1. Introduction

According to some current ideas in cosmology, in particular the chaotic inflationary scenario [1], the Universe may consist of infinitely many self-reproducing bubbles which are continuously nucleated quantum mechanically. Some of them expand and look like a Friedmann universe, others may collapse to form black holes, and some may be connected by wormholes. In principle, this dynamical network may exist at any scale of distance, or energy; at the Planck scale of energy, it is expected to arise from gravitational fluctuations which induce a foam-like structure over the spacetime manifold. Some work on the dynamics of the constituents of this network has already appeared in the literature [2,3,4,5,6], but the results so far obtained are not always consistent and seem to be model dependent. As a matter of fact, the very notion of spacetime foam, introduced by Wheeler nearly forty years ago [7], has periodically come under close scrutiny in the intervening years. However, a quantum theory of the above processes is still beyond our reach, and one must come up with some viable alternative.

The aim of this communication is to suggest an effective approach to the dynamics of spherically symmetric, self-gravitating objects that may arise, evolve and die in the spacetime foam. Among these ``objects'', somehow, there is the universe in which we live, and therefore the study of such fluctuations hardly needs a justification. In order to give some analytical substance to this qualitative picture, we envisage the spacetime foam as an ensemble of vacuum bubbles, or cells of spacetime, each characterized by its own geometric phase and vacuum energy density. In principle each cell may behave as a black hole, a wormhole, an inflationary bubble, etc., depending on the matching conditions on the neighboring cells [8]. Thus, a useful starting point for our present discussion, is the general matching equation between the internal and external metrics of a self-gravitating, spherically symmetric bubble [9]

 

$$\sigma _{in} R \, \sqrt{1 - \frac{\Lambda _{in}}{3} R ^{2} + \dot{R} ^{2}} +$$

$$\qquad \qquad \qquad - \sigma _{out} R \, \sqrt{ 1 - \frac{2 M G ... ...Lambda _{out}}{3} R ^{2} + \dot{R} ^{2} } = 4 \pi \rho \, G _{N} R ^{2} \quad .$$ (1)

In the above equation, $\Lambda _{ in , \, out} = 8 \pi G _{N} \, \epsilon _{in , \, out}$, are the cosmological constants representing the internal(in) and external(out) vacuum pressures; $\rho$ is the constant surface tension, and $\sigma _{in} (\sigma _{out}) = \pm 1$ depending on whether the radius of the bubble increases, or decreases, along the outward normal direction to the $2$-brane surface embedded in the interior (exterior) metric. Here we have assumed, for the sake of simplicity, the membrane equation of state $\rho = - p = \mathrm{constant}$. However, we should emphasize that other equations of state, such as $p = 0$ for dust, can be easily accomodated in our formalism. Following a suggestion first made in Ref. [4], we now propose to interpret Eq.(1) as the ``Hamiltonian constraint'', $${{\mathcal{K}} = 0}$$, for a system described by the effective hamiltonian, or super-hamiltonian,

$${\mathcal{K}} \equiv 4 \pi \rho \, R ^{2} - \sigma _{in} \frac{R}{G _{N}} \, \sqrt{1 - \frac{\Lambda _{in}}{3} R ^{2} + \dot{R} ^{2}}$$

$$+ \sigma _{out} \frac{R}{G _{N}} \, \sqrt{ 1 - \frac{2 M G _{N}}{R} - \frac{\Lambda _{out}}{3} R ^{2} + \dot{R} ^{2} } \quad .$$ (2)

Then, we can use the Hamilton equation, $${d {\mathcal{K}} = \dot{R} \, d P _{R}}$$, to deduce the momentum $P _{R}$ which is canonically conjugated to $\dot{R}$, and the corresponding lagrangian:

  

$$P _{R} = \int \frac{\partial {\mathcal{K}}}{\partial \dot{R}} \frac{d \dot{R}}{\dot{R}} \quad ,$$ (3)

$$L ^{\mathrm{eff}} = P _{R} \dot{R} - {\mathcal{K}} \quad .$$ (4)

Thus, from the condition (1), we find

$$P _{R} = \frac{R}{G _{N}} \ln \Bigg \vert \left( \frac{\... ...\sigma _{out} \sqrt{\dot{R} ^{2} + \beta _{out}}} \Bigg \vert$$ (5)

where $${\beta _{in} \equiv 1 - (\Lambda _{in} / 3) R ^{2}}$$ and $${\beta _{out} \equiv 1 - (\Lambda _{out} / 3) R ^{2} - 2 M G _{N} / R}$$.

