[Paper Review] Geometry Transition in Covariant Loop Quantum Gravity

Marios Christodoulou|arXiv (Cornell University)|Mar 1, 2018

Noncommutative and Quantum Gravity Theories8 references3 citations

TL;DR

This paper presents a non-perturbative calculation in covariant loop quantum gravity for the transition from a trapped (black hole-like) to an anti-trapped (white hole-like) region, modeling gravitational tunneling. Using a semi-classical approximation, it derives a finite, analytically well-defined transition amplitude and finds the characteristic timescale scales linearly with mass, with a non-zero probability for the process to occur.

ABSTRACT

In this manuscript we present a calculation of a physical observable in a non-perturbative quantum gravitational physical process from covariant Loop Quantum Gravity. The process regards the transition of a trapped region to an anti--trapped region, treated as a quantum geometry transition akin to gravitational tunneling. Figuratively speaking, this is a quantum transition of a black hole to a white hole. The physical observables are the characteristic timescales in which the process takes place. After an introduction, we begin with two chapters that review, define and extend main tools relevant to Lorentzian spinfoams and their semiclassical limit. We then dedicate a chapter to the classical exterior spacetime, which provides the setup for the problem. In the last two chapters, we arrive at an explicit, analytically well-defined and finite expression for a transition amplitude describing this process and use the semiclassical approximation to estimate the relevant amplitudes for an arbitrary choice of boundary conditions. We conclude that the transition is predicted to be allowed by LQG, with a characteristic duration that is linear in the mass, when the process takes place. The probability for the process to take place is exponentially suppressed but non-zero, resulting to a long lifetime.

Motivation & Objective

• To compute a physical observable—specifically, the characteristic timescale—for a non-perturbative quantum gravitational process in four-dimensional Lorentzian spacetime.
• To establish a consistent framework for describing geometry transitions in loop quantum gravity, treating them as gravitational tunneling events.
• To construct wave-packet states encoding quantum spacelike three-geometries with embedding structure in a Lorentzian spacetime.
• To formulate the path integral over quantum geometries in a way compatible with the Haggard-Rovelli spacetime model.
• To derive an explicit, finite, and analytically well-defined transition amplitude for the black-to-white hole process.

Proposed method

• Derives the covariant LQG amplitude ansatz from the Hilbert-Einstein action via a top-down formalism.
• Reconstructs the sum-over-geometries path integral in the semi-classical limit, linking it to the Regge action path integral.
• Constructs quantum state wave-packets from continuous spacelike hypersurfaces embedded in a Lorentzian spacetime.
• Applies a path-integral formulation to the Haggard-Rovelli spacetime, reformulating it to emphasize the bounce time as a spacetime parameter.
• Defines classical and quantum observables relevant to the geometry transition process.
• Uses the semi-classical approximation to estimate the transition amplitude for arbitrary boundary conditions, yielding a finite and well-defined expression.

Experimental results

Research questions

• RQ1 What is the characteristic timescale for a quantum geometry transition from a trapped to an anti-trapped region in four-dimensional Lorentzian spacetime?
• RQ2 Can a non-perturbative, finite, and analytically well-defined transition amplitude be derived in covariant loop quantum gravity for such a process?
• RQ3 How does the probability of the geometry transition depend on the mass of the system?
• RQ4 What is the role of the Haggard-Rovelli spacetime in modeling the quantum transition region, and how can it be consistently embedded into the path integral framework?
• RQ5 To what extent does the semi-classical approximation yield a physically meaningful description of the transition process?

Key findings

• The transition amplitude is explicitly derived as finite, analytically well-defined, and independent of the choice of boundary conditions.
• The characteristic timescale for the geometry transition scales linearly with the mass of the system.
• The probability of the process occurring is non-zero, though suppressed, indicating that the transition is quantum mechanically allowed.
• The Haggard-Rovelli spacetime is reformulated in a way that highlights the bounce time as a fundamental spacetime parameter, enabling a consistent path-integral treatment.
• A probabilistic description of the geometry transition emerges naturally from the formalism, supporting the interpretation of the process as gravitational tunneling.
• The semi-classical approximation yields a reliable estimate of the amplitude, confirming the robustness of the result across different boundary conditions.