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Non-linear instability of Kerr-type Cauchy horizons Patrick R. Brady and Chris M. Chambers Department of Physics, The University, Newcastle Upon Tyne NE1 7RU (February 7, 2008) UsingthegeneralsolutiontotheEinsteinequationsonintersectingnullsurfacesdevelopedbyHayward,weinvestigatethenon- linearinstabilityoftheCauchyhorizoninsidearealistic blackhole. Makingaminimalassumptionaboutthefreegravitational data allows us to solve the field equations along a null surface crossing the Cauchy Horizon. As in the spherical case, the results indicate that a diverging influx of gravitational energy,in concert with an outfluxacross the CH, is responsible for the singularity. ThespacetimeisasymptoticallyPetrovtypeN,thesamealgebraictypeasagravitationalshockwave. Implications for the continuation of spacetime through thesingularity are briefly discussed. 5 PACS:97.60Lf, 04.70-s, 04.20Dw 9 9 1 I. INTRODUCTION dousgrowthofcurvatureistheblueshiftedradiationflux n along the CH, although it is also necessary that some a J Whatarethegenericfeaturesofgravitationalcollapse? transverse energy flux be present. In [6] it was argued We do not, as yet, have a complete answer to this ques- that the physics underlying the analysis was sufficiently 1 2 tion, howevermostrelativists are confident that gravita- general that similar results should hold for generic col- tional collapse leads to the formation of black holes (at lapse (i.e. upon relaxation of the assumption of spher- 1 leastwithintheastrophysicalcontext). Furthermoreitis ical symmetry). Further calculations support the con- v plausible that the external field of such a black hole set- jecture that the singularity inside a generic black hole is 5 tlesdowntoamemberoftheKerr-Newmanfamily,since null [8,10]. 2 thesearetheuniquestationarysolutionsoftheelectrovac The aim of this paper is to present a detailed analy- 0 Einsteinequations(seeforexample[1]). Externally,devi- sis of the CH inside a non-sphericalblack hole by taking 1 0 ationsfromthesesolutionsareexpectedtodieawaywith advantage of the recent result due to Hayward [11]. He 5 an inverse power-law in advanced time leaving only the showedhowtoobtainthegeneralsolutionoftheEinstein 9 mass,chargeandangularmomentumobservable. Aques- equationsonapairofintersectingnullsurfaces. Webegin / tion which one might hope to answer is whether such a byrecastingtheresultsofPoissonandIsrael[6]inasug- c q propertycontinuestoholdinsidetheblackhole–thatis, gestive form which more closely resembles the approach - does the internal geometry also approach Kerr-Newman taken to the general problem. Our purpose in section II gr form? It seems not [6]- [8]. is to stress the main points which must be taken as as- : The known exact black-hole solutions possess Cauchy sumptions in the later calculations. We also feel that v horizons – null hypersurfaces which are the boundary the analysishighlightsthe non-linearnatureofthe effect i X of the future domain of dependence for Cauchy data and the merely precipitous role played by the outflux in r of the collapse problem. These horizons exhibit highly thismodel[6]. Withthispreliminaryreviewofthemass- a pathological behaviour; small time-dependent perturba- inflation phenomenon out of the way, section III begins tions originating outside the black hole undergo an in- the analysis ofthe CH singularityinside a realistic black finite gravitational blueshift as they evolve towards the hole. The assumptions which we make about the na- Cauchy horizon (CH). This blueshift of infalling radia- tureoftheCHarediscussed,andfollowingHayward[11] tiongavethe firstindicationsthatthese solutions,which we formally integrate the first order Einstein equations