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Tunnelling with wormhole creation

The description of quantum tunnelling in the presence of gravity shows subtleties in some cases. Here we discuss wormhole production in the context of the spherically symmetric thin-shell approximation. By presenting a fully consistent treatment based on canonical quantization, we solve a controversy present in literature.


I. INTRODUCTION
Quantum tunnelling plays various roles in cosmology. For instance, false vacuum decay through quantum tunnelling [1][2][3] is an important process for the universe to visit many vacua in the string landscape [4][5][6]. Also, the possibility of creation of an open universe through false vacuum decay has been extensively discussed [7][8][9][10]. Taking properly into account the effect of gravity can be quite non-trivial. Although in some cases the effect of gravity is secondary, there are in fact several cases in which gravity plays a crucial role, such as the upward quantum tunnelling from a lower to a higher energy vacuum [11,12].
Even when the effect of gravity is secondary, including gravity can make the treatment highly non-trivial. One example is the subtle issue raised by Lavrelashvili, Rubakov and Tinyakov [13] that fluctuations around bubble nucleation might cause an instability, which leads to explosive particle production. One prescription to cure this pathology was proposed in Ref. [14,15], in which it is shown that one can eliminate the instability, at least apparently, by an appropriate choice of the gauge.
Quantum tunnelling in connection with gravity has been discussed also in other contexts. One of them is wormhole formation [16][17][18][19][20][21][22][23][24], which is the main subject of this paper. Wormhole formation is a signature of what in the literature is also referred to as baby/child universe creation [25]. Spherical thin shells with various equations of state have been studied, as models of matter fields able to describe this process. Even in the simple case of a pure tension shell, the quantum mechanical formation of a wormhole seems possible. However, some inconsistencies between different prescriptions seem to exist in literature [26]. In this paper we will clarify that the origin of these apparent discrepancies is tightly related to the use of the time coordinate in the static chart. We then propose a plausible prescription based on a smooth time-slicing to tackle the problem. This paper is organized as follows. In Sec. II we briefly review the derivation of the standard result for the tunnelling amplitude based on the direct evaluation of the action, when the time slice of the static chart is used. In Sec. III we discuss the problem that arises when we try to apply the conventional formula to situations characterized by wormhole production. To overcome some difficulties that appear in this last case, in Sec. IV we then study the same problem using a canonical approach with smooth time slice: this allows us to derive the formula for the tunnelling rate without any ambiguity. In Sec. V we finally show how the same formula can be reproduced by the direct evaluation of the action if we carefully take the smooth time slice. Sec. VI is devoted to summary and discussion: we also elaborate on a remaining, more subtle, issue.

II. CONVENTIONAL APPROACH
In this paper we consider the simplest spherically symmetric domain wall model, whose Lagrangian is given by where R is the scalar curvature and m(R) is the radius dependent mass of the wall, e.g., m(R) =constant for a dust domain wall while m(R) = 4πσR 2 for a wall consisting of pure tension σ; moreover τ is the proper time along the wall, whileR denotes the circumferential radius of the wall. In general, quantities marked with a "ˆ" are considered to be evaluated at the position of the wall, e.g.,B =B(t) = B(t,r(t)) when B is a function of t and r, and r =r(t) is one possible parametrization of the wall trajectory. Depending on the model parameters, the wall motion can have some classically forbidden region for a range of the radius. We are interested in discussing the quantum tunnelling of the wall when it reaches a turning point, i.e. a boundary of the classically forbidden region, by explicitly taking into account gravity. In this section we derive a conventional but incorrect formula for the tunnelling rate of the wall. Although we mostly follow Ref. [27], we are not claiming that the result obtained in this reference is wrong. Indeed, our emphasis is about the fact that in this reference the authors clearly identified a discrepancy between the direct evaluation of the action that they propose and a naive canonical approach. Moreover, it was clearly emphasized in Ref. [27] that the proposed direct approach guarantees, instead, a continuous variation of the action as the parameters (the Schwarzschild mass, the Schwarzschild Sitter cosmological constant, the wall surface tension, in their model) are changed: on the contrary, the conventional canonical approach does not guarantee continuity of the action as a function of the parameters. At the same time the direct calculation of the action discloses the difficulties in the identification of the Euclidean manifold interpolating between the before-and after-tunnelling classical solutions in a consistent way: indeed, Farhi et al. associate what they call a pseudo-manifold to the instanton solution. The direct approach defines the pseudomanifold by weighting different volumes of the instanton along the classically forbidden trajectory by an integer number that counts how many times (and in which direction) the Euclidean volume is swept by the time slice. We will later show that the canonical approach, in full generality, can reproduce the same value for the tunnelling action given in the approach proposed by Farhi et al..
The direct evaluation of the action is possible because the solution is simply given by a junction of two spacetimes. Here, for simplicity, we assume that the inside and outside of the bubble are both empty, so that the inside can be taken as a piece of Minkowski spacetime and the outside as a piece of Schwarzschild spacetime. (In Ref. [27] the inside was equipped with vacuum energy density, i.e., a cosmological constant, but this does not change the treatment in any substantial way.) The method proposed by Farhi et al. was developed in coordinates adapted to the static and spherically symmetric nature of the spacetimes participating in the junction. With this, we mean that the Lagrangian was preferably considered in connection with the coordinate times in the static chart in both spacetime regions, which we denote by t S and t M in the simplified case that we are considering here. However, most of the calculations were performed using the proper time of an observer sitting on the junction, and therefore the result can be easily extended to a coordinate-independent expression, as we shall see in Sec. V.
The contributions to the action can be summarized as follows.
1. A matter term coming from the shell, I wall matter : this is nothing but the contribution from the stress-energy tensor localized on the bubble surface.

