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Physics
Letters
B
www.elsevier.com/locate/physletb
Black
hole
as
a
wormhole
factory
Sung-Won Kim
a,
Mu-In Park
b,
∗
aDepartment of Science Education, Ewha Womans University, Seoul, 120-750, Republic of Korea bResearch Institute for Basic Science, Sogang University, Seoul, 121-742, Republic of Korea
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Article history:
Received22September2015
Receivedinrevisedform15October2015 Accepted16October2015
Availableonline20October2015 Editor:M.Cvetiˇc
There havebeen lotsofdebates about thefinal fateofanevaporatingblack holeand the singularity hiddenbyaneventhorizoninquantumgravity.However,ongeneralgrounds,onemayarguethatablack hole stopsradiationatthe Planckmass (¯hc/G)1/2∼10−5g,where theradiatedenergy iscomparable to theblackhole’smass.Andalso, ithasbeenargued thatthere wouldbeawormhole-like structure, knownas“spacetime foam”,duetolargefluctuationsbelowthePlancklength(hG¯ /c3)1/2∼10−33 cm.In thispaper,asanexplicitexample,weconsideranexactclassicalsolutionwhichrepresentsnicelythose two properties in arecently proposed quantum gravity model basedon different scaling dimensions between spaceandtime coordinates.The solution,called “BlackWormhole”,consistsoftwo different states,dependingonitsmassparameterM andanIRparameter
ω
:Fortheblackholestate(withωM
2> 1/2),anon-traversable wormholeoccupiestheinterior regionoftheblack holearound thesingularity attheorigin,whereasforthewormholestate(withωM
2<1/2),theinteriorwormholeisexposedto an outsideobserver as theblack hole horizonis disappearing from evaporation. The blackhole state becomes thermodynamically stable as it approaches the merging point where the interior wormhole throat and theblackhole horizonmerges,and theHawking temperaturevanishesattheexact merge point(withωM
2=1/2).Thissolutionsuggeststhe“GeneralizedCosmicCensorship”bytheexistenceof awormhole-like structurewhichprotectsthe nakedsingularityeven afterthe blackholeevaporation. Onecouldunderstandthewould-bewormholeinsidetheblackholehorizonastheresult ofmicroscopic wormholescreatedby“negative”energyquantawhichhaveenteredtheblackholehorizoninHawking radiation process; thequantumblackholecouldbeawormholefactory! It is foundthat thisspeculative picturemaybeconsistentwiththerecent“ER=EPR”proposalforresolvingtheblackholeentanglement debates.©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
It is widely accepted that general relativity (GR) would not be appropriate fordescribing the small scale structure of space-time.Forexample,GR,whencombinedwithquantummechanics, provides a length scale lP
= (¯
hG/
c3)
1/2∼
10−33cm, which mayprovideanabsolutelimitationforthemeasurementsofspacetime distances[1].Actually,thisisthelengthscale onwhichquantum fluctuationsofthespacetime areexpectedtobeoforderofunity. Ontheother hand,thesingularity theorem,statingthenecessary existenceofsingularities,wheretheclassicalconceptofspaceand time breaks down, atcertain spacetime domains withsome rea-sonableassumptions inGR [2],maybe regardedasan indication oftheincompletenessofGR.
*
Correspondingauthor.E-mail addresses:sungwon@ewha.ac.kr(S.-W. Kim),muinpark@gmail.com (M.-I. Park).
Thesecircumstancesmayprovidestrongmotivation tofindthe quantum theory of gravity whichcan treat the above mentioned problems of GR. Actually,the necessity of quantizing the gravity has beenargued inorderto havea consistent interactionwitha quantum system[3].Moreover, ithas beenalsoshownthat even smallquantum gravitationaleffectsdramaticallychangethe char-acteristic features of a black hole so that it can emit radiation thoughthecausalstructure oftheclassicalgeometryisunchanged inthesemiclassicaltreatment[4].