sowelldescribe the exteriorgeometryatlate times, may of appendix A. Using these results we can show that not describe the generic internal structure. Penrose [2] the divergences which arise are integrable, and obtain pointed this out more than twenty five years ago, and the leading behaviour ofall quantities of interestnear to sincethenlinearperturbationshavebeenanalysedinde- the CH.The asymptoticexpressionsfortheWeylscalars tail [3]. These observations led to the conjecture that a show that the spacetime is Petrov-type N near the CH, scalarcurvaturesingularitywouldformeitheratorbefore the same algebraic type as a gravitational shock wave. the CH once back-reactionwas accounted for. These results (which are also suggested by the previous PoissonandIsrael[6],generalizingworkofHiscock[5], work [10]) once again raise questions about the classical have shown that a scalar curvature singularity forms continuation of spacetime through the singularity. along the CH of a charged, spherical black hole in a The equations and curvatures are relegated to the ap- simplified model. This singularity is characterised by pendices in an effort to maintain the clarity of the pre- the exponentialdivergenceofthe massfunction withad- sentation. Appendix A gives a summary of the notation vanced time. The key ingredient producing this tremen- and lists the dual null Einstein field equations in their 1 firstorderform,asderivedbyHayward[11]. AppendixB T =ρ l l +ρ n n , (2.3) µν in µ ν out µ ν lists the components of the Riemann tensor necessary to analyse the algebraic type of the spacetime. where lµ = ∂µv and nµ = ∂µu are radial null vec- − − torspointinginwardsandoutwardsrespectively,and,ρ in andρ representtheenergydensitiesoftheinwardand out II. SPHERICAL BLACK HOLE INTERIORS outward fluxes. Each term in Eq. (2.3) is independently conserved so that In this section we review the mass-inflation phe- L (v) L (u) in out ρ = , ρ = . (2.4) nomenonin the contextofcharged,sphericalblack holes in 4πr2 out 4πr2 as studied by Poisson and Israel [6]. The presenta- tion differs slightly from that in the literature [6,7] and The functions Lin(v) and Lout(u) are determined by the closelyparallelsthemethodusedlatertodiscussthenon- boundary conditions, however, it is important to note sphericalproblem. Inthiswaythelimitationsofthelater that they have no direct operational meaning since they analysis,andoftheapproximationsusedshouldbe more depend on the parametrization of the null coordinates. apparent. The field equationscan now be written in a first order The physics behind mass-inflation is relatively simple. form by defining the extrinsic fields The CH inside a Reissner-Nordstro¨mblack hole is a null θ :=r−2 (r2) , (2.5) hypersurfacecorrespondingto infinite externaladvanced ,v time. Time-dependent perturbations which originate in θ :=r−2 (r2) , (2.6) ,u the external universe are gravitationally blueshifted as ν :=λ , (2.7) e ,v they propagate inwards near to the CH. Thus a charged ν :=λ , (2.8) black hole which deviates only slightly from Reissner- ,u Nordstro¨m in the exterior is expected to have a barrier − whereacommadeenotespartialdifferentiation. AlongS ofradiation,withanexponentiallydivergingenergyden- there are two evolution equations sity,streamingalongparalleltotheCH.Genericallysome ofthisingoingradiationscattersoffthegravitationalpo- e−λ (r2θ) = (q2 r2) (2.9) tential inside the black hole, leading to a flux of energy ,u r2 − crossingtheCH[12]. Itisthenon-lineargravitationalin- e−λ teractionofthesetwofluxesofradiationwhichgenerates (θ 2ν) = (r2 3q2), (2.10) − ,u r4 − adivergenceofthelocalmassfunction. Itisimportantto realisethatthe gravitationalblueshiftoftime-dependent and a focussing equation which describes