12
White Holes as Remnants: A Surprising Scenario for the End of a Black Hole Eugenio Bianchi * Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Marios Christodoulou † CPT, Aix-Marseille Universit´ e, Universit´ e de Toulon, CNRS, F-13288 Marseille, France Dept. of Physics, Southern University of Science and Technology, Shenzhen 518055, P. R. China Fabio D’Ambrosio ‡ and Carlo Rovelli § CPT, Aix-Marseille Universit´ e, Universit´ e de Toulon, CNRS, F-13288 Marseille, France. Hal M. Haggard Physics Program, Bard College, 30 Campus Road, Annondale-On-Hudson, NY 12504, USA, Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, CAN (Dated: March 20, 2018) Quantum tunneling of a black hole into a white hole provides a model for the full life cycle of a black hole. The white hole acts as a long-lived remnant, solving the black-hole information paradox. The remnant solution of the paradox has long been viewed with suspicion, mostly because remnants seemed to be such exotic objects. We point out that (i) established physics includes objects with precisely the required properties for remnants: white holes with small masses but large finite interiors; (ii) non-perturbative quantum-gravity indicates that a black hole tunnels precisely into such a white hole, at the end of its evaporation. We address the objections to the existence of white-hole remnants, discuss their stability, and show how the notions of entropy relevant in this context allow them to evade several no-go arguments. A black hole’s formation, evaporation, tunneling to a white hole, and final slow decay, form a unitary process that does not violate any known physics. I. INTRODUCTION The conventional description of black hole evaporation is based on quantum field theory on curved spacetime, with the back-reaction on the geometry taken into ac- count via a mean-field approximation [1]. The approxi- mation breaks down before evaporation brings the black hole mass down to the Planck mass (m Pl = p ~c/G ∼ the mass of a 1 2 -centimeter hair). To figure out what happens next we need quantum gravity. A quantum-gravitational process that disrupts a black hole was studied in [2–6]. It is a conventional quantum tunneling, where classical equations (here the Einstein equations) are violated for a brief interval. This alters the causal structure predicted by classical general relativity [8–23], by modifying the dynamics of the local apparent horizon. As a result, the apparent horizon fails to evolve into an event horizon. Crucially, the black hole does not just ‘disappear’: it tunnels into a white hole [24–27] (from the outside, an object very similar to a black hole), which can then leak out the information trapped inside. The likely end of a * [email protected] † [email protected] ‡ [email protected] § [email protected] [email protected] black hole is therefore not to suddenly pop out of ex- istence, but to tunnel to a white hole, which can then slowly emit whatever is inside and disappear, possibly only after a long time [28–41]. The tunneling probability may be small for a macro- scopic black hole, but becomes large toward the end of the evaporation. This is because it increases as the mass decreases. Specifically, it will be suppressed at most by the standard tunneling factor p ∼ e -S E /~ (1) where S E is the Euclidean action for the process. This can be estimated on dimensional grounds for a stationary black hole of mass m to be S E ∼ Gm 2 /c, giving p ∼ e -(m/m Pl ) 2 , (2) which becomes of order unity towards the end of the evaporation, when m → m Pl . A more detailed deriva- tion is in [5, 6]. As the black hole shrinks towards the end of its evaporation, the probability to tunnel into a white hole is no longer suppressed. The transition gives rise to a long-lived white hole with Planck size horizon and very large but finite interior. Remnants in the form of geometries with a small throat and a long tail were called “cornucopions” in [42] by Banks et.al. and stud- ied in [34, 43–45]. As far as we are aware, the connection to the conventional white holes of general relativity was never made. arXiv:1802.04264v2 [gr-qc] 17 Mar 2018

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  • White Holes as Remnants: A Surprising Scenario for the End of a Black Hole

    Eugenio Bianchi

    Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USADepartment of Physics, The Pennsylvania State University, University Park, PA 16802, USA

    Marios Christodoulou

    CPT, Aix-Marseille Universite, Universite de Toulon, CNRS, F-13288 Marseille, FranceDept. of Physics, Southern University of Science and Technology, Shenzhen 518055, P. R. China

    Fabio DAmbrosio and Carlo Rovelli

    CPT, Aix-Marseille Universite, Universite de Toulon, CNRS, F-13288 Marseille, France.

    Hal M. Haggard

    Physics Program, Bard College, 30 Campus Road, Annondale-On-Hudson, NY 12504, USA,Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, CAN

    (Dated: March 20, 2018)

    Quantum tunneling of a black hole into a white hole provides a model for the full life cycle ofa black hole. The white hole acts as a long-lived remnant, solving the black-hole informationparadox. The remnant solution of the paradox has long been viewed with suspicion, mostly becauseremnants seemed to be such exotic objects. We point out that (i) established physics includesobjects with precisely the required properties for remnants: white holes with small masses but largefinite interiors; (ii) non-perturbative quantum-gravity indicates that a black hole tunnels preciselyinto such a white hole, at the end of its evaporation. We address the objections to the existenceof white-hole remnants, discuss their stability, and show how the notions of entropy relevant inthis context allow them to evade several no-go arguments. A black holes formation, evaporation,tunneling to a white hole, and final slow decay, form a unitary process that does not violate anyknown physics.

    I. INTRODUCTION

    The conventional description of black hole evaporationis based on quantum field theory on curved spacetime,with the back-reaction on the geometry taken into ac-count via a mean-field approximation [1]. The approxi-mation breaks down before evaporation brings the blackhole mass down to the Planck mass (mPl=

    ~c/G the

    mass of a 12 -centimeter hair). To figure out what happensnext we need quantum gravity.

    A quantum-gravitational process that disrupts a blackhole was studied in [26]. It is a conventional quantumtunneling, where classical equations (here the Einsteinequations) are violated for a brief interval. This alters thecausal structure predicted by classical general relativity[823], by modifying the dynamics of the local apparenthorizon. As a result, the apparent horizon fails to evolveinto an event horizon.

    Crucially, the black hole does not just disappear: ittunnels into a white hole [2427] (from the outside, anobject very similar to a black hole), which can then leakout the information trapped inside. The likely end of a

    [email protected] [email protected] [email protected] [email protected] [email protected]

    black hole is therefore not to suddenly pop out of ex-istence, but to tunnel to a white hole, which can thenslowly emit whatever is inside and disappear, possiblyonly after a long time [2841].

    The tunneling probability may be small for a macro-scopic black hole, but becomes large toward the end ofthe evaporation. This is because it increases as the massdecreases. Specifically, it will be suppressed at most bythe standard tunneling factor

    p eSE/~ (1)

    where SE is the Euclidean action for the process. Thiscan be estimated on dimensional grounds for a stationaryblack hole of mass m to be SE Gm2/c, giving

    p e(m/mPl)2

    , (2)

    which becomes of order unity towards the end of theevaporation, when m mPl. A more detailed deriva-tion is in [5, 6]. As the black hole shrinks towards theend of its evaporation, the probability to tunnel into awhite hole is no longer suppressed. The transition givesrise to a long-lived white hole with Planck size horizonand very large but finite interior. Remnants in the formof geometries with a small throat and a long tail werecalled cornucopions in [42] by Banks et.al. and stud-ied in [34, 4345]. As far as we are aware, the connectionto the conventional white holes of general relativity wasnever made.

    arX

    iv:1

    802.

    0426

    4v2

    [gr

    -qc]

    17

    Mar

    201

    8

    mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]

  • 2

    This scenario offers a resolution of the information-lossparadox. Since there is an apparent horizon but no eventhorizon, a black hole can trap information for a long time,releasing it after the transition to white hole. If we have aquantum field evolving on a black hole background metricand we call S its (renormalized) entanglement entropyacross the horizon, then consistency requires the metricto satisfy non-trivial conditions:

    (a) The remnant has to store information with entropyS m2o/~ (we adopt units G=c=1, while keeping ~ ex-plicit), where mo is the initial mass of the hole, beforeevaporation [46]. This is needed to purify Hawking radi-ation.

