There are at least two ways to encode gravity into geometry: Einstein's general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein--Cartan action, which were studied recently in the literature. The first discretization leads to the \emph{{loop gravity}} kinematical phase space where the zero torsion condition is enforced {first} and the other is the \emph{{dual loop gravity}} kinematical phase space where curvature is imposed to vanish {first}. Our argument is based on the observation that the GR first-order Einstein--Cartan action can also be seen as a first-order action for teleparallel gravity up to a boundary term. The results of our paper suggest that the \emph{dual loop gravity} framework is a natural discretization of teleparallel gravity, whereas \emph{loop gravity} is naturally related to the standard GR metric description.

中文翻译：

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远程平行重力和双环重力的一阶公式

至少有两种方法可以将重力编码为几何：爱因斯坦的广义相对论 (GR) 用于度量张量，以及遥平行重力，其中扭转而不是曲率编码了重力自由度的动力学。论文的主要目的是探讨环引力与远平行引力之间的关系。我们认为这两种引力公式与爱因斯坦-嘉当作用的两种不同离散化有关，最近在文献中对其进行了研究。第一个离散导致 \emph{{loop 重力}}运动学相空间，其中强制执行零扭转条件{first}，另一个是 \emph{{双环重力}}运动学相空间，其中强加曲率以消失{第一的}。我们的论点是基于观察到 GR 一阶爱因斯坦 - 嘉当作用也可以看作是直到边界项的远平行引力的一阶作用。我们论文的结果表明，\emph{双环重力}框架是远程平行重力的自然离散化，而\emph{环重力}自然地与标准GR度量描述相关。