We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was reduced to an expansion over metric variations between the tetrahedra and, in the main approximation, is a finite-difference form of the Hilbert–Einstein action. Using for the Schwarzschild problem a priori general non-spherically symmetrical ansatz, we get finite-difference equations for its discrete version. This defines a solution which at large distances is close to the continuum Schwarzschild geometry, and the metric and effective curvature at the center are cut off at the elementary length scale. Slow rotation can also be taken into account (Lense–Thirring-like metric). Thus, we get a general approach to the classical background in the quantum framework in zero order: it is an optimal starting point for the perturbative expansion of the theory, finite-difference equations are classical, and the elementary length scale has quantum origin. Singularities, if any, are resolved.

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关于Schwarzschild问题的离散版本

我们在路径积分方法中量化的广义相对论的离散Regge微积分公式中考虑Schwarzschild型解。之前，我们发现了Regge长度背景标度的松散固定机制。该基本长度尺度由普朗克尺度和该理论的这种量子扩展的一些自由参数定义。此外，Regge动作被简化为四面体之间度量变化的展开，并且在主要近似中是Hilbert-Einstein动作的有限差分形式。对于Schwarzschild问题，使用先验的一般非球形对称ansatz，我们得到其离散形式的有限差分方程。这就定义了一个解决方案，该解决方案在很长的距离上接近连续的Schwarzschild几何体，中心的公制和有效曲率在基本长度尺度上被切除。也可以考虑慢速旋转（类似“ Length-Thirring”的指标）。因此，我们可以从零级获得量子框架中经典背景的通用方法：这是理论扰动展开的最佳起点，有限差分方程是经典的，基本长度尺度具有量子起源。奇异点（如果有）得到解决。基本长度尺度具有量子起源。奇异点（如果有）得到解决。基本长度尺度具有量子起源。奇异点（如果有）得到解决。