    (b) Because of its small mass, the remnant can releasethe inside information only slowlyhence it must belong-lived. Unitarity and energy considerations imposethat its lifetime be equal to or larger than R m4o/~3/2[32, 47].

    (c) The metric has to be stable under perturbations, soas to guarantee that information can be released [4, 4850].

    In this paper we show that under simple assumptionsthe effective metric that describes standard black holeevaporation followed by a transition to a Planck-masswhite hole satisfies precisely these conditions. This resultshows that this scenario is consistent with known physicsand does not violate unitarity.

    One reason this scenario may not have been recognisedearlier is because of some prejudices (including againstwhite holes), which we discuss below. But the scenariopresented here turns out to be consistent with generalexpectations that are both in the AdS/CFT community(see for instance [51, 52]) and in the quantum gravitycommunity (see for instance the paradigm [14]).

    II. THE INTERNAL GEOMETRY BEFOREQUANTUM GRAVITY BECOMES RELEVANT

    We begin by studying the geometry before any quan-tum gravitational effect becomes relevant. The standardclassical conformal diagram of a black hole formed bycollapsing matter is depicted in Figure 1, for the case ofspherical symmetry.

    Classical general relativity becomes insufficient wheneither (a) curvature becomes sufficiently large, or (b) suf-ficient time has ellapsed. The two corresponding regions,A and B, where we expect classical general relativity tofail are depicted in the figure.

    Consider the geometry before these regions, namelyon a Cauchy surface that crosses the horizon at some(advanced) time v after the collapse. See Figure 1. Weare interested in particular in the geometry of the portioni of which is inside the horizon. Lack of focus on thisinterior geometry is, in our opinion, one of the sources ofthe current confusion. Notice that we are here fully inthe expected domain of validity of established physics.

    The interior Cauchy surface can be conveniently fixed

    as follows. First, observe that a (2d, spacelike) sphereS in (4d) Minkowski space determines a preferred (3d)ball i bounded by S: the one sitting on the same linearsubspacesimultaneity surfaceas S; or, equivalently,the one with maximum volume. (Deformations from lin-earity in Minkowski space decrease the volume). The firstcharacterisationlinearitymakes no sense on a curvedspace, but the secondextremized volumedoes. Fol-lowing [53], we use this characterization to fix i, which,incidentally, provides an invariant definition of the Vol-ume inside S. Large interior volumes and their possiblerole in the information paradox have also been consideredin [5458].

    The interior is essentially a very long tube. As timepasses, the radius of the tube shrinks, while its lengthincreases, see Figure 2.

    It is shown in [53, 5961], that for large time v thevolume of i is proportional to the time from collapse:

    V 3

    3 m2o v. (3)

    Christodoulou and De Lorenzo have shown [62] that thispicture is not changed by Hawking evaporation: towardthe end of the evaporation the area of the (apparent)horizon of the black hole has shrunk substantially, butthe length of the interior tube keeps growing linearly withtime elapsed from the collapse. This can be huge for ablack hole that started out as macroscopic (mo mPl),even if the horizon area and mass have become small.The key point is that (3) still hold, with mo being theinitial mass of the hole [62], see also [63].

    The essential fact that is often neglected, generatingconfusion, is that an old black hole that has evaporateddown to mass m has the same exterior geometry as ayoung black hole with the same mass, but not the sameinterior: an old, largely evaporated hole has an interiorvastly bigger than a young black hole with the samemass. This is conventional physics.

    To understand the end of a black holes evaporation,it is important to distinguish the phenomena concerning

    A B

    S

    v

    FIG. 1. Conformal diagram of a classical black hole. Thedashed line is the horizon. The dotted line is a Cauchy sur-face . In regions A and B we expect (distinct) quantumgravitational effects and classical GR is unreliable.

  • 3

    T.BANKSAND

    M. O'LOUGHLIN47

    u(r, n }=e(2.4)

    Interms

    ofthesefields

    theLagrangian

    is

    S=f & g[

    u R+2u(Ba) 4(Bu)

    2ue

    +Que(2.5)

    Theequations

    of motionforthis

    Lagrangianare

    givenin

    AppendixA.

    Ofcourse,to

    findinterior

    solutionswith

    zeromagnetic

    fieldtomatch

    ontothe

    exteriorextremal

    dilatonsolution,

    we setQ =0inthe

    aboveequation.

    Thepower

    seriesfor

    thefields

    oand

    (t, expandingfrom

    theshell towards

    theinterior,

    is

    2

    u(r, n) =R(~) 1

    R(~)

    [1+f, (r)n +f2(r)n~

    +f3(r)n+

    ],o(r, n

    ) =[ln(R

    (r)}+d, (r)n +d2(~)n+d3(~)n

    +],

    (2.6)

    (2.7)

    h(r, n)=1+h&(r)n+h2(r)n+h3(r)n

    +g(r, n)=1+g,(r)n+g2(r)n

    +g3(r)n+

    (2.8)

    (2.9)

    Theequations

    thatwe

    havetodescribe

    thissystem

    nowconsistofthe

    equationsfor

    uand

    o., andthe

    stresstensor

    equation.Atthe

    boundaryofthe

    collapsingshellthere

    isa nontrivial

    matchingequation

    forthestress

    tensorcom-

    ponentToo.

    Wewill

    assumethat

    theclassical

    Lagrangianfor

    thematter

    thatconstitutesthe

    shellisofthe

    form

    S =f &gu'[ (a~) 'm'~'+

    ].(2.10)

    Thatis, the

    matterinthe

    collapsingshell

    couplestothe

    dilatonlike

    somemassive

    modeofthe

    string.Inthe

    restframe

    ofthe

    collapsingshell,

    thematching

    equationreadsMu(r,0) =f

    Tor, dnE

    whichbecomes

    21/2

    (2. 11)

    wherethe

    coefficientsofthe

    leadingterms

    aredetermined

    bycontinuity

    ofPand

    cracrossthe

    shell.The

    metricis

    g&=diag[

    h(r,n),g(r, n)], andthe

    coefficientshave

    theexpansion,

    tion,but

    herethe

    dilatondynamics

    givesrise

    toan

    infiniteset

    ofspherically

    symmetricsolutions

    ofthe

    sourcefree

    fieldequations

    inafinite

    region.Wehave

    triedtorestrict

    thesolution

    byassuming

    acosmologicalform

    forthe

    metricds = dH+a(r)

    (dr+rdQ

    ) in-side

    theshell,

    butthis

    isinconsistentwith

    thefield

    equa-tions.

    Similarly,an

    attemptto

    keepthe

    three-dimensionally

    conformallyOatform

    ofthemetric,

    withconformal

    factortied

    tothe

    dilaton,isinconsistent.

    We

    havenot

    beenable

    tocome

    upwith

    anatural

    ansatz.Nonetheless,

    webelieve

    thatsmooth

    solutionsexist.

    Thereare

    manysmooth

    solutionsof the

    vacuumfield

    equationsrestricted

    toamanifold

    withthe

    topologyofa

    hemi-three-spherecross

    time.Our

    matchingconditions

    fixonly

    thevalues

    ofthemetric

    functionsand

    dilatonalong

    thetimelike

    worldlineofthe

    collapsingshell,

    leav-ing

    theirnormal

    derivativesundetermined.

    Thusthere

    seemstobe

    plentyofroom

    forpatchinginanonsingular

    vacuumsolution.

    Toobtain

    somefeeling

    forthemotion

    ofthecollapsing

    shellwe

    havemade

    thefairly

    arbitraryassumption

    that

    afi(r)

    (2.13)

    Thisgives

    usasingle

    first-orderordinary

    differentialequation

    forR(r).

    Thesolution

    soobtained

    behaveslikeR(r)=Q+e

    r',as ~~oo.

