In this paper, a Kerr-type solution in the Regge calculus is considered. It is assumed that the discrete general relativity, the Regge calculus, is quantized within the path integral approach. The only consequence of this approach used here is the existence of a length scale at which edge lengths are loosely fixed, as considered in our earlier paper. In addition, we previously considered the Regge action on a simplicial manifold on which the vertices are coordinatized and the corresponding piecewise constant metric is introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between four-simplices, this reduces to a finite-difference form of the Hilbert–Einstein action. The problem of solving the corresponding discrete Einstein equations (classical) with a length scale (having a quantum nature) arises as the problem of determining the optimal background metric for the perturbative expansion generated by the functional integral. Using a one-complex-function ansatz for the metric, which reduces to the Kerr–Schild metric in the continuum, we find a discrete metric that approximates the continuum one at large distances and is nonsingular on the (earlier) singularity ring. The effective curvature Rλννρ, including where Rλμ≠0 (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

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关于 Kerr 几何的离散版本

在本文中，考虑了 Regge 演算中的 Kerr 型解。假设离散广义相对论，Regge 演算，在路径积分方法中被量化。此处使用的这种方法的唯一结果是存在一个长度尺度，在该尺度上边长被松散地固定，正如我们之前的论文中所考虑的那样。此外，我们之前考虑了顶点在其上协调的单纯流形上的 Regge 动作，并引入了相应的分段常数度量，并发现对于最简单的周期性单纯形结构和四个单纯形之间的度量变化的领先顺序，这简化为希尔伯特-爱因斯坦作用的有限差分形式。求解具有长度尺度（具有量子性质）的相应离散爱因斯坦方程（经典）的问题是作为确定由泛函积分产生的微扰展开的最佳背景度量的问题而出现的。使用单复函数 ansatz 作为度量，它简化为连续统中的 Kerr-Schild 度量，我们找到了一个离散度量，它在大距离处近似连续统，并且在（早期）奇点环上是非奇异的。有效曲率 我们找到了一个离散度量，它在大距离处近似连续统，并且在（早期）奇点环上是非奇异的。有效曲率 我们找到了一个离散度量，它在大距离处近似连续统，并且在（早期）奇点环上是非奇异的。有效曲率Rλννρ，包括在哪里Rλμ≠0（重力源），重点分析奇点环附近。