Skip to main content
SpringerLink
Log in

A Gibbons–Penrose Inequality for Surfaces in Schwarzschild Spacetime

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.

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

Similar content being viewed by others

References

  1. Bray H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59, 177–267 (2001)

    MATH  MathSciNet  Google Scholar 

  2. Brendle S.: Constant mean curvature surfaces in warped product manifolds. Publications Mathématiques de l’IHÉS 117, 247–269 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski-type inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold ( arxiv:1209.0669)

  4. Gibbons, G.W.: Ph.D. thesis, Cambridge University (1973)

  5. Gibbons G.W.: Collapsing shells and the isoperimetric inequality for black holes. Class. Quantum Grav. 14, 2905–2915 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Guan P., Li J.: The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221, 1725–1732 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Huisken, G.: in preparation

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

    MATH  MathSciNet  Google Scholar 

  9. Mars, M.: Present status of the Penrose inequality. Class. Quantum Grav. 26, 193001, 59 pp (2009)

    Google Scholar 

  10. Mars M., Soria A.: On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension. Class. Quantum Grav. 29, 135005 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  11. Penrose R.: Naked singularities. Ann. New York Acad. Sci. 224, 125–134 (1973)

    Article  ADS  Google Scholar 

  12. Tod K.P.: Penrose quasilocal mass and the isoperimetric inequality for static black holes. Class Quantum Grav. 2, L65–L68 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Tod K.P.: More on Penrose’s quasilocal mass. Class Quantum Grav. 3, 1169–1189 (1986)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Tod K.P.: The hoop conjecture and the Gibbons–Penrose construction of trapped surfaces. Class. Quantum Grav. 9(6), 1581–1591 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Wang, M.-T.: Quasilocal mass and surface Hamiltonian in spacetime. (arxiv:1211.1407)

  16. Wang M.-T., Yau S.-T.: Quasilocal mass in general relativity. Phys. Rev. Lett. 102(2), 021101 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  17. Wang M.-T., Yau S.-T.: Isometric embeddings into the Minkowski space and new quasi-local mass. Commun. Math. Phys. 288(3), 919–942 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Simon Brendle

  2. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Mu-Tao Wang

Authors
  1. Simon Brendle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mu-Tao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mu-Tao Wang.

Additional information

Communicated by P. T. Chruściel

S. Brendle was supported in part by the National Science Foundation under grant DMS-1201924. M.-T. Wang was supported in part by the National Science Foundation under grant DMS-1105483. The authors would like to thank Gary Gibbons, Gerhard Huisken, Marc Mars, and Shing-Tung Yau for helpful discussions. In particular, we are grateful to Gerhard Huisken for pointing out the existence of umbilical slices in the Schwarzschild spacetime. M.-T. Wang would like to acknowledge the hospitality of National Center for Theoretical Sciences (Mathematics Division, Taipei Office) where part of this work was done during his visit.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brendle, S., Wang, MT. A Gibbons–Penrose Inequality for Surfaces in Schwarzschild Spacetime. Commun. Math. Phys. 330, 33–43 (2014). https://doi.org/10.1007/s00220-014-1972-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-1972-6

Keywords

Search

Navigation