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Electromagnetic Time

  • C. K. Raju
Part of the Fundamental Theories of Physics book series (FTPH, volume 65)

Abstract

In the previous chapter we saw that the introduction of the field creates more problems than it solves. Here we begin by disregarding the problems due to the field, and formulate the two-body problem of retarded electrodynamics, without radiation reaction. The resulting equations of motion are time-asymmetric, and fail to satisfy the ‘phase-flow’ hypothesis underlying the recurrence and reversibility paradoxes. We present a counter-example to show that the Lorentz-Dirac equation, resulting in preacceleration, may be invalid since it is derived by replacing a retarded ordinary differential equation (o.d.e.) by a higher-order standard o.d.e. obtained by Taylor approximation. The solutions of advanced o.d.e.’s branch into the future, implying in-principle unpredictability from the past and resolving Popper’s pond paradox. The branching and collapse of solutions of mixed o.d.e.’s suggests a resolution of the Wheeler-Feynman and grandfather paradoxes.

With a direct-action theory, or with Dirac’s definition of radiation damping, the elimination of advanced interactions is a serious problem. We present an exposition of (i) the Sommerfeld condition, pointing out its arbitrariness; (ii) the Wheeler-Feynman absorber theory, pointing out its internal inconsistency; and (iii) the Hogarth-Hoyle-Narlikar theory, pointing out its external inconsistency. The remaining absorber theory predicts the existence of rare advanced interactions. We compare this with the empirical results of Partridge, and suggest that experiment proposed by Heron and Pegg may now be revived.

Keywords

World Line Null Cone Retardation Vector Time Asymmetry Advanced Radiation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes and References