However,astheblackholebecomessmallerandsmallerby los-ing its mass from emitting particles, the semiclassical treatment becomes inaccurate andone cannot ignore the back reactions of theemitted particlesonthemetricandthequantumfluctuations on the metric itself anymore. Actually, regarding the back reac-tioneffects,onecanarguethatablackholestopsradiation atthe PlanckmassmP
= (¯
hc/
G)
1/2∼
10−5g, wheretheradiatedenergyis comparableto theblackhole’smass,since ablackholecannot
http://dx.doi.org/10.1016/j.physletb.2015.10.045
0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
lution,asan explicitexample,which representsnicelythose two propertiesinarecently proposedquantum gravitymodel,known asHoˇrava gravity,based ondifferentscalingdimensionsbetween spaceandtimecoordinates.Thesolution,called“BlackWormhole”, consistsoftwo differentstates depending onits mass parameter
M andanIR(infrared)parameter
ω
:Fortheblackholestate(withω
M2>
1/2),a non-traversable wormholeoccupiesthe interior re-gionoftheblackholearoundthesingularityattheorigin,whereas forthe wormholestate (withω
M2<
1/2), theinteriorwormhole is exposed to an outside observer as the black hole horizon is disappearedfromevaporation. Theblackholestate becomes ther-modynamicallystableasitapproaches tothemerge pointwhere theinteriorwormholethroat andtheblack holehorizonmerges, andtheHawking temperaturevanishes atthe exactmerge point (withω
M2=
1/2).The solution suggests that, in quantum gravity, the ‘conven-tional’cosmiccensorshipcanbegeneralizedevenafterblackhole evaporationbyformingawormholethroataroundtheused-to-be singularity.InGR,blackholeandwormholearequitedistinct ob-jects due to their completely different causal structures. But the claimed “Generalized Cosmic Censorship” suggests that the end stateofa blackholeisa wormhole, nota naked singularity.This maycorrespondtoafoam-likenatureofspacetime atshortlength scales.Furthermore,onecouldunderstandthewould-bewormhole insidetheblackholehorizonastheresultsofmicroscopic worm-holes created by “negative” energy quanta which have entered theblackholehorizoninHawkingradiationprocessessothatthe quantumblackholecouldbeawormholefactory. Itisfoundthatthis speculativepicturemaybe consistent withtherecent“ER
=
EPR”proposalforresolvingtheblackholeentanglementdebates. To see how this picture can arise explicitly, we consider the Hoˇrava gravity which has been proposed as a four-dimensional, renormalizable, higher-derivative quantum gravity without ghost problems,byadopting differentscalingdimensionsforspace and time coordinates in UV (ultraviolet) energy regime,
[
t]
= −
1,[
x] = −
z withthedynamical criticalexponents z≥
3,“atthe ex-penseofLorentzinvariance”[7].Weformally definethequantum gravitybyapathintegralZ
=
[
D
gi j][D
Ni][D
N]
ei S/h¯ (1)withtheproposed action(z
=
3 is considered,for simplicity),up tothesurfaceterms,S
=
dtd3x√
g N 2κ
2 Ki jKi j− λ
K2−
κ
2 2ν
4Ci jC i j+
κ
2μ
2ν
2i jkR(3) i
∇
jR (3) k−
κ
2μ
2 8 R (3) i j R(3)i j+
κ
2μ
2 8(
3λ
−
1)
4λ
−
1 4(R
(3))
2− W
R(3)+
32 W
+
κ
2μ
2ω
8(
3λ
−
1)
R (3),
(2) 4andcouplingconstants,
κ
,
λ,
ν
,
μ
,
W
,
ω
.Thelasttermintheac-tion(2)representsa “soft”violation,withtheIR parameter
ω
