Varying the gravitational Lagrangian produces a boundary contribution that has various physical applications. It determines the right boundary terms to be added to the action once boundary conditions are specified, and defines the symplectic structure of covariant phase space methods. We study general boundary variations using tetrads instead of the metric. This choice streamlines many calculations, especially in the case of null hypersurfaces with arbitrary coordinates, where we show that the spin-1 momentum coincides with the rotational 1-form of isolated horizons. The additional gauge symmetry of internal Lorentz transformations leaves however an imprint: the boundary variation differs from the metric one by an exact 3-form. On the one hand, this difference helps in the variational principle: gluing hypersurfaces to determine the action boundary terms for given boundary conditions is simpler, including the most general case of non-orthogonal corners. On the other hand, it affects the construction of Hamiltonian surface charges with covariant phase space methods, which end up being generically different from the metric ones, in both first and second-order formalisms. This situation is treated in the literature gauge-fixing the tetrad to be adapted to the hypersurface or introducing a fine-tuned internal Lorentz transformation depending non-linearly on the fields. We point out and explore the alternative approach of dressing the bare symplectic potential to recover the value of all metric charges, and not just for isometries. Surface charges can also be constructed using a cohomological prescription: in this case we find that the exact 3-form mismatch plays no role, and tetrad and metric charges are equal. This prescription leads however to different charges whether one uses a first-order or second-order Lagrangian, and only for isometries one recovers the same charges.

中文翻译：

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四分体变量广义相对论中的边界效应

改变引力拉格朗日量会产生具有各种物理应用的边界贡献。一旦指定了边界条件，它就会确定要添加到动作中的右边界项，并定义协变相空间方法的辛结构。我们使用四分体而不是度量来研究一般边界变化。这种选择简化了许多计算，特别是在具有任意坐标的零超曲面的情况下，我们表明自旋 1 动量与孤立层的旋转 1 形式重合。然而，内部洛伦兹变换的额外规范对称性留下了一个印记：边界变化与度量标准的不同之处在于精确的 3 型。一方面，这种差异有助于变分原理：粘合超曲面以确定给定边界条件的动作边界项更简单，包括非正交角的最一般情况。另一方面，它会影响使用协变相空间方法构建哈密顿表面电荷，最终与度量方法在一级和二级形式中一般不同。这种情况在文献中被处理——固定四分体以适应超曲面或引入微调的内部洛伦兹变换，取决于场。我们指出并探索了另一种方法，即修饰裸辛电位以恢复所有公制电荷的价值，而不仅仅是等距。表面电荷也可以使用上同调公式构建：在这种情况下，我们发现精确的 3 型不匹配不起作用，四元和公制电荷相等。然而，无论使用一阶拉格朗日函数还是二阶拉格朗日函数，这种规定都会导致不同的电荷，并且只有对于等距图才能恢复相同的电荷。