    Wecan

    thenuse

    thissolution

    tocheck

    thatthe

    othercoefficient

    functions,to

    leadingorder,

    arewell

    behavedfor

    allfinite

    valuesof~.

    Wecan

    continuethis

    procedureperturbatively,

    toverify

    thatthe

    coefficientsinthe

    expansioninpowers

    ofnare

    smoothfunctions

    of~.Ofcourse, thisdemonstration

    ofasmooth

    perturbationexpansion

    aroundthe

    shell, doesnotguarantee

    theexistence

    ofaneverywhere

    smoothsolu-

    tion.Wecontinue

    tosearch

    fora sensibleansatz

    thatwill

    enableus

    todemonstrate

    explicitlythe

    existenceof

    asmooth

    collapsingsolution,

    butwe

    feelconfident

    thatsuch

    a solutionexists.

    Thecollapsing

    solutionthat

    wehave

    described,begins

    asadimpleon

    Aatspace.Atany

    finitetime

    afterits for-

    mation,itwill

    havethe

    geometryshown

    inFig. 2.

    We

    willrefer

    tosuch

    anobject

    asafinite

    volumecornu-

    copion.Itisasolution

    ofthefield

    equationsthat

    isstaticover

    mostofspace.The

    timedependence

    occursonly

    inthe

    tip ofthehorn.

    Mu(r, 0) =R1

    2RR+

    1 R

    (2. 12)

    Atthispoint

    wemust

    be morespecific

    aboutthe

    fieldson

    theinterior

    oftheshell.

    InEinstein

    stheory,there

    isaunique

    sphericallysymmetric

    nonsingularvacuum

    solu-

    FIG.2.Instantaneoussnapshot

    ofacollapsingcornucopion.

    7SeeAppendix8fordetails.

    8Thefulldetails

    ofthederivation

    areinAppendix

    B.9Appendix

    C.

    T.BANKS ANDM. O'LOUGHLIN

    47

    u(r, n }=e(2.4)

    In terms ofthesefieldstheLagrangian

    is

    S=f & g[

    u R+2u(Ba) 4(Bu) 2u

    e

    +Que(2.5)

    Theequationsof motion

    forthisLagrangianaregiven

    inAppendix

    A. Ofcourse, tofind

    interiorsolutions

    withzero

    magneticfieldto

    matchonto

    theexterior

    extremaldilaton

    solution,we setQ =0intheabove

    equation.The

    powerseries

    forthefields

    oand

    (t, expandingfrom

    theshell towardstheinterior,

    is

    2

    u(r, n) =R(~) 1R(~)

    [1+f, (r)n +f2(r)n~

    +f3(r)n +],

    o(r, n) =[ln(R (r)}+d, (r)n +d2(~)n+d3(~)n +

    ],

    (2.6)

    (2.7)

    h(r, n)=1+h&(r)n+h2(r)n+h3(r)n

    +g(r, n)=1+g,(r)n+g2(r)n

    +g3(r)n+

    (2.8)

    (2.9)

    Theequationsthatwe havetodescribe

    thissystemnow

    consistoftheequationsforuand

    o., andthestresstensor

    equation.Attheboundary

    ofthecollapsingshellthere

    isa nontrivial

    matchingequation

    forthestresstensor com-ponent

    Too.We

    willassume

    thatthe

    classicalLagrangian

    forthematterthatconstitutes

    theshellisoftheformS =f &

    gu'[ (a~) 'm'~'+].

    (2.10)

    Thatis, thematterinthe

    collapsingshellcouples

    tothedilaton

    likesomemassive

    modeofthestring.Intherest

    frameofthe

    collapsingshell,

    thematching

    equationreadsMu(r,0) =f

    Tor, dnE

    whichbecomes

    21/2

    (2. 11)

    wherethecoefficientsoftheleading

    termsaredeterminedby continuity

    ofP andcracross

    theshell.

    The metricis

    g&=diag[

    h(r,n),g(r, n)], andthe

    coefficientshave

    theexpansion,

    tion,but

    herethe

    dilatondynamics

    givesrise

    toan

    infinitesetof

    sphericallysymmetric

    solutionsofthe

    sourcefree

    fieldequations

    inafinite

    region.We

    havetried

    to restrictthe

    solutionby

    assumingacosmological

    formforthe

    metricds = dH+a(r)

    (dr+rdQ) in-

    sidetheshell, butthisisinconsistent

    withthe

    fieldequa-

    tions.Similarly,

    anattempt

    tokeep

    thethree-

    dimensionallyconformally

    Oatformofthe

    metric,with

    conformalfactortiedtothe

    dilaton,isinconsistent.

    Wehave

    notbeenable

    tocome

    upwith

    anaturalansatz.

    Nonetheless,we

    believethat

    smoothsolutions

    exist.There

    aremany

    smoothsolutions

    of thevacuum

    fieldequations

    restrictedtoa manifold

    withthetopology

    ofahemi-three-sphere

    crosstime.Our

    matchingconditions

    fixonly

    thevalues

    ofthemetric

    functionsand

    dilatonalong

    the timelikeworld

    lineofthecollapsingshell, leav-

    ingtheir

    normalderivatives

    undetermined.Thus

    thereseemstobeplenty

    ofroomforpatching

    inanonsingularvacuum

    solution.Toobtain

    somefeeling

    forthemotionofthecollapsing

    shellwe havemadethefairly

    arbitraryassumption

    that

    afi(r)

    (2.13)

    Thisgives

    usasingle

    first-orderordinary

    differentialequation

    forR(r).The

    solutionso

    obtainedbehaves

    likeR(r)=Q+er',as ~~oo.

    Wecan

    thenuse

    thissolution

    tocheckthatthe

    othercoefficient

    functions,to

    leadingorder, are

    wellbehavedforallfinite

    valuesof~.Wecan

    continuethisprocedure

    perturbatively,toverify

    thatthecoefficients

    inthe

    expansioninpowers

    ofnaresmooth

    functionsof~.Ofcourse, thisdemonstration

    ofasmooth

    perturbationexpansion

    aroundtheshell, doesnot

    guaranteethe

    existenceofan

    everywheresmooth

    solu-tion.

    Wecontinuetosearch

    fora sensibleansatz

    thatwillenable

    usto

    demonstrateexplicitly

    theexistence

    ofasmooth

    collapsingsolution,

    butwe

    feelconfidentthat

    sucha solution

    exists.The

    collapsingsolution

    thatwehave

    described,begins

    asadimpleon

    Aatspace. Atanyfinite

    timeafterits for-mation,

    itwillhavethe

    geometryshown

    inFig. 2.Wewill

    referto

    suchan

    objectasa

    finitevolume

    cornu-copion. Itisasolution

    ofthefieldequations

    thatisstaticovermostofspace.

    Thetimedependence

    occursonly

    inthetip ofthehorn.

    Mu(r, 0) =R12R

    R+

    1 R(2. 12)

    Atthispointwe mustbe more

    specific aboutthefields

    ontheinterioroftheshell.InEinstein

    stheory, thereis

    auniquespherically

    symmetricnonsingular

    vacuumsolu-

    FIG.2.Instantaneoussnapshot ofacollapsing

    cornucopion.

    7SeeAppendix 8fordetails.8Thefulldetailsofthe derivation

    are inAppendixB.

    9AppendixC.

    T.BANKSAND

    M. O'LOUGHLIN47

    u(r, n }=e(2.4)

    In terms ofthesefieldstheLagrangian

    is

    S=f & g[

    u R+2u(Ba) 4(Bu)

    2ue

    +Que(2.5)

    Theequations

    of motionforthis

    Lagrangianare

    givenin

    AppendixA.