,of the“detailedbalance”conditionin[7]andthismodifiestheIR be-haviorssothatNewtoniangravitylimitexists[8–10].Theproposedactionisnotthemostgeneralformfora power-countingrenormalizablegravity,compatiblewiththeassumed fo-liation preserving Diff but it is general enough to contain all the known GR solutions, and the qualitative features of the so-lutionsareexpectedtobesimilar[8–11].Here,originally,the non-relativistic higher-derivative deformations were introduced from the technical reason of the necessity of renormalizable interac-tionswithouttheghostproblemwhichexistsinrelativistic higher-derivativetheories[7].Butwefurtherremarkthat,whichhasnot been well emphasized before, the (UV) Lorentz violation might
have a more fundamental reason in our quantum gravity
set-up since thismay be consistent withthe existence ofthe abso-lute minimumlengthlP whichdoesnotdepend onthereference
frames,violatingtheusualrelativistic lengthcontraction.1
For the simplest case of static, i.e., non-rotating, uncharged blackholes,whereonly thelast threetermsintheaction(2)are relevant,theexactsolutionshavebeenfoundcompletelyfor arbi-traryvaluesofcouplingconstants,
λ,
W,and
ω
[8–11].However,for the present purpose we only consider a simple example of
λ
=
1,W
=
0, ds2= −
N(r)2c2dt2+
dr 2 f(r)
+
r 2dθ2+
sin2θ
dφ2 (6) with N2=
f=
1+
ω
r2−
r[
ω
2r3+
4ω
M]
(7) so that the standard Einstein–Hilbert action and the asymptoti-callyflat,SchwarzschildblackholesolutionarerecoveredintheIR limit,i.e., N2=
f=
1−
2M/r+
O(
r−4)
withc2=
κ
4μ
2ω
/32,
G=
κ
2c2/32
π
[8,9].Here M denotesthe‘(G/
c2)
×
ADM mass’andthe positiveIR parameterω
controlsthestrengthofhigher-derivative corrections so that the limitω
→ ∞
,μ
→
0 with ‘μ
2ω
=
fixed’ correspondstoGRlimit.2One remarkable property of the solution is that there is an inner horizon r− as well as the outer horizon r+, which solves
f
(
r±)
=
0,at r±=
M 1±
1
−
1 2ω
M2 (8)as the result of higher (spatial) derivatives (Fig. 1): The higher derivative terms act like some (non-relativistic) effective matters
1 ItcouldbealsopossiblethattheLorentzviolationoccursonly“dynamically”at somelevelinquantumgravity.Forsomeextensivediscussionsaboutthispossibility, see[12].
2 Theimportanceofthislimitinamoreextensivecontextwillbediscussed else-where[13].
Fig. 1. Plotsof f(r)inHoˇravagravityforvarying
M with
afixedω(left)andforvaryingωwithafixedM (right).
Inparticular,intheleftweconsiderM
=0,0.25,0.5,1 (toptobottomsolidcurves)withω=2,andintherightω=0.25,0.5,2 (toptobottomsolidcurves)withM
=1,incontrastwithSchwarzschildsolutionforM
=1 inGR (dottedcurve).in the conventional Einstein equation so that there is some “re-pulsive”interactionatshortdistances.Moreover,eventhoughthe metric converges to the Minkowski’s flat spacetime at the ori-gin r
=
0,i.e., N2=
f=
1−
2√
ω
Mr+
ω
r2+
O(
r7/2),
its deriva-tive is not continuous so that there isa curvature singularity atr
=
0, which may be captured by the singularity of R∼
r−3/2,RμναβRμνα
β
∼
r−3.So, eventhough the singularity atr=
0 isa time-likeline(i.e.,time-likesingularity)andismilderthanthatof Schwarzschildblack hole, RμναβRμναβ
∼
r−6 (and alsoReissner– Nordström’s RμναβRμναβ
∼
r−4) and surrounded by the (two) horizonsprovidedω
M2≥
12
,
(9)thismightindicatethattheproposed gravitydoesnotcompletely resolvethesingularityproblemofGRstill,classically.