2.
A gravity term coming from the bubble wall, I wall gravity : this is, basically, the well-known extrinsic-curvature-tracejump term.
3. The bulk contributions vanish for classical solutions since there is no matter in the bulk. 4. Surface terms: although the appearance of surface terms is conceptually clear, the treatment of these terms may be non-trivial. As clearly discussed in Ref. [27] several contributions arise.
(a) In their treatment, a crucial contribution, I wall surface , comes from the bubble wall positions, where the normal to the constant time surface is discontinuous. However, this contribution does not appear if we adopt a smooth foliation of time across the wall. In Sec. IV we take this latter picture. (b) Another contribution comes from a surface at a large constant circumferential radius in the outside spacetime, I RBIG surface : this cut-off radius allows us to work with a (spatially) bounded volume, and the large radius limit has to be taken in the end. This limit naturally brings in divergences, which can be usually dealt with, e.g., by the Gibbons-Hawking prescription. The final regularized result is called I RBIG net below.
With the notation used above, and by setting (because of the square, the notation below differs from the one used in Ref. [27]) the above terms are [27] I wall where square brackets represent the jump of the bracketed quantities across the shell, i.e., Square brackets will be nowhere used with a different meaning. Moreover, the signs are unambiguously determined by the consistency with the junction condition [29] Gm Noticing that all the above contributions can be combined to give the Lagrangian Finally, adding a constant M to the Lagrangian so that the Lagrangian vanishes at the turning point,R ,τ = 0, we can evaluate L on a classical solution to obtain HereR ,τ is to be replaced with its classical solution, which is obtained from the junction condition (10) aŝ 1 The expression given in Ref. [27] looks slightly different, but it is equivalent to this one as long as we require that I wall surface is always real valued. As we shall explain later (see Eq. (19)), the sign flip of ǫ is only important in the Euclidean regime. As the argument of the logarithm has a jump there, we may have to add one more term proportional to a δ function at the sign flipping point to the right hand side of Eq. (5). However, the crucial point is that analyticity of I wall surface is broken at the sign flipping point. Therefore, it is difficult to find a consistent meaning for the analytic continuation of this expression to the Euclidean region.
As explicitly seen above, the action could in general contain second derivative terms. These second derivatives are removed by the "careful" inclusion of the boundary term, I wall surface . From Eq. (13), we identify the effective momentum conjugate toR as After Wick rotation to Euclidean time,τ = iτ , the Euclidean momentum,P eff = −iP eff and Eq. (14) becomē respectively. We indicates quantities after Wick rotation with "¯", if they are different from the Lorentzian ones. We also note thatP eff is real, since the modulus of the argument inside the logarithm is unity. Then, the tunnelling action may be evaluated asĪ to provide the tunnelling rate ∼ exp(−2Ī (tS) ).

