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Does Asymptotic Simplicity Allow for Radiation Near Spatial Infinity?

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Abstract.

A representation of spatial infinity based on the properties of conformal geodesics is used to obtain asymptotic expansions of the gravitational field near the region where null infinity touches spatial infinity. These expansions show that generic time symmetric initial data with an analytic conformal metric at infinity will give rise to developments with a certain type of logarithmic singularities at the points where null infinity and spatial infinity meet. These logarithmic singularities produce a non-smooth null infinity. The sources of the logarithmic singularities are traced back down to the initial data. It is shown that if the parts of the initial data responsible for the non-regular behaviour of the solutions are not present, then the initial data is static to a certain order. On the basis of these results it is conjectured that the only time symmetric data sets with developments having a smooth null infinity are those which are static in a neighbourhood of infinity. This conjecture generalises a previous conjecture regarding time symmetric, conformally flat data. The relation of these conjectures to Penrose’s proposal for the description of the asymptotic gravitational field of isolated bodies is discussed.

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References

  1. Ashtekar, A., Magnon-Ashtekar, A.: Energy-momentum in general relativity. Phys. Rev. Lett. 43, 181 (1979)

    Article  MathSciNet  Google Scholar 

  2. Chruściel, P.T., Delay, E.: On mapping properties of the general relativistic constraint operator in weighted function spaces, with applications. Mem. Soc. Math. France 94, 1–103 (2003)

    Google Scholar 

  3. Chruściel, P.T., Delay, E.: Existence of non-rivial, vacuum, asymptotically simple spacetimes. Class. Quantum Grav. 19, L71 (2002)

  4. Chruściel, P.T., MacCallum, M.A.H., Singleton, D.B.: Gravitational waves in general relativity XIV. Bondi expansions and the “polyhomogeneity” of Phil. Trans. Roy. Soc. Lond. A 350, 113 (1995)

    Google Scholar 

  5. Corvino, J.: Scalar curvature deformations and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137 (2000).

    MathSciNet  MATH  Google Scholar 

  6. Dain, S.: Private communication

  7. Dain, S.: Initial data for stationary spacetimes near spacelike infinity. Class. Quantum Grav. 18, 4329 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dain, S., Friedrich,H.: Asymptotically flat initial data with prescribed regularity at infinity. Commun. Math. Phys. 222, 569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dain, S. Valiente Kroon, J.A.: Conserved quantities in a black hole collision. Class. Quantum Grav. 19, 811 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Exton, A.R., Newman, E.T., Penrose, R.: Conserved quantites in the Einstein-Maxwell theory. J. Math. Phys. 10, 1566 (1969)

    Article  Google Scholar 

  11. Friedrich, H.: Radiative gravitational fields and asymptotically static or stationary initial data. To appear in the volume “50 years of the Cauchy problem in general relativity”, Birkhauser. Title changed to “Smothness at null infinity and the structure of initial data. http://xxx.lanl.gov/abs/gr-qc/0304003v2, 2003

  12. Friedrich, H.: On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Friedrich, H.: On static and radiative space-times. Commun. Math. Phys. 119, 51 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Friedrich, H.: Einstein equations and conformal structure: existence of anti-de Sitter-type space-times. J. Geom. Phys. 17, 125 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Friedrich, H.: Gravitational fields near space-like and null infinity. J. Geom. Phys. 24, 83 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Friedrich, H.: Einstein’s equation and conformal structure. In: The Geometric Universe. Science, Geometry and the work of Roger Penrose. S.A. Huggett, L.J. Mason, K.P. Tod, S.T. Tsou, N.M.J. Woodhouse (eds), Oxford: Oxford University Press, 1999, p. 81

  17. Friedrich, H.: Conformal Einstein evolution. In: The conformal structure of spacetime: Geometry, Analysis, Numerics. J. Frauendiener, H. Friedrich (eds), Lecture Notes in Physics, Berlin-Heddelberg-New York: Springer, 2002, p. 1

  18. Friedrich, H.: Spin-2 fields on Minkowski space near space-like and null infinity. Class. Quantum Grav. 20, 101 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Friedrich, H., Kánnár, J.: Bondi-type systems near space-like infinity and the calculation of the NP-constants. J. Math. Phys. 41, 2195 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Friedrich, H., Schmidt, B.: Conformal geodesics in general relativity. Proc. Roy. Soc. Lond. A 414, 171 (1987)

    MathSciNet  MATH  Google Scholar 

  21. Newman, E.T., Penrose, R.: New conservation laws for zero rest-mass fields in asymptotically flat space-time. Proc. Roy. Soc. Lond. A 305, 175 (1968)

    Google Scholar 

  22. Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. Roy. Soc. Lond. A 284, 159 (1965)

    MATH  Google Scholar 

  23. Sommers, P.: Space spinors. J. Math. Phys. 21, 2567 (1980)

    Article  MathSciNet  Google Scholar 

  24. Stewart, J.: Advanced general relativity. Cambridge: Cambridge University Press, 1991

  25. Valiente Kroon, J.A.: Polyhomogeneous expansions close to null and spatial infinity. In: The Conformal Structure of Spacetimes: Geometry, Numerics, Analysis. J. Frauendiner, H. Friedrich (eds), Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer, 2002, p. 135

  26. Valiente Kroon, J.A.: Early radiative properties of the developments of time symmetric conformally flat initial data. Class. Quantum Grav. 20, L53 (2003)

  27. Valiente Kroon, J.A.: A new class of obstructions to the smoothness of null infinity. Comm. Math. Phys. 244, 133 (2004)

    Article  MATH  Google Scholar 

  28. Valiente Kroon, J.A.: On the nonexistence of conformally flat slices for the Kerr and other stationary spacetimes. Phys. Rev. Lett. 92, 041101 (2004)

    Article  Google Scholar 

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  1. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090, Wien, Austria

    Juan Antonio Valiente Kroon

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  1. Juan Antonio Valiente Kroon
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Correspondence to Juan Antonio Valiente Kroon.

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Communicated by G.W. Gibbons

Acknowledgements I would like to thank H. Friedrich who introduced me to this research topic and has provided me with invaluable advice. I also acknowledge several enriching and helpful discussions with R. Beig, S. Dain and J. Winicour. I also thank an anonymous referee for a careful reading of the manuscript and an important observation leading to Lemma 4. This work is funded by a Lise Meitner fellowship (M690-N02 and M814-N02) of the Fonds zur Forderung der Wissenschaftlichen Forschung (FWF), Austria. The computer algebra calculations here described have been performed in the computer facilities of the Albert Einstein Institute, Max Planck Institute für Gravitationsphysik, Golm, Germany.

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Kroon, J. Does Asymptotic Simplicity Allow for Radiation Near Spatial Infinity?. Commun. Math. Phys. 251, 211–234 (2004). https://doi.org/10.1007/s00220-004-1174-8

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  • DOI: https://doi.org/10.1007/s00220-004-1174-8

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