Quantum Gravity and String Theory


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

Authors: Jan Helm

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. We get 32 partial differential equations for the 16 variables Emn ( E-tensor, inverse densitized tetrad of the metric gmn) and 16 variables Amn (A-tensor, 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. At the horizon, the E-tensor couples to the A-tensor in the 24 Hamiltonian equations and the singularity is removed, there is instead a peak in the metric . In the time-dependent case with a L- scaled wave ansatz for the A-tensor and the E-tensor we get a gravitational wave equation, which yields appropriate solutions only for quadrupole waves: as required by GR, the tetrad is exponentially damped , only the A-tensor carries the energy. The validity of the Einstein power formula for gravitational waves is shown for a binary black hole (binary gravitational rotator). From the horizon condition we derive the limit scale (Schwarzschild radius) of the gravitational quantum region: rgr=30μm , which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. We present the energy-momentum tensor, which is in agreement with the corresponding GR-expression for small wave amplitudes and is consistent with the Einstein power formula. In the quantum region r<= rgr , the Ashtekar-Kodama gravitation the theory becomes a gauge theory with the extended SU(2) (four generators) as gauge group and a corresponding covariant derivative. In the quantum region we derive the lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams at the example of the graviton-electron scattering.

Comments: 100 Pages.

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Submission history

[v1] 2018-07-01 14:28:12
[v2] 2018-07-23 05:57:29
[v3] 2018-12-04 05:57:38

Unique-IP document downloads: 58 times

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