Effective models of black holes interior have led to several proposals for regular black holes. In the so-called polymer models, based on effective deformations of the phase space of spherically symmetric general relativity in vacuum, one considers a deformed Hamiltonian constraint while keeping a non-deformed vectorial constraint, leading under some conditions to a notion of deformed covariance. In this article, we revisit and study further the question of covariance in these deformed gravity models. In particular, we propose a Lagrangian formulation for these deformed gravity models where polymer-like deformations are introduced at the level of the full theory prior to the symmetry reduction and prior to the Legendre transformation. This enables us to test whether the concept of deformed covariance found in spherically symmetric vacuum gravity can be extended to the full theory, and we show that, in the large class of models we are considering, the deformed covariance cannot be realized beyond spherical symmetry in the sense that the only deformed theory which leads to a closed constraints algebra is general relativity. Hence, we focus on the spherically symmetric sector, where there exist non-trivial deformed but closed constraints algebras. We investigate the possibility to deform the vectorial constraint as well and we prove that non-trivial deformations of the vectorial constraint with the condition that the constraints algebra remains closed do not exist. Then, we compute the most general deformed Hamiltonian constraint which admits a closed constraints algebra and thus leads to a well-defined effective theory associated with a notion of deformed covariance. Finally, we study static solutions of these effective theories and, remarkably, we solve explicitly and in full generality the corresponding modified Einstein equations, even for the effective theories which do not satisfy the closeness condition. In particular, we give the expressions of the components of the effective metric (for spherically symmetric black holes interior) in terms of the functions that govern the deformations of the theory.

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变形的广义相对论和量子黑洞内部

有效的黑洞内部模型已经提出了一些有关常规黑洞的建议。在所谓的聚合物模型中，基于真空中球对称广义相对论相空间的有效变形，人们在保持不变的矢量约束的同时考虑了变形的哈密顿约束，在某些情况下导致了变形协方差的概念。在本文中，我们将回顾并进一步研究这些变形重力模型中的协方差问题。特别是，我们为这些变形重力模型提出了Lagrangian公式，其中在对称性降低之前和Legendre变换之前，在完整理论的水平上引入了类似聚合物的变形。这使我们能够检验在球对称真空重力中发现的变形协方差的概念是否可以扩展到整个理论，并且表明，在我们正在考虑的一大类模型中，变形协方差不能实现超出球对称性的形式。导致封闭约束代数的唯一变形理论是广义相对论。因此，我们专注于球对称的扇形，其中存在非平凡的变形但封闭的约束代数。我们还研究了矢量约束变形的可能性，并证明了不存在约束代数保持封闭的条件下矢量约束的非平凡变形。然后，我们计算出最一般的变形哈密顿约束，它接受封闭约束代数，从而得出与变形协方差概念相关的定义明确的有效理论。最后，我们研究了这些有效理论的静态解，并且显着地，即使对于不满足紧密性条件的有效理论，我们也显式且全面地求解了相应的修正爱因斯坦方程。特别是，我们根据控制理论变形的函数，给出了有效度量（对于球形对称黑洞内部）的成分表示。即使对于不满足亲近性条件的有效理论，我们也可以明确，全面地解决相应的修正爱因斯坦方程。特别是，我们根据控制理论变形的函数，给出了有效度量（对于球形对称黑洞内部）的成分表示。即使对于不满足亲近性条件的有效理论，我们也可以明确，全面地解决相应的修正爱因斯坦方程。特别是，我们根据控制理论变形的函数，给出了有效度量（对于球形对称黑洞内部）的成分表示。