JHEP04(2019)106 . 3 Springer spacetimes. April 11, 2019 April 16, 2019 4 : : January 7, 2019 : Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP04(2019)106 [email protected] c , . 3 and Julio Oliva a,b 1811.03497 The Authors. c AdS-CFT Correspondence, Classical Theories of Gravity

In this letter we point out the existence of solutions to General Relativity , [email protected] Max-Planck-Institut f¨urGravitationsphysik (Albert-Einstein-Institut), Am 1, M¨uhlenberg D-14476 Potsdam, Germany Departamento de Universidad F´ısica, de Concepci´on, Casilla 160-C, Concepci´on,Chile E-mail: Departamento de Ciencias, Facultad deUniversidad Padre Artes Adolfo Hurtado Ib´a˜nez,Avenida Liberales, 750, Vi˜nadel Mar, Chile b c a Open Access Article funded by SCOAP ArXiv ePrint: we find a cosmologyfor that late is and everywhere early regular,spherical times topology has matches and either the an metric one exponential Friedmann-Lemaˆıtre-Robertson-Walker with or scale two factor. bounces andKeywords: that We compute the dualdifferent energy results. momentum tensor We atthroat also each and boundary show that showing that theysignal that these to it are go vacuum yields indeed from wormholes one traversablethe can boundary wormholes by to have to the computing more include other, the rotation as than and seen time one charge. by it a When geodesic takes the observer. cosmological for We constant a generalize is positive light Abstract: with a negative cosmologicalas constant traversable wormholes. in four The latterAt dimensions, connect every which constant two contain value asymptotically of locally solitons the AdS as radial coordinate well the spacetime is a spacelike warped AdS Andr´esAnabal´on Four-dimensional traversable wormholes and bouncingcosmologies in vacuum JHEP04(2019)106 2 3 ]. From 5 7 ]. The motivation was the description of 1 ], uses asymptotically AdS gravity to describe 4 – 1 – 5 4 6 3 3 8 ]. Holography [ 6 3 , 2 1 8 Indeed, a major open problem in physics is whether wormholes can take place in Traversable wormholes are spacetimes that causally connect two far away regions by which would spoil the stability of the system. We circunvent this dilemma by using AdS a conformal field theorythat Einstein (and wormholes its with deformations). athe boundary holographic of In point positive this of curvaturenegative context, do curvature. view not a This it not is careful is because existcoupled the study problematic [ scalar to to shows fields the in define scalar the a curvature. dual field field Therefore, theory theory they are conformally on have an a effective manifold negative of squared mass, of an accepted piece ofwith knowledge this within letter. theoretical physics. This situation should change physically sensible and simple circumstances.gravity It in is four well-established dimensions that requires asymptoticallyto exotic flat produce matter a fields wormhole or [ to go beyond General Relativity particles as gravitational andunderstand electromagnetic that fields the without Einstein-RosenSchwarzschild bridge singularities. metric and is that Nowadays just the we a non-singular part bridge of to other themeans universe Kruskal is of extension non-travesable. a of throat. the Despite the general interest in their existence, they are still not part 1 Introduction Wormholes have a longoriginally and introduced rich by history Einstein in and physics Rosen and [ the pop culture. The idea was 9 Bouncing cosmologies 10 Discussion 6 Holographic renormalization 7 Holographic stability 8 The charged and spinning generalization 3 Wormhole-like slicing of AdS 4 The wormhole solution 5 Absence of closed timelike curves Contents 1 Introduction 2 The spacelike warped AdS JHEP04(2019)106 ]. 