Abstract
We study pure D dimensional Einstein gravity in spacetimes with a generic null boundary. We focus on the symplectic form of the solution phase space which comprises a 2D dimensional boundary part and a 2(D(D − 3)/2 + 1) dimensional bulk part. The symplectic form is the sum of the bulk and boundary parts, obtained through integration over a codimension 1 surface (null boundary) and a codimension 2 spatial section of it, respectively. Notably, while the total symplectic form is a closed 2-form over the solution phase space, neither the boundary nor the bulk symplectic forms are closed due to the symplectic flux of the bulk modes passing through the boundary. Furthermore, we demonstrate that the D(D − 3)/2 + 1 dimensional Lagrangian submanifold of the bulk part of the solution phase space has a Carrollian structure, with the metric on the D(D − 3)/2 dimensional part being the Wheeler-DeWitt metric, and the Carrollian kernel vector corresponding to the outgoing Robinson-Trautman gravitational wave solution.
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Acknowledgments
We would like to thank authors of [53] for email exchanges and especially Luca Ciambelli for discussions which triggered this work. We would also like to thank Lars Andersson, Glenn Barnich, Luca Ciambelli, Daniel Grumiller and Shing-Tung Yau for discussions. MMShJ would like to thank ULB, Brussels where this work was started. Work of MMSHJ is partially supported by INSF SarAmadan grant. The work of HA is supported by Beijing Natural Science Foundation under Grant No. IS23018 and by the National Natural Science Foundation of China under Grant No. 12150410311. The work of HY is supported in part by Beijing Natural Science Foundation under Grant No. IS23013.
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Adami, H., Parvizi, A., Sheikh-Jabbari, M.M. et al. Carrollian structure of the null boundary solution space. J. High Energ. Phys. 2024, 73 (2024). https://doi.org/10.1007/JHEP02(2024)073
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DOI: https://doi.org/10.1007/JHEP02(2024)073