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A Quasi-Local Mass

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Abstract

We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown–York Hamilton–Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric embeddings into Minkowski space, although our notion of admissibility is different from that of Wang and Yau. Rigidity is also established, in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely the mass vanishes for any such surface. Furthermore, we show convergence to the ADM mass at spatial infinity, and provide the equation associated with optimal isometric embedding.

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Acknowledgements

The authors would like to thank Mu-Tao Wang for helpful comments

Author information

Authors and Affiliations

  1. Department of Mathematics, Clark University, Worcester, MA, 01610, USA

    Aghil Alaee

  2. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02138, USA

    Aghil Alaee

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

    Marcus Khuri

  4. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Shing-Tung Yau

  5. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China

    Shing-Tung Yau

Authors
  1. Aghil Alaee
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  2. Marcus Khuri
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  3. Shing-Tung Yau
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Correspondence to Aghil Alaee.

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Communicated by P. Chrusciel

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A. Alaee acknowledges the support of NSF Grant DMS-2316965, and an AMS-Simons travel grant. M. Khuri acknowledges the support of NSF Grant DMS-2104229.

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Alaee, A., Khuri, M. & Yau, ST. A Quasi-Local Mass. Commun. Math. Phys. 405, 111 (2024). https://doi.org/10.1007/s00220-024-04976-x

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