9 ]. As 8 boundary it 3 ]. Therefore, it 8 , (2.1) i 2 ) dt ) θ slicing to the sphere slicing can + sinh ( 3 du ]. When picking the AdS + ( 7 , ]. We use only the Einstein equations and 2 6 ] does not arise in our case. The wormhole 10 5 dθ + boundary. The wormhole spacetime arises thus 2 – 2 – 3 θdt 2 3 has a single boundary. Then we propose an ansatz to cosh 4 − h 2 4 λ superficially resembles a wormhole. However, the existence of = 3 3 can be written as 2 AdS λ ds in a smooth way and without introducing closed timelike curves [ by AdS 3 4 with radius The mathematics involved in our construction are fairly simple. We review some of Another major open problem in physics is to find a non-singular description of the 3 bouncing cosmology that can be de Sitter at2 late or early times. The spacelike warpedAdS AdS Later we study the hologrophic stabilityscalar of the field system. of We the show thatof putative a negatively CFT conformally curved coupled have boundaries perfectly consideredsolution well in is [ defined then dynamics embedded in andsolutions a the in general four unstability class dimensions. of This metricswe allow that us make contains to analogous all obtain its considerations Einstein charged black when and hole the spinning form. cosmological Finally, constant is positive, obtaining a construct a spacetime with a (warped)naturally. AdS It is then shown thatThe it time can that have either takes a a singlegive photon or an to two throats elegant go and argument from an one oncompute anti-throat. boundary the the to absence dual the of other energy closed is momentum time computed. tensor like We at curves then on each the boundary, spacetime. yielding different We results. constant value of theon radial an coordinate. integration constant The thatslicing boundary controls of can the be warping AdS at warpeda infinity. or globally Later not, we defined depending discuss changebe how of the used coordinates to from proof the that global AdS AdS cosmology of our Universe without the problem of thethe initial most singularity. interesting geometricalto ingredients in deform the AdS firstwe two shall sections. see First, our we Einstein show wormhole how is exactly a spacelike warped AdS spacetime at every a positive cosmological constant.is The crucial to step allow to for constructshow space this that, long anisotropies by sought at spacetime anof the the adequate bounce. spacetime election can of Notwithstandingde be these Sitter the chosen anisotropies spacetime. parameters to we in Thus, be our the exactly bouncing the metric, cosmologies everywhere the provide homogeneous late a and new evolution isotropic arena to explore the what is done in this paper. Big-Bang. A realization ofshown to this idea be is compatibleHowever, known with before as cosmological this bouncing data article, cosmologies, no andno ad-hoc simple which a matter example have viable fields been of or alternative a exotic for kinetic bouncing terms inflation cosmology [ [ was known with not introduce instabilities andalthough the negative, squared is always mass safe ofis in easy the this to conformally regard see [ coupled thatthat the scalar bulk the fields, solution surfaces presented of below constantis is radial a also coordinate smooth natural deformation are to in spacelike consider such warped wormholes a AdS where way [ the boundary itself is warped. This is exactly itself as the boundary of the wormhole. Indeed, in AdS, there are tachyonic masses that do JHEP04(2019)106 ], 12 (3.1) . (2.2)  i . (4.1) 2 2 ), known as ) ) = 0 and two 2 ) dt dt r 2.2 Einstein space. ) ) 2), is broken to dt θ θ 4 , ) ], and their black θ 8 [ 3 , SO(2 + sinh ( + sinh ( 3 , (4.3) + sinh ( 3 du du ( du − + ( 2 )( σ 2 + 3 r ν ) to standard global AdS with a ( 4 + 2 dθ f ν 3.1 submanifolds as follows + + + manifold it was pointed out that is . This is simply because the change of , (3.2) `mr 2 3