Thiscircumstancelooksinconsistentwiththecosmology solu-tion,wheretheinitialsingularitydoesnotexistwhencethereexist thehigher-derivativeeffects,i.e.,non-flatuniverses[8,9].Moreover, theblack holesingularity becomesnaked for
ω
M2<
1/2 sothat the cosmic censorship might not work in this edge of solution space,evenifM ispositivedefinite.But according to recent Botta-Cantcheff–Grandi–Sturla (BGS)’s construction of wormholes,it seems that there isanother possi-bilityforresolvingtheunsatisfactorycircumstance[14].Whatthey foundwasthatthereexitsawormholesolutionalso,inadditionto thenakedsingularitysolutionfor
ω
M2<
1/2,withoutintroducing additional(exotic)mattersatthethroat.Theirobtainedwormhole solution ds2= −
N±(r)
2c2dt2+
dr 2 f±(r)
+
r 2dθ2+
sin2θdφ
2 (10) with N2±=
f±=
1+
ω
±r2−
r[
ω
2 ±r3+
4ω
±M±]
(11)is madeof two coordinate patches, each one covering the range
[
r0,
+∞)
in one universe and the two patches joining at the wormholethroatr0,whichisdefinedastheminimumoftheradial coordinater.Intheconventionalapproachfor(traversable)wormholes,there are basically two unsatisfactoryfeatures [15,16]. First,we do not knowmuchaboutthemechanismforawormholeformation.This isin contrasttothe blackhole case,where thegravitational col-lapseofordinary matters, like stars,can forma black hole,ifits mass is enough. Second, the usual steps of ‘constructing’ worm-holesaretooartificialassummarizedbythefollowingthreesteps: (1) Preparetwoorseveraluniverses;(2) Connecttheuniversesby
cut andpaste oftheir throats; (3) Put the needed“exotic” mat-ters, which violate the energy conditions, to the throats so that Einstein’sequationsaresatisfied.
However, inthenewapproach,theproblemofartificialityofa wormhole construction isavoided by observing that the solution is smoothly joined at thethroat, so that the additional compen-sating (singular or non-singular) matters are not needed, if the metric and its derivatives are continuous at the throat. But, if we further consider the reflection symmetric two universes, i.e., f+
(
r)
=
f−(
r),
N+2(
r)
=
N−2(
r),
withω
+=
ω
−≡
ω
,M+=
M−≡
M,theonlypossiblewayofsmoothpatchingatthethroatis3
df± dr
r0=
0 (12)andthethroatradiusisobtainedas r0
=
M 2ω
1/3.
(13)Here, itis importantto note thatthe throatis locatedalways inside theblackholehorizon,i.e., r−
<
r0<
r+ forω
M2>
1/2 so that it is unobservableto an outer observer,whereas the worm-holethroatemergesforω
M2<
1/2,insteadofanakedsingularity (Fig. 2).Fora fixedω
,thethroatradius,after emergingfromthe coincidencewiththeextremalblackholeradius,r0=
r+=
r−=
M,decreases monotonically as M decreases and finally vanishes for
M
=
0,i.e., Minkowskivacuum (Fig. 2,left). Thisis the situation thathasbeenassumedinBGS’spaper[14].But,ontheotherhand, for a fixed M and varyingω
, one finds that the throat radius increases again indefinitely asω
decreases. This means that the wormholesize can bequite largewhen thecouplingconstantω
, whichcouldflowunderrenormalizationgroup,becomessmallerat quantumgravity regime,likethePlancksizeblackholeor worm-hole (Fig. 2, right).4 In the actual quantum gravity process, like black hole evaporations, M, due to Hawking radiations, as well asω
,duetorenormalizationgroupflowwiththechangeofenergy scales,canvarysothatweneedtoconsidersomecombinationsof situationsoftheleft andrightinFig. 2,by extendingtheoriginal interpretation.3 InBGS’spaper[14],itisclaimedthatthejunctioncondition(12)isnotalways necessaryforλ=1 butmoregeneralclassofwormholesolutionscouldbepossible. But,thisisonlyforthecaseofsingular,δ-functiondiscontinuitiesintheequations ofmotions.Whereas,forthenon-singulardiscontinuitiesatthethroat,which can-not beproperlytreatedintheBGS’sanalysis,thecondition(12)isstillessentialfor ourwormholeconstruction.