III. WORMHOLE PRODUCTION
The framework discussed in the preceding section is generically applicable to the tunnelling problem. However, analytic continuation brings up situations that are technically and conceptually more involved. To see this, first we notice that ǫ = ±1 flips sign when vanishes, where R g := 2GM . We denote the value ofR at the sign changing point byR c . In the Lorentzian regime, the sign flip of ǫ does not occur in regions outside horizons: it can happen behind horizons, but in these cases no pathology arises [26]. In any case, in this work, because of our definitions (2), we are implicitly excluding regions behind horizons. This is certainly non restrictive for our current purpose, because it is possible to prove that tunnelling must always begin and end in regions that are not behind horizons, and it is always true that P eff is continuous during the time evolution. However, in the Euclidean regime, not only the sign flip can happen, but the argument of the logarithm (and hence the logarithm itself) in P eff also has a jump at the point where the sign of ǫ flips: this can not be avoided if one consistently requires that the effective momentum should vanish at both turning points. (In fact, the discontinuity cannot be avoided if we require that P eff analytically continued back to the Lorentzian regime is real, both, before and after the tunnelling.) This happens because the expression for P eff is essentially non-analytic. For this reason, it is hard to justify the use of analytic continuation of an action that contains P eff . In the present case, from Eq. (19), we find that the sign flip can happen for ǫ + only. From the analytic continuation of Eq. (11), we find that dt S /dτ also vanishes at the sign flip point. This means that the trajectory of the wall becomes purely radial. At this point there is a jump of the logarithm inP eff . We draw a schematic picture of the wall trajectory when there is a sign flip in Fig. 1. In this picture the center corresponds to R = 2GM , the radial direction is the rescaled radius and the angular direction is the Euclidean time,t S . As a concrete example, let us consider the case of the pure tension wall with m = 4πσR 2 . In this case, from Eq. (14), we find that the turning points corresponding toR ,τ = 0 are given by the solutions of where we have introducedσ := 4πGσ. It is easy to see that f (R g ) ≥ 0 and the equality holds forσ = 1/R g . At the minimum of f (R) whereR = 4/(3σ), we have f (4/(3σ)) = R g − 32/(27σ). Therefore, we find that there is a classically forbidden region forσ < 32/(27R g ). A wormhole can be produced when the critical radius, where the discontinuity appears,R is in the classically forbidden region. As mentioned above, this critical radius does not result in pathologies in the classically allowed region. Therefore, ifR c > R g , the critical radius is under the potential barrier. This means that wormhole production is possible whenσ < 1/R g . Now, we discuss the key issue of this paper. As long as we use the foliation by the Schwarzschild time, it is problematic to consistently define the Euclidean manifold interpolating between the configurations before and after the tunnelling. As a concrete example, let us consider the case shown in Fig. 1 (for this case, plots of the effective momentum along the tunnelling trajectory and of the potential barrier can be found in Fig. 2). When dt S /dτ is positive, the wall is located atR <R c and the Schwarzschild spacetime is relevant forR < R < ∞. Minkowski spacetime is connected beyond the wall. After passing through the pointR =R c , dt S /dτ becomes negative. Then, the wall is present forR >R c and the Schwarzschild spacetime is relevant for R <R. Again the Minkowski spacetime is connected beyond the wall. Then, one may wonder where the asymptotic region with R → ∞ is. The asymptotic region is on the other side extending beyond the center, corresponding to R = 2GM . The time slice cannot terminate at the center (bifurcation point) of the Schwarzschild spacetime. We then see that the geometry on this time slice suddenly changes at the sign flip point. Namely, the final configuration contains a wormhole, corresponding to the Full spacetime before the tunnelling Full spacetime after the tunnelling Here it is also clear that after the tunnelling the slice contains R = Rg. This was not the case for the slice before the tunnelling shown in Fig. 3. existence of a minimum circumferential radius. At the same timeP eff is discontinuous there. As long as we stick to this time slice, it is difficult to obtain a satisfactory and consistent prescription. Figures 3 and 4 show the situation before and after the tunnelling, respectively. By comparing the slice before the tunnelling (thick horizontal line in the Penrose diagram for the configuration before the tunnelling in Fig. 3.C), with the slice after the tunnelling (thick horizontal line in the Penrose diagram for the configuration after the tunnelling in Fig. 4.C) we can also have a clear example of the situation discussed just above for the Euclidean spacetime that should interpolate between these two configurations. In the next section we discuss the same process in the canonical formalism without specifying the gauge, which makes it possible to overcome these difficulties.

