Minisuperspace, WKB, Quantum States of General Relativistic Extended Objects

S. Ansoldi¹

Dipartimento di Matematica e Informatica, Università di Udine,
and I.N.F.N. Sezione di Trieste
via delle Scienze, 206 - I-33100 Udine (UD) - ITALY

 

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)²thin 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 [Israel(1966),Israel(1967),Barrabes and Israel(1991)]. Starting from this toehold 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 [Berezin(1990)] and Visser [Visser(1991)], that date back to the early nineties, or the more recent [Berezin(2002),Corichi et al.(2002)] 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 [Ansoldi(2002)].

Let us then consider a spherical shell (we refer the reader to [Barrabes and Israel(1991)] for very concise/clear background material and for definitions). For our purpose the relevant result is equation (4) in [Barrabes and Israel(1991)], i.e. the junction conditions³ $K ^{-} _{ij} - K ^{+} _{ij} \sim S _{ij} - g _{ij} S / 2 .$ $K _{ij}$ is the extrinsic curvature of the shell and can have different values on the two sides ($+$ and $-$ spacetime regions) of it. $S _{ij}$ 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 condition

$$\epsilon _{-} ( \dot{R} ^{2} + f _{-} (R) ) ^{1/2} - \epsilon _{+} ( \dot{R} ^{2} + f _{+} (R) ) ^{1/2} = M (R) / R ,$$ (1)

where $f _{\pm} (r)$ are the metric functions in the static coordinate systems adapted to the spherical symmetry of the $4$-dimensional spacetime regions joined across the shell, $\epsilon _{\pm} = \pm 1$ are signs and $R$ and $M (R)$ are the the radius (a function of the proper time $\tau$ of a co-moving observer⁴) and the matter content (what remains of $S _{ij}$) of the spherical shell, respectively. As shown for example in [Farhi et al.(1990),Ansoldi et al.(1997),Ansoldi(1994)] the above equation can be obtained from an effective Lagrangian⁵ $L _{\mathrm{EFF}} (R , \dot{R})$, as a first integral of the second order Euler-Lagrange equation. From $L _{\mathrm{EFF}} (R , \dot{R})$ the momentum, $P (R, \dot{R}) = \partial L _{\mathrm{EFF}} (R , \dot{R}) / (\partial \dot{R})$, conjugated to the only degree of freedom $R$ can also be obtained. We are not interested in the explicit form of $P$ here, we just point out that it is a function of $R$ and $\dot{R}$ highly non-linear in $\dot{R}$. This non-linearity spoils the natural and simple interpretation of the momentum than we know from classical of analytical mechanics. Nevertheless we can still solve (1) for the classically allowed trajectories of the shell, using a standard analogy with the motion of a unitary mass particle with zero energy in an effective potential [Berezin et al.(1987),Blau et al.(1987),Aurilia et al.(1989)]. This gives $\dot{R}$ as a function of $R$ and, substituting this expression for $\dot{R}$ in $P (R, \dot{R})$, we obtain the conjugated momentum on a solution of the classical equations of motion. In what follows we are going to indicate the momentum evaluated on a classical trajectory as $P (R ; {\mathcal{S}})$: this emphasizes that it is a function of $R$, that it depends on the set ${\mathcal{S}}$ of the other parameters of the problem, but, of course, it is not a function of $\dot{R}$.

 

Figure 1: Plot of some WKB levels for the example discussed in the text. The curves a, b, c, d, e, f, g, h, i, j correspond to the quantum numbers 1, 3, 3, 4, 5, 10, 20, 30, 40, 50, respectively.

 

By integrating the expression for $P (R ; {\mathcal{S}})$ on a classically allowed trajectory with turning points $R _{\mathrm{MIN}}$ and $R _{\mathrm{MAX}}$, we can compute the value of the classical action for that trajectory. It is a function of the set of parameters characterizing the matter content and of the geometry, ${\mathcal{S}}$, and WKB quantum states of the system can now be defined as states for which the above action is a multiple of the quantum

$$S ( {\mathcal{S}} ) = 2 \int _{R _{\mathrm{MIN}}} ^{R _{\... ...\mathcal{S}}) d R \sim n, \quad n = 0 , 1 , 2 , 3 , \dots{}.$$ (2)

In quantum gravity we expect to have a theory that selects some geometries from the set of all possible ones consistently with quantum dynamics. In our discussion we have limited our treatment to a minisuperspace approximation: indeed we see that the quantization condition (2) selects only some of the possible geometries, those in which the parameters of the model are related by the quantum number $n$ by (2). We can see this more explicitly in the following simple model: a dust shell ($M (R) = m$) separating two domains of anti-de Sitter spacetime with the same cosmological constant ( $f _{-} (r) = f _{+} (r) = f (r) = 1 + r ^{2} / H ^{2}$): it is possible to prove that a non-trivial junction of the two spacetimes can be performed, although we are not going to discuss this aspect here nor we are going to describe the resulting global spacetime structure. In this case the set of parameters is ${\mathcal{S}} = \{ m , H \}$ and the quantization condition (2) becomes $S ( m , H ) \sim n$, $n = 0 , 1 , 2 , 3 , \dots{}$. Some levels are plotted in figure 1 and clearly show that given one of the parameters (say $m$) only a discrete set of values for the other $H$ is allowed: thus the quantization condition restricts the possible values that can be given at the parameters that characterize the geometry and/or the matter content of the shell. This is consistent with the general picture presented above.

Other applications of this general approach are presently under study. In particular, generalizations to higher dimensions [Ansoldi et al.(In preparation)] could be relevant in the brane cosmological scenario.