Skip to main content
SpringerLink
Log in

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper, we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in \(L^p\), \(1\le p<2\). In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Allen, B.: IMCF and the stability of the PMT and RPI under \(L^2\) convergence. Ann. Henri Poincaré 19(4), 1283–1306 (2018)

    Article  MathSciNet  Google Scholar 

  2. Andersson, L., Metzger, J.: The area of horizons and the trapped region. Commun. Math. Phys. 290, 941–972 (2009)

    Article  MathSciNet  Google Scholar 

  3. Andersson, L., Mars, M., Simon, W.: Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes. Adv. Theor. Math. Phys. 12(4), 853–888 (2008)

    Article  MathSciNet  Google Scholar 

  4. Andersson, L., Eichmair, M., Metzger, J.: Jang’s equation and its applications to marginally trapped surfaces, Complex analysis and dynamical systems IV, Part 2. Contemp. Math. 554, 13–45 (2011)

    Article  Google Scholar 

  5. Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(5), 661–693 (1986)

    Article  MathSciNet  Google Scholar 

  6. Beig, R., Chruściel, P.: Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem. J. Math. Phys. 37(4), 1939–1961 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bray, H., Khuri, M.: A Jang equation approach to the Penrose inequality. Discrete Contin. Dyn. Syst. 27(2), 741–766 (2010)

    Article  MathSciNet  Google Scholar 

  8. Bryden, E.: Stability of the positive mass theorem for axisymmetric manifolds. Pac. J. Math. 305, 89–152 (2020)

    Article  MathSciNet  Google Scholar 

  9. Bryden, E., Khuri, M., Sokolowsky, B.: The positive mass theorem with angular momentum and charge for manifolds with boundary. J. Math. Phys. 60(5), 052501–10 (2019)

    Article  MathSciNet  Google Scholar 

  10. Chruściel, P., Maerten, D.: Killing vectors in asymptotically flat space-times. II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions. J. Math. Phys. 47(2), 022502–10 (2006)

    Article  MathSciNet  Google Scholar 

  11. Eichmair, M.: Existence, regularity, and properties of generalized apparent horizons. Commun. Math. Phys. 294(3), 745–760 (2010)

    Article  MathSciNet  Google Scholar 

  12. Eichmair, M.: The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight. Commun. Math. Phys. 319(3), 575–593 (2013)

    Article  MathSciNet  Google Scholar 

  13. Eichmair, M., Huang, L.-H., Lee, D., Schoen, R.: The spacetime positive mass theorem in dimensions less than eight. J. Eur. Math. Soc. (JEMS) 18(1), 83–121 (2016)

    Article  MathSciNet  Google Scholar 

  14. Han, Q., Khuri, M.: Existence and blow-up behavior for solutions of the generalized Jang equation. Commun. Partial Differ. Equ. 38, 2199–2237 (2013)

    Article  MathSciNet  Google Scholar 

  15. Hayward, S.: Gravitational energy in spherical symmetry. Phys. Rev. D 53, 1938–1949 (1996)

    Article  MathSciNet  Google Scholar 

  16. Hidayet, A., Akbar, F., Gunara, B.: Higher dimensional Penrose inequality in spherically symmetric spacetime. Chin. J. Phys. 54(4), 582–586 (2016)

    Article  MathSciNet  Google Scholar 

  17. Huang, L.-H., Lee, D.: Equality in the spacetime positive mass theorem. Commun. Math. Phys. 376, 2379–2407 (2020)

    Article  MathSciNet  Google Scholar 

  18. Huang, L.-H., Lee, D., Sormani, C.: Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. J. Reine Angew. Math. 727, 269–299 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)

    Article  MathSciNet  Google Scholar 

  20. Jauregui, J., Lee, D.: Lower semicontinuity of ADM mass under intrinsic flat convergence (2019). arXiv:1903.00916

  21. Khuri, M.: The Hoop Conjecture in spherically symmetric spacetimes. Phys. Rev. D 80, 124025 (2009)

    Article  MathSciNet  Google Scholar 

  22. Lakzian, S., Sormani, C.: Smooth convergence away from singular sets. Commun. Anal. Geom. 21(1), 39–104 (2013)

    Article  MathSciNet  Google Scholar 

  23. Lee, D., Sormani, C.: Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds. J. Reine Angew. Math. 686, 187–220 (2014)

