Minisuperspace, WKB and Quantum States of General Relativistic Extended Objects

Abstract

Abstract. The dynamics of relativistic thin shells is a recurrent topic in the literature about the classical theory of gravitating systems and the still ongoing attempts to obtain a coherent description of their quantum behavior. Certainly, a good reason to make this system a preferred one for many models is the clear, synthetic description given by Israel junction conditions. Using some results from an effective Lagrangian approach to the dynamics of spherically symmetric shells, we show a general way to obtain WKB states for the system; a simple example is also analyzed.The study of the dynamics of an (infinitesimally) 1 thin surface layer separating two domains of spacetime is an interesting problem in General Relativity. The system can be described in a very concise and geometrically flavored way using Israel's junction conditions [1,2,3]. Starting from these toeholds many authors have then tackled the problem of finding some hints about the properties of the still undiscovered quantum theory of gravitational phenomena using shells as convenient models. In this context, just as examples of what can be found in the literature, we quote the seminal works of Berezin [5] and Visser [6], that date back to the early nineties, or the more recent [7,8] and references therein.What we are going to shortly discuss in the present contribution is set in this last perspective and suggests a semiclassical approach to define WKB quantum states for spherically symmetric shells. This method has already been used in [4].Let us then consider a spherical shell (we refer the reader to [3] for very concise/clear background material and for definitions). For our purpose the relevant result is equation (4) is the extrinsic curvature of the shell and can have different values on the two sides (+ and − spacetime regions) of it. S i j is the stress energy tensor describing the energy/matter content of the shell (S is its trace). For a spherical shell these equations reduce to the single conditionwhere f ± (r) are the metric functions in the static coordinate systems adapted to the 1 Far from being only idealizations of more realistic situations, shells have been extensively used to build concrete models of many astrophysical and cosmological scenarios (for a detailed bibliography, please, see the references in [4]).2 Conventions are as in [9] and the definition of the (quantities relevant to the concept of) extrinsic curvature can be found in [3,9].

Stability, Singularities and Mass Thresholds in Child Universe Production: a concise survey including some recent results and prospects

“…In the case which we discussed above, in which a de Sitter-Schwarzschild junction is performed by a shell having a uniform tension that equals the opposite of its energy density, the generic form of the potential diverges to minus infinity at both small and large value of r. The addition of a gauge field content on the bubble surface, modifies the small scale behavior, adding a potential well corresponding to a stationary classical configuration. An analogous result can be obtained invoking quantum effects for the stabilization process: in particular the bounded solution can be also considered in a quantum regime, and quantized semiclassically, using for instance the Bohr-Sommerfeld quantization condition [5,6]. Although the junction performed in [5] is not the most relevant in connection with cosmological applications, to reiterate the analysis in phenomenologically more sound situations does not pose any new technical problems.…”

“…If this condition is met for values of the parameters for which solutions of type 1 above can be realized, i.e. for which classical solutions starting from zero radius and expand-ing to infinity do exist, then child universes are formed 6 . Summarizing, two requirements have to be met for child universes to be realized as false vacuum bubbles using the thin-wall approximation: requirement 1: there must be a process by which a very small bubble can become big enough, so that, both, the early and late time evolution of our universe can be described; requirement 2: at late time, the evolution must guarantee that ǫ (+) = −1, so that the universe is, effectively, classically disconnected from the parent universe and expands by creating its own space.…”

Braneworlds and Quantum States of Relativistic Shells

“…We then see that, already in the very simple and natural semiclassical approach discussed above (of which the toy model in Ref. [11] is a preliminary test), the quantization condition S({k, l, m}, {ρ m , ρ r , ρ Λ , ρ ? }) ∼ n , n = 1, 2, .…”