The inverse Legendre transform (4) leads to the effective lagrangian we are looking for

 

$$L ^{\mathrm{eff}} = \frac{R}{G _{N}} \left( \, \sigma _{in} \sqrt{\dot{R} ^2 + \beta ... ...igma _{out} \sqrt{\dot{R} ^2 + \beta _{out}} - 4 \pi \rho \, G _{N} R \right) +$$

$$- \frac{R \dot{R}}{G _{N}} \left[ \sigma _{in} \sinh ^{-1} \left( \... ... \sinh ^{-1} \left( \frac{\dot{R}}{\sqrt{\beta _{out}}} \right) \right] \quad .$$ (6)

From here, we deduce the proper time effective hamiltonian in canonical form,

$${\mathcal{H}} = 4 \pi \rho \, R ^{2} - \mathrm{sign} (\rh... ...left( \frac{G _{N} P _{R}}{R} \right) \right] ^{1/2} \quad .$$ (7)

At this point, we should mention that the above Lagrangian and Hamiltonian can also be deduced, in a more fundamental way, directly from the Einstein-Hilbert action plus a boundary term, under the assumption of spherical symmetry. This can be verified, for instance, by extending the derivation of Ref. [3] to our general case. The variational principle, in this case, leads precisely to the equations of motion derived from the Hamiltonian constraint $${{\mathcal{K}} = 0}$$. It is also worth observing that, by our reformulation, we have transformed the initial problem, i.e., the motion of a spherically symmetric self-gravitating membrane, into an equivalent, one dimensional, non-linear problem involving the dynamics of a single degree of freedom. In this ``point-particle'' interpretation of our effective lagrangian, one expects that ``pair production'' takes place, and in section 2 we discuss an explicit example of this process. Finally, we should mention that the formulation given in Ref. [10] is conceptually close to ours, but is based on the use of the Schwarzschild coordinate time as evolution parameter along the bubble trajectory; the drawback of this choice of coordinate is that, even assuming a vanishing external pressure, it is impossible to Legendre transform the effective lagrangian and obtain the corresponding hamiltonian as a function of the canonical pair $( R , \, P _{R})$.

Note that the hamiltonian (7) involves a square root operation in analogy to the familiar expression of the energy of a relativistic point particle. In our case the coefficient of the square root depends on $\mathrm{sign} (\rho)$ in order to be consistent with the classical equation of motion (1). The opposite sign is classically meaningless. However, since we are dealing with a relativistic system, both positive and negative energies become physically relevant at the quantum level. This leads us to a broader interpretation of spacetime foam as a ``Dirac sea of extended objects'', in which not only wormholes, but also black holes and vacuum bubbles are continuously created and destroyed as zero-point energy fluctuations in the gravitational quantum vacuum. As a matter of fact, the effective lagrangian (6), (or the effective hamiltonian (7)), encodes the dynamics of a four-parameter ($\rho$, $\Lambda_{in}$, $\Lambda_{out}$, $M$) family of spherically symmetric, classical solutions of the self-gravitating bubble equation of motion. Furthermore, the results (6) and (7) can be extended in a straightforward manner to an even larger family of solutions by endowing the internal geometry with a non-vanishing Schwarzschild mass term $M _{in} G _{N} / R$, or the external geometry with an electric charge.

The spacetime foam models currently available in the literature focus essentially on the Schwarzschild metric (see, however, Ref. [2]), and correspond to the sector $M > 0$ of our family of classical solutions. For instance, in the sub-sector $\rho > 0$, $\Lambda _{in} > 0$, $\Lambda _{out} = 0$, one finds the vacuum bubbles discussed in Ref. [11]. In particular, we recall the type E trajectories with $M > M _{cr}$, listed in the same reference, because they give rise to baby-universes connected through wormholes to the parent universe. The characteristics of those trajectories are instrumental to our discussion in Section 4.

Other types of ``foam-like'' solutions belong to the subsector $\Lambda _{in} = \Lambda_{out} = 0$. They include the ``surgical'' Schwarzschild-Schwarzschild wormholes [3], and the ``hollow'' Minkowski-Schwarzschild wormholes [4], [12], while the region $\rho < 0$ of the same subsector contains the traversable wormholes, i.e. wormholes whose throats can be crossed by timelike observers. In this connection, note that in our membrane model a negative tension plays the same role as the negative energy density of the more conventional wormholes made out of ``dust'': it provides the ``repulsive force'' required to oppose the gravitational collapse of the throat [13]. The explicit correspondence between negative energy density and surface tension is provided by the simple relation: $${\rho \leftrightarrow m / 4 \pi R ^{2}}$$. However, while the negative energy density of dusty wormholes is ascribed to some kind of exotic matter [13], or to gravitational vacuum polarization [14], we suggest to interpret membranes with negative tension as boundary layers between different physical vacua as suggested, for instance, by the existence of normal and confining vacua in QCD. More about negative energy density later.

In section 2 we show that in the flat spacetime limit, i.e. for $G _{N} \rightarrow 0$, the resulting lagrangian can be used to compute the false vacuum decay amplitude in ordinary quantum field theory.

In sections 3 and 4, as an explicit application of our general method, we shall study two examples of gravitational fluctuations in the sector $\Lambda _{in} = \Lambda _{out} = M = 0$, which correspond to vacuum bubbles. Their discussion and comparison with the existing literature on the subject provide an excellent testing ground for the validity of our approach.