Theories with compact extra dimensions are sometimes unstable to decay into a bubble of nothing -- an instability resulting in the destruction of spacetime. We investigate the existence of these bubbles in theories where the moduli fields that set the size of the extra dimensions are stabilized at a positive vacuum energy -- a necessary ingredient of any theory that aspires to describe the real world. Using bottom-up methods, and focusing on a five-dimensional toy model, we show that four-dimensional de Sitter vacua admit bubbles of nothing for a wide class of stabilizing potentials. We show that, unlike ordinary Coleman-De Luccia tunneling, the corresponding decay rate remains non-zero in the limit of vanishing vacuum energy. Potential implications include a lower bound on the size of compactified dimensions.
Within the framework of flux compactifications, we construct an instanton describing the quantum creation of an open universe from nothing. The solution has many features in common with the smooth 6d bubble of nothing solutions discussed recently, where the spacetime is described by a 4d compactification of a 6d Einstein-Maxwell theory on S^2 stabilized by flux. The four-dimensional description of this instanton reduces to that of Hawking and Turok. The choice of parameters uniquely determines all future evolution, which we additionally find to be stable against bubble of nothing instabilities…
I present analytic time symmetric initial data for five dimensions describing ``bubbles of nothing'' which are asymptotically flat in the higher dimensional sense, i.e. there is no Kaluza-Klein circle asymptotically. The mass and size of these bubbles may be chosen arbitrarily and in particular the solutions contain bubbles of any size which are arbitrarily light. This suggests the solutions may be important phenomenologically and in particular I show that at low energy there are bubbles which expand outwards, suggesting a new possible instability in higher dimensions. Further, one may find bu…
Using simple 6D junction conditions, we describe two surprising geometries. First in a case of transitions between dS4xS2 vacua, the S2 can be stretched significantly larger than the vacuum values both before and after the transition. Then we discover that the naive instability to decompactification is actually a bubble of nothing instead.
We construct a simple AdS_4 x S^1 flux compactification stabilized by a complex scalar field winding the extra dimension and demonstrate an instability via nucleation of a bubble of nothing. This occurs when the Kaluza -- Klein dimension degenerates to a point, defining the bubble surface. Because the extra dimension is stabilized by a flux, the bubble surface must be charged, in this case under the axionic part of the complex scalar. This smooth geometry can be seen as a de Sitter topological defect with asymptotic behavior identical to the pure compactification. We discuss how a similar cons…
The rate and manner of vacuum decay are calculated in an explicit flux compactification, including all thick-wall and gravitational effects. For landscapes built of many units of a single flux, the fastest decay is usually to discharge just one unit. By contrast, for landscapes built of a single unit each of many different fluxes, the fastest decay is usually to discharge all the flux at once, which destabilizes the radion and begets a bubble of nothing. By constructing the bubble of nothing as the limit in which ever more flux is removed, we gain new insight into the bubble's appearance. Fina…
We construct exact solutions with the bubble of nothing in the Dvali-Gabadadze-Porrati braneworld model. The configuration with a single brane can be constructed, unlike in the Randall-Sundrum braneworld model. The geometry on the single brane looks like the Einstein-Rosen bridge. We also discuss the junction of multibranes. Surprisingly, even without any artificial matter fields on the branes such as three-dimensional tension of the codimension-two objects, two branes can be connected in certain configurations. We investigate solutions of multibranes too. The presence of solutions may indicat…
Nothing---the absence of spacetime---can be either an endpoint of tunneling, as in the bubble of nothing, or a starting point for tunneling, as in the quantum creation of a universe. We argue that these two tunnelings can be treated within a unified framework, and that, in both cases, nothing should be thought of as the limit of anti-de Sitter space in which the curvature length approaches zero. To study nothing, we study decays in models with perturbatively stabilized extra dimensions, which admit not just bubbles of nothing---topology-changing transitions in which the extra dimensions pinch …