4 Inorderthatthisfeaturecanbeseenexplicitly,itisimportanttoconsiderthe correctmassparameter
M
[17],ratherthananotherformofanintegrationconstant β=4ωG M/c2[9,11].Fig. 2. Blackholehorizons(top(r+)andbottom(r−)curves)andwormholethroat(middle(r0)curve)radiiforvarying
M with
afixedω=2 (left),andforvaryingωwith afixedM
=1 (right).Fig. 3. Ablackholepairwhichareconnectedbyawormholethroat
r
0insidethe blackholehorizonsr+
.Thearrows representthe time-likegeodesicsinside the horizons.So, the wormhole solution is obtained, without the prob-lem of artificiality, when the black hole horizon disappears for
ω
M2<
1/2.Whereas,thesolutionforω
M2>
1/2 withthethroat inside the horizon is not the “traversable” wormhole since the throatwill be still inside the blackhole horizon of the “mirror” black hole in another universe. This is similar to the Einstein– Rosen bridge [18] but the difference is that the throat may not coincidewiththeblackholehorizongenerallyinthepresentcase. Moreover, in the present case the throat is located at the fixed “time”r0 sothatanytime-liketrajectoriesshouldmeetthethroat ifexits(Fig. 3). Thismeansthattheblackholesolution(10),(11)withthe “time-like”throat for
ω
M2>
1/2 should be considered asaphysicallydistinctobjectfromtheblackholesolution(6),(7)having the (hidden) singularity at r
=
0. And in order to avoid a rather strange situation that the wormhole throat “suddenly” emerges fromtheextremal blackholewhich hasa singularityatr
=
0 still,it wouldbe natural toconsiderthetime-like throatin(10), (11)for
ω
M2>
1/2 as the “would-be” wormholethroat.In orderto be distinguished from theusual blackhole solution(6),(7), wemay callthe solution(10),(11)asthe “BlackWormhole” solution.
Now,we have a completely regular vacuum solution,without thecurvature singularities,which interpolates betweenthe black hole state for
ω
M2>
1/2 and wormhole state forω
M2<
1/2.This seems to support Wheeler’s foam picture of the quantum
spacetime inquantumgravitybutasa realstatic, notasa virtual time-dependent, wormhole. And in our quantum gravity theory, where the concept of horizon emerges at low energies, escap-ing thehorizons isnot impossibleforhighenergy particles with Lorentz-violatingdispersion relations.Butforlowenergypointof view,withoutprobingtheinteriorstructureofrealblackholes,the
observableconsequencesoftheblackwormholesolutionwouldbe expressedintheformofa“GeneralizedCosmicCensorship”,5 sug-gestingthat“thenakedsingularitydoesnotappearstillbyforming wormholesevenafterthehorizonsdisappearinquantumgravity” though, before evaporation, there would be no naked singularity bytheexistenceofhorizonsasusual.
Anotherimportantnewimplicationoftheblackwormhole so-lutiontolow energyobserversisthat therewouldbe transforma-tionsbetweenblackholesandwormholes,6 whichisknowntobe
impossiblein GR,dueto theno-go theoremfor topology change
[20].In other words, oncea (primordial) wormholeis formed in the quantum gravity regime, due to quantum fluctuation in the early universe,itmay evolveinto ablack holestate by the com-binationsofFig. 2(right)fromrenormalizationgroupflows toGR and Fig. 2(left) fromaccretion of matters. It would be interest-ing toseewhetherthiscouldbe amechanismfortheprimordial black holes andsupermassive black holes, which are believedto be formed very early inthe universeanddistinguished from the stellar-mass black holes which are (or believed to be) generated fromcollapsingstars.
Onthe other hand,accordingtotheblack wormholesolution, once a black hole is formed in our quantum gravity, it always has the would-be wormhole inside the horizon but this inside wormholeisexposedto outerobserverswhen thehorizon disap-pearsafterthecompleteevaporation.Butweknowthattheinside wormholeisabsentintheGRlimit,ascanbeseeninFig. 2(right).
Then, where does the inside wormhole come from in quantum
gravityregime?Thisisthequestionaboutthephysicalmechanism of a wormholeformation, which hasbeen lacking. It seems that the only possible answer to thisquestion could be found inthe Hawking radiation process, which involves virtual pairs of parti-clesneartheeventhorizon, oneofthe pairentersintothe black holewhiletheother escapes:The escapedparticleisobservedas arealparticlewithapositive energywithrespecttoanobserverat infinity andthen,the absorbedparticlemusthave anegative
en-ergyinorderthattheenergyisconserved[4].Thisimpliesthatthe negativeenergyparticlesthatfallintoblackholecouldbean
“ex-5 In[14],this notionwas consideredasa “complementary”betweenthe two mechanismstocensurethesingularities.
