Quantum Gravity and String Theory


A Covariant Formulation of the Ashtekar-Kodama Quantum Gravity and Its Solutions

Authors: Jan Helm

Abstract This article consists of two parts. In the first part A we present in a concise form the present approaches to the quantum gravity, with the ADM formulation of GR, the Ashtekar and the Kodama ansatz at the center, and we also derive the 3-dimensional Ashtekar-Kodama constraints. In the second part B , we introduce a 4-dimensional covariant version of the 3-dimensional (spatial) Hamiltionian, Gaussian and diffeomorphism constraints of the Kodama state with positive cosmological constant L in the Ashtekar formulation of quantum gravity. In chapter B1-4 we present the equations and their solutions. We get 32 partial differential equations for the 16 variables Emn (inverse densitized tetrad of the metric gmn) and 16 variables Amn (gravitational wave tensor). We impose the boundary condition: for r->inf g(Emn)->gmn i.e. in the classical limit of large r the Kodama state generates the given asymptotic spacetime (normally Schwarzschild-spacetime) . For L->0 in the static (time independent) the tetrad decouples from the wave tensor and the 24 Hamiltonian equations yield for Amn the constant background solution. The diffeomorphism becomes identically zero, and the tetrad can satisfy the Schwarzschild spacetime and the Gaussian equations for all {r,θ}, i.e. it the Einstein equations are valid everywhere outside the horizon. In the time-dependent case with a L- scaled wave ansatz for Amn and Emn we get a gravitational wave equations , which yields appropriate solutions only for angular momentum lx>=2 (quadrupole wave) the tetrad is locally damped with exp(-4 sqrt(r/3)), only the wave tensor Amn carries the energy. In chapter B5 and 6 numerical solutions for special cases of the time-independent and of time-dependent equations are discussed. In chapter B7 the energy tensor of the Ashtekar-Kodama gravity is introduced. In chapter B8 we present the quantum field version of the Ashtekar-Kodama gravity and demonstrate the calculation of cross-sections. Finally, in chapter B9 we give as an outlook an overview of the QFT including the Ashtekar-Kodama gravity. All derivations and calculations were carried out in Mathematica-programs, so the results can be considered with high probability as error-free, the programs are cited in the literature index. In the chapters B1-4, which deal with the solutions, every subchapter consists of a flow-diagram, which gives the overview and a text part, which describes the corresponding program in detail and can be skipped at first reading.

Comments: 88 Pages.

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Submission history

[v1] 2018-07-01 14:28:12

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