Bubbles of nothing are a class of vacuum decay processes present in some theories with compactified extra dimensions. We investigate the existence and properties of bubbles of nothing in models where the scalar pseudomoduli controlling the size of the extra dimensions are stabilized at positive vacuum energy, which is a necessary feature of any realistic model. We map the construction of bubbles of nothing to a four-dimensional Coleman-De Luccia problem and establish necessary conditions on the asymptotic behavior of the scalar potential for the existence of suitable solutions. We perform deta…
We construct instanton solutions describing the decay of flux compactifications of a $6d$ gauge theory by generalizing the Kaluza-Klein bubble of nothing. The surface of the bubble is described by a smooth magnetically charged solitonic brane whose asymptotic flux is precisely that responsible for stabilizing the 4d compactification. We describe several instances of bubble geometries for the various vacua occurring in a $6d$ Einstein-Maxwell theory namely, AdS_4 x S^2, R^{1,3} x S^2, and dS_4 x S^2. Unlike conventional solutions, the bubbles of nothing introduced here occur where a {\em two}-s…
We discuss non-perturbative instabilities, mediated by bubbles of nothing, in the context of the AdS/CFT correspondence. By exploring the phase diagram of such decays we give an interpretation of the process in terms of an effective potential in a quantum mechanics of a single variable, a winding mode condensate. The decay corresponds to a conventional first order transition, the true vacuum being a non-geometrical "nothing" state.
Applying the inverse scattering method to static and bi-axisymmetric Einstein equations, we construct a non-rotating black lens inside bubble of nothing whose horizon is topologically lens space L(n,1)=S^3/Z_n. Using this solution, we discuss whether a static black lens can be in equilibrium by the force balance between the expansion and gravitational attraction.
We argue that expanding bubbles of nothing are a widespread feature of systems of black holes with multiple or non-spherical horizons, appearing as a limit of regions that are narrowly enclosed by the horizons. The bubble is a minimal cycle that links the Einstein-Rosen bridges in the system, and its expansion occurs through the familiar stretching of space in black hole interiors. We demonstrate this idea (which does not involve any Wick rotations) with explicit constructions in four and five dimensions. The geometries of expanding bubbles in these dimensions arise as a limit of, respectively…
In 3+1 dimensions there are anti-de quotients which are black holes with toroidal event horizons. By analytic continuation of the Schwarzschild-anti-de Sitter solution (and appropriate identifications) one finds two one parameter families of spacetimes that contain these quotient black holes. One of these families consists of B-metrics ("bubbles of nothing"), the other of black hole spacetimes. All of them have vanishing conserved charges.
In this contribution we review the ideas and results presented in the articles arXiv:2002.01764 and arXiv:2110.08276 where the decay of AdS vacua into bubbles of nothing is studied. We firstly consider AdS vacua in General Relativity with a cosmological constant and in arbitrary dimensions. In this context we study a class of gravitational instantons generalizing the Witten's bubble of nothing and we discuss the derivation of the euclidean action. Secondly we consider a "stringy" non-supersymmetric AdS$_4$ vacuum arising in a compactification of massive IIA supergravity on a $S^6$ and preservi…
We consider ``bubbles of nothing'' constructed by analytically continuing black hole solutions in Anti-de Sitter space. These provide interesting examples of smooth time-dependent backgrounds which can be studied through the AdS/CFT correspondence. Our examples include bubbles constructed from Schwarzschild-AdS, Kerr-AdS and Reissner-Nordstrom AdS. The Schwarzschild bubble is dual to Yang-Mills theory on three dimensional de Sitter space times a circle. We construct the boundary stress tensor of the bubble spacetime and relate it to the properties of field theory on de Sitter.
