arXiv 2103.055982021-03-09gr-qc

关于离散形式的Kerr几何

On the discrete version of the Kerr geometry

V. M. Khatsymovsky

备注：: 34 pages, 3 figures
发表信息：: Int. J. Mod. Phys. A, Volume No. 36, Issue No. 20, Article No. 2150130, Year 2021
DOI：: 10.1142/S0217751X2150130X

摘要：

考虑了Regge演算中的一个Kerr型解。假设离散的广义相对论，Regge演算，是量化的路径积分的方法。这里使用的这种方法的唯一后果是存在一个长度尺度，在这个尺度上，边长是松散固定的，正如我们在前面的论文中所考虑的那样。此外，我们以前考虑了Regge作用在一个单形流形上的顶点是协调的，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形结构和在4-单形之间的度量变化的领先顺序，这减少到希尔伯特-爱因斯坦作用的有限差分形式。求解相应的离散爱因斯坦方程（经典）与长度尺度（具有量子性质）的问题，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出现。使用一个复杂的函数anastrophic的度量，这减少了克尔-Schild度量的连续，我们发现一个离散的度量，近似的连续一个在大的距离，是非奇异的（早期）奇异环。分析了有效曲率$R_{λννρ}$，包括其中的$R_{λμ} \neq 0$（重力源），重点是奇异环附近。

英文摘要：

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 introduced, and found that for the simplest periodic simplicial structure and in the leading order over metric variations between 4-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 an 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 $R_{λννρ}$, including where $R_{λμ} \neq 0$ (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

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