6 Haywardhassuggestedasimilarblackhole–wormholetransformationonthe groundsof“trapping”horizons whichmaydescribeblack holesand wormholes unifiedly[19].Butinhisframework,assumingthe
bifurcating black
holehorizons isessentialanditisnotclearhowtoextendhisframeworktoourcaseof non-bifurcatinghorizonsofextremalstatewherewormholethroatandextremalblack holehorizoncoincides.Moreover,hedidnotsuggestthatinteriorstructureofa blackholemightbechangedafterthetransformationfromawormholesothatthe singularitymightnotoccurintheblackholestate,also.Fig. 4. Plotsofthe
negative mass solution
off(r)inHoˇravagravityforvaryingM with
afixedω(left)andforvaryingωwithafixedM (right).
Inparticular,intheleftwe considerM
= −0.25,−0.5,−1 (bottomtotopsolidcurves)withω= −2,andintherightω= −0.25,−0.5,−2 (bottomtotopsolidcurves)withM
= −1,incontrastwith thenegativemassSchwarzschildsolutionforM
= −1 inGR(dottedcurve).otic matter”source fora wormholeformationinside thehorizon. Butonecaneasilydiscoverthatthiswouldbequiteimplausiblein GRsincenegativemasses(orenergies)repel(i.e.,producethe out-wardaccelerationsof),aspositivemassesattract(i.e.,producethe inward accelerationsof),all otherbodies regardlessoftheir (pos-itive ornegative) massesfrom the equivalenceprinciple. We have never observed the negative mass object yet and it could
pro-duce some strange phenomena, known as“the runawaymotion”
ofapairofpositiveandnegative massparticles,buttherepulsive natureofgravityfornegativemassesisanotherremarkable conse-quence ofGR[21]. Inthelanguage ofthe Newtonian approxima-tion,where thegravitationalpotential ofa sphericallysymmetric body is given by
ϕ
(
r)
= (
f(
r)
−
1)/2, that property is indicated bythefactthatdf/
dr,whichisrelatedtothe(radial)accelerationa
= −
dϕ
/
dr= −(
1/2)(df/
dr),
is always positive forthe potential ofapositivemassM>
0,whereasdf/
dr isalwaysnegativeforthe potentialofanegativemassM<
0.Thispropertyimpliesthat neg-ativemasscouldnotforma(stable)structure,naturally.Thiscould explainwhyitwouldnotbepossibletoformawormholeby col-lapsing of exotic matters, in particular, negative energy matters, includingthenegativeenergyparticlesthat fallintoblackholein GR,incontrasttothecaseofablackholeformation.Actually, pre-viouslythenegative energyparticlehasbeenthoughttoresultin justreducingtheblackhole massbysome compensationprocess withpositive energysources,whichmayexistinsidethehorizon, forthepositiveblackholemass.However,thesituationisquitedifferentinourquantumgravity context,wherethegravity becomes weakeratshortdistanceand changesits(attractiveorrepulsive)natureafterpassingthesurface ofdf
/
dr=
0, whichwe call the“zero-gravity surface”for conve-nience.Inotherwords,forthepositiveblackholemassM>
0,the gravity becomes repulsive inside the zero-gravity surface (Fig. 1, solidcurves). Thisnowimpliesthat eventhepositive masscould notformastructureinsidethezero-gravitysurface,naturally,in con-trastto theoutsidethezero-gravitysurface wherethe itsgravity isstillattractivesothatastablestructurecanbeformed,asinGR. In orderto see how this problemcould be resolved dramati-cally by thenegative massesin ourquantum gravity context,we now turn to consider the negativemass solution in Hoˇrava grav-ity,whichhasneverbeenstudiedintheliterature.Inordertofind a negative mass solution, one might first try to consider M<