The non-perturbative instabilities of hot Kaluza-Klein spacetime are investigated. In addition to the known instability of hot space (the nucleation of 4D black holes) and the known instability of KK space (the nucleation of bubbles of nothing by quantum tunneling), we find two new instabilities: the nucleation of 5D black holes, and the nucleation of bubbles of nothing by thermal fluctuation. These four instabilities are controlled by two Euclidean instantons, with each instanton doing double duty via two inequivalent analytic continuations; thermodynamic instabilities of one are shown to be …
We investigate the non-perturbative stability of supersymmetric compactifications with respect to decay via a bubble of nothing. We show examples where this kind of instability is not prohibited by the spin structure, i.e., periodicity of fermions about the extra dimension. However, such "topologically unobstructed" cases do exhibit an extra-dimensional analog of the well-known Coleman-De Luccia suppression mechanism, which prohibits the decay of supersymmetric vacua. We demonstrate this explicitly in a four dimensional Abelian-Higgs toy model coupled to supergravity. The compactification of t…
A bubble of nothing is a spacetime instability where a compact dimension collapses. After nucleation, it expands at the speed of light, leaving "nothing" behind. We argue that the topological and dynamical mechanisms which could protect a compactification against decay to nothing seem to be absent in string compactifications once supersymmetry is broken. The topological obstruction lies in a bordism group and, surprisingly, it can disappear even for a SUSY-compatible spin structure. As a proof of principle, we construct an explicit bubble of nothing for a $T^3$ with completely periodic (SUSY-c…
In arXiv:1407.8203, we introduced the idea of s-sourcery, a general formalism for building many-body quantum ground states using renormalization-group-inspired quantum circuits. Here we define a generalized notion of s-sourcery that applies to mixed states, and study its properties and applicability. We prove a number of theorems establishing the prevalence of mixed s-source fixed points. For our examples we focus on thermal states of local Hamiltonians. Thermal double states (also called thermofield double states) and the machinery of approximate conditional independence are used heavily in t…
An interesting idea is that the universe could be spontaneously created from nothing, but no rigorous proof has been given. In this paper, we present such a proof based on the analytic solutions of the Wheeler-DeWitt equation (WDWE). Explicit solutions of the WDWE for the special operator ordering factor p=-2 (or 4) show that, once a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum, it can expand exponentially no matter whether the bubble is closed, flat or open. The exponential expansion will end when the bubble becomes large and thus the early univer…
We study non-perturbative instabilities of AdS spacetime in General Relativity with a cosmological constant in arbitrary dimensions. In this simple setup we explicitly construct a class of gravitational instantons generalizing Witten's bubble of nothing. We calculate the corresponding Euclidean action and show that its change is finite. The expansion of these bubbles is described by a lower-dimensional de Sitter geometry within a non-compact foliation of the background spacetime. Moreover we discuss the existence of covariantly constant spinors as a possible topological obstruction for such de…
The perturbative vacuum of type 0 string theory is unstable due to the existence of the closed string tachyon. This instability can be removed by S^1 compactification with twisted boundary condition for the tachyon field. We show that even in this situation unwrapped NS5-branes are unstable and decay to bubbles of nothing smoothly without tunneling any potential barrier. We discuss a relation between the closed string tachyon condensation and the instability of NS5-branes.
We construct an instanton describing the pair production of non-Kaluza Klein bubbles of nothing in higher odd dimensional de Sitter spaces. In addition to showing that higher dimensional de Sitter spaces have a nonzero probability to become topologically nontrivial, this process provides direct evidence for the association of entropy with cosmological horizons and that non-Kaluza Klein bubbles of nothing are a necessary ingredient in string theory or any other consistent quantum theory of gravity in higher dimensions.
We consider five dimensional deSitter spacetimes with a deficit angle due to the presence of a closed 2-brane and identify one dimension as an extra dimension. From the four dimensional viewpoint we can see that the spacetime has a structure similar to a Kaluza-Klein bubble of nothing, that is, four dimensional spacetime ends at the 2-brane. Since a spatial section of the full deSitter spacetime has the topology of a sphere, the boundary surface surrounds the remaining four dimensional spacetime, and can be considered as the celestial sphere. After the spacetime is created from nothing via an …
We give an explicit formulation of the time dependent AdS/CFT correspondence when there are multiple vacua present in Lorentzian signature. By computing sample two point functions we show how different amplitudes are related by cosmological particle production. We illustrate our methods in two example spacetimes: (a) a ``bubble of nothing'' in AdS space, and (b) an asymptotically locally AdS spacetime with a bubble of nothing on the boundary. In both cases the alpha vacua of de Sitter space make an interesting appearance.