2 2 r dt + αβ + 1 2 dθ dθ 2 ∞ ) g 2 r θ 2 + r 3 + ) ) are connected has been remarked in [ = ` ). 2 in AdS 2 σ r − 4 , (4.2) 3.1 2.2 . The isometry of AdS dt – 3 – − θdt cosh( 3 = )
2 θ R 4 − ( h αβ 2 ∈ + 1 + (6
R , this is just an artifact of the coordinates. There is cosh ) 2 4 r r . There is also a pathological version of ( − + 1 cosh 2 ν 2  ∞ λ ` 2 2 3 σ σ − ` 4 r t, θ, u = ) as the boundary of an asymptotically AdS = 2 ) + 3 2 r r λ 4 ) are satisfied provided ` µ 2 ( ) = ) = 2.2 ν ) to global AdS stop existing as a by-product of the identification. g r r ( ( + 3.2 g + f ) in the warped metric: = 3.1 which does contain closed timelike curves. In this paper, we shall 2 ) R j 2 3 r , + 1 dr radius, i.e. ( dr 2 dx 2 ) is a smooth manifold, free of closed timelike curves. It is a Lorentzian 4 f i ` 4 ). While this slicing seems to have a wormhole throat at r 2 ` 4 σ dx 2.2 = 3.1 ij AdS β = γ = GL(2 2 dx = R α ds 3 × ], have been extensively studied and they arise as solutions of topologically massive is the dx ) ` 11 R αβ , 2 W AdS The fact that the two boundaries of ( g ds The Einstein equations ( It is our interest toHence, obtain ( it is natural to propose the following ansatz coordinates that maps ( In the next sectioninstead we of show how topology. is possible to have a wormhole4 by resorting to geometry The wormhole solution a well-known global change ofround coordinates sphere that at maps the ( boundary. where by performingpossible identifications to disconnect in the the two boundaries fixed- at where for the metric ( disconnected boundaries at 3 Wormhole-like slicing ofAs AdS is well known, it is possible to slice AdS holes [ gravity with graviton mass timelike warped AdS only focus on the physically relevant case ( SL(2 The spacetime ( version of the squashed three pseudosphere. Spacelike warped AdS where the coordinates satisfy ( JHEP04(2019)106 ) t, r (4.7) has no f 6 to have i ≤ 2 ` 2 σ m − 2 all curves intersect at 4) = 1. The plots are for m r − 3 . (4.4) ) as a wormhole is now σ m` − = 11 with two throats and an 2.1 2 4 ( σ σ . (4.6) 6 √  h . ) 2
σ r ) 1 r + 1) ( − 16 − 2 g 1 r 2 − never vanishes provided ) ( r . (4.5) r (12 2 dr 4 3 = 0. For a given ( f
` 3 f
2 m √ 6 3 −∞ 3 + 1 p = 0, the spacetime is everywhere constant 2 − coincides with the proper time of a geodesic 0 8 6 4 2 det < r ∞ 10 + 1) 2 | m 2 – 4 – + r 2 /  −∞ r 1 > r >
Z ( `m | 1 3 2 rm ` ∞ ` σ − `t/σ 2 -1 4) 2 ). The interpretation of ( coordinate. All the plots have r = − = 4 and r t 3.1 coordinate is 3 , σ σ 12 ∆ − there. . According to this observer, the time it takes for a light ray r θ σ − 96 ( -2 4 + , the wormhole goes from having a single throat for ` > σ > 24 1 = 12 = 0 = at both asymptotic regions. The crossing time is given by 6. The two throats must have a local maximum in between that we 4 versus the r 4 / 4 − are integration constants. We are interested in the case where µναβ ` σ > R 3 m σ = 4 that have a single throat (from down up) and ) r σ is independent of ( µναβ g f and ) R . Here we plot the dimensionless determinant of the spatial sections with constant, ( r ( σ f 2 and 1 as . As shown by figure The scaled timelike coordinate  = 3 = observer located at to go from oneasymptotically boundary AdS to the other is finite, which is expected since the spacetime is straightforward. two throats for call an anti-throat. It is possiblecurvature to and see coincides that with for ( The Kretschmann invariant is real zero. An straightforward analysis shows that Thus, for these ranges ofregular the and parameters, the the range metric of functions are the everywhere positive and σ anti-throat. As expected,r det is asymmetric unless Where Figure 1 det = JHEP04(2019)106 t ) r ( ) in f r . The ( . = 0 is 2 1 ξ ) f 2 dt ξ dτ − − 3 2 must come 2 1 ξ ξ , (5.1) < t + − 2 1 2 3 2 ξ )(  1+ < ξ ). +1 2 dr dτ 2 1 ξ dτ du , ξ  ( 1 ) ξ dt , (4.8) dτ r = 1 ( 3 and 0 ) = 0. Taking into account ) ζ f θ ∗ 2 √ z dt dτ − < − 2 and 1 r  ζ ]. We note that the coordinates ) sinh ( < ξ )( r √ dθ dτ 2 ( 13 I z f  . This is the region we have plotted. 