0 forthesolution(7)butonecan easilyfindthatthiscouldnot be therightsolution: The solutionis notdefined belowa certain ra-dius, wherethe quantity inside the square rootbecome zero forω
>
0 or showsa differentasymptotic(r→ ∞
) behavior, i.e., notasymptoticallyflat,N2
=
f=
1+
2ω
r2+
2M/r+
O(
r−4),
forω
<
0. However, by notingthat the solutionof the metric ansatz (6) isuniqueuptothe(
±
)signinfrontofthesquare rootin(7) gener-ally,wefindthatthenegativemasssolutionasN2
=
f=
1+
ω
r2“+
”r[
ω
2r3+
4ω
M]
(14) forthedifferentsignofthesquarerootterm,comparedtothe so-lution(7).ThissolutionrecoverstheSchwarzschildsolutioninthe IR limit forω
<
0, as(7)doesfor M>
0 andω
>
0 (Fig. 4);forω
>
0,the solution(14)neither recovers theSchwarzschild solu-tion nor the metric is defined for the whole space region. [But in this case, in order that GR is recovered, we need to considerμ
2<
0 asinasymptoticallydeSitterspace[9]andwewilldiscuss aboutthisagainlater.]One canprovethat thenewsolution(14)doesnot have hori-zons,ascanbeeasilyobservedinFig. 4alsobutithasacurvature singularity atr
=
0, wherethemetric isfinite butdiscontinuous, withthesamedegreesofdivergencesasthe solution(7).So,this solutionitselfrepresentsanakedsingularityasthenegative mass Schwarzschild solution does in GR. Actually, this may be one of main obstacleforconsidering thenegative masssolutionasa vi-ablesolutioninGRandthisisnotmuchimprovedforthesolution(14)alone,inourHoˇravagravitycontext.
But,asinthe positive masscase, wehaveanother (exact) so-lutionofawormhole, forthereflectionsymmetrictwo universes, with thesame formula(13) forthe throatr0. Inother words,in Hoˇrava gravity, thereexists a regular, i.e.,singularityfree, negative masssolutionsothatitcould beviablebutonlyintheformofa wormholegeometry.Butherewewillnotconsideraboutthe neg-ativemass wormholesolutionfurther sinceits formation mecha-nismisunclearatpresent.
Rather,weconsiderthenegativemasssolution(14)inorderto seehowdoesthenegative masses,likethenegativeenergy parti-cles that fall intoblack holecould interactand forma structure, likethe(black)wormholewithapositivemass.Actually,the solu-tion(14)indicatesthatthegravityofanegativemassisrepulsive atlarge distancebutbecomes weaker,i.e., less-repulsive, atshort distance andmoreoverbecomes attractive insidethe zero-gravity surface ofdf
/
dr=
0.Thisimpliesthat thenegative massescould formastructurenaturally,atshortdistanceinsidethezero-gravity surface,incontrasttothecaseofpositivemasseswhichcouldnot formastructure insidetheir own zero-gravitysurface aswehave explained already. Thisis the remarkable consequenceof Hoˇrava gravity whichcould justifyour picturethat wormholesarecreated andsustainedbythecontinuousinflowofnegativeenergyparticlesvia theHawkingradiationprocess,whichisimpossibleinGR.Thisisthe mainclaimofthispaperandinamorecompactform,thiscanbe expressedas:Thequantumblackholecouldbeawormholefactory!whichhaveenteredtheblackholehorizonwithitsHawkingradiated,positive en-ergy,partnerquantaoutside thehorizon.Thisdescribesthequantumblackhole asafactoryofmicroscopicwormholeswhichwouldmergetoamacroscopicblack wormholesolution,inconformitywith“ER=EPR” proposal.