    Ofcourse,to

    findinterior

    solutionswith

    zeromagnetic

    fieldtomatch

    ontothe

    exteriorextremal

    dilatonsolution,

    we setQ =0intheabove

    equation.The

    powerseries

    forthe

    fieldsoand

    (t, expandingfrom

    theshell towards

    theinterior,is

    2

    u(r, n) =R(~) 1R(~)

    [1+f, (r)n +f2(r)n~

    +f3(r)n+

    ],o(r, n

    ) =[ln(R (r)}+d, (r)n +d2(~)n+d3(~)n

    +],

    (2.6)

    (2.7)

    h(r, n)=1+h&(r)n+h2(r)n+h3(r)n

    +g(r, n)=1+g,(r)n+g2(r)n

    +g3(r)n+

    (2.8)

    (2.9)

    Theequationsthatwe havetodescribe

    thissystemnow

    consistoftheequations

    foruand

    o., andthe

    stresstensor

    equation.Atthe

    boundaryofthe

    collapsingshellthere

    isa nontrivial

    matchingequation

    forthestress

    tensorcom-

    ponentToo.

    Wewill

    assumethat

    theclassical

    Lagrangianforthe

    matterthatconstitutes

    theshellisoftheform

    S =f &gu'[ (a~) 'm'~'+

    ].(2.10)

    Thatis, the

    matterinthe

    collapsingshell

    couplestothe

    dilatonlike

    somemassive

    modeofthestring.

    Inthe

    restframe

    ofthe

    collapsingshell,

    thematching

    equationreadsMu(r,0) =f

    Tor, dnE

    whichbecomes

    21/2

    (2. 11)

    wherethe

    coefficientsoftheleading

    termsaredetermined

    bycontinuity

    ofPand

    cracrossthe

    shell.The

    metricis

    g&=diag[

    h(r,n),g(r, n)], andthe

    coefficientshave

    theexpansion,

    tion,but

    herethe

    dilatondynamics

    givesrise

    toan

    infiniteset

    ofspherically

    symmetricsolutions

    ofthe

    sourcefree

    fieldequations

    inafinite

    region.We

    havetried

    to restrictthe

    solutionby

    assumingacosmological

    formforthe

    metricds = dH+a(r)

    (dr+rdQ) in-

    sidetheshell, butthis

    isinconsistentwith

    thefield

    equa-tions.

    Similarly,an

    attemptto

    keepthe

    three-dimensionally

    conformallyOatform

    ofthemetric,

    withconformal

    factortied

    tothe

    dilaton,isinconsistent.

    Wehave

    notbeen

    abletocome

    upwith

    anatural

    ansatz.Nonetheless,

    webelieve

    thatsmooth

    solutionsexist.

    Thereare

    manysmooth

    solutionsof the

    vacuumfield

    equationsrestricted

    toa manifoldwith

    thetopology

    ofahemi-three-sphere

    crosstime.

    Ourmatching

    conditionsfix

    onlythe

    valuesofthe

    metricfunctions

    anddilaton

    alongthe

    timelikeworld

    lineofthecollapsing

    shell, leav-ing

    theirnormal

    derivativesundetermined.

    Thusthere

    seemstobeplentyofroom

    forpatchinginanonsingular

    vacuumsolution.

    Toobtainsome

    feelingforthemotion

    ofthecollapsing

    shellwe havemade

    thefairly

    arbitraryassumption

    that

    afi(r)

    (2.13)

    Thisgives

    usasingle

    first-orderordinary

    differentialequation

    forR(r).The

    solutionso

    obtainedbehaves

    likeR(r)=Q+er',

    as ~~oo.We

    canthen

    usethis

    solutiontocheck

    thatthe

    othercoefficient

    functions,to

    leadingorder,

    arewellbehaved

    forallfinitevalues

    of~.Wecan

    continuethis

    procedureperturbatively,

    toverify

    thatthe

    coefficientsinthe

    expansioninpowers

    ofnaresmooth

    functionsof~.Ofcourse, thisdemonstration

    ofasmooth

    perturbationexpansion

    aroundthe

    shell, doesnotguarantee

    theexistence

    ofaneverywhere

    smoothsolu-

    tion.Wecontinue

    tosearchfora sensible

    ansatzthat

    willenable

    usto

    demonstrateexplicitly

    theexistence

    ofasmooth

    collapsingsolution,

    butwe

    feelconfident

    thatsuch

    a solutionexists.

    Thecollapsing

    solutionthat

    wehavedescribed,

    beginsasadimple

    onAatspace.

    Atanyfinite

    timeafterits for-

    mation,itwill

    havethe

    geometryshown

    inFig. 2.Wewill

    referto

    suchan

    objectasafinite

    volumecornu-

    copion.Itisasolution

    ofthefield

    equationsthatisstatic

    overmostofspace.The

    timedependence

    occursonly

    inthetip ofthehorn.

    Mu(r, 0) =R1

    2RR+

    1 R

    (2. 12)

    Atthispoint

    we mustbe morespecific

    aboutthefields

    onthe

    interiorofthe

    shell.InEinstein

    stheory,there

    isaunique

    sphericallysymmetric

    nonsingularvacuum

    solu-

    FIG.2.Instantaneoussnapshot

    ofacollapsingcornucopion.

    7SeeAppendix 8fordetails.8The

    fulldetailsofthe derivationare

    inAppendixB.

    9AppendixC.

    T.BANKS ANDM. O'LOUGHLIN

    47

    u(r, n }=e(2.4)

    In terms ofthesefieldstheLagrangian

    is

    S=f & g[

    u R+2u(Ba) 4(Bu) 2u

    e

    +Que(2.5)

    Theequationsof motion

    forthisLagrangianaregiven

    inAppendix

    A. Ofcourse, tofind

    interiorsolutions

    withzero

    magneticfieldto match

    ontothe

    exteriorextremal

    dilatonsolution,

    we setQ =0intheaboveequation.The

    powerseries

    forthefields

    oand

    (t, expandingfrom

    theshell towardstheinterior,

    is

    2

    u(r, n) =R(~) 1R(~)

    [1+f, (r)n +f2(r)n~

    +f3(r)n +],

    o(r, n) =[ln(R (r)}+d, (r)n +d2(~)n+d3(~)n +

    ],

    (2.6)

    (2.7)

    h(r, n)=1+h&(r)n+h2(r)n+h3(r)n

    +g(r, n)=1+g,(r)n+g2(r)n

    +g3(r)n+

    (2.8)

    (2.9)

    Theequationsthatwe havetodescribe

    thissystemnow

    consistoftheequationsforuand

    o., andthestresstensor

    equation.Attheboundary

    ofthecollapsingshellthere

    isa nontrivial

    matchingequation

    forthestresstensor com-ponent

    Too.We

    willassumethat

    theclassical

    Lagrangianforthe

    matterthatconstitutestheshellisoftheform

    S =f &gu'[ (a~) 'm'~'+

    ].(2.10)

    Thatis, thematterinthe

    collapsingshellcouples

    tothedilaton

    likesomemassive

    modeofthestring.Intherest

    frameofthe

    collapsingshell,

    thematching

    equationreadsMu(r,0) =f

    Tor, dnE

    whichbecomes

    21/2

    (2. 11)

    wherethecoefficientsoftheleading

    termsaredeterminedby continuity

    ofP andcracrossthe

    shell.The metric

    isg&

    =diag[ h(r,n),g(r, n)], and

    thecoefficients

    havetheexpansion,

    tion,but

    herethe

    dilatondynamics

    givesrise

    toan

    infinitesetofspherically

    symmetricsolutions

    ofthesource

    freefield

    equationsinafinite

    region.We

    havetried

    to restrictthesolutionby

    assumingacosmological

    formforthe

    metricds = dH+a(r)

    (dr+rdQ) in-

    sidetheshell, butthisisinconsistent

    withthe

    fieldequa-

    tions.Similarly,

    anattempt

    tokeep

    thethree-

    dimensionallyconformally

    Oatformofthe

    metric,with

    conformalfactortiedtothedilaton,

    isinconsistent.We

    havenotbeen

    abletocome

    upwith

    anaturalansatz.

