AI 中文总结
该研究构造了无焦散零线段上广义相对论的体运动学泊松结构，通过三阶段狄拉克约化得到零引力数据的主相空间及相关括号，为研究零约束代数等提供了经典正则舞台。
AI 中文摘要
我们在施加Raychaudhuri约束和Damour约束之前，构造了无焦散零线段上广义相对论的体运动学泊松结构。我们首先证明，Ehresmann联络（也称为Carroll boost）的位移是引力中的纯规范自由度，这使得Ehresmann联络可被固定为背景结构，进而得到零引力数据的主相空间。该相空间包含：自旋0对（包含面积及其共轭量）、自旋1对（包含零方向及其共轭量），以及编码在 unimodular 横截度量中的自旋2部分。随后，我们通过三阶段狄拉克约化构造括号，该约化过程实现了归一化、水平性和剪切约束。我们发现，自旋2括号是非局域的，涉及零生成元上一阶传输算子的反对称格林核；自旋0和自旋1动量的自括号，分别包含与自旋2传播子缩并的横截度量的垂直导数和水平导数。最后，我们通过辛形式哈密顿向量场的显式实现，独立推导了这些括号。这些结果为研究零约束代数、可观测量及量子化提供了经典正则舞台。
英文摘要
We construct the bulk kinematical Poisson structure of general relativity on a caustic-free null segment, prior to imposing the Raychaudhuri and Damour constraints. We first show that shifts of the Ehresmann connection, also known as Carroll boosts, are pure gauge in gravity. This allows the Ehresmann connection to be fixed as a background structure and leads to the prime phase space of null gravitational data. The resulting phase space comprises of a spin-0 pair that includes the area and its conjugate, a spin-1 pair including the null direction and its conjugate, and the spin-2 sector encoded in the unimodular transverse metric. We then construct the brackets through a three-stage Dirac reduction implementing a normalization, horizontality, and shear constraints. We find that the spin-2 bracket is non-local and involves an antisymmetric Green kernel for a first-order transport operator along the null generators. The self-brackets of the spin-0 and spin-1 momenta involve, respectively, the vertical and horizontal derivatives of the transverse metric contracted with the spin-2 propagator. We finally provide an independent derivation of the brackets from the explicit realization of the Hamiltonian vector fields of the symplectic form. These results provide the classical canonical arena for studying the null constraint algebra, observables, and quantization.
Comments45 pages, v1
