Abstract
Wheeler–Feynman electrodynamics (WF) is an action-at-a-distance theory about world-lines of charges that in contrary to the textbook formulation of classical electrodynamics is free of ultraviolet singularities and is capable of explaining the irreversible nature of radiation. In WF, the world-lines of charges obey the so-called Fokker–Schwarzschild–Tetrode (FST) equations, a coupled set of nonlinear and neutral differential equations that involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation of this theory in terms of Maxwell–Lorentz electrodynamics without self-interaction that we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These conditional solutions solve the FST equations on any finite time interval with prescribed continuations outside of this interval. As a byproduct, we also prove existence and uniqueness of solutions to the Synge equations on the time half-line for a given history of charge world-lines.
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References
Anderson, J.L.: Principles of Relativity Physics. Academic Press Inc., London. ISBN: 0120584506 (1967)
Angelov V.G.: On the Synge equations in a three-dimensional two-body problem of classical electrodynamics. J. Math. Anal. Appl. 151(2), 488–511 (1990)
Bauer, G.: Ein Existenzsatz für die Wheeler–Feynman-Elektrodynamik. Herbert Utz Verlag. ISBN: 3896751948 (1997)
Bauer, G., Deckert, D.-A., Dürr, D.: Maxwell–Lorentz dynamics of rigid charges. arXiv:1009.3105 (2010)
Deckert, D.-A.: Electrodynamic Absorber Theory. Der Andere Verlag, Osnabrück. ISBN: 978-3-86247-004-4 (2010)
Deckert, D.-A., Dürn, D., Vona, N.: Delay equations of the Wheeler–Feynman type. Contemp. Math. Fundam. Dir. (to appear)
Dirac P.A.M.: Classical theory of radiating electrons. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 167(929), 148–169 (1938)
Driver R.D.: A ”backwards” two-body problem of classical relativistic electrodynamics. Phys. Rev. 178(5), 2051 (1969)
Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York. ISBN: 0387902317 (1977)
Driver R.D.: Can the future influence the present?. Phys. Rev. D 19(4), 1098 (1979)
Diekmann, O., van Gils, S.A., Lunel, S.M.V., Walther, H.-O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis: v. 110, 1st edn. Springer, Berlin (1995)
Van Dam H., Wigner E.P.: Classical relativistic mechanics of interacting point particles. Phys. Rev. 138(6B), B1576–B1582 (1965)
Fokker A.D.: Ein invarianter Variationssatz für die Bewegung mehrerer elektrischer Massenteilchen. Zeitschrift für Physik A Hadrons and Nuclei 58(5), 386–393 (1929)
Gauß C.: A letter to W. Weber in March 19th, 1845. Gauß: Werke 5, 627–629 (1877)
Komech A., Spohn H.: Long-time asymptotics for the coupled Maxwell–Lorentz equations. Commun. Partial Differ. Equ. 25(3), 559 (2000)
Lieb, E.H., Loss, M.: Analysis, 2nd edn. Oxford University Press, Oxford (2001)
De Luca J.: Variational principle for the Wheeler–Feynman electrodynamics. J. Math. Phys. 50(6), 062701–062724 (2009)
Myškis, A.D.: General theory of differential equations with a retarded argument. Am. Math. Soc. Translation. no. 55, 62. MR 0046551 (13,752a) (1951)
Pauli, W.: Relativitätstheorie. Encykl. d. Math. Wissenschaften, V (IV-19)
Schwarzschild K.: Zur Elektrodynamik. II. Die elementare elektrodynamische Kraft. Nachr. Ges. Wis. Gottingen 128, 132 (1903)
Schild A.: Electromagnetic two-body problem. Phys. Rev. 131(6), 2762 (1963)
Siegel C.L., Moser J.K.: Lectures on Celestial Mechanics. Springer, Berlin (1971)
Spohn H.: Dynamics of Charged Particles and their Radiation Field. Cambridge University Press, Cambridge (2004)
Synge J.L.: On the electromagnetic two-body problem. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 177(968), 118–139 (1940)
Synge J.L.: Relativity. Series in Physics. North-Holland, Amsterdam (1976)
Tetrode H.: Über den Wirkungszusammenhang der Welt. Eine Erweiterung der klassischen Dynamik. Zeitschrift für Physik A 10(1), 317–328 (1922)
Wheeler J.A., Feynman R.P.: Interaction with the absorber as the mechanism of radiation. Rev. Mod. Phys. 17(2–3), 157 (1945)
Wheeler J.A., Feynman R.P.: Classical electrodynamics in terms of direct interparticle action. Rev. Mod. Phys. 21(3), 425 (1949)
Whitehead A.N.: The Concept of Nature: Tarner Lectures. Cambridge University Press, Cambridge (1920)
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Bauer, G., Deckert, D.A. & Dürr, D. On the existence of dynamics in Wheeler–Feynman electromagnetism. Z. Angew. Math. Phys. 64, 1087–1124 (2013). https://doi.org/10.1007/s00033-012-0293-x
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DOI: https://doi.org/10.1007/s00033-012-0293-x