TraversableWormholes

JuanMaldacenaInstituteforAdvancedStudy

October,2018

Introduction/Motivation

• Specialrelativityisbasedontheideaofamaximumspeedforpropagationofsignals.

• Ingeneralrelativitywehaveageneralcurvedgeometry.

• Istherealsoamaximalspeedforpropagationofsignals?

• Inprinciple,yes.Itisgivenbythelightconesinthatcurvedgeometry.

• Couldwehaveacurvedgeometrythatallowsa``shortcut’’?

• Couldquantumfluctuationsproducethesegeometries?

Therearecurvedgeometrieswherenaivelyfarawaypointsarerelativelycloseby.

AretheysolutionsofEinsteinequations?

FullSchwarzschildsolution

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

ER

Eddington,Lemaitre,Einstein,Rosen,Finkelstein,Kruskal

Vacuumsolution.Twoexteriors,sharingtheinterior.

Rightexterior

Leftexterior

singularity

Wormholeinterpretation.

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

L R

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Nontraversable

Nosignals

Nocausalityviolation

Fuller,Wheeler,Friedman,Schleich,Witt,Galloway,Wooglar

L R

Verylargedistance

Nosciencefictiontraversablewormholes

• Einsteinequations

• Butthestresstensorhastoobeysomeconstraints,itisnottotallyarbitrary.

• Inparticular,itisbelieveditshouldobeytheAANEC.

Rµ⌫ �1

2gµ⌫R = Tµ⌫

AANEC

• Achronal AverageNullEnergyCondition.

• Achronal (fastestline)

• Note:inQFT,wecanhaveinsomeregions.

Z 1

�1dx

�T�� � 0

FlatspaceQFT:Faulkner,Leigh,Parrikar,Wang.Hartman,Kundu,Tajdini

T�� < 0

Duetothispropertyofmatter,ambientcausalityispreserved.

ButwestillhavethefullSchwarzschildsolutiontointerpret.

Blackholesasquantumsystems

• AblackholeseenfromtheoutsidecanbedescribedasaquantumsystemwithSdegreesoffreedom(qubits).

=

Wormholeandentangledstates

Figure 2: Maximally extended Schwarzschild spacetime. There are two asymptotic regions.The blue spatial slice contains the Einstein-Rosen bridge connecting the two regions.

not in causal contact and information cannot be transmitted across the bridge. This can

easily seen from the Penrose diagram, and is consistent with the fact that entanglement

does not imply non-local signal propagation.

(a)(b)

Figure 3: (a) Another representation of the blue spatial slice of figure 2. It contains a neckconnecting two asymptotically flat regions. (b) Here we have two distant entangled blackholes in the same space. The horizons are identified as indicated. This is not an exactsolution of the equations but an approximate solution where we can ignore the small forcebetween the black holes.

All of this is well known, but what may be less familiar is a third interpretation of the

eternal Schwarzschild black hole. Instead of black holes on two disconnected sheets, we

can consider two very distant black holes in the same space. If the black holes were not

entangled we would not connect them by a Einstein-Rosen bridge. But if they are somehow

created at t = 0 in the entangled state (2.1), then the bridge between them represents the

entanglement. See figure 3(b). Of course, in this case, the dynamical decoupling is not

7

Connectedthroughtheinterior

Entangled

=W.IsraelJ.M.J.M.Susskind

|TFDi =X

n

e��En/2|ĒniL|EniR(inaparticularentangledstate)

• Isitdifficulttogettwoblackholesintothisstate?Orclosetothisstate?

• Isit”stable”inanysense?

• Orcanwedismissthesesolutionsasmathematicalcuriosities…?

• Cantheyteachusinterestinglessonsaboutquantumgravity?

FirstanalyzethisprobleminNearly-AdS2 gravity

N-AdS2xS2

horizon

Nearextremalblackholes

M � QM ⇠ Q

Itisacriticalsystem!

ThesurprisinglysimplegravitationaldynamicsofN-AdS2

AdS2

NAdS2=QFTonAdS2 +locationofboundary

(HLBdy ⇥Hbulk ⇥HRBdy)/SL(2, R)

Allgravitationaleffects

Propertimealongtheboundary=timeoftheasymptoticallyflatregion=timeofthequantumsystem

Addaconstantinteractionbetweenthedualquantumsystems

• Nearly-AdS2 gravity• Plusmatter• Plusboundaryconditionsconnectingthetwosides

• Thisgeneratesnegativenullenergyandallowsforaneternallytraversablewormhole

u isproperlengthalongtheboundary,orboundarytime.

Sint = µ

Zdu�L(u)�R(u)

NAdS2 gravity+Interaction

Interactions

Boundariesnowmove“straightup”

Signalscannowpropagatefromoneboundarytotheother.

