3 Classical Dynamics

The results presented above are valid for arbitrary values of the four parameters entering the problem, namely the mass $M$ and the charge $Q$ of the external Reißner-Nordström spacetime, the de Sitter radius $H$ of the interior geometry and the total mass-energy of the dust shell, $m$. We now specialize them to a more particular setting (which has the advantage of removing the second line in expression (10) for the effective potential):

1. we take the external Reißner-Nordström spacetime to be extremal, i.e. with $\vert Q \vert = M$;
2. we assume that the total mass energy of the shell is $m = \vert Q \vert$.

The Penrose diagrams for the full de Sitter and extremal Reißner-Nordström [80] spacetimes are shown in figures 1 and 2 respectively.

Figure 1: Maximally extended Penrose diagram of the de Sitter spacetime.

Figure 2: Maximally extended Penrose diagram of the extremal Reißner-Nordström spacetime.

Before studying the possible shell trajectories in the two geometries to identify the bounded ones, which we are interested in, we take full advantage of the parameter reduction implicit in the assumptions above, by passing to adimensional variables: this will be more convenient also for the subsequent numerical treatment. We thus choose to parametrize all the variables and constants in terms of the de Sitter cosmological horizon $H$ by setting

$$x = \frac{R}{H} \quad , \quad t = \frac{\tau}{H} \quad ,... ...uad , \quad \vert \Theta \vert = \frac{M}{H} = \frac{m}{H} .$$ (14)

Then the quantities which are functions of $R$, become functions of $x$, all retaining their numerical values but $P$ and $S$, which are rescaled by $H$ and $H ^{2}$ respectively:

$$\bar{P} ( x ; \Theta ) = \frac{P ( R ; H , Q )}{H} \quad , \quad \bar{S} ( \Theta ) = \frac{S ( H , Q )}{H ^{2}} .$$ (15)

We thus have for the quantities evaluated along a classical trajectory, which are relevant in the study of the classical dynamics⁴

$$\bar{f} _{\mathrm{in}} (x) = 1 - x ^{2}$$ (16)

$$\bar{f} _{\mathrm{out}} (x) = 1 - \frac{2 \vert \Theta \vert}{x} + \frac{\Theta ^{2}}{x ^{2}}$$ (17)

$$\bar{\sigma} _{\mathrm{in}} (x) = - \mathrm{Sign}\left( x ^{3} - 2 \vert \Theta \vert \right)$$ (18)

$$\bar{\sigma} _{\mathrm{out}} (x) = - \mathrm{Sign}\left( x ^{4} - 2 \vert \Theta \vert x + 2 \Theta ^{2} \right)$$ (19)

$$\bar{V} (x) = - \frac{x ^{2} \left( x ^{4} - 4 \vert \Theta \vert x + 4 \Theta ^{2} \right)}{4 \Theta ^{2}}$$ (20)

$$\bar{P} (x) = - x \tanh ^{-1} \left\{ \left( \frac{x \sqrt{x ^{4} - 4 \vert \Th... ...x ^{2} \right) \left( x - \vert \Theta \vert \right) ^{2} \right\} } \right\} ;$$ (21)

of course all can be expressed as functions of the single adimensional parameter $\Theta$.

Following [64,75] we can study the classical dynamics in a compact way by means of a comprehensive graphical method fully exploiting the handy relation (9). It consists in plotting the potential $\bar{V} (x)$ together with the metric functions $\bar{f} _{\mathrm{in}}$, $\bar{f} _{\mathrm{out}}$. Then the allowed trajectories with the corresponding turning points can be determined looking at the segments of the $x$-axis (corresponding to zero energy), that are above the graph of the potential. In this diagram the points where the metric functions vanish, quickly help in determining if a classical path crosses the horizons of the external/internal geometry. Moreover the rescaled values of the radial coordinate $x$ for which $\sigma _{\mathrm{in/out}}$ changes sign are given, if they exist, by the values at which the metric function plots are tangent to the graph of the potential. This graphical information can be completed by the following analytical results.

Turning points of the potential:

from (20) we see that the potential for $\Theta \neq 0$ has a (double) zero at $x = 0$, so that it is regular at the origin, which is thus a trivial turning point of classical trajectories. Other turning points may, or may not, be present, depending on the value assumed by the parameter $\Theta$. Two cases are possible, as shown in figure 3.

