Uptunneling to de Sitter

We propose a Euclidean preparation of an asymptotically AdS$_2$ spacetime that contains an inflating dS$_2$ bubble. The setup can be embedded in a four dimensional theory with a Minkowski vacuum and a false vacuum. AdS$_2$ approximates the near horizon geometry of a two-sided near-extremal Reissner-Nordstr\"{o}m black hole, and the two sides can connect to the same Minkowski asymptotics to form a topologically nontrivial wormhole geometry. Likewise, in the false vacuum the near-horizon geometry of near-extremal black holes is approximately dS$_2$ times 2-sphere. We interpret the Euclidean solution as describing the decay of an excitation inside the wormhole to a false vacuum bubble. The result is an inflating region inside a non-traversable asymptotically Minkowski wormhole. We speculate about the possibility of a brief period of causal connection when light matter fields are present.


Introduction
False vacua are accessible in classical and quantum field theory, given enough time and energy. By a careful preparation of the initial state, an experimentalist could, in principle, create an arbitrarily big region of false vacuum in the laboratory. For instance, this could be done by applying time-reversal to the decay products of a state that contains such a region to begin with.
In gravity, the situation is qualitatively different. A big region of a false vacuum with positive cosmological constant (CC) can eternally inflate, and as a result, change the asymptotic structure of the spacetime. In fact, in classical gravity Penrose singularity theorem [1] forbids the formation of such an inflating bubble in a Minkowski vacuum with no singularity in the past [2]. It is tantalizing to ask whether this can happen quantum mechanically. It would be quite remarkable if quantum gravity completely excised macroscopic domains of de Sitter vacua from Minkowski and Anti-de Sitter physics.
But if it didn't, one could hope to use the false vacuum bubble to shed some light on the notoriously hard to understand quantum mechanics of de Sitter (dS) spacetime. We have a better handle on quantum gravity in asymptotically Minkowski spacetime via scattering amplitudes and in asymptotically Anti-de Sitter (AdS) spacetime via holography. If dS arises in the excitations of the Minkowski or AdS vacua, those frameworks can, at least in principle, be employed to also explore dS quantum gravity.
On the other hand, dS can help formulate new problems in the asymptotically Minkowski or AdS setups. For instance, because of the negative interior pressure, dS bubbles are generically hidden behind black hole horizons. This provides an interesting interplay between black hole microstate counting and that of de Sitter. 1 But perhaps the best motivation is coming from the cosmological observations. They suggest that we live in a dS false vacuum with a tiny CC, and that our universe has gone through a period of inflation at very early times. One might wonder: Could it all have started inside the laboratory of a dedicated experimentalist?
This question was raised 30 years ago in [4,5,6]. To go around the singularity theorem, they invoke quantum tunneling to a spacetime that if classically extended it would contain a singularity in the past. However, the solution of [4,5,6] is degenerate, and to date there is no consensus on whether or not it describes a valid tunneling process. See [7] and [8] for two recent papers with contrasting views and for further references. To break the impasse, finding an alternative solution seems necessary.
Here we propose an alternative. One that is a byproduct of our solution (in 2d gravity) to a similarly interesting question raised in [8]: Is it possible to prepare, via a Euclidean path integral, an asymptotically AdS state that contains an inflating bubble?
The two questions are connected if we take advantage of simplifications that arise in the near horizon geometry of near-extremal black holes. Specifically, our focus is on the near extremal Reissner-Nordström (RN) black hole in the Minkowski vacuum and on the near extremal Schwarzschild-de Sitter (SdS) black hole in the false vacuum. Both geometries have a long throat with an approximately constant radius and hence one can dimensionally reduce to obtain a 2d gravity model called Jackiw-Teitelboim (JT) gravity [9,10]. The 2d geometry is AdS 2 in the RN case and dS 2 in the SdS case. These connections have been the subject of extensive research in recent years. Two papers that have been particularly inspiring to us are [11,12], though for completeness a brief review will be given in section 2.
In section 3, we find a Euclidean solution in which a brane emanates from the AdS 2 boundary, it decays into a dS 2 bubble (a piece of a 2-sphere in Euclidean signature), and bounces back to reach the AdS boundary (see figure 1). Cutting this bounce solution at the moment of time-reflection symmetry and Wick rotating gives the desired asymptotically   figure   1 and analytic continuation to the past. Left: A two sided AdS 2 black hole with an elongated throat due to the presence of the massive brane. Right: An inflating bubble that is nucleated inside the AdS 2 throat. The domain walls (green) fall though the black hole horizons. In the 4d embedding, the inflating region has the spatial topology of R × S 2 . At first, the R factor expands exponentially and the 2-sphere slowly. Eventually, the expansion becomes isotropic and we obtain a locally dS 4 solution. As seen, the classically extended geometry is singular in the past as dictated by the Penrose singularity theorem [1,13].
AdS 2 geometry with an inflating bubble in the middle (see figure 2). Once embedded in four dimensions, the two AdS asymptotics match to the mouths of a near-extremal RN wormhole. The dS 2 region describes the near-extremal SdS geometry. And the domain walls carry opposite magnetic charges to break the magnetic field lines.
In section 4, we will speculate about the tunneling probability by comparing the bounce action to the Euclidean action for the free propagation of the brane in AdS 2 . We will conclude by further remarks on the viability of this scenario, as well as alternative solutions that allow (temporary) causal communication with the inflating region.

