Dark bubbles and black holes
UUUITP-05/21
Dark bubbles and black holes
Souvik Banerjee, a Ulf Danielsson, b Suvendu Giri c,d a Institut f¨ur Theoretische Physik und Astrophysik, Julius-Maximilians-Universit¨at W¨urzburg,AmHubland, 97074 W¨urzburg, Germany b Institutionen f¨or fysik och astronomi, Uppsala Universitet, Box 803, SE-751 08 Uppsala, Sweden c Dipartimento di Fisica, Universit`a di Milano-Bicocca, I-20126 Milano, Italy d INFN, sezione di Milano-Bicocca, I-20126 Milano, Italy
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In this paper we study shells of matter and black holes on the expandingbubbles realizing de Sitter space, that were proposed in arXiv:1807.01570. The explicitsolutions that we find for the black holes, can also be used to construct Randall-Sundrumbraneworld black holes in four dimensions. a r X i v : . [ h e p - t h ] F e b ontents To construct a model of de Sitter space and dark energy in string theory is a great challenge.Over the years, evidence has accumulated suggesting that many, possibly all, attemptsmade so far suffer from instabilities [1, 2]. For a review, see [3]. It is therefore of greatimportance to find alternative routes towards finding dS space. In [4], we proposed thatdark energy can be realized through an expanding bubble of true vacuum in a metastableAdS . Contrary to previous attempts, our focus is not on obtaining a time independent,metastable string vacuum with a positive vacuum energy. Instead, our model makes explicituse of an unstable higher dimensional AdS space with our universe riding on a bubble oftrue vacuum that mediates the decay of the unstable AdS. Our model has much in commonwith the braneworlds of Randall and Sundrum, [5, 6], but there are also crucial differences.Our model was further studied in [7], where we showed, using a five dimensional bulkwith stretched strings, how this will, through junction conditions generate an effectivetheory of four dimensional Einstein gravity on our dark bubble. In subsequent works,[8, 9], we worked out how to imprint a Schwarzschild geometry on the dark bubble andthe backreaction thereof on the bulk. For other aspects of these dark bubbles see, e.g.,[10, 11, 12, 13].In this paper, we work out in detail, the embedding of certain four dimensional structureson the dark bubble into AdS for two illuminating examples. We first examine a thin shellof matter, stabilized by its internal pressure, and then move on to consider a black hole.The new five dimensional metric that we find can also be used to realize black holes in fourdimensional Randall-Sundrum braneworlds. A key observation in our paper is that thefive dimensional spacetimes above (outside) and below (inside) our dark bubble are verydifferent. We show how these two different views, one from above and the other from below, we call this a dark bubble since it is a bubble that gives rise to dark energy in four dimensions. – 1 –roject the same effective four dimensional theory of gravity on the dark bubble in a rathermiracalous way.The rest of the paper is organized as follows. In section 2, we discuss the first exampleof a four dimensional rigid spherical shell of matter and demonstrate the uplifting of thisconstruction to the five dimensional bulk spacetimes above and below our shellworld. Whileabove the shell, we have a solution sourced by a stringy distribution, there is no sourcebelow and therefore, the junction condition across demands a non trivial choice of boundaryconditions imposed on the brane from the bulk spacetime on either side. In section 3, wemove to our second example, namely, exploring the possibility of having a black hole onour shellworld. We show that it is possible in our construction by considering differentnon-linear corrections in gravitational perturbation theory. Finally, we conclude with aninvitation to upcoming research activities in this direction. To better understand the physics of the four dimensional spacetime on the dark bubble, itis illustrative to consider a thin, rigid shell of matter with a radius r much larger than itsSchwarzschild radius. This could, for instance, be a shell made of ordinary baryonic mattersuch as iron. We showed in [4, 7] that a point mass in four dimensions is the end pointof a string located at a fixed radial distance, stretching in the fifth dimension. Followingthis