    Nonetheless,we

    believethat

    smoothsolutions

    exist.There

    aremany

    smoothsolutions

    of thevacuum

    fieldequations

    restrictedtoa manifold

    withthetopology

    ofahemi-three-sphere

    crosstime.Our

    matchingconditions

    fixonly

    thevalues

    ofthemetric

    functionsand

    dilatonalong

    the timelikeworld

    lineofthecollapsingshell, leav-

    ingtheir

    normalderivatives

    undetermined.Thus

    thereseemstobeplenty

    ofroomforpatching

    inanonsingularvacuum

    solution.Toobtain

    somefeelingforthemotion

    ofthecollapsingshellwe havemade

    thefairlyarbitrary

    assumptionthat

    afi(r)

    (2.13)

    Thisgives

    usasingle

    first-orderordinary

    differentialequation

    forR(r).The

    solutionso

    obtainedbehaves

    likeR(r)=Q+er',as ~~oo.

    Wecan

    thenuse

    thissolution

    tocheckthattheother

    coefficientfunctions,

    toleading

    order, arewellbehaved

    forallfinitevaluesof~.

    Wecancontinue

    thisprocedureperturbatively,

    toverify

    thatthecoefficients

    intheexpansion

    inpowersofnare

    smoothfunctions

    of~.Ofcourse, thisdemonstrationofa

    smoothperturbation

    expansionaround

    theshell, doesnotguarantee

    theexistence

    ofaneverywhere

    smoothsolu-

    tion.Wecontinuetosearch

    fora sensibleansatz

    thatwillenable

    usto

    demonstrateexplicitly

    theexistence

    ofasmooth

    collapsingsolution,

    butwe

    feelconfidentthat

    sucha solutionexists.

    Thecollapsingsolution

    thatwehavedescribed,

    beginsasadimple

    onAatspace. Atany

    finitetimeafterits for-

    mation,itwillhave

    thegeometry

    showninFig. 2.We

    willreferto

    suchan

    objectasa

    finitevolume

    cornu-copion. Itisasolution

    ofthefieldequationsthatisstatic

    overmostofspace. Thetimedependence

    occursonly

    inthetip ofthehorn.

    Mu(r, 0) =R12R

    R+

    1 R(2. 12)

    Atthispoint we mustbe morespecific aboutthe

    fieldsontheinterioroftheshell.

    InEinsteinstheory, there

    isaunique

    sphericallysymmetric

    nonsingularvacuum

    solu-

    FIG.2.Instantaneoussnapshot ofacollapsing

    cornucopion.

    7SeeAppendix 8fordetails.8Thefulldetailsofthe derivation

    are inAppendixB.

    9AppendixC.

    FIG. 2. The interior geometry of an old black hole: a verylong thin tube, whose length increases and whose radius de-creases with time. Notice it is finite, unlikely the Einstein-Rosen bridge.

    the two regions A and B where classical general relativitybecomes unreliable.

    Region A is characterised by large curvature and coversthe singularity. According to classical general relativitythe singularity never reaches the horizon. (N.B.: Twolines meeting at the boundary of a conformal diagramdoes not mean that they meet in the physical spacetime.)

    Region B, instead, surrounds the end of the evapora-tion, which involves the horizon, and affects what hap-pens outside the hole. Taking evaporation into account,the area of the horizon shrinks progressively until reach-ing region B.

    The quantum gravitational effects in regions A and Bare distinct, and confusing them is a source of misun-derstanding. Notice that a generic spacetime region inA is spacelike separated and in general very distant fromregion B. By locality, there is no reason to expect thesetwo regions to influence one another.

    The quantum gravitational physical process happeningat these two regions must be considered separately.

    III. THE A REGION: TRANSITIONINGACROSS THE SINGULARITY

    To study the A region, let us focus on an arbitraryfinite portion of the collapsing interior tube. As we ap-proach the singularity, the Schwarzschild radius rs, whichis a temporal coordinate inside the hole, decreases andthe curvature increases. When the curvature approachesPlanckian values, the classical approximation becomesunreliable. Quantum gravity effects are expected tobound the curvature [811, 1319, 2224, 27, 29, 64, 65].Let us see what a bound on the curvature can yield. Fol-lowing [66], consider the line element

    ds2 = 4(2 + l)2

    2m 2d2+

    2m 2

    2 + ldx2+(2+l)2d2, (4)

    where lm. This line element defines a genuine Rieman-nian spacetime, with no divergences and no singularities.Curvature is bounded. For instance, the Kretschmann

    FIG. 3. The transition across the A region.

    invariant K RR is easily computed to be

    K() 9 l2 24 l2 + 48 4

    (l + 2)8m2 (5)

    in the large mass limit, which has the finite maximum

    K(0) 9m2

    l6. (6)

    For all the values of where l 2 < 2m the lineelement is well approximated by taking l = 0 which gives

    ds2 = 44

    2m 2d2 +

    2m 2

    2dx2 + 4d2. (7)

    For < 0, this is the Schwarzschild metric inside theblack hole, as can be readily seen going to Schwarzschildcoordinates

    ts = x, and rs = 2. (8)

    For > 0, this is the Schwarzschild metric inside a whitehole. Thus the metric (4) represents a continuous transi-tion of the geometry of a black hole into the geometry ofa white hole, across a region of Planckian, but boundedcurvature.

    Geometrically, = constant (space-like) surfaces foli-ate the interior of a black hole. Each of these surfaceshas the topology S2 R, namely is a long cylinder. Astime passes, the radial size of the cylinder shrinks whilethe axis of the cylinder gets stretched. Around = 0the cylinder reaches a minimal size, and then smoothlybounces back and starts increasing its radial size andshrinking its length. The cylinder never reaches zero sizebut bounces at a small finite radius l. The Ricci tensorvanishes up to terms O(l/m).

    The resulting geometry is depicted in Figure 3. Theregion around = 0 is the smoothing of the central blackhole singularity at rs = 0.

    This geometry can be given a simple physical interpre-tation. General relativity is not reliable at high curva-ture, because of quantum gravity. Therefore the pre-diction of the singularity by the classical theory has noground. High curvature induces quantum particle cre-ation, including gravitons, and these can have an effec-tive energy momentum tensor that back-reacts on the

  • 4

    classical geometry, modifying its evolution. Since the en-ergy momentum tensor of these quantum particles canviolate energy conditions (Hawking radiation does), theevolution is not constrained by Penroses singularity the-orem. Equivalently, we can assume that the expectationvalue of the gravitational field will satisfy modified effec-tive quantum corrections that alter the classical evolu-tion. The expected scale of the correction is the Planckscale. As long as l m the correction to the classicaltheory is negligible in all regions of small curvature; aswe approach the high-curvature region the curvature issuppressed with respect to the classical evolution, andthe geometry continues smoothly past = 0.

    One may be tempted to take l to be Planckian lPl =~G/c3

    ~, but this would be wrong. The value of l

    can be estimated from the requirement that the curvatureis bounded at the Planck scale, K(0) 1/~2. Using thisin (6) gives

    l (m ~) 13 , (9)

    or, restoring for a moment physical units

    l lPl(

    m

    mPl

    ) 13

    , (10)

    which is much larger than the Planck length when mmPl [2]. The three-geometry inside the hole at the tran-sition time is

    ds23 =2m

    ldx2 + l2d2. (11)

    The volume of the Planck star [2], namely the minimalradius surface is

    V = 4l2

    2m

    l(xmax xmin). (12)

    The range of x is determined by the lifetime of the holefrom the collapse to the onset of region B, as x = ts. Ifregion B is at the end of the Hawking evaporation, then(xmax xmin) m3/~ and from Eq. (9), l (m~)1/3,leading to an internal volume at crossover that scales as

    V m4/~. (13)

    We observe that in the classical limit the interior volumediverges, but quantum effects make it finite.