JT gravity
Our goal is to study the motion of codimension-1 domain walls and branes in the RN and SdS geometries. Since we are mainly concerned with the Euclidean solution, it is enough to look at the static patch metric: where dΩ 2 is the line element on unit 2-sphere S 2 , and where Λ is vacuum energy of the false minimum, and The horizons correspond to the zeros of f (r). The extremal limit is when the two zeroes coincide. In the SdS case this is at Suppose we tune M i , M e and Q such that both geometries are near extremality and with approximately equal horizon areas. Then the near horizon geometry can be studied by dimensionally reducing over S 2 and working with the 2d model. Ignoring the KK modes and setting r 0 = 1, we have the spherically symmetric metric ansatz where µ, ν run over 0 and 1. Dropping topological terms, the 4d action reduces to 2d JT gravity coupled to 2d matter fields ψ: where in terms of 4d Newton's constant C = 1/(4G). φ is called the dilaton field. At φ = φ b ∼ 1 the 2d theory has to be matched with the higher dimensional one. k is the geodesic curvature of this boundary, and d its line element. Having set the extremal radius to one, the dilaton potential is To leading order in φ the 2d metric in the false vacuum is the dS 2 metric with unit radius of curvature. Working in the static patch and Wick rotating we get the sphere metric This can be derived from the variation of (2.6) with respect to φ, or directly from (2.2) by expanding in the near-horizon, near-extremal limit. The φ solution is given by The points θ = 0 and θ = π correspond, respectively, to the cosmological and the black hole horizons of the SdS geometry. Beyond the cosmological horizon, φ expands and one eventually recovers isotropic 4d inflation. The form of the solution (2.9) can also be derived by varying the effective 2d action (2.6) with respect to g µν , which gives and setting T µν = 0. Matter perturbations back-react on φ, and in order for the approximation φ 1 to hold true, we need (restoring factors of r 0 ) We neglect the small effect of these perturbations on the curvature of the 2d metric g µν . In the RN case, the extremal radius of a charge Q black hole is We take r e 1. In the near extremal limit the 2d metric is approximately a unit-curvature AdS 2 . In Euclidean signature as follows from the variation of (2.6) with respect to φ, or directly expanding (2.3) and Wick rotating. Allowing for a small offset between the extremal radii r e and r 0 = 1, corresponds to including a small 2d CC on the r.h.s. of (2.10). The φ solution is then We neglect O(φ) corrections to the 2d geometry.

Domain wall motion
The motion of a domain wall follows from the junction condition between the two geometries it connects, and it is particularly simple in the spherically symmetric case [14,15]. In our approximation, it follows from (2.10): where L/R label the two sides, ξ µ is the normal to the domain wall trajectory, pointing from right to left, and κ is related to the brane tension σ via Our bounce solution consists of a brane that emanates from the AdS 2 boundary and bifurcates into the dS-AdS domain walls as in figure 1. 2 Below we will discuss each part separately.