intuition, the five dimensional structure corresponding to the rigid shell on the darkbubble would be expected to be located at a fixed coordinate radius and stretching in thefifth dimension. The proper radius r p ≡ krz then increases towards the boundary, givinga conical structure as sketched in figure 1a. The matter imprint on the dark bubble willthen be interpreted as the holographic projection of this five dimensional object into fourdimensions. Just as a shell of matter in four dimensions has a complicated equation of state(so as to remain stable against collapse but still possible to deform), the same would beexpected to be true for the five dimensional extended structure.The five dimensional picture below the dark bubble is, however, quite different. Thereare no matter sources in this region, and the spacetime must be a solution of the Einstein’sequations with just a negative cosmological constant. The existence, and form, of thissolution is quite surprising. As we will show in the next section, the solution contains aholographic “shadow” of the structure above the dark bubble, and the resulting picturewill look like figure 1b. We will also see how the construction nicely generalizes the simplerexamples studied in [8, 9], where we made use of some techniques introduced in [14, 15]. Let us first consider the five dimensional bulk spacetime above the brane as sketched infigure 1a. Above the brane, it is convenient to work in a gauge in which the brane isflat. Spacetime outside the structure is expected to be that of a matter source. Since thisstructure lies far outside its own Schwarzschild radius, the metric just outside it should be– 2 –dS CHR k + k − (a) k + k − (b) Figure 1 : A thin, rigid shell of four dimensional matter on the dark bubble seen (a) fromoutside, and (b) from inside. The gray region indicates the inside of the dark bubble. given by the CHR metric [16]d s = k z (cid:20) − (cid:18) − Mr (cid:19) d t + d r − M/r + r dΩ (cid:21) + d z k z , (2.1)while the spacetime inside is empty AdS . The structure that supports the rigid shell onthe brane is located at r = r = constant. In terms of the proper time ( τ ) on this structure,the induced metric is simply d s = k z (cid:0) − d τ + r dΩ (cid:1) + d z /k z . The stress tensor onit ( S ij ) is determined by the thin-shell junction conditions matching five dimensional AdSto CHR and reads S ττ = − k + r z (cid:32) − (cid:114) − Mr (cid:33) ,S θθ = S φφ = 1 k + r z (cid:32) − M/r (cid:112) − M/r − (cid:33) ,S zz = 2 k + r z (cid:32) − M/ r (cid:112) − M/r − (cid:33) . (2.2)When this structure ends on the dark bubble, Bianchi identities, which require the stresstensor to be covariantly conserved, produce a delta function contribution on the dark bubbleat z = z . To see this explicitly, let us write down the covariant conservation equation ∇ µ T µν ! = 0 for ν = z ( T µν refers to the five dimensional stress tensor). Explicitly, this gives ∇ µ T µν = ∂ µ T µν + Γ σγσ T γν + Γ νσµ T σµ ! = 0 ⇒ z∂ z T zz + 4 T zz ! = T tt + T rr + T θθ + T φφ . (2.3) This is the metric sourced by a neutral black string in five dimensions and induces four dimensionalSchwarzschild geometry on constant z slices. Symmetry arguments similar to Birkhoff’s theorem would leadto the conclusion that the metric outside a cylindrically symmetric matter distribution is given by the CHRmetric outside the four dimensional Schwarzschild radius of such a structure. – 3 –he stress tensor obtained in equation (2.2), dressed up with a properly normalized deltafunction localized at r = r gives T ij = S ij (cid:112) − M/r k + z δ ( r − r ) , T rr = 0 . (2.4)As a consistency check, this indeed satisfies equation (2.3). For this structure to end on thebrane at z = z , T zz must be further dressed up with a step function Θ( z − z ), T zz = 2 (cid:112) − M/r k r z (cid:32) − M/ r (cid:112) − M/r − (cid:33) δ ( r − r )Θ( z − z ) , (2.5)which gives T tt + T θθ + T φφ = z∂ z T zz + 4 T zz = 4 (cid:112) − M/r k r z (cid:32) − M/ r (cid:112) − M/r − (cid:33) δ ( r − r )Θ( z − z )+ 2 (cid:112) − M/r k r z (cid:32) − M/ r (cid:112) − M/r − (cid:33) δ ( r − r ) δ ( z − z ) . (2.6)The delta function in z arises from the derivative of the step function and induces matteron the dark bubble that exactly corresponds to the two dimensional shell embedded in fourdimensions. There