IV. CANONICAL APPROACH WITH SMOOTH TIME SLICE
We consider the canonical approach in this section, following Ref. [28]. The spherically symmetric metric is specified in the 3 + 1 decomposition as where, with standard notation, dΩ 2 is the spherically symmetric part of the line element. Then, the action in the canonical formalism is obtained as with where p, π L and π R are the conjugate momenta tor, L and R, respectively. As for derivatives, we adopt the following standard convention:Ḃ We stress that the values of all the metric functions are assumed to be continuous across the wall, although their derivatives can be discontinuous. The constraint equations H t = 0 and H r = 0 are solved in the bulk as where we introduce the following definitions By integrating the constraint equations across the wall, we obtain the junction conditions, which in the present notation can be written as In the WKB approximation, the wave function is written as ∝ exp(iI(r, L, R)) and the conjugate momenta are identified as Hence, the action relevant to discuss the WKB wave function is We should notice that in this approach removing the last term in Eq. (24) is absolutely unambiguous. To handle the above expression (31) without specifying the gauge, a key observation is the existence of a function Φ = Φ(L, R, R ′ ) that satisfies where ∼ means that the equality holds neglecting total derivative terms. It is then possible to integrate the above equation to obtain In the above expression there is an arbitrariness, as a total derivative of an arbitrary function of R with respect to r can be added, which, of course, does not affect the final result. Then, the action becomes where we define In the first equality of Eq. (34), we removed the contribution of ΨṘ at r → ∞, assuming that the time slice is asymptotically identical to the one in the static chart of the Schwarzschild spacetime, in which Ψ vanishes because β = 0 and X = A. UsingṘ we can rewrite the last term in the parentheses on the right hand side of Eq. (34) as where in the last equality we have extracted outṘ andṙ from the square brackets since their values evaluated on both sides of the junction are identical. Thus, we obtain As we have ΨR ′ − Φ = −RLβ, the first term in Eq. (38) vanishes using the junction condition (29), and we finally obtain the gauge unfixed action relevant for the WKB wave function as Let us now examine the motion of the shell, dR/dt, in more detail. The part of the action related to the shell takes the form From this expression, the conjugate momentum tor turns out to be given by from which we getL From the normalization of the four velocity, we also find which is further simplified using Eq. (42) asN Now, we are ready to rewrite dR/dτ . Using the equation of motion for R, Then we obtain where in the third equality, we have used Eqs. (42) and (44). Substitutingβ = (X 2 −Â 2 ) 1/2 , this equation can be solved forX asX Remembering that p andR ,τ do not have a jump across the junction, from Eq. (46) and the junction condition (29), we recover exactly Eq. (10). Furthermore, substituting Eq. (46) into Eq. (45), we obtain and henceX Therefore, we can finally write the jump of Ψ as After Euclideanization, Eq. (39) can be then rewritten using the above results, and it gives and This expression is identical to Eq. (18) obtained in Sec. II for the tunnelling that does not produce a wormhole. First, since A = X on the initial and final surfaces, where the time slices coincide with the ones witht S =constant andt M =constant,Φ vanishes there. Since ǫ = +1 in this case, as mentioned earlier, the difference betweenP eff and [Ψ] does not arise. By contrast, in the case with wormhole production the first term in Eq. (50) does not vanish because X is negative in the region between R = R g and the wall in Schwarzschild, and hence X = −A there. Namely, the first term contributes as where r g is the value of r at R = R g on the final surface. Hence, the difference between Eq. (18) and (50) is evaluated asĪ if we assume thatP eff in Eq. (18) has a discrete jump atR = R c . Of course, this discrepancy is not strange at all, since the naive extension of the validity range of the formula (18) cannot be justified.

V. CONSISTENT DIRECT EVALUATION
As we anticipated, we will now show that the method using a pseudo-manifold for the description of the instanton solution gives the same result that we derived by using the canonical approach in the preceding section. Although this equivalence might seem almost trivial because both approaches are based on the same smooth foliation of Euclidean spacetime, its explicit proof would be pedagogically useful.
We return then to the discussion in Sec. II. The first key observation is that the contribution from the carefully included I wall surface should not be included when we adopt a smooth foliation. The second point is that we have rewritten a term in Eq. (4) as We have then subtracted M (t f S − t i S ) from the total action. In the computation of Sec. II half of this subtraction was compensated by I RBIG surface and the rest by the above contribution (55). However, we find when we use a smooth foliation for the tunnelling solution with wormhole formation. This shows that an additional contribution πM R g to the Euclidean action arises. Gathering all, we find that the Euclidean action evaluated by using a smooth foliation is given bȳ which is precisely identical toĪ.

VI. SUMMARY AND DISCUSSION
In this work we studied the wormhole production for the simplest spherically symmetric shell model in asymptotically flat spacetime. In this simple setup, the instanton solution can be generically described by the junction of Euclideanized Minkowski and Schwarzschild spacetimes. This solution, however, is not a Riemannian manifold in the sense that the existence of the domain wall may depend on the path taken to reach the possible location of the wall in spacetime. The term pseudo-manifold was used in [27] to denote this solution. A key point that we have emphasized here, is that in this case the ordinary constant-time surfaces associated with the static chart do not foliate the instanton smoothly. As a result, methods based on this time slicing inevitably become conceptually ambiguous.
We have here discussed, however, that even in these cases, if we choose a smooth time slicing to connect the configurations before and after the tunnelling, it is still possible to find the WKB wave function along an interpolating path of configurations with a bubble wall. In this way, we can identify an appropriate expression for the tunnelling rate without any ambiguity. The result agrees with the direct evaluation of the Euclidean action once we properly subtract the zero-point energy and count how many times each region in the instanton solution is swept when we consider a smooth foliation.
It is possible to trace the subtle nature of the pseudo-manifold to the fact that the time lapse in the Euclidean region is not positive everywhere. Indeed, the sign of the time lapse has to be opposite between the center and the asymptotic infinity, for at least some range during the time evolution. This is a feature that is common to the upward tunnelling in the case of bubble nucleation. It would be worth investigating whether or not this negative lapse causes any problem when we take into account fluctuations around the WKB trajectory.