2020/08/31 by V. M. Khatsymovsky, Vladimir Khatsymovsky
Physics and Astronomy · #Black Holes and Theoretical Physics #Curvature #Deriving the Schwarzschild solution #General relativity #Loop quantum gravity #Noncommutative and Quantum Gravity Theories #Planck length #Quantum gravity #Relativity and Gravitational Theory #Schwarzschild geodesics #Schwarzschild metric #Schwarzschild radius #gr-qc #msc:83C27 #msc:83C57
paper · pdf · doi:10.3390/universe6100185
published as Universe, Vol. 6, No.10, 185 (2020) · 24 pages, 1 figure; Special issue "Selected Papers from the 17th Russian Gravitational Conference -- International Conference on Gravitation, Cosmology and Astrophysics (RUSGRAV-17)"; explanations and refs added
openalex created_date 2020/09/08 · openalex publication_date 2020/10/17 · arxiv created 2020/10/21 · arxiv updated 2020/10/22 · openalex updated_date 2026/08/05
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.