Figure: Graph of the potential $\bar{V} (x)$ for different values of the parameter $\Theta$. Depending on the value of $\Theta$, the classical trajectory can have two non-vanishing turning points, or no non-vanishing turning points, as shown in the first and second figure, respectively. The in between case, which occurs for $\Theta = \Theta _{\mathrm{crit.}} = (3/4) ^{3/2}$, is depicted in the third diagram.

Either there can be no other turning points, so that only a so called ``bounce'' classical trajectory exists (as is the case in figure 3 for $\Theta = 0.7$), or there can be two more turning points so that in addition to the bounce trajectory there is also a bounded one (this is also shown in figure 3 for $\Theta = 0.5$). As explicitly proved in A, the critical value for the parameter $\Theta$, $\Theta _{\mathrm{crit.}} = ( 3 / 4 ) ^{3/2}$ gives the in between case, when the potential is tangent to the $x$ axis (third plot, again in figure 3). We thus see that only for $0 < \Theta \leq \Theta _{\mathrm{crit.}}$ there are two non-vanishing turning points $x _{\mathrm{min}}$, $x _{\mathrm{max}}$ (actually with $x _{\mathrm{min}} \equiv x _{\mathrm{max}}$ if $\Theta = \Theta _{\mathrm{crit.}}$) and thus bounded solutions are allowed, the classical path being represented by the segment $[0 , x _{\mathrm{min}} ]$. We note that it is possible to find $x _{\mathrm{min}}$ and $x _{\mathrm{max}}$ in closed form solving the quartic equation that gives the non-vanishing solutions of $\bar{V} (x) = 0$, and this (not very enlightening) expressions are reported in A.

Horizon positions with respect to the classical path:

to correctly understand the spacetime geometry we also need the relative positions of the horizons with respect to the classical trajectories. This is also briefly discussed later on, but it is useful to report here a general result that can be deduced from figure 4, where for $0 < \Theta \leq \Theta _{\mathrm{crit.}}$ the turning points, $0$, $x _{\mathrm{min}}$, $x _{\mathrm{max}}$ and the horizon of the exterior metric ( $x (\Theta) = \Theta$) are plotted as functions of $\Theta$.

Figure 4: Graph of the non-negative roots ($x = 0$, $x = x _{\mathrm{min}} (\Theta)$ and $x = x _{\mathrm{max}} (\Theta)$) of the potential $\bar{V} (x)$ together with the horizon $x = \Theta$ of the ``out'' Reißner-Nordström spacetime. Bounded trajectories are delimited by the $x = 0$ line and the $x = x _{\mathrm{min}} (\Theta)$ curve, so that they always cross the horizon of the Reißner-Nordström spacetime, which the graph shows to be always smaller than $x = x _{\mathrm{min}} (\Theta)$.

It can be seen that the classical bounded trajectory, corresponding to the region between the $x$ axis and the $x _{\mathrm{min}}$ dashed curve, crosses for all values of $\Theta$ in the considered range the exterior horizon at $x = \Theta$. The horizon of the internal de Sitter domain (which is not plotted and corresponds to the horizontal line $x \equiv 1$ in rescaled variables) is instead never crossed by a bounded trajectory⁵.

Asymptotic behaviour:

quite generally, from (20) we also see that

$$\lim _{x \to + \infty} \bar{V} (x) = - \infty .$$ (22)

Regularity at the origin:

the first and second derivatives of the potential are vanishing at $x = 0$, $\mathrm{d} \bar{V} (x) / \mathrm{d} x = \mathrm{d} ^{2} \bar{V} (x) / \mathrm{d} x ^{2} = 0$, and the third derivative is positive, $\mathrm{d} ^{3} \bar{V} (x) / \mathrm{d} x ^{3} = 6 / \vert \Theta \vert$, so that $x = 0$ is a local maximum for $\bar{V} (x)$.

