Boundary effects in General Relativity with tetrad variables

TL;DR

This work analyzes boundary variations in General Relativity using tetrad variables, revealing a 2d boundary mismatch with the metric formalism that is captured by an exact 3-form (the DPS dressing form). It shows that while the mismatch does not affect the variational principle for closed regions, it significantly influences covariant phase space charges, prompting a dressing procedure that recovers metric charges for diffeomorphisms and eliminates internal Lorentz charges. The paper then extends the analysis to null boundaries, canonical pairs on boundaries, and corner terms, and contrasts covariant phase space charges with Barnich-Brandt charges across both second-order and first-order (with Barbero-Immirzi) formalisms, including Yang-Mills as a parallel example. A key takeaway is that the choice of formalism (second-order vs first-order) and gauge fixing (adapted tetrads, time gauge, Kosmann derivative) crucially determines the agreement between tetrad and metric charges, with explicit prescriptions to restore equivalence in many physically relevant cases. These results illuminate the structure of boundary terms, corner contributions, and charges in tetrad GR, with implications for black hole thermodynamics, edge modes, and potential quantum formulations.

Abstract

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.

Figures (1)

• Figure 1: The two standard settings for the variational problem considered in this section. Left panel: The boundary of the four-dimensional domain of integration consists of a pair of space-like hypersurfaces $\Sigma_{1,2}$ and a time-like one $\cal T$, joined at the space-like corners ${\cal C}_{1,2}$. The figure shows the two basis $(n,\hat{r})$ and $(r,\hat{n})$, where $n$ is the (time-like) unit-norm normal to $\Sigma_2$, $r$ is the (space-like) unit-norm normal to $\cal T$, while $\hat{n}$ and $\hat{r}$ are, respectively, the unit-norm projections of $n$ and $r$ in T$\cal T$ and T$\Sigma$. Right panel: The boundary of the four-dimensional domain of integration consists of a pair of space-like hypersurfaces $\Sigma_{1,2}$ and a converging section of a past light-cone $\cal N$, joined at the space-like corners ${\cal C}_{1,2}$. The figure shows the two basis $(\tau,\hat{r})$ and $(n,l)$, where $\tau$ is the (time-like) unit-norm normal to $\Sigma_2$, $n$ is the (null) normal to $\cal N$, while $\hat{r}$ and $l$ are, respectively, the unit-norm projections of $n$ in T$\Sigma$ and the transverse null vector to $n$.