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Worldtube conservation laws for the null-timelike evolution problem

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Abstract

I treat the worldtube constraints which arise in the null-timelike initial-boundary value problem for the Bondi-Sachs formulation of Einstein’s equations. Boundary data on a worldtube and initial data on an outgoing null hypersurface determine the exterior spacetime by integration along the outgoing null geodsics. The worldtube constraints are a set of conservation laws which impose conditions on the integration constants. I show how these constraints lead to a well-posed initial value problem governing the extrinsic curvature of the worldtube, whose components are related to the integration constants. Possible applications to gravitational waveform extraction and to the well-posedness of the null-timelike initial-boundary value problem are discussed.

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References

  1. Goldberg, J.N.: Strong conservation laws and equations of motion in covariant field theories. Phys. Rev. 89, 263 (1953)

    Article  ADS  MATH  Google Scholar 

  2. Goldberg, J.N.: Conservation laws in general relativity. Phys. Rev. 111, 315 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Goldberg, J.N.: Conservation equations and equations of motion in the null formalism. Gen. Relativ. Gravit. 5, 183 (1974)

    Article  ADS  Google Scholar 

  4. Bondi H., van der Burg M.J.G., Metzner A.W.K.: Proc. R. Soc. A 269, 21 (1962)

    Article  ADS  MATH  Google Scholar 

  5. Sachs R.K.: Proc. R. Soc. A 270, 103 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Penrose R.: Phys. Rev. Lett. 10, 66 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  7. Newman E.T., Posadas R.: J. Math. Phys. 12, 2319 (1971)

    Article  ADS  Google Scholar 

  8. Lind R.W., Messmer J., Newman E.T.: J. Math. Phys. 13, 1884 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  9. Stewart, J.M.: The cauchy problem and the initial boundary value problem in numerical relativity Class. Quan. Grav. 15, 2865 (1998)

    Article  ADS  MATH  Google Scholar 

  10. Friedrich H., Nagy G.: Commun. Math. Phys. 201, 619 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Kreiss H.-O., Winicour J.: Class. Quan. Grav. 23, S405 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space Time. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  13. Winicour J.: J. Math. Phys. 24, 1193 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Winicour J.: J. Math. Phys. 25, 2506 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  15. Tamburino L.A., Winicour J.: Phys. Rev. 150, 1039 (1966)

    Article  ADS  Google Scholar 

  16. Kreiss, H.-O., Lorenz, J.: Initial-Boundary Value Problems and the Navier-Stokes Equations, 1989. Reprint SIAM CLASSICS (2004)

  17. Bishop N.T., Gómez R., Lehner L., Szilágyi B., Winicour J., Isaacson R.A.: Cauchy-characteristic matching. In: Iyer, B., Bhawal, B. (eds) Black Holes, Gravitational Radiation and the Universe, Kluwer Academic Publishers, Dordrecht (1998)

    Google Scholar 

  18. Bishop N.T., Gómez R., Lehner L., Winicour J.: Phys. Rev. D 54, 6153 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  19. Bartnik R.: Class. Quan. Grav. 14, 2185 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Rendall A.D.: Proc. Roy. Soc. Lond. A 427, 221 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Jeffrey Winicour

  2. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476, Golm, Germany

    Jeffrey Winicour

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  1. Jeffrey Winicour
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Correspondence to Jeffrey Winicour.

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Winicour, J. Worldtube conservation laws for the null-timelike evolution problem. Gen Relativ Gravit 43, 3269–3288 (2011). https://doi.org/10.1007/s10714-011-1241-3

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  • DOI: https://doi.org/10.1007/s10714-011-1241-3

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