A note on mass-minimising extensions

Research Article
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Abstract

A conjecture related to the Bartnik quasilocal mass, is that the infimum of the ADM energy, over an appropriate space of extensions to a compact 3-manifold with boundary, is realised by a static metric. It was shown by Corvino (Commun Math Phys 214(1):137–189, 2000) that if the infimum is indeed achieved, then it is achieved by a static metric; however, the more difficult question of whether or not the infimum is achieved, is still an open problem. Bartnik (Commun Anal Geom 13(5):845–885, 2005) then proved that critical points of the ADM mass, over the space of solutions to the Einstein constraints on an asymptotically flat manifold without boundary, correspond to stationary solutions. In that article, he stated that it should be possible to use a similar construction to provide a more natural proof of Corvino’s result. In the first part of this note, we discuss the required modifications to Bartnik’s argument to adapt it to include a boundary. Assuming that certain results concerning a Hilbert manifold structure for the space of solutions carry over to the case considered here, we then demonstrate how Bartnik’s proof can be modified to consider the simpler case of scalar-flat extensions and obtain Corvino’s result. In the second part of this note, we consider a space of extensions in a fixed conformal class. Sufficient conditions are given to ensure that the infimum is realised within this class.

Keywords

Static extensions Quasilocal mass Bartnik mass  Conformal metric 

Notes

Acknowledgments

The author gratefully acknowledges many useful comments from two anonymous reviewers.

References

  1. 1.
    Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62(20), 845–885 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bartnik, R.: Phase space for the Einstein equations. Commun. Anal. Geom. 13(5), 845–885 (2005)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bartnik, R., Isenberg, J.: The constraint equations. In: Chruściel, P., Friedrich, H. (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 1–38. Birkhäuser, Basel (2004)CrossRefGoogle Scholar
  5. 5.
    Beig, R., Chrusciel, P.T.: Killing initial data. Class. Quantum Gravity 14(1A), A83 (1997). http://stacks.iop.org/0264-9381/14/i=1A/a=007
  6. 6.
    Brown, J.D., York Jr, J.W.: Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D 47(4), 1407 (1993)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Cantor, M., Brill, D.: The laplacian on asymptotically flat manifolds and the specification of scalar curvature. Compos. Math. 43(3), 317–330 (1981). http://eudml.org/doc/89504
  8. 8.
    Choquet-Bruhat, Y., Christodoulou, D.: Elliptic systems in spaces on \(H_{s,\delta }\) manifolds which are Euclidean at infinity. Acta Math. 146(1), 129–150 (1981)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000)MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Dain, S.: Proof of the angular momentum-mass inequality for axisymmetric black holes. J. Differ. Geom. 79, 33–67 (2008)MATHMathSciNetGoogle Scholar
  11. 11.
    Fischer, A.E., Marsden, J.E.: Deformations of the scalar curvature. Duke Math. J. 42(3), 519–547 (1975). doi:10.1215/S0012-7094-75-04249-0 MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hille, E., Phillips, R.: Functional Analysis and Semi-Groups. American Mathematical Society, Providence (1957)Google Scholar
  13. 13.
    Maxwell, D.: Initial data for black holes and rough spacetimes. Ph.D. thesis, University of Washington (2004)Google Scholar
  14. 14.
    McCormick, S.: First law of black hole mechanics as a condition for stationarity. Phys. Rev. D 90, 104,034 (2014). doi:10.1103/PhysRevD.90.104034 CrossRefGoogle Scholar
  15. 15.
    McCormick, S.: The phase space for the Einstein–Yang–Mills equations, black hole mechanics, and a condition for stationarity. Ph.D. thesis, Monash University (2014)Google Scholar
  16. 16.
    Miao, P.: Variational effect of boundary mean curvature on ADM mass in general relativity. arXiv preprint math-ph/0309045 (2003)
  17. 17.
    Miao, P., Tam, L.F.: Static potentials on asymptotically flat manifolds. Annales Henri Poincaré 16(10), 2239–2264 (2015). doi:10.1007/s00023-014-0373-x MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations. I. J. Math. Phys. 16(3), 493–498 (1975)MATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia

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