is no matter below the brane and the five dimensional metric is simply a vacuumsolution to Einstein’s equations with a negative cosmological constant. However, the fivedimensional structure extending upwards will result in bending of the brane. One could,in principle, find a smooth global coordinate transformation that straightens out the bentbrane (corresponding to a gauge choice in which the brane is located at a constant z = z ),but we will not do this here, and will stick to the bent gauge.The spacetime below the brane (inside the dark bubble) deviates from empty AdS inresponse to the bending of the brane and can be computed order by order in the bending,which is proportional to the mass induced on the dark bubble. Let us write this perturbedmetric asd s = d z k − z + k − z η ab d x a d x b + perturbation (cid:122) (cid:125)(cid:124) (cid:123) χ ab d x a d x b = d z k − z + k − z (cid:2) − (1 + h t ( r, z )) d t + (1 + h r ( r, z )) d r + (1 + h a ( r, z )) r dΩ (cid:3) . (2.7)Choosing the perturbation to be traceless ( h ab η ab = 0), Einstein’s equations at linear orderin the perturbation give the following differential equation for the time component h t : ∂ rr h t + 2 r ∂ r h t + 5 k z ∂ z h t + k − z ∂ zz h t = 0 . (2.8) the properly normalized delta function is δ ( r − r ) / √ g rr = δ ( r − r ) (cid:112) − M/r/kz , so that it integratesto unity along the radial direction i.e. , (cid:82) √ g rr δ ( r − r ) / √ g rr = 1. – 4 –o solve this equation, we can use the fact the time-time component of the metric gives thegravitational potential of the solution to linear order in the perturbation χ/r . It was shownin [8] that in vacuum, this is given by the Bessel function K in momentum space. To getto position space, we need to superimpose the Bessel functions by integrating their Fouriertransform over the shell at r = r (given by (cid:82) dΩ below) which sources the solution. Thisgives h t = (cid:90) d (cid:126)p (cid:90) dΩ e i(cid:126)p · ( (cid:126)r − (cid:126)r ) K ( p ) = 2 π ∞ (cid:90) −∞ d p π (cid:90) d θ sin θ e ipr cos θ p K ( p ) (cid:90) dΩ e ipr cos θ = 16 π ∞ (cid:90) −∞ d p sin prpr sin pr pr p K ( p ) = 8 π rr k − z ( r + r ) (cid:113) k − z ( r + r ) − k − z ( r − r ) (cid:113) k − z ( r − r ) . (2.9)Einstein’s equations further give a differential equation for the radial component of theperturbation in terms of the time component:3 ∂ z h r + r∂ rz h r + ∂ z h t = 0 , (2.10)which can be solved to give (we have dropped the constant 8 π from h t ) h r = 13 r k − z r (cid:20)(cid:0) − k − (2 r − r ) ( r + r ) z (cid:1) (cid:113) k − z ( r + r ) − ( r (cid:55)→ − r ) (cid:21) . (2.11)The angular piece h a can be simply found from the tracelessness of the perturbation h t + h r + 2 h a = 0 to give h a = 13 r k − z r k − z (cid:0) r − rr + 2 r (cid:1) + k − z ( r − r ) (cid:0) r + rr + r (cid:1)(cid:113) k − z ( r − r ) − ( r (cid:55)→ − r ) . (2.12)Note that in the limit r → i.e. , when the spherical shell on the brane shrinks to apoint source at r = 0, the metric perturbation reduces to the result for a point sourceobtained in [8]. This is simply a consistency check, which ensures that the metric inducedon the bent brane is, to leading order in M/r , given by the Schwarzschild metric. In fact,the solution we have found here is exact up to the leading order in
M/r and all orders in1 /k − zr ∼ R AdS /r p , where r p is the proper radius.This is an interesting solution, which despite the parameter r , is a vacuum solution withno matter in the bulk. At small radius, the gravitational potential given by h t approachesa constant, which smoothly transitions into a 1 /k − r behavior far away as shown in figure 2.The width of the transition can be determined by taking the second derivative of the metricperturbation, which approaches a Gaussian of width ∼ /k − z close to r . Expressed interms of proper length, the width of the shadow is given by k − z × /k − z = k − − , which is theAdS-length. The transition becomes sharper as the AdS length becomes microscopic, butcontinues to be smooth. Another way to think of this spacetime is that it is a solution to five– 5 –imensional Einstein’s equations with a negative cosmological constant, with the boundarycondition that it induces Schwarzschild geometry on a brane that bends in response toinduced four dimensional