Brane in AdS 2
In the absence of the brane the metric and dilaton are given by (2.13) and (2.14) respectively. Next we add a brane that stretches to the AdS boundary and impose Z 2 symmetry across it. When the two sides of the brane are identified, this is often called an end of the world brane. It is considered as a model for a one-sided black hole formed from the collapse of a pure state [16]. See also [17,18] for possible connections to cosmology. We do not make this identification. The normal vector to the brane is where over-dot indicates derivative with respect to the proper length. From (3.1), we finḋ where the normalized brane tension is called κ 0 . This is the equation of a geodesic in AdS 2 , consistent with the fact that in the presence of a soft source (the brane) the 2d curvature has to remain finite. Denoting by η the normal coordinate to the brane, the scalar curvature near the brane is related to its geodesic curvature via The Z 2 symmetry implies k R = −k L , hence we must have k L = k R = 0. Strictly speaking, the curve has a small O(φ) curvature that is irrelevant for our discussion. The angular position of the brane at radial coordinate ρ as measured from the closest approach to the origin ρ = ρ m can also be calculated using (3.4), Finally, the brane meets the AdS 2 boundary at ρ = ρ c with an angle with respect to the normal. This contributes to the boundary term in the JT action because k = 2α b δ(u)+regular, if u is the proper length along ∂AdS 2 measured from this point. Without any decay, this solution can be Wick rotated at ρ = ρ m to give a two-sided eternal black hole with a brane inside as in figure 2-Left.

Brane decay
Suppose at ρ = ρ 1 the brane branches into two domain walls separating Euclidean AdS 2 from a Euclidean dS 2 (i.e. S 2 ) region. Let us first show that the 2d geometry is smooth along the domain wall. This follows from the derivative of (3.1) along the domain wall. On the right we get u ν ∂ ν κ and on the left The first term can be written in terms of the geodesic curvature, and in the second, we can use (2.10) and the fact that u · ξ = 0 to find By energy-momentum conservation the second term on the left is the same as the term on the right. Therefore 3 k R = k L ⇒ k AdS = k dS . (3.10) This holds also at the branching point, where the brane can be smeared over some width and be thought of as a flux of Euclidean energy from the AdS side in (3.9) that is absorbed by the domain wall. Equation (3.10) implies that there is no conical singularity at that point and the sum of the three exterior angles is 2π. Note that if the boundaries of the left AdS region, the right AdS region and the dS part were all smooth curves, the sum of the angles would be 3π. Therefore, there has to be a break in the boundary trajectories (see figure 3). Denote the break of the right AdS boundary by α and the ρ velocity after the breakρ + , then On the dS side, using Z 2 symmetry and imposing that the bubble includes the cosmological horizon at θ = 0, we findθ = − cos α, at bifurcation (3.12) whereθ is measured in the direction away from the branching point. In addition, (3.10) forbids formation of a dS bubble that is carved out of AdS without a deficit angle, because otherwise d k dS < 2π < d k AdS no angular deficit. (3.13) Indeed, if the AdS region in the bounce solution of figure 1 is continued beyond the bubble walls it either encounters a piece of the AdS boundary or the brane trajectories collide at an angle, implying a conical deficit. Finally, momentum conservation relates α, κ 0 and the domain wall tension κ: (3.14) Hence α < π/2, andθ < 0 at the branching point.