  1. 1.
    J.L. Synge, Proc. R. Soc., A177, 118–139 (1940); R.D. Driver, Phys. Rev., 178, 2051–57 (1969); D.K. Hsing, Phys. Rev., D16, 974–82 (1977). In more than one dimension, the equations are not only more complicated, but are of a different type,being ‘mixed’ or ‘neutral’.Google Scholar
  2. 2.
    H. Van Dam and E.P. Wigner, Phys. Rev., 138B, 1576 (1965); 142, 838 (1966). These equations were considered in the context of relativistic point dynamics.Google Scholar
  3. 3.
    For more details, see, e.g., R.D. Driver, Introduction to Differential and Delay Equations (Springer, Berlin, 1977); L.E. El’sgol’ts, Introduction to the Theory of Differential Equations with Deviating Arguments,(Tr.) R.J. McLaughlin (Holden-Day, San Francisco, 1966). The theorem does not apply to equation (3) which admits unbounded delays.Google Scholar
  4. 4.
    P.A.M. Dirac, Proc. R. Soc., A167, 148 (1938), equations (55) and (56). Essentially the same derivation maybe found in, e.g., F. Rohrlich, Classical Charged Particles (Addison-Wesley, Reading, Mass., 1965) p 142, who manages to avoid another Taylor expansion used by Dirac.Google Scholar
  5. 5.
    L. Page, Phys. Rev., 12, 371 (1918); 24, 296 (1924); A. Einstein, L. Infeld and B. Hoffmann, Ann. Math., 39, 65 (1938); H. P. Robertson, Ann. Math., 39, 101 (1938); L. Infeld, Phys. Rev., 53, 836 (1938); A. Eddington and G. L. Clark, Proc. R. Soc., A166, 465(1938); A. Einstein and L. Infeld, Can. J. Math., 1, 209(1949); G. L.Clark, Proc. R. Soc. (Edinb.), A64, 49 (1954); B. Bertotti, Nuovo Cim., 12, 226 (1954). A more detailed list may be found in the article by P. Havas in Statistical Mechanics of Equilibrium and Non-Equilibrium, ed. J. Meixmer (Amsterdam: North Holland, 1965). More recently, somewhat similar approximations have been attempted by L. P. Grishchuk and S.M. Kopejkin, in: J. Kovalevsky and V.A. Brumberg (eds), Relativity in Celestial Mechanics and Astronomy, IAU (1986) pp 19–34; V.I. Zhdanov, J. Phys. A: Math. Gen., 24, 5011–27 (1991).Google Scholar
  6. 6.
    One may prove this using the standard theory of linear differential equations, and proceeding in steps from past data,x =‘p on [—r,0], say.Google Scholar
  7. 7.
    In his book cited in Ref. 3, p 22.Google Scholar
  8. 8.
    E. Hairer, S.P. N¢rsett, and G. Wanner, Solving Ordinary Differential Equations, Springer Series in Computational Mathematics, Vol. 8 ( Springer, Berlin, 1987 ).Google Scholar
  9. 9.
    An additional problem is that discontinuities at the ends of delay intervals, which are smoothed out for retarded equations are not smoothed out for mixed-type equations, as noted by P. Hui, in Schulman (Ref. 15), while purely advanced equations progressively lose smoothness. Numerical solution of delay equations is, of course, possible, in some cases, and the figures obtained here have used numerical techniques, with adaptive step-size control, and discontinuity-handling.Google Scholar
  10. 10.
    C.J. Eliezer, Proc. R. Soc., A194, 543 (1948); W.B. Bonnor, Proc. R. Soc, A337, 591 (1974); T.C. Mo and C.H. Papas, Phys. Rev., D 4, 3566 (1971); J.C. Herera, Phys. Rev., D 15, 453 (1977).Google Scholar
  11. 11.
    J. Huschilt and W.E. Bayles, Phys. Rev., D 9, 2479 (1974); E. Conay, Phys. Lett., A 125,155 (1987); E. Conay, Int. J. Theor. Phys., 29, 1427 (1990).Google Scholar
  12. 12.
    A. Schild, Phys. Rev., 131, 2762 (1963).MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    C. M. Anderssen and H. C. von Baeyer, Phys. Rev., D 5, 802 (1972); Phys. Rev., D 5, 2470 (1972).ADSGoogle Scholar
  14. 14.
    R. D. Driver, Phys. Rev., D 19, 1098 (1979).Google Scholar
  15. 15.
    L. S. Schulman, J. Math. Phys., 15, 295–8 (1974).MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    K. L. Cooke and D. W. Krumme, J. Math. Anal. Appl., 24, 372–87 (1968).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    A discontinuity develops because the characteristics of a nonlinear hyperbolic partial differential equation may intersect. Such a phenomenon is well known in the theory of shocks in fluids. While one can deny that the Euler equations provide a true model of fluid flow, one can hardly deny ‘hyperbolicity’, which more or less amounts to a belief in finite propagation speeds, and well-posedness of the Cauchy problem; see, e.g. Bers et al, Partial Differential Equations (New York: Wiley-Interscience, 1968 ). The Hawking-Penrose theorems, for example, may be interpreted in this way.Google Scholar
  18. 18.
    J. L. Friedmann and M.S. Morris, Phys. Rev. Lett., 66 (4) (1991).Google Scholar
  19. 19.
    The assumption (i) might seem a bit dubious if stated honestly: Physicists (i.e., human beings) are free to perform any experiment, and physical laws are no ‘serious’ constraint. We argue later on, in Chapter VIII, that the precise assumption here is an assumption about the nature of time, derived from mundane experience.Google Scholar
  20. 20.
    A. Einstein and W. Ritz, Phys. Z., 10, 323 (1909).Google Scholar
  21. 21.
    A possible cosmological reason for this choice has been put forward by D.W. Sciama, Proc. R. Soc., A273, 484 (1963). The argument is that if Fin x 0, it could diverge, as in Olbers’ paradox. For a resolution of Olbers’ paradox, see Chapter V II.Google Scholar
  22. 22.
    J. A. Wheeler and R.P Feynman, Rev. Mod. Phys., 17, 157 (1945); 21, 425 (1949).MathSciNetzbMATHGoogle Scholar
  23. 23.
    J. G. Cramer, Rev. Mod. Phys., 58, 647 (1986).MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    C. K. Raju, J. Phys. A: Math. Nucl. Gen., 13, 3303–17 (1980).ADSCrossRefGoogle Scholar
  25. 25.
    For example, Sciama, Ref. 21, has slightly reformulated the argument: the sum of the retarded fields in equation (29) is an incoherent sum, and hence vanishes, whereas the sum in equation (33) is a coherent sum and compensates by just the right amount to produce anti-damping. That is, the conclusion rests on time-asymmetric (thermodynamic) assumptions similar to those in Popper’s pond paradox.Google Scholar
  26. 26.
    J. E. Hogarth, Proc. R. Soc., A267, 365–383 (1962).MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    P. C. W. Davies, ‘Is the universe transparent or opaque’, J. Phys. A: Gen. Phys., 5, 1722–37 (1974).ADSCrossRefGoogle Scholar
  28. 28.
    F. Hoyle and J. V. Narlikar, Proc. R. Soc., A277, 1–23 (1964).MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    The hypothesis of finite size seems ad hoc,and does not seem to augur well for quantum mechanical generalizations. However, an interpretation of quantum mechanics was proposed on this basis [C. K. Raju, Int. J. Theor. Phys., 20, 681–96 (1981)], and it was proved that [C. K. Raju, J. Phys. A, 16, 3551 (1983); Hadronic J. Suppl., 1 (1986)] one could modify the propagators of quantum field theory, on the same hypothesis, to obtain a Lorentz covariant finite field theory for all polynomial lagrangians.Google Scholar
  30. 30.
    R. B. Partridge, Nature, 244, 263–65 (1973).ADSCrossRefGoogle Scholar
  31. 31.
  32. 32.
    M. L. Heron and D.T. Pegg, J. Phys. A: Math. Nucl. Gen., 7, 1975–9 (1974).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • C. K. Raju
    • 1
    • 2
  1. 1.Indian Institute of Advanced StudyRashtrapati NivasShimlaIndia
  2. 2.Centre for Development of Advanced ComputingNew DelhiIndia

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