matter. h t r r (a) h t r r (b) Figure 2 : Gravitational potential below the brane as a function the coordinate distance r .This approaches a constant at small r and falls off as /k − r at large r , with a transitionaround r = r . The width of the transition is proportional to /k − z . This can be seen inthe figures above where (b) is plotted for larger values of k − as compared to (a). We therefore have a metric below the brane that transitions smoothly from AdS at r (cid:28) r to that of a point source at r (cid:29) r with no matter localized at r = r . Thetransition, which takes place in vacuum, is made possible by five dimensional gravitationalbackreaction of the spherical shell and the corresponding structure on the outside.This solution is also valid for a Randall-Sundrum braneworld, in which case reflectionsymmetry across the brane would imply the same vacuum solution on either side of thebrane, without any material structure to support the four dimensional spherical shell. Thebrane would bend simply due to matter placed on it. To summarize, we have generated a structure that extends outwards from the shell inthe fifth dimension and its endpoint on the dark bubble induces a macroscopic shell offour dimensional matter. We have performed the analysis in the limit where the shell ofmatter in the four dimensional braneworld is much bigger than its Schwarzschild radius r (cid:29) M , but we expect the shadow shell to be the structure that persists to all orders.The matter source on the brane causes it to bend, which in turn sources the geometrybelow the brane (inside of the dark bubble), given in equations (2.9), (2.11) and (2.12).Pictorially one can imagine how the two pieces in figure 1a and figure 1b can be stitchedtogether. Whether we come from above or from below, the induced metric on the darkbubble will be Schwarzschild (at linear order) and all junction conditions will be satisfied.It is worth stressing here again that the two different choices of gauges on the two sidesof he dark bubble make the analysis simpler in this construction. While above the brane,where there is a source, we specifically worked with the straight gauge, it was convenient– 6 –o choose bent gauge in the construction of the five dimensional bulk spacetime belowthe brane, where there is no source. These two different gauges are useful to have, for afully consistent five dimensional picture on either side of the brane while producing thesame effective four dimensional gravity on the brane. This makes the analysis of junctionconditions across the brane extremely interesting. In [8] we discussed the computation ofthe four dimensional gravitational propagator in momentum space using straight gauge,and argued that in order to achieve the correct behavior we needed modes in the formof the Bessel function K on one side while a specific combination of Bessel functions K and I on the other side. This combination was responsible for a purely non-normalizablebehaviour of the modes on our shellworld. Furthermore, we also mentioned that in the caseof a bent gauge, to reproduce the same four dimensional gravity on the braneworld, weneed only K on both sides of the brane.Note that while considering the metric below the brane in the present work, we actuallyonly considered pure K . Therefore our present construction relying on two different choicesof gauges on two sides of the shell is interesting from the perspective of mode mixing. It ispossible that one could have a bit of I modifying higher order corrections of the metricoutside of the dark bubble, and possibly on the braneworld as well. However, we leave adetailed analysis of this for a future work. If we want to consider black holes on our dark bubbles, we must go beyond linearizedgravity. In the preceding section, we discussed a solution to leading order in
M/r and toall orders in 1 /k zr ∼ R AdS /r p . The question is whether it is possible to find this solutionto all orders in M/r . We will try to answer this question in this section.Let us first discuss the global properties of the spacetime below the brane. The firstorder expression is valid as long as
M/r is small. Is it possible to move away from the braneon the inside (small z ) and pass through r = 0 using just the first order corrections? To seewhether this is possible, let us note that the proper Schwarzschild radius on the brane is ∼ kM z . The Schwarzschild radius is much larger than the AdS radius, so k M z (cid:29) kz (cid:29) k M z (cid:29)