Comment

• Westartedwithtwonearly-criticalsystems.• Addedarelevantinteraction.• Gotasystemwithgap.• Butwhosepropertiesaregovernedbythepropertiesofthecriticalsystem.(eg spectrumoflowenergyexcitationssetbythespectrumofanomalousdimensionsofthecriticalmodel.)

Wheredoesthatinteractioncomefrom?

Exchangeofbulkfieldscanleadtotheinteractionbetweenthequantumsystemsthedescribetheblackhole

Analogy:VanderWaalsinteractionTwoneutralatomsexchangingphotons.

Hint /~dL.~dRd3

dsmallenoughsothat1/dislargerthanthegapbetweenthegroundstateandthenextstates.

1/d

Entanglethetwoatoms.

d

Thisreasoninginspiredthefollowingsolution

Wormholesin4dimensions,intheStandardModel+gravity

AlexeyMilekhin Fedor Popov

Basedonworkwith:

RelatedtopreviousworkwithXiaoliang Qi

InspiredbyworkbyGao,Jafferis andWallon“Traversablewormholes”

DrawingbyJohnWheeler,1966

Chargewithoutcharge.Masswithoutmass

Spatialgeometry.Traversablewormhole

Recallaclassicresult

ItisaLongwormhole• Ittakeslongertogothroughthewormholethanthroughtheambientspace.

• NotpossibleinclassicalphysicsduetotheNullEnergyCondition.BecausetheNECistrue.

• NeedquantumeffectstoviolatetheANEC.Casimirenergy.

• Canwedoitinacontrollableway?

E / � 1L

NegativeCasimirenergy

Quantumeffect

T++ < 0

Thenullenergyconditiondoesnotholdfornulllinesthatarenotachronal!

time

Circle

Eg.Twospacetime dimensions

NegativenullenergyinQFT

Somenecessaryelements

• WeneedsomethinglookinglikeacircletohavenegativeCasimirenergy.

• Largenumberofbulkfieldstoenhancethesizeofquantumeffects.

• Wewillshowhowtoassembletheseelementsinafewsteps.

Thetheory

Einstein+U(1)gaugefield+masslesschargedfermion

CouldbetheStandardModelatverysmalldistances,withthefermionseffectivelymassless.TheU(1)isthehypercharge.SU(3)xSU(2)xU(1).

S =

Zd

4x

M

2plR�

1

g

2F

2 + i ̄ 6 D �

Thefirstsolution:ExtremalblackholeMagneticchargeq

AdS2 ⇥ S2

lPlanck = 1

Z

S2F = q = integer

rere ⇠ q

βisthe‘’length”ofthethroat.Redshiftfactorbetweenthetopandthebottom

horizon

AdS2 ⇥ S2

Verysmall

lPlanck = 1

re

M = re + r3eT

2 = re +r3e�2

Thenextsolution:NearExtremalblackholere ⇠ q

Motionofchargedfermions

• Magneticfieldonthesphere.• ThereisaLandaulevelwithpreciselyzeroenergy.• Orbitalandmagneticdipoleenergiespreciselycancel.

BMasslessfermionsà U(1)chiralsymmetry

4danomalyà 2danomalyà thereshouldbemasslessfermionsin2d.

(HereweviewFasnon-dynamical).

Motionofchargedfermions• Degeneracy=q=fluxofthemagneticfieldonthesphere.Formaspinj,representationofSU(2),2j+1=q.

• Weeffectivelygetqmasslesstwodimensionalfermionsalongthetimeandradialdirection.

• Wecanthinkofeachofthemasfollowingamagneticfieldline.

q masslesstwodimensionalfields,alongfieldlines.

AdS2

ds2 =�dt2 + d�2

sin2 �ds2 = �(r2 � 1)dt2 + dr

2

(r2 � 1)Global Thermal/Rindler

NearlyAdS2

ds2 =�dt2 + d�2

sin2 �

Connectthemtoflatspace,sothattisanisometry.Theyacquirenon-zeroenergywhenthethroathasfinitelength

ds2 = �(r2 � 1)dt2 + dr2

(r2 � 1)Global Thermal/Rindler

M = re + r3eT

2 = re +r3e�2

Connectapairblackholes

connectandinglobalAdS2

Approximateconfiguration

Wedescribethesolutionbyjoiningthreeapproximatesolutions.Joinedviaoverlappingregionsofvalidity.

Onehaspositivemagneticcharge,theothernegative.

AdS2xS2

Extremalchargedblackholethroat

Flatspace+pointparticles

Fermiontrajectories

Positivemagneticcharge

Negativemagneticcharge

Chargedfermionmovesalongthemagneticfieldlines.Closedloop.

CasimirenergyAssume:“Lengthofthethroat”islargerthanthedistance.