2 + ) which does not represent a wormhole. ∆ − 2 r 1 ξ 4 ( r − ξ g + 1 2 − )( − 2 ∗  1 3 − r z = ). Outside of the AdS point the metric has two grows unboundedly. . As suggested in the plot, if one approaches the 2 dt dτ < 2 θ π − t  z 2 1  r – 5 – ,  dτ 2 du = 2 )( ), ∆ 2 ξ < ξ 1 2 )  = 0 = z AdS √ θ ) , ξ min I r r t 1 − ( ξ + f r ( 1 = 4, i.e. global AdS − ξ 2 2 ) sinh ( σ 0 and 0 ` r σ 4 = ( > f 1 2 z as a function of the parameters 0 ξ − ) = , ). t is the time it takes to a photon to go from one asymptotic region to the r ζ 2 ( ) t = 0 and 5.1 f θ − m 2 ξ − ) provide a global covering of the manifold. A closed timelike curve satisfies ) cosh ( r are everywhere positive functions, it is straightforward to see that ( + 6 4.1 g 2 1 g  ξ yields a good parametrization of the curve. If the curve is closed, . Crossing time ∆ < depicts the crossing time as a function of the parameters ( ) in ( = 2 and 0 τ 2 σ f t, u, θ, r where back to its original value.that Hence, there must be ain point contradiction where with ( This argument is a( slight generalization of the one in [ has no real zeroFigure provided 5 Absence of closed timelike curves term of its roots, namely with asymptotic regions and ∆ other, as seen by aboundaries geodesic of observer the at domain on the plot ( To make this integral, and plot it, it is conveniente to write the metric function Figure 2 red dot corresponds to has a global minimum at that point (∆ JHEP04(2019)106 . . ν rr m m , g N and µ (6.1) (6.2)  √ . The ) 2 r N µ σ ` γ h δ → − → − ( − ], is 0 will be = = . (6.4) m R m µν µ 15 ` 2 , g  2 +3 r < N 2 λ 2 + ) 14 = and ν ` 2 dt r µν )  h θ h changing − → − ) with + √ ij r x T 2.2 3 . + sinh ( d the case with K , du 0 ∂M ( µν Z h m 1 its Ricci curvature. κ σ + 4 r > , (6.3) ) . (7.2) R + µν θ ij ( K −
T 2 h H r = 0 . (7.1) − K ) − dθ φ √ ωt ] µν + 0 is the same than x , and is given by ( γ h −→∞ [ 3 2 + ij r ` 2 d h dt R – 6 – = lim 2 ku r < ) − 1 8 1 r θ ∂M ) E ( Z − h + 2 ( ij 1 κ φ T hypersurfaces and . The scalar field equation is −→∞ µν = cos(  D where we have used the standard holographic dictionary r + γ r G cosh with the rank of the gauge groups of the ABJ(M) theory, φ 2 `  / − 2 3 N 6 = ` N = lim 2 2 + m µν / T 1 ij − R k κ γ   2 π / g 2 1 12 ` 2 σκ is the extrinsic curvature of the boundary metric, − √ = ]. = µν x 2 j 4 κ 16 K ` d , dx i M with the level and Z πG dx k κ 1 E . The dual enery momentum tensor is 2 = 8 + 1 ij 3 − κ T σ ] = D The bulk solution is everywhere regular. Therefore, the dual energy momentum tensor g [ = I 2 The following ansatz separates the field equation 7 Holographic stability Let us now deal withfirst the thing one issue would of like the tofield check stability is in of what the three happens CFT with dimensions aare on when simple defined the conformally propagating in boundary coupled on terms scalar metric of this the geometry. metric All objects in this section see for instance [ is continuous along thedifferent energy boundary momentum of tensor is the a physicist’s spacetime. proof that the The boundary fact is not that connected. each boundary has a The wormhole spacetime is invariantHence, under the the energy-momentum combined tensor changes We for note that the factortheory in variables front of the energyto identify momentum tensor can be translated to field which yields The boundary metric is ν where is the induced metric onis the the fixed outward pointing normal,discussed below. where we Varying the assume action that gives the energy momentum tensor: We now pass to find what isThe the procedure dual is energy momentum as tensor follows. associated to The this action, spacetime. including boundary counterterms [ 6 Holographic renormalization JHEP04(2019)106 (8.7) (8.1) (7.4) (7.5) can be , 3  , (8.6) ) 2 p , (8.5) (  2 4 = 0 , (8.3) dp