Thanks to the above properties, we can now perform the study of the classical dynamics for bounded trajectories by restricting the parameter $\Theta$ to the range

$$0 < \vert \Theta \vert \leq \left( \frac{3}{4} \right) ^{3/2} ,$$ (23)

where we have the general situation shown in figure 5. The figure shows, for $\Theta = 0.55$, the potential $\bar{V} (x)$ together with $\bar{f} _{\mathrm{in}} (x)$ and $\bar{f} _{\mathrm{out}} (x)$. The classically allowed path is the thicker light gray segment on the $x$-axis.

Figure: The graphical method to study the classical shell trajectories consists in plotting the potential $\bar{V} (x)$, together with the curves for the metric functions of the interior and of the exterior. The light gray path is a classically allowed bounded trajectory: as discussed in the text it crosses the horizon of the external geometry but no changes in the orientation of the normals occur along it.

Let us consider the dynamics in the de Sitter spacetime: the shell expands from vanishing radius up to a maximum (grayed path in the figure), which remains inside the de Sitter cosmological horizon, since, as can be seen, it is not crossed by the trajectory; then the shell shrinks back to zero radius. The sign of $\sigma _{\mathrm{in}}$ does not change along the trajectory, since as we can see always from figure 5, there are no points on the trajectory in which the plot of $\bar{f} _{\mathrm{in}} (x)$ is tangent to $\bar{V} (x)$: moreover, as can be easily verified, it is always positive, so that the trajectory crosses the left part of the de Sitter Penrose diagram. This is shown in figure 6, where the interior region is the shaded area, since for $\sigma _{\mathrm{in}} = + 1$ the exterior normal is pointing to the right.

Figure 6: Penrose diagram of the de Sitter interior geometry with the bubble trajectory. The interior domain is the shaded region in the diagram.

In the same way we can perform the analysis in the Reißner-Nordström domain: we can see (in figure 5, but also from the above discussion about figure 4) that the bounded trajectory during the expansion from a vanishing radius, as well as during the following collapse, crosses the horizon of the exterior geometry. As before the sign of $\sigma _{\mathrm{out}}$ does not change along the trajectory (in the zoomed region of figure 5 we can more clearly see that the metric function graph is not tangent to the potential) and thus $\sigma _{\mathrm{out}} = - 1$ always. If we draw the associated Penrose diagram, the exterior region is the shaded one in figure 7.

Figure 7: Penrose diagram of the Reißner-Nordström exterior geometry with the bubble trajectory. The exterior domain is the shaded region in the diagram.

The results shown above rest on two hypotheses, that we implicitly took for granted but which deserve a more detailed treatment.

The first one concerns the stability of the expanding and recollapsing shell against single particle decay: if the shell were not stable at the moment of time symmetry, then it would become thick and its trajectory would not be approximated by a sharp line (as depicted in figures 6 or 7). It is possible to show, following the treatment of [8] (please see B for details), that configurations stable against single particle decay of charged and/or uncharged particles actually exist.

The second issue is about possible changes of signs in $\sigma _{\mathrm{in}}$ or $\sigma _{\mathrm{out}}$, which would require a different analysis with respect to the one performed above. We also devote an appendix (C) to show that bounded trajectories are not affected by changes of sign in $\sigma _{\mathrm{in}}$ or $\sigma _{\mathrm{out}}$, so that the analysis performed above in a particular case is indeed valid in general.

With these remarks in mind, the complete spacetime manifold can now be confidently obtained by joining the interior with the exterior along the shell trajectory, i.e. joining the two shaded regions in figure 6 and figure 7 to get the final result shown in figure 8.

Figure 8: Penrose diagram of the de Sitter interior and the Reißner-Nordström exterior geometries joined along the bubble trajectory. It is obtained by joining the shaded regions of the two previous figures. The classical dynamics of this compound spacetime is described by Israel's junction conditions and is discussed in detail in the text.

An observer inside the shell detects a non-vanishing cosmological constant. He lives as an observer inside a cosmological horizon. But as soon as he crosses the shell trajectory, he experiences a completely different situations: the cosmological constant suddenly vanishes and he can now detect a non-vanishing electric and gravitational field, as if outside a body with mass $M$ and charge $Q$ (with $M = \vert Q \vert$, because of our simplifying assumptions). We are now interested in studying the properties of this gravitational configuration when the system can be considered to be in a semi-classical quantum regime. For this we need an evaluation of the classical action along the classical trajectory of the shell.