6 Discussion

We have presented a model in which it is analyzed a general relativistic system composed of two spacetime domains, a de Sitter interior with cosmological horizon $H$ and an extremal Reißner-Nordström exterior with $M = \vert Q \vert$, joined across a thin shell of mass-energy $m = \vert Q \vert$. A semiclassical quantization of classically bounded trajectories can be performed using a scheme a la Bohr-Sommerfeld. In this way it is found that the quantum states can be characterized by a quantum number $n$, which is responsible for restricting the allowed values of $Q$ and $H$. In particular the quantum dynamics of the system constrains the values of $H$ as functions $H = H (Q ; n)$, which we have approximated after a numerical analysis of the problem.

It is interesting to note that this is a sensible result in a semiclassical approximation to a quantum gravitational situations. Indeed a full quantum theory of gravity would treat the three-metric as a dynamical infinite-dimensional variable to be determined, say, by the Wheeler-deWitt equation in superspace. From our point of view, under the assumption we made, we are only in a minisuperspace approximation, since the functional form of the metric functions is fixed from the very beginning, leaving $H$ and $Q$ as the only free parameters: quantum gravity will thus impose condition on these quantities, i.e. determine the only residual degrees of freedom in the three geometry. We can also see from the enlarged plot of figure 11, that the quantization condition selects a minimum value for the allowed charge of the semiclassical quantized shell. At the same time, the cosmological constant of the interior region, cannot exceed a maximum value, which is also a sensible consequence of quantization. For a given fixed value of the ratio $Q/H = \bar{\kappa}$, which means that we are ``moving'' across the graphs of figures 10 and 11 along a line through the origin, we have discrete allowed values for $Q$ and $H$: these point are located on a parabola, since from (25) we exactly have $S (\bar{\kappa} H , H ) = H ^{2} \bar{S} (\bar{\kappa})$.

We also note that by the final result, systems characterized by different scales in the parameters can be described: among these, as expected, we find small scale systems (with a charge which is a small multiple of the elementary electron charge). Moreover an external asymptotic observer measures a total mass energy for the shell given by

$$E = m + \frac{1}{2} \left( \frac{Q ^{2}}{r ^{2}} - \frac{M ^{2}}{r ^{2}} \right)$$

which in our case is nothing but $E = m$, since $M ^{2} = Q ^{2}$: thus we effectively see in the outside domain an object with rest mass $m = \vert Q \vert$ and charge $Q$, which is the exterior manifestation of a bounded interior containing a part of spacetime characterized by a non-vanishing vacuum energy. Due to the form of the potential this bound semiclassical state is metastable, and will decay into an infinitely expanding shell after a finite time (a similar situation occurs in [81]), which in principle could be calculated (as proper time), studying the process of tunnelling across the classical effective potential barrier⁷.