arXiv 1912.024592019-12gr-qc

变形广义相对论与量子黑洞内部

Deformed General Relativity and Quantum Black Holes Interior

Denis Arruga, Jibril Ben Achour, Karim Noui

备注：: 29 pages, invited contribution to special Issue: "Probing New Physics with Black holes", Matched the published version in Universe

摘要：

黑洞内部的有效模型已经导致了几个关于规则黑洞的建议。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，考虑变形的哈密顿约束，同时保持非变形的矢量约束。在这篇文章中，我们重新审视和进一步研究这些变形重力模型的协方差问题。特别是，我们提出了一个拉格朗日公式，这些变形的重力模型，在对称性降低和勒让德变换之前，在完整的理论水平上引入聚合物变形。这使我们能够测试是否变形协方差的概念中发现球对称真空引力可以扩展到完整的理论，我们表明，在大类的模型，我们正在考虑，变形协方差不能实现超越球对称的意义上说，唯一的变形理论，导致一个封闭的约束代数是广义相对论。因此，我们专注于球对称部门，存在非平凡变形，但封闭的约束代数。我们调查的可能性变形的向量约束，以及我们证明，非平凡变形的向量约束的条件下，约束代数保持封闭不存在。然后，我们计算最一般的变形哈密顿约束，它承认一个封闭的约束代数，从而导致一个定义良好的有效理论与变形协方差的概念。最后，我们研究这些有效的理论的齐次解，值得注意的是，我们明确地和完全一般性地解决了相应的修改后的爱因斯坦方程，即使是不满足封闭条件的有效理论。

英文摘要：

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. 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 can not 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 homogeneous 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.

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