L=timeittakestogothroughthethroatasmeasuredfromoutside

L>>d

Casimirenergyisoftheorderof

Lout

L

d

L � Lout

> d

FullenergyalsoneedstotakeintoAccounttheconformalanomalybecauseAdS2 hasawarpfactor.Thatjustchangesthenumericalfactor.

E / � qL+ d

⇠ � qL

Findingthesolution

Balancetheclassicalcurvature+gaugefieldenergyvstheCasimirenergy.

Nowthethroatisstabilized.Negativebindingenergy.

Verysmall.Onlylowenergywavescanexploreit

M � q = q3

L2� q

L,

@M

@L= 0 �! L ⇠ q2

Ebinding = M � q = �1

q= � 1

rs

SolveEinsteinequationsinthethroatregionwiththenegativequantumstresstensor=

Thisisnotyetasolutionintheoutsideregion:

Thetwoobjectsattractandwouldfallontoeachother

Addingrotation

dre ⇠ q

⌦ =

rred3

Keplerrotationfrequency

d � re

Throatisfragile• Rotationà radiationà effectivetemperature:T=Ω

• WeneedthatΩ issmallerthantheenergygapofthethroat

• Theconfigurationwillonlyliveforsometime,untiltheblackholesgetcloser..

• TheseissuescouldbeavoidedbygoingtoAdS4 …

⌦ ⌧ 1L

⌦ =

rred3

Keplerrotationfrequency

Somenecessaryinequalities

Fromstabilizedthroatsolution

BlackholescloseenoughtothatCasmirenergycomputationwascorrect.

Blackholesfarenoughsothattheyrotateslowlycomparedtotheenergygap.

Unruh-liketemperaturelessthanenergygapKeplerrotationfrequency

Theyarecompatible

Othereffectswecouldthinkoffarealsosmall:canallowsmalleccentricity,addelectromagneticandgravitationalradiation,etc.Hasafinitelifetime.

L ⇠ q2

d ⌧ L �! d ⌧ q2

rq

d3= ⌦ ⌧ 1

L�! q 53 ⌧ d

q53 ⌧ d ⌧ q2

Lengthscales

rs ⇠ q ⌧ d ⌧ L ⇠ q2 ⌧ ⌦�1

Sizeofeachblackholeandinverseofbindingenergyofthewormhole

Distancebetweenblackholes

”length”ofthroatortimeittakestogothroughthewormhole.

Inverseenergygap.Redshiftedenergyforexcitationsdeepinthewormhole.

Inverserotationfrequency

Finalsolution

Looksliketheexterioroftwonearextremalblackholes.Buttheyconnected.Butthereisnohorizon!.Zeroentropysolution.Ithasasmallbindingenergy.

Entropyandentanglement

• Totalspacetime hasnoentropyandnohorizon.

• Ifweonlylookatoneobjectà entanglemententropy=extremalblackholeentropy

• Wormhole=twoentangledblackholes

• TotalHamiltonian H = HL +HR +Hint

Generatedbyfermionsinexterior

• Ifweweretodisconnecttheblackholesintheexteriorà theinteriorwouldevolveintothegeometryoftheeternalnear-extremalblackhole.

Rotationà temperature

Temperaturedoesnotcreateparticlesinthethroat

T ⇠ ⌦ ⌧ Egap ⇠1

L

WormholeisthestablethermodynamicphaseforT<1/Q3

Two Black Holes : F = �TQ2

Wormhole : F = �Ebinding = �1

Q

Forthesolutionwedescribedsofar:Wormholeismetastable.

LengthLasdincreases

dd d

q2

q2 q5/2 d

L

Ec / +1

4

1

L� 1

L+ d

CasimircylinderConformalanomaly

Stopsbeingclassical

WormholesintheStandardModel

IfnatureisdescribedbytheStandardModelatshortdistancesanddissmallerthantheelectroweakscale,

Ifthestandardmodelisnotvalidà similaringredientsmightbepresentinthetruetheory.

Thatitcan exist,doesnotmeanthatitiseasily producedbysomenaturalorartificialprocess.

1 ⌧ q ⌧ 108

Distancedsmallerthanelectroweakscale.

Theyareconnectedthroughawormhole!

MuchsmallerthantheonesLIGOortheLHCcandetect!

Pairofentangled blackholes.

Conclusions

• WedisplayedasolutionofanEinsteinMaxwelltheorywithchargedfermions.

• Itisatraversablewormholeinfourdimensionsandwithnoexoticmatter.

• Itbalancesclassicalandquantumeffects.• Ithasanon-trivialspacetime topology,whichisforbiddenintheclassicaltheory.

• Canbeviewedasapairofentangledblackholes.• Butithasnohorizonandnoentropy.

Questions

• Ifwestartfromdisconnectednearextremalblackholes:Cantheybeconnectedquicklyenough?à topologychange.