t X p α
dτ µν , (8.4) + 4 2 + F y 4 ) µ 2 q 2 ξp u q 4 ( 4 A ∇ 1 y dq α + = 0 . (7.3) is  Y − + θ . (8.2) + 2  the scalar field will always 2 H dφ 3 ` nσ
− y ,
q . k 2 ` 2 3 dφ 2 2 p y p 2  2 ... ξ 2 4) σk p ) = ) = q ) + q 2 2 y αβ + 4, the wormholes are stable under q 2 q − ( ξ ( 2 + ξ 2 F σ t, u Qξpq p + 4 q θ ≥ 3 σ ( Y ξ 2 √ 2 ( p φ αβ y − σ − k 3 1 + 2 1 + F 2 ∆ σ + P 1 + σ e 2 Ay µν σ + + q k − g − 2 qA √ − φ . For 1 4 +
1 + 2 1 2  y + p σ − ) for large values of make the following change of coordinates dφ − dτ 2 – 7 – σ 2 − σ , 2 y θ · p σ ν √
p 7.1 4 and large enough ξq 4 F 2 + −  y q = 0 ), ∆ + p − ] family of spacetimes in four dimensions 2 µσ P ξpq σ e 1 ξ + A F y  + − 18 σ < + dτ αu 2 T  φ + Q = dθ κ Q 1 + dH 2
= p + 2 p 4  2 2 y t + ) = 2 H q ( − ∆ = p 2 ξ is then ( + 2 ξ σ H 2 2 µν B 2 2 2 T X , are satisfied provided ` g , dθ A H d n µ . This solution is known to contain all spinning black holes in four 2 Q 3 ` 2 1 + −  B 1 − rn ν 2 2 σ − − ∂ − κP − 4  ` ξ R 2 − + ) = ) = ) = = r ν 2 µν ( g B t, u Ap Y q X 1 1 2 ( µ κQ ]. ∂ − τ − = q 17 = ( 2 T µν µν Q = R F 2 ds the acceleration parameter to zero, where dimensions as special limits. To retrieve the wormhole we found that is necessary to set The Einstein-Maxwell equations with with the gauge field So far we havegeneralize studied the the spacetime simplestconsider to case the where introduce Plebanski-Demianski the [ charge wormhole and is spin. static. An It is educated natural guess to lead us to scalar field fluctuations offound the in CFT. [ More details of the stability of8 warped AdS The charged and spinning generalization where It can be readilydevelop an noticed unstability that due for to the complex ∆ The asymptotic form of The leading order of the field equation ( JHEP04(2019)106 (9.1) 6= 0 the , (8.8) T 2 T τ Q . Q (1) (9.2) 2 − ) O
+ dφ ) σ i θ 2 ) +  dφ 2 = 0. When . (8.9) ) α T θ 3 3 + sin ( Q 3 − mn dψ − 1 σ = 4 yields for large 2 − )( σ  + sin ( t ` σ otherwise the spacetime describe 2 ( 1 and = 1 the curvature is positive. It + = 0 limit which exactly coincides , α f = − dψ 1  3 α − + µt − = y controls the topology of the boundary. 3 + (   +  = 2 2 + 1 2 2 ), when  t 2 dθ ` n τ dθ ) , t 2 σ + − + ` 2 − 2 boundaries. The black hole flows into the bulk − nα – 8 – dφ dφ = exp( = = 2 ) provided 2 ) 1 . At every constant radial coordinate there is a t ) + (6 4 ξ ∞ = 0 limit is well defined provided a singular gauge θ θ 4 − y 4.1 t = 0 a singularity is developed. ). Therefore, the same analysis applies here regarding α = 2 2 r = )( r ` , cos ( cos ( 4.3 σ θ 4 y h 
)  t σ , ( ) = τ ` g t 2
− (  = sinh( +  f  2 2 ) y 2 α α t α = 0 and ( dt exp + is now r f 4 2 2 , ` 4 l 4 + 4 σ σ αβ − + + 1) and − g 2 2 2 2 3 ` 6 + t = coordinate. Only at − ( dτ ` 2 2 2 r σ = 2 l − n n ds . The change of coordinates 2 2 αβ = 6= 0 then the interpretation of the spacetime as a wormhole is less simple. The − 1 ) = ` − R 2 t α ( = 0 the boundary has no curvature and when = = ds g  If 2 1 y y ous section. 