(Peer-Reviewed) Covariant phase space with null boundaries
Kai Shi ¹, Xuan Wang ¹, Yihong Xiu ¹, Hongbao Zhang 张宏宝 ¹ ²
¹ Department of Physics, Beijing Normal University, Beijing 100875, China
中国 北京 北京师范大学物理学系
² Theoretische Natuurkunde, Vrije Universiteit Brussel, and The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium
Abstract
By imposing the boundary condition associated with the boundary structure of the null boundaries rather than the usual one, we find for Einstein's gravity that the variational principle works only in its submanifold with the null boundaries given by the expansion-free and shear-free hypersurfaces rather than in the whole covariant phase space.
This implies that the key requirement in Harlow–Wu's algorithm for the timelike boundaries is too restrictive for the null ones. To incorporate more generic situations into Harlow–Wu's algorithm, we relax such a requirement. As a result, we successfully reproduce the Hamiltonian obtained previously by Wald–Zoupas' prescription for Einstein's gravity.
Integrated metasurface-freeform system enabled multi-focal planes augmented reality display
Shifei Zhang, Lina Gao, Yidan Zhao, Yongdong Wang, Bo Wang, Junjie Li, Jiaxi Duan, Dewen Cheng, Cheng-Wei Qiu, Yongtian Wang, Tong Yang, Lingling Huang
Opto-Electronic Science
2026-01-23