1, which implies that M (cid:28) /k . The proper Schwarzschild radius awayfrom the brane (in units of AdS radius) at some z is kM z . It turns out that the first ordercorrections obtained in [8] are finite when r = 0, and proportional to k M z . For k M z (cid:28)
M/r , we need that z is small enough.Hence, the metric goes to the AdS metric for small enough z , which is to be expected sincethis is a mode designed to vanish as z goes to zero. Therefore, if the spacetime below thebrane were to have a horizon, it will be capped off and contained above some finite andsmall z .It is furthermore possible to systematically calculate corrections to all orders in 1 /k zr ,order by order in χ/r . Note that the orders are coupled and care is needed when truncatingthe expansions. This can be done in Gauss normal coordinates, where the metric can be– 7 –ritten as d s = d z k z + k z (cid:0) − g tt d t + g rr d r + g θθ r dΩ (cid:1) . (3.1)To linear order in χ/r and quadratic order in 1 /k zr , Einstein’s equations with a negativecosmological constant give g tt = 1 − χr , g rr = 1 + χr (cid:18) − k r z (cid:19) , g θθ = 1 + χr (cid:18) k r z (cid:19) . (3.2)At leading (zeroth) order in 1 /k rz , this leads to the familiar expression from [8]d s = d z k z + k z (cid:20) − (cid:18) − χr (cid:19) d t + (cid:18) χr (cid:19) d r + (cid:18) χr (cid:19) r dΩ (cid:21) . (3.3)To second order in 1 /k rz and second order in χ/r , we get g tt = 1 − χr + χ r (cid:18) − c + 78 k r z (cid:19) ,g rr = 1 + χr (cid:18) − k r z (cid:19) + χ r (cid:18) − c − c + 2 + 3 c k r z (cid:19) ,g θθ = 1 + χr (cid:18) k r z (cid:19) + χ r (cid:16) c + c k r z (cid:17) , (3.4)with a number of free parameters c i . We can continue to solve Einstein’s equationsperturbatively to higher orders in χ/r and 1 /k zr , with more free parameters. However,in practice, we are mainly interested in the metric to all orders in χ/r , and leading orderin 1 /k rz . To find this, it is convenient to choose a gauge where the determinant of themetric is independent of the perturbations at leading order in 1 /k rz . Furthermore, we canimpose that there are no corrections beyond linear order in the angular part of the metric.Remarkably, we then find that the perturbative solution can be resummed at zeroth orderin 1 /k rz to give a metric that is an exact solution to all orders in χ/r d s = k z (cid:20) − − χ/r (1 + χ/r ) d t + 1(1 − χ/r ) d r + (cid:18) χr (cid:19) r dΩ (cid:21) + d z k z . (3.5)A brane embedded in this geometry at z ( r ) = z (cid:114) χr , (3.6)has an induced metric that is exactly Schwarzschild (to all orders in χ/r and zeroth orderin 1 /k rz ) d s = k z (cid:20) − (cid:18) − Mρ (cid:19) d t + d r − M/ρ + ρ dΩ (cid:21) , (3.7)where ρ = r + χ is the new radial coordinate. We also note that M = 3 χ/ z / (cid:0) k z (cid:1) is sub-leading– 8 –orizon k + k − Figure 3 : A dark bubble with a black hole with its interior shown in dark gray. The horizonextends below the brane, into the dark bubble and is smoothly capped off at small z . and does not contribute at this order. We can also resum the perturbation series at thenext order (linear) in 1 /k rz , to obtain an exact solution to all orders in χ/r given by g tt = 1 − χ/r (1 + χ/r ) + 18 k r z · χ r · (1 − χ/r ) (7 − χ/r )(1 + χ/r ) ,g rr = 1(1 − χ/r ) − k r z · χr · − χ/r − χ /r (1 − χ/r ) (1 + χ/r ) ,g θθ = (cid:18) χr (cid:19) + 18 k r z · χr · (1 − χ/r ) (4 + χ/r )(1 + χ/r ) , (3.8)which gives an exact (in χ/r ) embedding of the Schwarzschild brane at z ( r ) = z (cid:114) χr (cid:18) − k r z · χ r · − χ/r (1 + χ/r ) (cid:19) . (3.9)The induced metric is Ricci flat and represents a Schwarzschild geometry to quadratic orderin 1 /k rz and all orders in χ/r . A coordinate transformation can be performed to changeto Schwarschild coordinates as in equation (3.7), but we will not do that here.The metric obtained in equation (3.8) is quite remarkable in that it has a coordinatesingularity at r = 2 χ , but has no curvature singularities. Moreover, there is no source belowthe brane at r = 0. The horizon at r = 2 χ is therefore a shadow horizon and arises dueto the bending of the brane that gets capped off at a finite but small z . This is shownin figure 3. This is a remarkable geometry, and as discussed in section 2, can representblack holes on our dark bubbles as well as braneworld black holes for a Randall-Sundrumgeometry. This could be relevant for the old problem of finding an explicit metric of afour dimensional braneworld black hole mentioned in, e.g., [17], and generalizing recentresults for three dimensional BTZ black holes in, e.g., [18]. Let us reiterate that for aRandall-Sundrum braneworld, the brane can bend in response to matter on the brane andthis is the geometry on either side of the brane.