Interestingly,thismayprovideaphysicaloriginoftheEinstein– Rosenbridge in therecent “ER
=
EPR” proposal forresolving the issue of entanglement in a black hole spacetime [22] claiming that “the black hole and its Hawking radiation are entangled viaEinstein–Rosenbridges”(Fig. 5).Followingtheir proposal, one could understandthe would-be wormholes inside theblack hole horizonastheresults ofmicroscopic wormholescreatedby neg-ative energy quanta which have entered the black hole horizon butentangled withitsHawkingradiated, positive energy,partner quanta, which now lives in a single universe, widely separated fromits mother blackhole. Thismay be consistent withour re-flectionsymmetricconfiguration,butitisnotclearhowtoextend thispicturetomoregeneralconfigurationswhichcannot be inter-pretedasaconfiguration inasingle universe.Moreover, itis not cleareitherhowthethroatsofEinstein–Rosen’sbridgesevolveinto ourwould-bewormholeindetail.In conclusion, we have considered a vacuum and staticblack wormhole solutionwhichisregular, i.e.,singularity-free,and inter-polatesbetweentheblackholestatefor
ω
M2>
1/2 andwormhole stateforω
M2<
1/2 throughthecoincidencestateofanextremal blackholeanda (kind of) Einstein–Rosenbridge forω
M2=
1/2.From this, we have suggested the transformation between the
blackholeandwormholestates,anditsresultinggeneralized cos-miccensorship.Andfurthermore,wehavearguedthatthe would-bewormholes insidetheblack holehorizoncouldbe understood astheresultsofmicroscopicwormholescreatedby“negative” en-ergyquantawhichhaveenteredtheblackholehorizoninHawking radiationprocessessothatthequantumblackholecouldbea worm-holefactory.
But then, “Can the transformation really occur dynamically?” In orderto answer to thisquestion, we first consider the trans-formationfromablackholestatetoa wormholestate.Ofcourse, thereissome obstacle inGRcontext since theblackhole should pass7theextremalblackholetobecomeawormholestatebutthe extremalblackholehasvanishingHawkingradiationand temper-aturesothat itisbelievedasthestablegroundstateintheblack hole states.However recently, certain classical instability, known asAretakisinstability,forextremalblackholeshavebeenfoundin GRsothattheusualbeliefmaynotbequitecorrect[25].A heuris-ticargumentaboutthisinstabilitysuggeststhattheinnerhorizon instability of near-extremal black holes [26] might cause an in-stability ofthecoincided inner andouter horizons[27].Ifthisis
7 Recently,therehavebeenquiteactiveresearchesanddebatesaboutwhether anon-extremalblackholecan“jumpover”theextremality,bycapturinga parti-clewithappropriateparameters[23].Inparticular,aquantumjumportunneling seemstobeaquitepromisingmechanismforbreakingextremalblackholesinour quantumgravityset-up[24].
though linearlystable, exitsforourwormholestate toosothatit transformsto theextremalblackhole state,a seedoflarge black holes.
Finally,twofurtherremarksareinorder.
First,eventhoughwehaveobtainedthesolutioninaparticular quantum gravity model which is power-counting renormalizable without ghostproblems,the features ofthesmall scalestructure seemstobequitegenericifthevanishingHawkingtemperaturefora Planckmassblackhole,whichimplyingtheexistenceofthe worm-holethroatr0 satisfyingdf
/
dr|
r0=
0,isconsideredasaverifiable predictionthatanytheoryofquantumgravitymakes.Forexample, so-called “renormalizationgroupimprovedblackholespacetimes” would have the similar wormhole structure and our discussion maynotbelimitedtoHoˇravagravity[5].Butwecaneasilycheck thatitisnotapplicabletoKerrnorReissner–Nordströmblackholes inGR:Inthesecases,therearemoremetricfunctionsoradditional matterfieldsbutthereisnosolutionforthethroatwhereall the metricfunctionsorfieldsjoinsmoothly.9Second, wehave foundthat, inorder todescribe thenegative masssolution (14),we needto considerthecouplingconstant of
μ
2<
0 so that one can recover the well-defined GR parametersc2
,
G>
0 intheIR limit.Thisdoesnotaffectthecouplingsofthe second-derivative terms inthe action (2) (notethat we are con-sideringthecaseofW
=
0)butonlyaffectsother UV couplings [11], which differ from those for the positive mass solution (7). Thisindicatesthatthe(weak)equivalenceprinciple maybeviolated by UV effectsgenerally,depending onthesignofparticle’s mass. Itwouldbeachallengingproblemthattherunawaymotionsofa pairofpositiveandnegativemassparticlescouldbeavoidedfrom theinequivalent motionsofthepositive andnegative mass parti-clesinUV.Acknowledgements
This work was supported by Basic Science Research
Pro-gram through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education, Science and Technology
(2010-0013054)(S.-W.K),(2-2013-4569-001-1)(M.-I.P.).
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