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Acta Physica Slovaca 68, No.1&2, 1 – 185 (2018) (185 pages)
GEOMETRY TRANSITION IN COVARIANT LOOP QUANTUM GRAVITY Marios Christodoulou Centre National de la Recherche Scientifique, Centre de Physique Théorique Aix-Marseille Université, Case 907, F-13288 Marseille, France Full text: ::pdf :: (Received 11 September 2018, accepted 23 October 2018) Abstract: In this manuscript we present a calculation from covariant Loop Quantum Gravity, of a physical observable in a non-perturbative quantum gravitational physical process. The process regards the transition of a trapped region to an anti–trapped region, treated as a quantum geometry transition akin to gravitational tunneling. The physical observable is the characteristic timescale in which the process takes place. Focus is given to the physically relevant four–dimensional Lorentzian case. We start with a top–to–bottom formal derivation of the ansatz defining the amplitudes for covariant Loop Quantum Gravity, starting from the Hilbert-Einstein action. We then take the bottom–to–top path, starting from the Engle-Perreira-Rovelli-Livine ansatz, to the sum–over–geometries path integral emerging in the semi-classical limit, and discuss its close relation to the naive path integral over the Regge action. We proceed to the construction of wave–packets describing quantum spacelike three-geometries that include a notion of embedding, starting from a continuous hypersurface embedded in a Lorentzian spacetime. We derive a simple estimation for physical transition amplitudes describing geometry transition and show that a probabilistic description for such phenomena emerges, with the probability of the phenomena to take place being in general non-vanishing. The Haggard-Rovelli (HR) spacetime, modelling the spacetime surrounding the ge- ometry transition region for a black to white hole process, is presented and discussed. We give the HR-metric in a form that emphasizes the role of the bounce time as a spacetime parameter and we give an alternative path for its construction. We de- fine the classical and quantum observables relevant to the process, propose an interpretation for the transition amplitudes and formulate the problem such that a path integral over quantum geometries procedure can be naturally applied. We proceed to derive an explicit, analytically well-defined and finite expression for a transition amplitude describing this process. We then use the semi–classical approximation to estimate the amplitudes describing the process for an arbitrary choice of boundary conditions. We conclude that the process is predicted to be allowed by LQG, with a crossing time that is linear in the mass. The probability for the process to take place is suppressed but non-zero. We close by discussing the physical interpretation of our results and further directions. Keywords: General Theory of Relativity, Holonomy, Spinors, Path-integral Representations, Covariant Loop Quantum Gravity, Black Holes, White holes, Spin-networks |
ISSN 1336-040X (online) ISSN 0323-0465 (printed) For authors: ActaStyle.cls |
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