    MathSciNet  MATH  Google Scholar 

  24. LeFloch, P., Sormani, C.: Nonlinear stability of rotationally symmetric spaces with low regularity. J. Funct. Anal. 268(7), 2005–2065 (2015)

    Article  MathSciNet  Google Scholar 

  25. Malec, E., Murchadha, N.Ó.: The Jang equation, apparent horizons and the Penrose inequality. Class. Quantum Gravity 21(24), 5777–5787 (2004)

    Article  MathSciNet  Google Scholar 

  26. Mars, M.: Present status of the Penrose inequality. Class. Quantum Gravity 26(19), 193001–59 (2009)

    Article  MathSciNet  Google Scholar 

  27. Metzger, J.: Blowup of Jang’s equation at outermost marginally trapped surfaces. Commun. Math. Phys. 294(1), 61–72 (2010)

    Article  MathSciNet  Google Scholar 

  28. Misner, C., Sharp, D.: Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev. 136, B571–B576 (1964)

    Article  MathSciNet  Google Scholar 

  29. Paetz, T., Simon, W.: Marginally outer trapped surfaces in higher dimensions. Class. Quantum Gravity 30(23), 235005–14 (2013)

    Article  MathSciNet  Google Scholar 

  30. Parker, T., Taubes, C.: On Witten’s proof of the positive energy theorem. Commun. Math. Phys. 84(2), 223–238 (1982)

    Article  MathSciNet  Google Scholar 

  31. Petersen, P.: Riemannian Geometry. Springer, New York (2006)

    MATH  Google Scholar 

  32. Portegies, J.: Semicontinuity of eigenvalues under intrinsic flat convergence. Calc. Var. PDE 54(2), 1725–1766 (2015)

    Article  MathSciNet  Google Scholar 

  33. Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)

    Article  MathSciNet  Google Scholar 

  34. Schoen, R., Yau, S.-T.: Proof of the positive mass theorem II. Commun. Math. Phys. 79, 231–260 (1981)

    Article  MathSciNet  Google Scholar 

  35. Schoen, R., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities (2018). arXiv:1704.05490

  36. Sormani, C.: Measure theory in non-smooth spaces. In: Nicola G. (ed.) De Gruyter, Chapter 9, pp. 288–338 (2017)

  37. Sormani, C., Stavrov, I.: Geometrostatic manifolds of small ADM mass. Commun. Pure Appl. Math. 72(6), 1243–1287 (2019)

    Article  MathSciNet  Google Scholar 

  38. Sormani, C., Wenger, S.: The intrinsic flat distance between Riemannian manifolds and integral current spaces. J. Differ. Geom. 87(1), 117–199 (2011)

    Article  MathSciNet  Google Scholar 

  39. Wald, R.: General Relativity. University of Chicago Press, Chicago (1984)

    Book  Google Scholar 

  40. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Dan Lee for helpful conversations, and Walter Simon for comments. Christina Sormani gratefully acknowledges office space in the Simons Center for Geometry and Physics, Stony Brook University at which most of the research for this paper was performed, and also support from a CUNY Fellowship Leave. Edward Bryden would like to thank the CUNY Graduate Center for allowing him to visit in Spring 2019 during which this paper was completed.

Author information

Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Edward Bryden & Marcus Khuri

  2. CUNY Graduate Center and Lehman College, New York, NY, 10016, USA

    Christina Sormani

Authors
  1. Edward Bryden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcus Khuri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christina Sormani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcus Khuri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

E. Bryden acknowledges the support of NSF Grant DMS-1612049. M. Khuri acknowledges the support of NSF Grant DMS-1708798. C. Sormani acknowledges the support of NSF Grant DMS-1612049.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bryden, E., Khuri, M. & Sormani, C. Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry. J Geom Anal 31, 4191–4239 (2021). https://doi.org/10.1007/s12220-020-00431-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-020-00431-0

Keywords

Mathematics Subject Classification

Search

Navigation