dS-AdS domain wall
The Euclidean dS 2 metric and dilaton solutions are given by (2.8) and (2.9). The continuity of φ at the domain wall implies To simplify equations, we assume the bubble forms at ρ 1 1, and therefore, there is a hierarchy Then, we get from (3.15)ρ The opposite sign ofρ andθ is because we are keeping the cosmological horizon at θ = 0, rather than black hole horizon θ = π. Therefore, larger values of ρ and hence φ correspond to smaller values of θ. The fact thatθ < 0 at the branching point implies thatρ + > 0, and hence, bifurcation can happen only after the AdS brane has passed ρ = ρ m . Using (3.17), the junction condition at the dS-AdS domain wall can be simplified to a one-dimensional motion in a potentialθ where The Euclidean bounce solution will be in the region −1 ≤ V (θ) ≤ 0, which corresponds to To ensure that these condition are satisfied for a finite range of θ, we impose See figure 4 for a sketch of the potential. Wick rotation to Lorentzian signature has to be done at one of the turning points: The potential for the Lorentzian motion is −V (θ). Hence, if the rotation is done at the larger root θ + , the walls are guaranteed to fall through the black hole horizon (θ = π) and the cosmological region of the SdS geometry will inflate. The starting point of the bounce solution θ 1 is obtained from (3.11) and (3.12). Using (3.17) this can be written as whereθ 1 = − −V (θ 1 ), and ρ 1 andρ 1 are implicitly functions of θ 1 and they also depend on A and κ 0 . It would be convenient to eliminate A and κ 0 , and instead, consider ρ 1 and ρ m as independent parameters. It is then easy to see that this equation has a solution when ρ m ρ 1 , and that this solution is close to θ − . In the limit ρ m ρ 1 We have so there is a solution. In fact, it is easy to see that at this solutioṅ and The solution we are interested in first bounces at θ − (right after the bifurcation point) then at θ + and for a second time at θ − (right before meeting the second bifurcation point). The point of time-reflection symmetry is θ = θ + . In order for the interior to be free from conical singularity, we have to make sure that the bounce is completed over ∆ϕ dS = π. In the above approximation, This imposes one constraint on B, B 0 , κ, which we numerically solved for and plotted in figure 5. connected to the Minkowski asymptotics. The angular size of this piece must satisfy where using (3.6) ∆ 1 = arccos(tanh ρ m ), ∆ 2 = arccos tanh ρ m tanh ρ 1 (3.30) and .
All ∆'s can be made much less than 1 by choosing ρ 1 ρ m 1.

Speculations
Relative probabilities are obtained by comparing the norm of various branches of the wavefunction. Here the comparison is between the two-sided RN geometry containing a brane in the middle, and the same geometry but with the brane replaced by an expanding dS bubble. A common approach is to apply saddle-point approximation to the Euclidean gravity path integral to calculate the square of the norms (as in the standard example of Coleman-De mentalist. Moreover, if the tension is too low the false vacuum region will entirely collapse (e.g. via percolating CDL bubbles). To prevent the latter, we need the critical size of the CDL bubble r CDL ∼ σ Λ (4.5) to be sufficiently big compared to the CC scale, and to ensure the former, we need it to be small compared to r 0 : Hence, we need the CDL down-tunneling to be microscopic and unlikely. Even though this is a mild requirement, the full setup is admittedly exotic, and almost certainly useless for creating a Universe in the lab. Firstly, because it requires constructing the topologically nontrivial RN wormhole with a brane in the throat. And secondly, because the domain walls are treated as external objects, whose microscopic dynamics is unspecified, very much in the spirit of the original work [4]. There is no guarantee that the brane excitation and its bifurcation into the domain walls actually take place in any realistic model.
However, there are reasons to believe that it is not completely fictitious. It has been argued in [22] that a traversable magnetic wormhole can be constructed in a QED-like model, and that the formation time is not exponentially suppressed in S BH as suggested by [23,24]. This wormhole is kept open using the negative Casimir energy. Starting from there, it is a trivial task to get a non-traversable RN wormhole (which is all that we need here) by adding mass to the system. In fact, our finding motivates the search for a more interesting (but also more challenging) scenario, namely, to engineer the tunneling inside the traversable wormhole. This would circumvent Penrose singularity theorem by using quantum violation of null energy condition. Unlike the RN wormhole, there is a time translation symmetry in this case. Hence, one can place a brane inside the throat and wait for it to decay into dS. Assuming this happens (work in progress), there is a short period of causal connection between the exterior and the bubble. Eventually, as the wormhole stretches the Casimir energy dilutes and horizons will form.
Lastly, our effective treatment of branes and domain walls guarantees that gravity by itself does not forbid formation of the false vacuum bubble. If the Maxwell theory in the Minkowski vacuum emerges from Higgsing a non-abelian gauge theory inside the false vacuum, the discharge of the magnetic field lines is also automatic and electromagnetism does not forbid the process either. Given that in non-gravitational theories false vacua are probed in scattering processes, it is not unimaginable that our toy model can indeed be an approximation to a scattering taking place inside a well-prepared wormhole.