Skip to main content
SpringerLink
Log in

Measuring Mass via Coordinate Cubes

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

Abstract

Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat 3-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov’s scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss–Bonnet theorem.

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. Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(3), 997–1006 (1961)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Bray, H., Kazaras, D., Khuri, M., Stern, D.: Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds. arXiv:1911.06754

  4. Bray, H., Stern, D.: Scalar curvature and harmonic one-forms on three-manifolds with boundary. arXiv:1911.06803

  5. Carlotto, A., De Lellis, C.: Min-max embedded geodesic lines in asymptotically conical surfaces. J. Differ. Geom. 112(3), 411–445 (2019)

    Article  MathSciNet  Google Scholar 

  6. Chruściel, P.: Boundary Conditions at Spatial Infinity from a Hamiltonian Point of View, Topological Properties and Global Structure of Space-Time, pp. 49–59. Plenum Press, New York (1986)

    Book  Google Scholar 

  7. Fan, X.-Q., Shi, Y., Tam, L.-F.: Large-sphere and small-sphere limits of the Brown–York mass. Commun. Anal. Geom. 17(1), 37–72 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  8. Gromov, M.: Dirac and Plateau billiards in domains with corners. Cent. Eur. J. Math. 12(8), 1109–1156 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Li, C.: A polyhedron comparison theorem for 3-manifolds with positive scalar curvature. Invent. Math. 219, 1–37 (2020). https://doi.org/10.1007/s00222-019-00895-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Li, C., Mantoulidis, C.: Positive scalar curvature and skeleton singularities. Math. Ann. 374, 99–131 (2019)

    Article  MathSciNet  Google Scholar 

  11. 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  ADS  MathSciNet  Google Scholar 

  12. Stern, D.: Scalar curvature and harmonic maps to \({\mathbb{S}}^1\). arXiv:1908.09754

  13. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

I want to thank Hubert Bray for encouraging me in the pursuit of the use of (3). I also want to thank anonymous reviewers for their insightful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Pengzi Miao

Authors
  1. Pengzi Miao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pengzi Miao.

Additional information

Communicated by P. Chrusciel

Publisher's Note

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

Pengzi Miao research was partially supported by NSF Grant DMS-1906423.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miao, P. Measuring Mass via Coordinate Cubes. Commun. Math. Phys. 379, 773–783 (2020). https://doi.org/10.1007/s00220-020-03811-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03811-3

Search

Navigation