    The l 0 limit of the line element (4) defines a met-ric space which is a Riemannian manifold almost every-where and which can be taken as a solution of the Ein-steins equations that is not everywhere a Riemannianmanifold [66]. Geodesics of this solution crossing the sin-gularity are studied in [66]: they are well behaved at = 0 and they cross the singularity in a finite propertime. The possibility of this natural continuation ofthe Einstein equations across the central singularity ofthe Schwarzschild metric has been noticed repeatedly by

    many authors. To the best of our knowledge it was firstnoticed by Synge in the fifties [67] and rediscovered byPeeters, Schweigert and van Holten in the nineties [68].A similar observation has recently been made in the con-text of cosmology in [69].

    As we shall see in the next section, what the ~ 0limit does is to confine the transition inside an event hori-zon, making it invisible from the exterior. Reciprocally,the effect of turning ~ on is to de-confine the interior ofthe hole.

    IV. THE TRANSITION AND THE GLOBALSTRUCTURE

    The physics of the B region concerns gravitationalquantum phenomena that can happen around the hori-zon after a sufficiently long time. The Hawking radiationprovides the upper bound m3o/~ for this time. Afterthis time the classical theory does not work anymore. Be-fore studying the details of the B region, let us considerwhat we have so far.

    A B

    FIG. 4. Left: A commonly drawn diagram for black hole evap-oration that we argue against. Right: A black-to-white holetransition. The dashed lines are the horizons.

    The spacetime diagram utilized to discuss the blackhole evaporation is often drawn as in the left panel ofFigure 4. What happens in the circular shaded region?What physics determines it? This diagram rests on anunphysical assumption: that the Hawking process pro-ceeds beyond the Planck curvature at the horizon andpinches off the large interior of the black hole from therest of spacetime. This assumption uses quantum fieldtheory on curved spacetimes beyond its regime of valid-ity. Without a physical mechanism for the pinching off,this scenario is unrealistic.

    Spacetime diagrams representing the possible forma-tion and full evaporation of a black hole more realisticallyabound in the literature [811, 1319, 2224, 29] and theyare all similar. In particular, it is shown in [3, 4] that thespacetime represented in the right panel of Figure 4, canbe an exact solution of the Einstein equations, except forthe two regions A and B, but including regions withinthe horizons.

    If the quantum effects in the region A are simply thecrossing described in the previous section, this deter-

  • 5

    mines the geometry of the region past it, and shows thatthe entire problem of the end of a black hole reduces tothe quantum transition in the region B.

    The important point is that there are two regions insidehorizons: one below and one above the central singular-ity. That is, the black hole does not simply pop out ofexistence: it tunnels into a region that is screened insidean (anti-trapping) horizon. Since it is anti-trapped, thisregion is actually the interior of a white hole. Thus, blackholes die by tunneling into white holes.

    Unlike for the case of the left panel of Figure 4, nowrunning the time evolution backwards makes sense: thecentral singularity is screened by an horizon (time re-versed cosmic censorship) and the overall backward evo-lution behaves qualitatively (not necessarily quantitively,as initial conditions may differ) like the time-forward one.

    Since we have the explicit metric across the centralsingularity, we know the features of the resulting whitehole. The main consequence is that its interior is whatresults from the transition described in the above section:namely a white hole born possibly with a small horizonarea, but in any case with a very large interior volume,inherited from the black hole that generated it.

    If the original black hole is an old hole that startedout with a large mass mo, then its interior is a very longtube. Continuity of the size of the tube in the transi-tion across the singularity, results in a white hole formedby the bounce, which initially also consists of a very longinterior tube, as in Figure 5. Subsequent evolution short-ens it (because the time evolution of a white hole is thetime reversal of that of a black hole), but this processcan take a long time. Remarkably, this process results ina white hole that has a small Planckian mass and a longlife determined by how old the parent black hole was.In other words, the outcome of the end of a black holeevaporation is a long-lived remnant.

    FIG. 5. Black hole bounce, with a sketch of the inside geome-tries, before and after the quantum-gravitational transition.

    The time scales of the process can be labelled as inFigure 5. We call vo the advanced time of the collapse,v and v+ the advanced time of the onset and end ofthe quantum transition, uo the retarded time of the fi-nal disappearance of the white hole, and u and u+ theretarded times of the onset and end of the quantum tran-sition. The black hole lifetime is

    bh = v vo. (14)

    The white hole lifetime is

    wh = uo u+. (15)

    And we assume that the duration of the quantum tran-sition of the B region satisfies u+u = v+ v .

    Disregarding Hawking evaporation, a metric describingthis process outside the B region can be written explic-itly by cutting and pasting the extended Schwarzschildsolution, following [3]. This is illustrated in Figure 6:two Kruskal spacetimes are glued across the singularityas described in the previous section and the shaded re-gion is the metric of the portion of spacetime outside acollapsing shell (here chosen to be null).

    FIG. 6. Left: Two Kruskal spacetimes are glued at the singu-larity. The grey region is the metric of a black to white holetransition outside a collapsing and the exploding null shell.Right: The corresponding regions in the physical spacetime.

    While the location of the A region is determined by theclassical theory, the location of the B region, instead, isdetermined by quantum theory. The B process is indeeda typical quantum tunneling process: it has a long life-time. A priori, the value of bh is determined probabilis-tically by quantum theory. As in conventional tunneling,in a stationary situation (when the horizon area variesslowly), we expect the probability p per unit time for thetunneling to happen to be time independent. This im-plies that the normalised probability P (t) that the tun-neling happens between times t and t+dt is governed bydP (t)/dt = pP (t), namely is

    P (t) =1

    bhe tbh , (16)

    which is normalised (0P (t)dt = 1) and where bh sat-

    isfies

    bh = 1/p. (17)

    We note parenthetically that the quantum spread inthe lifetime can be a source of apparent unitarity vio-lation, for the following reason. In conventional nuclear

  • 6

    decay, a tunneling phenomenon, the quantum indeter-mination in the decay time is of the same order as thelifetime. The unitary evolution of the state of a particletrapped in the nucleus is such that the state slowly leaksout, spreading it over a vast region. A Geiger counterhas a small probability of detecting a particle at thetime where it happens to be. Once the detection hap-pens, there is an apparent violation of unitarity. (In theCopenhagen language the Geiger counter measures thestate, causing it to collapse, loosing information. In theMany Worlds language, the state splits into a continuumof branches that decohere and the information of a sin-gle branch is less than the initial total information.) Ineither case, the evolution of the quantum state from thenucleus to a given Geiger counter detection is not uni-tary; unitarity is recovered by taking into account thefull spread of different detection times. The same mustbe true for the tunneling that disrupts the black hole. Iftunneling will happen at a time t, unitarity can only berecovered by taking into account the full quantum spreadof the tunneling time, which is to say: over different fu-ture goemetries. The quantum state is actually given bya quantum superposition of a continuum of spacetimesas in Figure 5, each with a different value of v and v+.We shall not further pursue here the analysis of this ap-parent source of unitarity, but we indicate it for futurereference.

    V. THE B REGION: THE HORIZON AT THETRANSITION

    The geometry surrounding the transition in the B re-gion is depicted in detail in Figure 7. The metric of

    FIG. 7. The B region. Left: Surfaces of equal Schwarzschildradius are depicted. Right: The signs of the null Kruskalcoordinates around B.

    the entire neighbourhood of the B region is an extendedSchwarzschild metric. It can therefore be written in nullKruskal coordinates

    ds2 = 32m3

    re

    r2m dudv + r2d2, (18)

    where (1 r

    2m

    )er

    2m = uv. (19)

    On the two horizons we have respectively v = 0 andu = 0, and separate regions where u and v have different

    signs as in the right panel of Figure 7. Notice the rapidchange of the value of the radius across the B region,which yields a rapid variation of the metric componentsin (18).

