Strong field behavior of Wang-Yau Quasi-local energy
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909980385345536 |
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| author | Zhao, Bowen Andersson, Lars Yau, Shing-Tung |
| author_facet | Zhao, Bowen Andersson, Lars Yau, Shing-Tung |
| contents | We look at the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike $2$-surface $Σ$ approaches an apparent horizon from outside. Assuming that coordinate functions of the isometric embedding are bounded in $W^{2,1}$ and mean curvature vector of the image surface remains spacelike, we find that the limit falls in two exclusive cases: 1) If the horizon cannot be isometrically embedded into $R^3$, the Wang-Yau quasi-local energy blows up as $Σ$ approaches the horizon while the optimal embedding equation is not solvable for $Σ$ near the horizon; 2) If the horizon can be isometrically embedded into $R^3$, the optimal embedding equation is solvable up to the horizon with the unique solution at the horizon corresponding to isometric embedding into $R^3$ and the Wang-Yau quasi-local mass admits a finite limit at the horizon. We discuss the implications of our results in the conclusion section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong field behavior of Wang-Yau Quasi-local energy Zhao, Bowen Andersson, Lars Yau, Shing-Tung General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry We look at the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike $2$-surface $Σ$ approaches an apparent horizon from outside. Assuming that coordinate functions of the isometric embedding are bounded in $W^{2,1}$ and mean curvature vector of the image surface remains spacelike, we find that the limit falls in two exclusive cases: 1) If the horizon cannot be isometrically embedded into $R^3$, the Wang-Yau quasi-local energy blows up as $Σ$ approaches the horizon while the optimal embedding equation is not solvable for $Σ$ near the horizon; 2) If the horizon can be isometrically embedded into $R^3$, the optimal embedding equation is solvable up to the horizon with the unique solution at the horizon corresponding to isometric embedding into $R^3$ and the Wang-Yau quasi-local mass admits a finite limit at the horizon. We discuss the implications of our results in the conclusion section. |
| title | Strong field behavior of Wang-Yau Quasi-local energy |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2406.10751 |