10 Discussion In this paper wesolutions have to constructed four dimensional the Einstein first gravity geometrically with non-trivial a negative family cosmological of constant. wormhole The which is just theThere metric Friedmann-Lemaˆıtre-Robertson-Walker is with a spherical straightforward topology. generalization of this cosmology along the lines of the previ- with exactly the same formregularity. than ( The number ofby bounces figure and anti-bounces the spacetime can have is described with 9 Bouncing cosmologies The existence of wormholes whenfor the bouncing cosmological cosmologies constant is when negative themetric, motivate cosmological us constant to is look positive. The relevant Einstein AdS solitons. metric is singular at black hole, which is regularthrough at the the with the wormhole with static wormhole is chargedtransformation and is the substracted from the gaugeFor field. can be seen that there are wormholes only when Then the metric and the gauge field have a smooth and the reparemeterization JHEP04(2019)106 3 r ]. It is 11 ]. It was recently ]. The holographic spacetimes with the 20 3 19 ], and the size of the throat increases with ) yields warped AdS black holes [ to eliminate the conical singularity at the 21 u 2.2 – 9 – . These boundary conditions have been used to ) yields black holes together with the flow of the 3 4.1 ]. ) have zeroes, it is possible to cut the spacetime at 23 4.3 ]. Our findings imply that such settings can also take place ) in ( . This procedure yields a soliton if the zero is of order one. r 22 ( u ∂ f ]. This geometry seems to be locally the same than the geometry of our wormhole. 24 The bouncing cosmology presented in the last section yields a smooth description of The introduction of identifications in ( It is worth noting that the “real-world” physical relevance of spacelike warped AdS When the function Wormhole geometries in four dimensional, asymptotically AdS spacetimes have re- We thank Laura Andrianopoli,Jean-Luc Lehners Sebastian and Bramberger, Marioby Elena Trigiante the for Caceres, Chilean discussions. Bianca FONDECYT[Newton-Picarte This Cerchiai, Grants [Grant work DPI20140053 No. and was DPI20140115] 1181047, supported andfoundation. the in 1170279, Alexander part von 1161418], Humboldt CONYCT-RCUK compatible with the data if the spherical Universe is large enough. Acknowledgments hole [ the evolution of thean Universe. standard homogeneous What and isexperimental isotropic remarkable Universe data there for favours is late a and that flat early it universe. times. is Hence, Nowadays, possible this the to cosmological recover model would only be likely that the same identificationwarping parameter in into ( the radial direction. stems from the fact that it arises is the near horizon geometry of the extremal Kerr black If the zero isspacetime. of The order interior asymptotic twocan region then yields be one in thought simply certain asboundary cases finds metrics an a another RG contain new asymptotic flow. (warped) vacuumholographically region AdS The describe of in soliton graphene general the [ relativity bulk when the conformal class of the the rotation in thein bulk vacuum. [ the first zerodegeneration and surface identify of the coordinate ceived large attention recentlydual due of to a highly holography, entangled see statehas of for two two instance non-interacting asymptotically CFTs [ AdS isfound an regions eternal that that black the hole, are which inclusionbulk causally which of disconnected causally an [ connects interaction the boundaries between [ the two CFTs opens a throat in the warping that is runningmetric can along be this either coordinate.like warped curves. or The We not. have asymptoticmore generalized The form general the wormhole geometry is of charged and traversable Plebanski-Demianski thestudied and shown spacetime. at constant- that free this it of new These is closed light. spacetimes a time should special now limit be of the hypersurfaces perpendicular to the radial coordinate are warped AdS JHEP04(2019)106 ]. , ] , ] JHEP SPIRE Int. J. ] JHEP , , , IN Phys. Rev. ][ , Annals Phys. , Commun. , (2008) 127 Class. 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