    To fix the region B, we need to specify more preciselyits boundary, which we have not done so far. It is possibleto do so by identifying it with the diamond (in the 2d dia-gram) defined by two points P+ and P with coordinatesv, u both outside the horizon, at the same radius rP ,and at opposite timelike distance from the bounce time,see Figure 8.

    FIG. 8. The B transition region.

    The same radius rP implies

    v+u+ = vu (

    1 rP2m

    )erP2m . (20)

    The same time from the horizon implies that the lightlines u = u and v = v+ cross on ts = 0, or u + v = 0,hence

    u = v+. (21)

    This crossing point is the outermost reach of the quantumregion, with radius rm determined by

    v+u (

    1 rm2m

    )erm2m . (22)

    The region is then entirely specified by two parameters.We can take them to be rP and = v+v u+u.The first characterizes the radius at which the quan-tum transition starts. The second its duration. (Strictlyspeaking, we could also have v+ v and u+ u ofdifferent orders of magnitude, but we do not explore thispossibility here.)

    There are indications about both metric scales inthe literature. In [3, 70], arguments where given forrP 7/3 m. Following [5], the duration of the tran-sition has been called crossing time and computedby Christodoulou and DAmbrosio in [6, 7] using LoopQuantum Gravity: the result is m, which can betaken as a confirmation of earlier results [26, 71, 72] ob-tained with other methods. The two crucial remainingparameters are the black hole and the white hole life-times, bh and wh.

    The result in [6] indicates also that p, the probabilityof tunneling per unit time, is suppressed exponentially

    by a factor em2/~. Here m is not the initial mass mo

  • 7

    of the black hole at the time of its formation, rather, itis the mass of the black hole at the decay time. This isin accord with the semiclassical estimate that tunnelingis suppressed as in (1) and (2). As mentioned in theintroduction, because of Hawking evaporation, the massof the black hole shrinks to Planckian values in a timeof order m3o/~, where the probability density becomes oforder unit, giving

    bh m3o/~ (23)

    and

    ~. (24)

    We conclude that region B has a Planckian size.We notice parenthetically that the value of p above is

    at odds with the arguments given in [3] for a shorter life-

    time bh m2o/~. This might be because the analysis

    in [6] captures the dynamics of only a few of the relevantdegrees of freedom, but we do not consider this possibil-ity here. The entire range of possibilities for the blackto white transition lifetime, m2o/

    ~ bh m3o/~, may

    have phenomenological consequences, which have beenexplored in [7377]. (On hypothetical white hole obser-vations see also [78]).

    VI. INTERIOR VOLUME AND PURIFICATIONTIME

    Consider a quantum field living on the backgroundgeometry described above. Near the black hole hori-zon there is production of Hawking radiation. Its back-reaction on the geometry gradually decreases the areaof the horizon. This, in turn, increases the transitionprobability to a white hole. After a time bh m3o/~,the area of the black hole reaches the Planckian scaleAbh(final) ~, and the transition probability becomes oforder unity. The volume of the transition surface is huge.

    To compute it with precision, we should compute theback-reaction of the inside component of the Hawkingradiation, which gradually decreases the value of m asthe coordinate x increases. Intuitively, the inside com-ponents of the Hawking pairs fall toward the singularity,decreasing m. Since most of the decrease is at the endof the process, we may approximate the full interior ofthe hole with that of a Schwarzschild solution of mass moand the first order estimate of the inside volume shouldnot be affected by this process. Thus we may assumethat the volume at the transition has the same order asthe one derived in Eq. (13), namely

    Vbh(final) ~mo bh m4o/

    ~. (25)

    Using the same logic in the future of the transition, weapproximate the inside metric of the white hole withthat of a Schwarzschild solution of Planckian mass, sincein the future of the singularity, the metric is again ofKruskal type, but now for a white hole of Plankian mass.

    The last parameter to estimate is the lifetime wh =u0u+ of the white hole produced by the transition. Todo so, we can assume that the internal volume is con-served in the quantum transition. The volume of the re-gion of Planckian curvature inside the white hole horizonis then

    Vwh(u) l2m

    lwh, (26)

    where now l m ~, and therefore

    Vwh(initial) ~ wh. (27)

    Gluing the geometry on the past side of the singularityto the geometry on the future side requires that the twovolumes match, namely that (26) matches (13) and thisgives

    wh m4o/~3/2. (28)

    This shows that the Planck-mass white hole is a long-lived remnant [62].

    With these results, we can address the black hole infor-mation paradox. The Hawking radiation reaches futureinfinity before u, and is described by a mixed state withan entropy of order m2o/~. This must be purified by cor-relations with field excitations inside the hole. In spite ofthe smallness of the mass of the hole, the large internalvolume (25) is sufficient to host these excitations [79].This addresses the requirement (a) of the introduction,namely that there is a large information capacity.

    To release this entropy, the remnant must be long-lived. During this time, any internal information thatwas trapped by the black hole horizon can leak out. In-tuitively, the interior member of a Hawking pair can nowescape and purify the exterior quantum state. The longlifetime of the white hole allows this information to es-cape in the form of very low frequency particles, thusrespecting bounds on the maximal entropy contained ina given volume with given energy.

    The lower bound imposed by unitarity and energy con-siderations is R m4o/~3/2 [32, 46, 47] and this is pre-cisely the white hole lifetime (28) deduced above; hencewe see that they satisfy the requirement (b) of the in-troduction. Therefore white holes realize precisely thelong-lived remnant scenario for the end of the black holeevaporation that was conjectured and discussed mostlyin the 1990s [29, 31, 33, 34, 4245].

    The last issue we should discuss is stability. Generi-cally, white holes are known to be unstable under per-turbations (see for instance Chapter 15 in [48] and ref-erences therein). The instability arises because modesof short-wavelength are exponentially blue-shifted alongthe white hole horizon. In the present case, however,we have a Planck-size white hole. To run this argumentfor instability in the case of a planckian white hole, it isnecessary to consider transplanckian perturbations. As-suming no transplanckian perturbations to exist, there

  • 8

    are no instabilities to be considered. This addresses therequirement (c). Alternatively: a white hole is unstablebecause it may re-collapse into a black hole with simi-lar mass; therefore a Planck size white hole can at mostre-collapse into a Planck size black hole; but this hasprobability of order unity to tunnel back into a whitehole in a Planck time.

    Therefore the proposed scenario addresses the consis-tency requirements (a), (b), and (c) for the solution ofthe information-loss paradox and provides an effectivegeometry for the end-point of black hole evaporation: along-lived Planck-mass white hole.

    VII. ON WHITE HOLES

    Notice that from the outside, a white hole is indistin-guishable from a black hole. This is obvious from theexistence of the Kruskal spacetime, where the same re-gion of spacetime (region I) describes both the exteriorof a black hole and the exterior of a white hole. Forrs>2m, the conventional Schwarzschild line element de-scribes equally well a black hole exterior and a white holeexterior. The difference is only what happens at r = 2m.

    The only locally salient difference between a white anda black hole is that if we add some generic perturba-tion or matter on a given constant ts surface, in (theSchwarzschild coordinate description of) a black hole wesee matter falling towards the center and accumulatingaround the horizon. While in (the Schwarzschild coor-dinate description of) a white hole we see matter accu-mulated around the horizon in the past, moving awayfrom the center. Therefore the distinction is only one ofnaturalness of initial conditions: a black hole has spe-cial boundary conditions in the future, a white hole hasspecial boundary conditions in the past.

    This difference can be described physically also as fol-lows: if we look at a black hole (for instance when theEvent Horizon Telescope [80] examines Sagittarius A*),we see a black disk. This means that generic initial condi-tions on past null infinity give rise on future null infinityto a black spot with minimal incoming radiation: a spe-cial configuration in the future sky. By time reversalsymmetry, the opposite is true for a white hole; genericinitial conditions on future null infinity require a blackspot with minimal incoming radiation from past null in-finity: a special configuration in the past.