We describe the semiclassical decay of a class of orbifolds of AdS space via a bubble of nothing. The bounce is the small Euclidean AdS-Schwarzschild solution. The negative cosmological constant introduces subtle features in the conservation of energy during the decay. A near-horizon limit of D3-branes in the Milne orbifold spacetime gives rise to our false vacuum. Conversely, a focusing limit in the latter produces flat space compactified on a circle. The dual field theory description involves a novel analytic continuation of the thermal partition function of Yang-Mills theory on a three-sphe…
We extend the study of time-dependent backgrounds in the AdS/CFT correspondence by examining the relation between bulk and boundary for the smooth 'bubble of nothing' solution and for the locally AdS black hole which has the same asymptotic geometry. These solutions are asymptotically locally AdS, with a conformal boundary conformal to de Sitter space cross a circle. We study the cosmological horizons and relate their thermodynamics in the bulk and boundary. We consider the alpha-vacuum ambiguity associated with the de Sitter space, and find that only the Euclidean vacuum is well-defined on th…
We discover that a class of bubbles of nothing are embedded as time dependent scaling limits of previous spacelike-brane solutions. With the right initial conditions, a near-bubble solution can relax its expansion and open the compact circle. Thermodynamics of the new class of solutions is discussed and the relationships between brane/flux transitions, tachyon condensation and imaginary D-branes are outlined. Finally, a related class of simultaneous connected S-branes are also examined.
The lightlike supersymmetric solutions of N=2, D=4 gauged supergravity coupled to an arbitrary number of abelian vector multiplets are classified using spinorial geometry techniques. The solutions fall into two classes, depending on whether the Killing spinor is constant or not. In both cases, we give explicit examples of supersymmetric backgrounds. Among these BPS solutions, which preserve one quarter of the supersymmetry, there are gravitational waves propagating on domain walls or on bubbles of nothing that asymptote to AdS_4. Furthermore, we obtain the additional constraints obeyed by half…
We demonstrate the existence of gravitational critical phenomena in higher dimensional electrovac bubble spacetimes. To this end, we study linear fluctuations about families of static, homogeneous spherically symmetric bubble spacetimes in Kaluza-Klein theories coupled to a Maxwell field. We prove that these solutions are linearly unstable and posses a unique unstable mode with a growth rate that is universal in the sense that it is independent of the family considered. Furthermore, by a double analytical continuation this mode can be seen to correspond to marginally stable stationary modes of…
We describe new bubble decays in pure D+1 dimensional Einstein theory with two compact directions. The instanton solution is constructed by analytic continuation of the Kaluza-Klein electrically charged black hole solution. We show that the instanton describes the decay of a Kaluza-Klein vacuum M^{D-1} x T^2 with a non-vanishing torus tilt parameter. The decay is produced by the creation of a bubble of nothing which expands with time. We compute the instanton action, which shows that this Kaluza-Klein vacuum becomes more stable as the torus tilt parameter is increased. As an application, we co…
Morse-Bott inequalities, Topology Change and Cobordisms to Nothing
Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid（马德里自治大学-西班牙国家研究委员会理论物理研究所）；Departamento de Física Teórica, Universidad Autónoma de Madrid（马德里自治大学理论物理系）
In this paper we construct a stringy embedding of the dark bubble model of an expanding 4D cosmology with the help of branes rotating in extra dimensions. The universe rides a bubble which has nucleated in an unstable higher dimensional background. Our construction is therefore a string theoretical realization of Vilenkin's quantum cosmology. While the cosmological constant vanishes at lowest order, loop corrections predicted by the Weak Gravity Conjecture can induce a phenomenologically viable value. The model predicts the existence of large extra dimensions compatible with observational cons…
