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Communications in Mathematical Physics

, Volume 331, Issue 2, pp 755–803 | Cite as

Hertz Potentials and Asymptotic Properties of Massless Fields

  • Lars AnderssonEmail author
  • Thomas Bäckdahl
  • Jérémie Joudioux
Article

Abstract

In this paper we analyze Hertz potentials for free massless spin-s fields on the Minkowski spacetime, with data in weighted Sobolev spaces. We prove existence and pointwise estimates for the Hertz potentials using a weighted estimate for the wave equation. This is then applied to give weighted estimates for the solutions of the spin-s field equations, for arbitrary half-integer s. In particular, the peeling properties of the free massless spin-s fields are analyzed for initial data in weighted Sobolev spaces with arbitrary, non-integer weights.

Keywords

Initial Data Wave Equation Cauchy Problem High Spin Minkowski Space 
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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References

  1. 1.
    Asakura F.: Existence of a global solution to a semilinear wave equation with slowly decreasing initial data in three space dimensions. Commun. Partial Differ. Equ. 11(13), 1459–1487 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bäckdahl T., Valiente Kroon J.A.: On the construction of a geometric invariant measuring the deviation from Kerr data. Ann. Henri Poincaré 11(7), 1225–1271 (2010)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartnik R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(5), 661–693 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beig, R.: TT-tensors and conformally flat structures on 3-manifolds. In: Mathematics of Gravitation, Part I (Warsaw, 1996), pp. 109–118. Polish Academy of Sciences, Warsaw (1997)Google Scholar
  5. 5.
    Benn I.M., Charlton P., Kress J.: Debye potentials for Maxwell and Dirac fields from a generalization of the Killing-Yano equation. J. Math. Phys. 38, 4504–4527 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Branson T.: Stein–Weiss operators and ellipticity. J. Funct. Anal. 151(2), 334–383 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cantor M.: Elliptic operators and the decomposition of tensor fields. Am. Math. Soc. Bull. N. Ser. 5(3), 235–262 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford Mathematical Monographs. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  9. 9.
    Christodoulou D., Klainerman S.: Asymptotic properties of linear field equations in Minkowski space. Commun. Pure Appl. Math. 43(2), 137–199 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space, volume 41 of Princeton Mathematical Series. Princeton University Press, Princeton (1993)Google Scholar
  11. 11.
    D’Ancona P., Georgiev V., Kubo H.: Weighted decay estimates for the wave equation. J. Differ. Equ. 177(1), 146–208 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Evans, L.C.: Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
  13. 13.
    Fayos F., Llanta E., Llosa J.: Maxwell’s equations in the Debye potential formalism. Ann. Inst. H. Poincaré Phys. Théor. 43(2), 195–209 (1985)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Friedlander F.G.: The Wave Equation on a Curved Space-Time. Cambridge University Press, Cambridge (1975)zbMATHGoogle Scholar
  15. 15.
    Gasqui, J., Goldschmidt, H.: Déformations Infinitésimales des Structures Conformes Plates, volume 52 of Progress in Mathematics. Birkhäuser Boston Inc., Boston (1984)Google Scholar
  16. 16.
    Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations, volume 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin (1997)Google Scholar
  17. 17.
    Kato T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)Google Scholar
  18. 18.
    Klainerman S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38(3), 321–332 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 (Santa Fe, N.M., 1984), pp. 293–326. American Mathematical Society, Providence (1986)Google Scholar
  20. 20.
    Klainerman S.: Remarks on the global Sobolev inequalities in the Minkowski space \({\mathbb{R}^{n+1}}\). Commun. Pure Appl. Math. 40(1), 111–117 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Klainerman S., Nicoló F.: Peeling properties of asymptotically flat solutions to the Einstein vacuum equations. Class. Quant. Gravity 20, 3215–3257 (2003)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Leray, J.: Hyperbolic Differential Equations. The Institute for Advanced Study, Princeton (1953)Google Scholar
  23. 23.
    Lockhart R.B., McOwen R.C.: On elliptic systems in \({\mathbb{R}^n}\). Acta Math. 150(1-2), 125–135 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lockhart, R.B., McOwen, R.C.: Correction to: ”On elliptic systems in \({\mathbb{R}^n}\)” [Acta Math. 150 (1983), no. 1-2, 125–135; MR0697610 (84d:35048)]. Acta Math. 153(3–4):303–304 (1984)Google Scholar
  25. 25.
    Mason L.J., Nicolas J.-P.: Peeling of Dirac and Maxwell fields on a Schwarzschild background. J. Geom. Phys. 62(4), 867–889 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    McOwen R.C.: On elliptic operators in \({\mathbb{R}^n}\). Commun. Partial Differ. Equ. 5(9), 913–933 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Penrose, R.: Twistors as charges for spin-3/2 in vacuum. Twistor Newsl. 32, 1–5 (1991)Google Scholar
  28. 28.
    Penrose, R.: Twistors as spin-3/2 charges continued: \({{\rm SL}_3(\mathbb{C})}\)-bundles. Twistor Newsl. 33, 1–6 (1991)Google Scholar
  29. 29.
    Penrose R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. Ser. A 284, 159–203 (1965)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Penrose R., Rindler W.: Spinors and Space-Time I & II. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1986)Google Scholar
  31. 31.
    Sachs R.: Gravitational waves in general relativity. VI. The outgoing radiation condition. Proc. R. Soc. Ser. A 264, 309–338 (1961)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Shu W.-T.: Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space. Commun. Math. Phys. 140(3), 449–480 (1991)ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Shubin, M.A. (ed): Partial differential equations. VIII, volume 65 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (1996)Google Scholar
  34. 34.
    Sommers P.: Space spinors. J. Math. Phys. 21(10), 2567–2571 (1980)ADSCrossRefMathSciNetGoogle Scholar
  35. 35.
    Stewart J.M.: Hertz-Bromwich-Debye-Whittaker-Penrose potentials in general relativity. R. Soc. Lond. Proc. Ser. A 367, 527–538 (1979)ADSCrossRefGoogle Scholar
  36. 36.
    Weck N., Witsch K.J.: Generalized spherical harmonics and exterior differentiation in weighted Sobolev spaces. Math. Method Appl. Sci. 17(13), 1017–1043 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Woodhouse N.M.J.: Real methods in twistor theory. Class. Quant. Gravity 2(3), 257–291 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Lars Andersson
    • 1
    Email author
  • Thomas Bäckdahl
    • 1
    • 2
  • Jérémie Joudioux
    • 1
    • 3
  1. 1.Albert Einstein InstitutePotsdamGermany
  2. 2.The School of MathematicsThe University of EdinburghEdinburghUK
  3. 3.Fakultät für PhysikUniversität WienWienAustria

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