    We close this section by briefly discussing the no tran-sition principle considered by Engelhardt and Horowitzin [51]. By assuming holographic unitarity at infinityand observing that consequently information cannot leakout from the spacetime enclosed by a single asymptoticregion, these authors rule out a number of potential sce-narios, including the possibility of resolving generic sin-gularities inside black holes. Remarkably, the scenariodescribed here circumvents the no transition principleand permits singularity resolution in the bulk: the reasonis that this singularity is confined in a finite spacetime

    region and does not alter the global causal structure.

    VIII. ON REMNANTS

    The long-lived remnant scenario provides a satisfac-tory solution to the black-hole information paradox. Themain reason for which it was largely discarded was thefact that remnants appeared to be exotic objects extra-neous to known physics. Here we have shown that theyare not: white holes are well known solutions of the Ein-stein equations and they provide a concrete model forlong-lived remnants.

    Two other arguments made long-lived remnants un-popular: Pages version of the information paradox; andthe fact that if remnants existed they would easily beproduced in accelerators. Neither of these arguments ap-plies to the long-lived remnant scenario of this paper. Wediscuss them below.

    In its interactions with its surroundings, a black holewith horizon area A behaves thermally as a system withentropy Sbh = A/4~. This is a fact supported by alarge number of convincing arguments and continues tohold for the dynamical horizons we consider here. TheBekenstein-Hawking entropy provides a good notion ofentropy that satisfies Bekensteins generalized secondlaw, in the approximation in which we can treat the hori-zon as an event horizon. In the white hole remnant sce-nario this is a good approximation for a long time, butfails at the Planck scale when the black hole transitionsto a white hole.

    Let us assume for the moment that these facts implythe following hypothesis (see for instance [46])

    (H) The total number of available states for aquantum system living on the internal spatialslice i of Figure 1 is Nbh = e

    Sbh = eA/4~.

    Then, as noticed by Page [81], we have immediately aninformation paradox regardless of what happens at theend of the evaporation. The reason is that the entropyof the Hawking radiation grows with time. It is natu-ral to interpret this entropy as correlation entropy withthe Hawking quanta that have fallen inside the hole, butfor this to happen there must be a sufficient number ofavailable states inside the hole. If hypothesis (H) aboveis true, then this cannot be, because as the area of thehorizon decreases with time, the number of available in-ternal states decreases and becomes insufficient to purifythe Hawking radiation. The time at which the entropysurpasses the area is known as the Page time. This haslead many to hypothesize that the Hawking radiation isalready purifying itself by the Page time: a consequenceof this idea is the firewall scenario [82].

    The hypothesis (H) does not apply to the white-holeremnants. As argued in [79], growing interior volumes to-gether with the existence of local observables implies thatthe number of internal states grows with time instead ofdecreasing as stated in (H). This is not in contradiction

  • 9

    with the fact that a black hole behaves thermally in itsinteractions with its surroundings as a system with en-tropy S = A/4~. The reason is that entropy is notan absolute concept and the notion of entropy must bequalified. Any definition of entropy relies on a coarsegraining, namely on ignoring some variables: these couldbe microscopic variables, as in the statistical mechani-cal notion of entropy, or the variables of a subsystemover which we trace, as in the von Neumann entropy.The Bekenstein-Hawking entropy correctly describes thethermal interactions of the hole with its surroundings,because the boundary is an outgoing null surface andSbh counts the number of states that can be distinguishedfrom the exterior; but this is not the number of statesthat can be distinguished by local quantum field opera-tors on i [79]. See also [83].

    Therefore there is no reason for the Hawking radiationto purify itself by the Page time. This point has beenstressed by Unruh and Wald in their discussion of theevaporation process on the spacetime pictured in the leftpanel of Figure 4, see e.g. [84]. Our scenario differsfrom Unruh and Walds in that the white hole transitionallows the Hawking partners that fell into the black holeto emerge later and purify the state. They emerge slowly,over a time of order m4o/~3/2, in a manner consistent withthe long life of the white hole established here.

    The second standard argument against remnants isthat, if they existed, it would be easy to produce them.This argument assumes that a remnant has a smallboundary area and little energy, but can have a verylarge number of states. The large number of states wouldcontribute a large phase-space volume factor in any scat-tering process, making the production of these objects inscattering processes highly probable. Actually, since inprinciple these remnants could have an arbitrarily largenumber of states, their phase-space volume factor wouldbe infinite, and hence they would be produced sponta-neously everywhere.

    This argument does not apply to white holes. The rea-son is that a white hole is screened by an anti-trappinghorizon: the only way to produce it is through quantumgravity tunneling from a black hole! Even more, toproduce a Planck mass white hole with a large interiorvolume, we must first produce a large black hole and letit evaporate for a long time. Therefore the thresholdto access the full phase-space volume of white holes ishigh. A related argument is in [33], based on the factthat infinite production rate is prevented by locality.In [45] Giddings questions this point treating remnantsas particles of an effective field theory; the field theory,however, may be a good approximation of such a highlynon-local structure as a large white hole only in theapproximation where the large number of internal statesis not seen. See also [34].

    IX. CONCLUSION

    As a black hole evaporates, the probability to tunnelinto a white hole increases. The suppression factor for

    this tunneling process is of order em2/m2Pl . Before reach-

    ing sub-Planckian size, the probability ceases to be sup-pressed and the black hole tunnels into a white hole.

    Old black holes have a large volume. Quantum grav-itational tunneling results in a Planck-mass white holethat also has a large interior volume. The white hole islong-lived because it takes awhile for its finite, but large,interior to become visible from infinity.

    The geometry outside the black to white hole transi-tion is described by a single asymptotically-flat space-time. The Einstein equations are violated in two regions:The Planck-curvature region A, for which we have givenan effective metric that smoothes out of the singularity;and the tunneling region B, whose size and decay prob-ability can be computed [6]. These ingredients combineto give a white hole remnant scenario.

    This scenario provides a way to address the informa-tion problem. We distinguish two ways of encoding in-formation, the first associated with the small area of thehorizon and the second associated to the remnants in-terior. The Bekenstein-Hawking entropy Sbh = A/4~ isencoded on the horizon and counts states that can onlybe distinguished from outside. On the other hand, awhite hole resulting from a quantum gravity transitionhas a large volume that is available to encode substantialinformation even when the horizon area is small. Thewhite hole scenarios apparent horizon, in contrast to anevent horizon, allows for information to be released. Thelong-lived white hole releases this information slowly andpurifies the Hawking radiation emitted during evapora-tion. Quantum gravity resolves the information problem.

    CR thanks Ted Jacobson, Steve Giddings, GaryHorowitz, Steve Carlip, and Claus Kiefer for very usefulexchanges during the preparation of this work. EB andHMH thank Tommaso De Lorenzo for discussion of timescales. EB thanks Abhay Ashtekar for discussion of rem-nants. HMH thanks the CPT for warm hospitality andsupport, Bard College for extended support to visit theCPT with students, and the Perimeter Institute for The-oretical Physics for generous sabbatical support. MC ac-knowledges support from the SM Center for Space, Timeand the Quantum and the Leventis Educational GrantsScheme. This work is supported by Perimeter Institutefor Theoretical Physics. Research at Perimeter Instituteis supported by the Government of Canada through In-dustry Canada and by the Province of Ontario throughthe Ministry of Research and Innovation.

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    White Holes as Remnants: A Surprising Scenario for the End of a Black HoleAbstractI IntroductionII The internal geometry before quantum gravity becomes relevantIII The A region: Transitioning Across the SingularityIV The transition and the global structureV The B region: the Horizon at the TransitionVI Interior Volume and Purification TimeVII On White HolesVIII On remnantsIX Conclusion References


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