– 9 –he metric above, leads to the intriguing geometry sketched in figure 3. The braneonly bends gently before entering through the horizon, with z increasing by a factor ofonly (cid:112) /
2. Inside of the horizon, however, the metric diverges towards r = 0 where thesingularity sits as z → ∞ . Note how this is consistent with the fact that our metric wasconstructed, in the Poincare patch corresponding to the limit of a large dark bubble, underthe assumption that there were no sources. The horizon that we find requires a singularitybut it sits at infinity. What would an observer on the dark bubble observe entering throughthe horizon? Inside of a Schwarzschild black hole, using Schwarzschild coordinates, youfind a time dependent geometry that contracts towards the future singularity in a finitetime. r is time, while t is an infinite spacelike coordinate. Regardless of when you enterthe black hole the same story will play out. The only difference is where in the spacialcoordinate t that you enter. The dark bubble adds another twist to the story. While theobservations within the four dimensional world are identical, there is also a five dimensionalinterpretation. What happens when you enter through the horizon is that you step intoan elevator that brings you up in z so that you hit the singularity at infinity in a finitetime. At what ever time that you enter, the elevator will be ready to bring you along on itsjourney. In this paper we have demonstrated that our model proposed in [4], which can incorporatedark energy into string theory, is also capable of describing non-trivial gravitational phe-nomena such as black holes. In the process, we have also discussed another very interestingtoy example of a thin shell of matter in four dimensions and its evolution in the fifthdimension leading to two completely different bulk extensions above and below our darkbubble. The two different bulk extensions were necessitated by the fact that the fivedimensional spacetime above the dark bubble is sourced by physical strings attached toit, while there is no source below the dark bubble. Our challenge was to combine thesetwo five dimensional pictures in such a way that they reproduce the same gravitationaldescription on our shellworld. In particular, if we look from above, the end points of thestrings result in an effective Schwarzschild geometry on the dark bubble. Therefore, it wasnatural to expect that the same geometry should be induced on the brane also from thefive dimensional bulk geometry inside the dark bubble. In this work we showed that thiscan actually be achieved by imposing different boundary conditions from either side of thedark bubble.We then proceed further with the vacuum solution below the brane and add perturbativecorrections in the form of expansions in
M/r and R AdS /r p . We are able to obtain a solution,exact in all orders in M/r and at linear order in R AdS /r p , which induces a black holesolution on our dark bubble. Although it is immensely encouraging to find such a blackhole solution, it is far from clear whether such a configuration yields a realistic outcome ofgravitational collapse. While the solution formally exists, as we have shown, we believe thatthere are new possibilities that distinguish our model from Randall-Sundrum braneworlds.– 10 –n particular, there is the intriguing possibility of realizing the black shell alternative to ablack hole proposed in [19, 20]. We will address this question in an upcoming work. Acknowledgements
SG would like to thank Uppsala University for their kind hospitality where this workwas completed. The work of SB is supported by the Alexander von Humboldt postdoc-toral fellowship. SG is partially supported by the INFN and the MIUR-PRIN contract2017CC72MK003AZ.
