On the discrete version of the Kerr geometry

Abstract

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 $R_{\lambda \nu \nu \rho }$, including where $R_{\lambda \mu }\ne 0$ (gravity sources), is analyzed with a focus on the vicinity of the singularity ring.

arXiv comment34 pages, 3 figures

References in corpus (21)

• The Dynamics of General Relativity
• Nonperturbative Quantum Gravity
• Quantum Transfiguration of Kruskal Black Holes
• Quantum Extension of the Kruskal Space-time
• A new form of the Kerr solution
• Quantum Gravity on the Lattice
• Newtonian Potential in Quantum Regge Gravity
• On the Measure in Simplicial Gravity
• Cartesian Kerr-Schild variation on the Newman-Janis ansatz
• Regge calculus models of closed lattice universes
• Unit-lapse versions of the Kerr spacetime
• A cosmological solution of Regge calculus
• CDT and Cosmology
• On the discrete Christoffel symbols
• A numerical study of the Regge Calculus and Smooth Lattice methods on a Kasner cosmology
• An Einstein-Bianchi system for Smooth Lattice General Relativity. I. The Schwarzschild spacetime
• On the non-perturbative graviton propagator
• Defining integrals over connections in the discretized gravitational functional integral
• On the discrete version of the black hole solution
• On the discrete version of the Schwarzschild problem
• On the Kerr metric in a synchronous reference frame

Cited by in corpus (1)

• On the discrete version of the Kerr-Newman solution