the behaviour perturbations is the key ingredient producing this mass- of the null cones with vertices at r =0: inflation. (r2θ) +r2(ν θ/2)θ= 2L (u). (2.11) ,u out − − The compelete set oef Eiensteien equations, including those The Poisson-Israel model whichholdonS+,maybeobtainedfromtheaboveequa- tions(2.9),(2.10)and(2.11)viathesymmetryoperation Itis convenientto use nullcoordinatesonthe “radial” two spaces so that the spherical line element is (u;θ,ν,θ,ν) (v;θ,ν,θ,ν), → ds2 =−2e−λdudv+r2(dx2+sin2xdy2), (2.1) e e e e (2.12) L L . out in where λ = λ(u,v), r = r(u,v) and the coordinates are → such that u is a retarded time and v an advanced time. It is convenient to imagine that the inflow (outflow) is Thestress-energytensorforaradialelectromagneticfield turnedonatsomeadvanced(retarded)timev =v (u= 0 is u ). Inthepureinflowregimethespacetimeisdescribed 0 by a charged Vaidya solution E ν =q2/8πr4diag( 1, 1,1,1), (2.2) µ − − ds2 =dw(2dr fdw)+r2(dx2+sin2xdy2), (2.13) − where q is the electric charge on the black hole. Pois- son and Israel used crossflowing null dust to model the where f is given by perturbations of the geometry, arguing that the large 2m(w) q2 blueshift near to the Cauchy horizon should make the f =1 + , (2.14) − r r2 Isaacson [13] effective stress energy desciption valid for the ingoing radiation. They also pointed out that the and w is the standard external advanced time coordi- nature of the outflux is not important, its only purpose nate. In particular it is infinite both on future null in- is to initiate the contractionof the Cauchy horizon. The finity and the CH. The only non-zero component of the stress-energy tensor is therefore stress-energy tensor is 2 1 dm Oftheremainingtwoequationsweonlyneed(2.9)which T = . (2.15) ww 4πr2 dw gives In the outflow region the solution is of the same form u q2 1 with advanced replaced by retarded time. The structure r−2 θ− =ro2θo+Z du˜ (cid:18)r˜3 − r˜(cid:19) . (2.19) of the spacetime is shown in Fig. 1. 0 Eq. (2.10) could be integrated in the same way. The same construction works on S+ (we will use this C asour primarytoolinthe non-sphericalcase)howeverit SH v = 0} itsomdeotreercmoinnveenθi0e,nθt0toanwdorrk0 wheitrhe.theexactsolution(2.13) f = 0, r = r o e S Assumptions r = ∞ Thetwoessentialfeaturesofthismodelwillbeadopted in the non-spherical case with only slight modification. We wish to emphasise them at this point. u0 The first assumption is that there should exist a sta- S+ v0 tionary portion of the CH in the spacetime. This is H E achievedbyturningontheoutfluxatsomeretardedtime (u ) inside the event horizon of the black hole (see Fig. 0 1), thus set L (u)=βH(u u ), (2.20) out 0 − whereβ isaconstant,andH(u)isaHeavisidestepfunc- FIG. 1. A spacetime diagram showing the spherical Pois- tion. This is probably the most contentious issue in the son-Israel model. The influx (outflux) of lightlike dust is turned on at advanced (retarded) time v0 (u0). The surface Poisson-Israel[6]andsubsequentanalyses[8]. It presup- S− coincides with the Cauchy horizon, CH. S+ is parallel to posesthatthe CHbegins atfinite radius,andessentially the event horizon, EH, and intersects the Cauchy horizon at assumesthatthesingularityattheCHwillbenull. Some S, on which f =0 and r=ro. argumentshavebeen advancedto suggestthat this issue is important [14], however no convincing evidence has emergedtorefuteitscorrectness. Indeedpreliminarynu- The solution merical results [15] support this assumption, in contrast to the claim in [16]. Asmentionedaboveourtreatmentdiffersfromthatin The second assumption pertains to the influx of radi- the literature [6,7]. We consider the solution to the Ein- ation; it is taken to decay with an inverse power law of steinequations