Adiabatic Solutions in General Relativity as Null Geodesics on the Space of Boundary Diffeomorphisms
Emine Şeyma Kutluk
备注：
This article summarizes and slightly expands the works in arXiv:2106.13268 and arXiv:1904.12869. To appear in the Proceedings of the 16th Marcel Grossmann Meeting (5-10 July 2021). Based on the talk presented in the parallel session "Mathematical Problems of Relativistic Physics: Classical and Quantum4;
We use a trick similar to Weinberg's for adiabatic modes, in a Manton approximation for general relativity on manifolds with spatial boundary. This results in a description of the slow-time dependent solutions as null geodesics on the space of boundary diffeomorphisms, with respect to a metric we prove to be composed solely of the boundary data. We show how the solutions in the bulk space is determined with the constraints of general relativity.
To give our description a larger perspective, we furthermore identify our resulting Lagrangian as a generalized version of the covariantized Lagrangian for continuum mechanics. We study the cases of 3+1 and 2+1 dimensions and show for the solutions we propose, the Hamiltonian constraint becomes the real homogeneous Monge-Ampere equation in the special case of two spatial dimensions.
In arXiv:1904.12869 it was shown how in an adiabatic limit the vacuum Einstein equations on a compact spatial region can be re-expressed as geodesic equations on the group of diffeomorphisms of the boundary. This is reminiscent of the program initiated by V. Arnold to reformulate models of continuum mechanics in terms of geodesic motion on diffeomorphism groups. We revisit some of the results of arXiv:1904.12869 in this light, pointing out parallels and differences with the typical examples in geometric continuum mechanics. We work out the case of 2 spatial dimensions in some detail.
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School of Physics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究院物理学院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学及应用研究院）
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Carrollian Worldsheets on Carrollian Targets, When does ILST describe a null string theory?
School of Physics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所物理学院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用数学研究所）
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We consider General Relativity (GR) on a space-time whose spatial slices are compact manifolds $M$ with non-empty boundary $\partial M$. We argue that this theory has a non-trivial space of `vacua', consisting of spatial metrics obtained by an action on a reference flat metric by diffeomorpisms that are non-trivial at the boundary. In an adiabatic limit the Einstein equations reduce to geodesic motion on this space of vacua with respect to a particular pseudo-Riemannian metric that we identify. We show how the momentum constraint implies that this metric is fully determined by data on the boun…