References [1] U. H. Danielsson and T. Van Riet,
What if string theory has no de Sitter vacua? , Int. J. Mod.Phys.
D27 (2018), no. 12, 1830007 [ ].[2] G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa,
De Sitter Space and the Swampland , .[3] E. Palti, The Swampland: Introduction and Review , Fortsch. Phys. (2019), no. 6, 1900037[ ].[4] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri and M. Schillo, Emergent de Sitter Cosmologyfrom Decaying Anti–de Sitter Space , Phys. Rev. Lett. (2018), no. 26, 261301 [ ].[5] L. Randall and R. Sundrum,
A Large mass hierarchy from a small extra dimension , Phys. Rev.Lett. (1999) 3370–3373 [ hep-ph/9905221 ].[6] L. Randall and R. Sundrum, An Alternative to compactification , Phys. Rev. Lett. (1999)4690–4693 [ hep-th/9906064 ].[7] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri and M. Schillo, de Sitter Cosmology on anexpanding bubble , JHEP (2019) 164 [ ].[8] S. Banerjee, U. Danielsson and S. Giri, Dark bubbles: decorating the wall , JHEP (2020) 085[ ].[9] S. Banerjee, U. Danielsson and S. Giri, Bubble needs strings , .[10] I. Koga and Y. Ookouchi, Catalytic Creation of Baby Bubble Universe with Small PositiveCosmological Constant , JHEP (2019) 281 [ ].[11] I. Basile and S. Lanza, de Sitter in non-supersymmetric string theories: no-go theorems andbrane-worlds , .[12] G. Dibitetto, N. Petri and M. Schillo, Nothing really matters , JHEP (2020) 040[ ].[13] H. C. Rosu, S. C. Mancas and C. C. Hsieh, Superfluid Rayleigh-Plesset extension of FLRWcosmology , .[14] S. B. Giddings, E. Katz and L. Randall, Linearized gravity in brane backgrounds , JHEP (2000) 023 [ hep-th/0002091 ]. – 11 –
15] A. Padilla,
Infra-red modification of gravity from asymmetric branes , Class. Quant. Grav. (2005), no. 6, 1087–1104 [ hep-th/0410033 ].[16] A. Chamblin, S. W. Hawking and H. S. Reall, Brane world black holes , Phys. Rev. D (2000)065007 [ hep-th/9909205 ].[17] R. Emparan, G. T. Horowitz and R. C. Myers, Exact description of black holes on branes ,Journal of High Energy Physics (Jan, 2000) 007–007.[18] R. Emparan, A. M. Frassino and B. Way,
Quantum BTZ black hole , Journal of High EnergyPhysics (Nov, 2020).[19] U. H. Danielsson, G. Dibitetto and S. Giri,
Black holes as bubbles of AdS , JHEP (2017) 171[ ].[20] U. Danielsson and S. Giri, Observational signatures from horizonless black shells imitatingrotating black holes , JHEP (2018) 070 [ ].].