We consider compactifications of massive IIA supergravity on a six-sphere. This setup is known to give rise to non-supersymmetric AdS$_4$ vacua preserving SO$(7)$ as well as G$_2$ residual symmetry. Both solutions have a round $S^6$ metric and are supported by the Romans' mass and internal $F_6$ flux. While the SO$(7)$ invariant vacuum is known to be perturbatively unstable, the G$_2$ invariant one has been found to have a fully stable Kaluza-Klein spectrum. Moreover, it has been shown to be protected against brane-jet instabilities. Motivated by these results, we study possible bubbling solut…
False vacuum states are metastable in quantum field theories, and true vacuum bubbles can be nucleated due to the quantum tunneling effect. It was recently suggested that an evaporating black hole (BH) can be a catalyst of bubble nucleations and dramatically shortens the lifetime of the false vacuum. In particular, in the context of the Standard Model valid up to a certain energy scale, even a single evaporating BH may spoil the successful cosmology by inducing the decay of our electroweak vacuum. In this paper, we reinterpret catalyzed vacuum decay by BHs, using an effective action for a thin…
We investigate the formation via tunneling of inflating (false-vacuum) bubbles in a true-vacuum background, and the reverse process. Using effective potentials from the junction condition formalism, all true- and false-vacuum bubble solutions with positive interior and exterior cosmological constant, and arbitrary mass are catalogued. We find that tunneling through the same effective potential appears to describe two distinct processes: one in which the initial and final states are separated by a wormhole (the Farhi-Guth-Guven mechanism), and one in which they are either in the same hubble vol…
We consider the noncommutative Abelian-Higgs theory and construct new types of exact multi-vortex solutions that solve the static equations of motion. They in general do not follow from the BPS equations; only for some specific values of parameters, they satisfy the BPS equations saturating the Bogomol'nyi bound. We further consider the Abelian-Higgs theory with more complicated scalar potential allowing unstable minima and construct exact solutions of noncommutative false vacuum bubble with integer magnetic flux. The classical stability of the solutions is discussed.
This paper examines the classical dynamics of false vacuum regions embedded in surrounding regions of true vacuum, in the thin-wall limit. The dynamics of all generally relativistically allowed solutions -- most but not all of which have been previously studied -- are derived, enumerated, and interpreted. We comment on the relation of these solutions to possible mechanisms whereby inflating regions may be spawned from non-inflating ones. We then calculate the dynamics of first order deviations from spherical symmetry, finding that many solutions are unstable to such aspherical perturbations. T…
We present a non-singular instanton describing the creation of an open universe with a compactified extra dimension. The four dimensional section of this solution is a singular instanton of the type introduced by Hawking and Turok. The ``singularity'' is viewed in five dimensions as a smooth bubble of ``nothing'' which eats up a portion of spacetime as it expands. Flat space with a compact extra dimension is shown to be gravitationally metastable, but sufficiently long lived if the size of the extra dimension is large compared with the Planck length.
I show that cosmological bubble walls in the thin wall approximation are unstable to the creation of "barnacles" -- loci of different wall tension adjacent to regions filled with a third vacuum. Barnacle formation leads to the same observational consequences as the extensively studied bubble collision scenario, but occurs exponentially more often. The process is described by a saddle point of the thin wall action with two negative modes.
We consider the fate of AdS vacua connected by tunneling events. A precise holographic dual of thin-walled Coleman--de Luccia bounces is proposed in terms of Fubini instantons in an unstable CFT. This proposal is backed by several qualitative and quantitative checks, including the precise calculation of the instanton action appearing in evaluating the decay rate. Big crunches manifest themselves as time dependent processes which reach the boundary of field space in a finite time. The infinite energy difference involved is identified on the boundary and highlights the ill-defined nature of the …
