TUNNELLING WITH WORMHOLE CREATION
S. Ansoldia'b, T. Tanakac'd*
"National Institute of Nuclear Physics (INFN). 1-34149, Trieste, Italy b University of Udine, 1-33100, Udine, Italy '"Department of Physics, Kyoto University, 606-8502, Kyoto, Japan d Yukawa Institute for Theoretical Physics, Kyoto University, 606-S502, Kyoto, Japan
Received October 18, 2014
The description of quantum tunnelling in the presence of gravity shows subtleties in some cases. 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 the literature.
Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday
DOI: 10.7868/S0044451015030143
1. INTRODUCTION
Quantum tunnelling plays various roles in cosmology. For instance, false vacuum decay through quantum tunnelling fl 3] is an important process for the universe to visit many vacua in the string landscape [4 6]. Also, the possibility of creation of an open universe through false vacuum decay has been extensively-discussed [7 10]. Properly taking the effect of gravity into account can be quite nontrivial. Although the effect of gravity is secondary in some cases, there are in fact several cases where 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 nontrivial. 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 Refs. [14, 15], where it is shown that the instability can be eliminated, 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 24], which is the main subject
E-mail: tanaka'fl'yukawa.kvoto-u.ac.jp
of this paper. Wormhole formation is a signature of what is also referred to in the literature 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 the literature [26]. In this paper, we show 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. 2, 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. 3, 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. 4 we then study the same problem using the canonical approach with a smooth time slice: this allows us to derive the formula for the tunnelling rate without any ambiguity. In Sec. 5, 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. Section 6 is devoted to a summary and discussion: we also elaborate on a remaining, more subtle, issue.
2. CONVENTIONAL APPROACH
In this paper, we consider the simplest spherically symmetric domain-wall model, whose Lagrangian is given by
S = J <1t,„(R). (1)
where R. is the scalar curvature and m(R) is the radius-dependent IIlclSS of the wall, e.g., rn(R) = const for a dust domain wall, while m(R) = 4naR2 for a wall consisting of pure tension a\ moreover, r is the proper time along the wall, and R denotes the circumferential radius of the wall. In general, quantities marked with a hat are assumed to be evaluated at the position of the wall, e.g., B = B(t) = B(t, f(t)) if B is a function of t and r, and r = f(t) is one possible parameterization 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 gravity into account.
In this section, we derive a conventional but incorrect formula for the tunnelling rate of the wall. Although we mostly follow Ref. [27], we do not claim that the result obtained there is wrong. Indeed, our emphasis is about the fact that the authors of Ref. [27] 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 de Sitter cosnio-logical constant, the wall surface tension in their model) are changed: on the contrary, the conventional canonical approach does not guarantee the continuity of the action as a function of the parameters. At the same time, the direct calculation of the action reveals difficulties in the identification of the Euclidean manifold interpolating between the before- and after-tunnelling classical solutions in a consistent way: indeed, Farlii et al. associate what they call a pseudo-manifold to the instant on solution. The direct approach defines the pseudo-manifold by weighing different volumes of the instant on 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 show in what follows that the canonical approach, in full generality, can reproduce the same
value for the tunnelling action given in the approach proposed in [27].
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 both the inside and outside of the bubble are empty, and hence 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 a vacuum energy density, i. e., a cosmological constant, but this does not change the treatment in any substantial way.) The method proposed in [27] 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 is and ¿m in the simplified case that we consider 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 see in Sec. 5.
The contributions to the action can be summarized as follows.
1. A matter term coming from the shell, I*™1,1lpr: 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, ^gravity: this is, basically, the well-known extrinsic-cutvature-trace-jump 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 nontrivial. As clearly discussed in Ref. [27], several contributions arise.
(a) The crucial contribution in [27], I™*1f'u.r, conies 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. 4, we take this last picture.
(b) Another contribution conies from a surface at a large constant circumferential radius in the outside spacetime, /^?fac P: cllt~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 Cbelow.
With the notation used above and by setting (because
of the square, the notation below differs from the one used in Ref. [271 )
.41 = 1. .« = 1 2GM
R
the above terms can be written as [27]1^
rwall
rwall _ / i I gravity J I 2("'
R2
inil!) dr.
2Re(R2T + A2)1/'2
(2)
(3)
rwall 1 surface
e(R2T + A2)1/'2
i r d ~2gJt1 dTTrX
' f(R2T + A 2)i/2+eR, R log ——
(4)
.4
2 G J 'i
2 RRT log
\R2t+A2)1'2+(R,
R-
f(i?fT + .42)1/2 V"'" ' 2 iî2f(iî2T + .42)1/2
riÏBiG = ( -^BIG surface I Q
2,4 2
3 Ma
(-4 2U
riÎBIG _ jRBIG _ ( T „ \ — net — surface lJsurfaceiO —
■^ul-'D-of 1
V ^BIG
(5)
(6)
(7)
where square brackets represent the jump of the bracketed quantities across the shell, i.e.,
= lim (B(f - 6) - B(r + .
(8)
11 The expression for given in R.ef. [2i] looks slightly
different, but it is equivalent to this one as long as we require that ^surfaro always real valued. As we explain later (see Eq. (19)), the sign flip of e is only important in the Euclidean regime. Because the argument of the logarithm has a jump there, we may have to add one more term proportional to a A" function at the sign flipping point to the right-hand side of Eq. (o). However, the crucial point is that the analyticity of is broken at the
sign flipping point. Therefore, it is d
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