3. ``Minkowski pair'' creation

The simplest example of gravitational vacuum fluctuation is given by the spontaneous nucleation of a vanishing mass-energy shell in Minkowski spacetime. Despite the triviality of both the internal and external geometries, the dynamics of such a process is nonetheless affected by the unavoidable ambiguities and technical problems of quantum gravity. To circumvent all these difficulties, one usually employs a semi-classical model obtained by gluing together two Minkowski metrics along the bubble trajectory .

In our approach, the steps are as follows. The signs of $\sigma _{in}$, $\sigma _{out}$ are fixed by the matching condition

$$\sigma _{in} R \sqrt{1 + \dot{R} ^{2}} - \sigma _{out} R \sqrt{1 + \dot{R} ^2} = 4 \pi \rho \, G _{N} R ^{2} \quad .$$ (14)

For a positive surface tension we have $${\sigma _{in} = - \sigma _{out} = 1}$$, otherwise the two spacetime domains cannot be glued together. The resulting geometry represents the limiting configuration of vanishing cosmological constant and Schwarzschild mass discussed in appendix D of Ref. [11]. This is a closed universe formed by two compact spherical regions of flat spacetime. The shell equation of motion obtained by squaring (14) is

$$\dot{R} ^{2} = - 1 + 4 \pi ^{2} \rho ^{2} G _{N} ^{2} R ^{2} \quad ,$$ (15)

and admits only bounce solutions irrespective of the sign of $\rho$. In our case

   

$$L ^{\mathrm{eff}} = - 4 \pi \rho R ^{2} + \frac{2 R}{G _{N}} \sqrt{1 + \dot{R} ^{2}} - \frac{2 R \dot{R}}{G _{N}} \sinh ^{-1} \dot{R} \quad ,$$ (16)

$$P _{R} = \frac{\partial L ^{\mathrm{eff}}}{\partial \dot{R}} = - \frac{2 R}{G _{N}} \sinh ^{-1} \dot{R} \quad ,$$ (17)

$${\mathcal{H}} = 4 \pi \rho R ^{2} - \frac{2 R}{G _{N}} \cosh \left( \frac{G _{N} P _{R}}{2 R} \right) \quad .$$ (18)

Note that, in order to recover Eq.(18) from the general formula (7), one must carefully assign the phases of the complexified functions $\sqrt{\beta _{in}}$, $\sqrt{\beta _{out}}$ in the analytically extended Schwarzschild manifold, and only then one may consider the limit of vanishing Schwarzschild mass. This is because the effective hamiltonian (7) is actually a complex function when considered on the maximally analytic extension of the underlying spacetime manifold. This and other technical problems will be discussed elsewhere. Presently, however, it is worth observing that the kinetic term, obtained by expanding ``$\cosh$'' up to second order, is negative. Therefore, ${\mathcal{H}} = 0$ is the classical equation of motion of a (positive kinetic energy) particle in the reversed potential $V (R) \sim - 4 \pi \rho R ^{3} + 2 R ^{2} / G _{N}$. This potential constitutes an effective barrier for the ``particle'' motion and explains the absence of a discrete spectrum of stationary quantum states. The classical dynamics of this type of domain wall has been discussed in Ref. [15] with special emphasis on the repulsive character of the resulting gravitational field. In our formulation, this repulsive effect is plainly exhibited by the potential above. We propose to associate the expected quantum mechanical ``leakage'' through the potential barrier with the process of ``universe creation'' by quantum tunneling from nothing. In order to give substance to this interpretation, one needs to compute the corresponding transmission coefficient, and, in order to perform this calculation, we rotate the dynamical quantities to imaginary time:

$$P _{E} = \frac{2 R}{G _{N}} \arccos \left( \frac{R}{R _{G}... ...here} \qquad R _{G} = \frac{1}{2 \pi \rho \, G _{N}} \quad .$$ (19)

This gives, for the nucleation coefficient

$$B = \frac{4}{G _{N}} \int _{0} ^{R _{G}} d R \, R \, \a... ...G}} \right) = \frac{1}{8 \pi \rho ^{2} G _{N} ^{3}} \quad ,$$ (20)

in agreement with the bounce calculation of Ref. [12].

In the case of negative surface tension the calculation proceeds along the same steps outlined above, but with $\sigma _{in} = -1$, $\sigma _{out} = +1$. The resulting geometry is the limiting case of a ``surgically'' constructed Schwarzschild wormhole, when the mass is sent to zero [3]. However, any constant time section of the resulting spacetime has infinite volume. Thus, this second type of ``Minkowski pair'' does not correspond to a compact object and cannot be nucleated quantum mechanically from ``nothing''.