.Y.: Phys. Essays 11(2), 264–272 (1998)MathSciNetCrossRefGoogle Scholar
  34. Lo, C.Y.: Phys. Essays 12(2), 226–241 (1999a)Google Scholar
  35. Lo, C.Y.: Phys. Essays 12(2), 508–526 (1999b)MathSciNetGoogle Scholar
  36. Lo, C.Y.: Phys. Essays 13(1), 109–120 (March, 2000a)MathSciNetGoogle Scholar
  37. Lo, C.Y.: Phys. Essays 13(4), 527–539 (December, 2000b)MathSciNetCrossRefGoogle Scholar
  38. Lo, C.Y.: Phys. Essays 15(3), 303–321 (2002)MathSciNetCrossRefGoogle Scholar
  39. Lo, C.Y.: Phys. Essays 16(1), 84–100 (March, 2003a)MathSciNetGoogle Scholar
  40. Lo, C.Y.: Chinese J. Phys. 41(4), 332–342 (August, 2003b)MathSciNetGoogle Scholar
  41. Lo, C.Y.: Chinese Phys. 13 (2), 159–167 (2004)CrossRefADSGoogle Scholar
  42. Lo, C.Y.: Phys. Essays 18(4) (2005)Google Scholar
  43. Lo, C.Y.: Prog. Phys. 2, 6–8 (2006)MathSciNetGoogle Scholar
  44. Lo, C.Y., Chan, D.P., Hui, R.C.Y.: Phys. Essays 15(1), 77–86 (2002)Google Scholar
  45. Lorentz, H.A.: Proc. K. Ak. Amsterdam 8, 603 (1900)Google Scholar
  46. Low, F.E.: Center for theoretical physics, MIT, Mass., private communications (1997)Google Scholar
  47. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman, San Francisco (1973)Google Scholar
  48. Norton, J.: What was Einstein's principle of equivalence? In: Howard, D., Stachel, J. (eds.), Einstein's Studies Vol. 1: Einstein and the History of General Relativity. Birkhäuser (1989)Google Scholar
  49. Ohanian, H.C., Ruffini, R.: Gravitation and spacetime. Norton, New York (1994)Google Scholar
  50. Pais, A.: Subtle is the Lord. Oxford University Press, New York (1996)Google Scholar
  51. Pauli, W.: Theory of relativity. Pergamon Press, London (1958)Google Scholar
  52. Peng, P.: Commun. Theor. Phys. (Beijing, China) 31, 13–20 (1999)Google Scholar
  53. Penrose, R.: Rev. Mod. Phys. 37(1), 215–220 (1965)CrossRefADSMathSciNetzbMATHGoogle Scholar
  54. Peres, A.: Phys. Rev. Lett. 3, 571 (1959)CrossRefADSzbMATHGoogle Scholar
  55. Rosen, N.: Bull. Res. Council Israel 3, 328 (1953)Google Scholar
  56. Rosen, N.: Phys. Z. Sowjet 12, 366 (1937)zbMATHGoogle Scholar
  57. Scheidigger, A.E.: Revs. Modern Phys. 25, 451 (1953)CrossRefADSGoogle Scholar
  58. Straumann, N.: General relativity and relativistic astrophysics. Springer, New York (1984)Google Scholar
  59. Thorne, K.S.: Black holes & time warps, Norton, New York, p. 488, (1994)Google Scholar
  60. Thorne, K.S.: Gravitational radiation. In: Hawking & Israel (ed.) 300 Years of Gravitation. Cambridge University Press, New York (1987)Google Scholar
  61. Tolman, R.C.: Relativity, thermodynamics, and cosmology. Dover, New York (1987)Google Scholar
  62. Wald, R.M.: General relativity. The University of Chicago Press, Chicago (1984)Google Scholar
  63. Weber, J., Wheeler, J.A.: Rev. Modern Phys. 29(3), 509 (1957)Google Scholar
  64. Weinberg, S.: Gravitation and Cosmology, John Wiley, New York, p. 273, (1972)Google Scholar
  65. Whitehead, A.N.: The principle of relativity. Cambridge University Press, Cambridge (1922)Google Scholar
  66. Will, C.M.: Theory and Experiment in Gravitational Physics. Cambridge University Press (1981)Google Scholar
  67. Yilmaz, H.: Hadronic J. 2, 997–1020 (1979)Google Scholar
  68. Zel'dovich, Ya.B., Novikov, I.D.: Stars and relativity, Dover, New York pp. 7–16 (1996)Google Scholar
  69. Zhou (Chou), P.-Y.: Scientia Sinica (Series A) XXV(6), 628–643 (1982)Google Scholar
  70. Zhou (Chou), Pei-Yuan.: On coordinates and coordinate transformation in Einstein’s theory of gravitation. In: Proc. of the Third Marcel Grossmann Meetings on Gen. Relativ. Hu, Ning (ed.) Science Press & North Holland (1983) 1–20Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • C. Y. Lo
    • 1
  1. 1.Applied and Pure Research InstituteNashuaUSA

Personalised recommendations