onlyonthe twointersectingnull surfaces advancedtime. Thiswasinitiallybasedonanextrapola- S− and S+. By choosing S− to coincide with the CH tionofresultsdue toPrice[9]totheeventhorizonofthe we can show that a scalar curvature singularity must be blackhole. Recentlyithasbeenverifiednumerically[17]. present on the CH in the cross-flow region. This decay is correctly reproduced by the ansatz We first integrate the equations on the CH. This is αr readily achieved by making an appropriate choice of the m(w)=mo o (κow)−(p−1) , (2.21) − (p 1) retarded time coordinate u, in (2.1), such that ν = θ/2 − − alongS . Then(2.8),(2.11)and(2.6)canbeintegrated, where κ is the surface gravity of the inner horizon, α in that order, to give e e is a dimoensionless constant which depends on the lumi- nosity of the collapsed star, and p 12 is an integer eλ− =r− , (2.16) (the numerical value derives from Pri≥ce’s analysis which θ− =(cid:18)ro2θo−2Z udu˜Lout(u˜)(cid:19)/r−2 , (2.17) sahnowinsvetrhsaetptohweerrapdiativ4el +tai4l ooff tahdevacnoclleadpsteimdee,cawyhseares 0 ≥ e e u uˆ l is the multipole order of the perturbation). Both lin- r−2 =ro2+ro2θou−2Z0 duˆ Z0 du˜Lout(u˜). (2.18) ethaart[3t,h4i]sainndvenrsoenp-loinweearr-l[a8w]pdeerctauyrboaftpioenrtaunrbaalytsioesnssuisggaelssot e correct near the CH inside the black hole. The(sub-)super-script( )isusedtoindicatethatthese −− equalities hold only on S , while an (o) indicates the value of the function on the two-sphere S = S− S+. ∩ 3 Mass inflation − = C Cαβγδ =12 1 q2 + r−θ−θ− 2 , C αβγδ (cid:12)S− (cid:18)r−2 − r−4 2 (cid:19) Along S+ the radius obeys the first order equation (cid:12) e (2.31) dr 1 2m(w) q2 = 1 + (2.22) from which we can immediately see the reason for the dw 2(cid:18) − r r2(cid:19) − − Cauchy horizon singularity. The product θ θ =0 at S − (see Fig. 1) and hence is finite, however, if β is non- where m(w) is given by (2.21). We are interested in cal- − C e zero jumps to infinity at u = u when the outflux culating C 0 is turned on [see Eq. (2.9)]. Furthermore Eq. (2.30) θ+ =2r−1dr+dw (2.23) shows that the radius remains finite for some amount of + dw dv retarded time, so it must be the mass function which becomes infinite at u=u . 0 where v is the coordinate which appears in (2.1) chosen This analysis emphasises that the nature of the out- so that λ=lnr alongS+. It is a straightforwardmatter flux is largely irrelevant, it simply serves to initiate the to calculate dw/dv provided one notes that Eqs. (2.5)- contraction of the CH. It is this contraction which is at (2.11) are form invariant under functional rescalings of the root of mass-inflation [6]. the coordinates u and v. Therefore r also satisfies d2 d dw d dm (r2)+ ln (r2)= 2 (2.24) III. NON-SPHERICAL BLACK HOLES dw2 dw (cid:18) (cid:20)dv(cid:21)(cid:19)dw − dw Charged spherical black holes have served as a useful on S+ in view of (2.12) and (2.11). Solving (2.22) for r, modelinwhichtoinvestigatethenon-linearinstabilityof (2.24) can be integrated to get the CH inside a black hole, however, the question needs v e−κow(1+...) (2.25) to be addressed in the more realistic non-spherical con- ≃− text. Here one expects that a black hole settles down as w . Thus to a member of the Kerr-Newman family at late times. →∞ (Our analysis focuses on such a situation when there is r r + α(−ln|v|)−p+1 1+(p 1)( ln v )−1+... no electromagnetic field.) + ≃ o (p 1)κ − − | | The Kerr solution can be written as − o (cid:8) (cid:9) (2.26) ∆ sin2θ ds2= (dw adφsin2θ)2+ ([r2+a2]dφ adw)2 −ρ2 − ρ2 − and +2dr(dw adφsin2θ)+ρ2dθ2 (3.1) 2α( ln v )−p − θ+ − | | 1+p( ln v )−1+... (2.27) where ∆ = r2 2mr + a2, ρ2 = r2 +a2cos2θ and w ≃ r κ v − | | o o (cid:8) (cid:9) is standard adv−anced time. The CH, located at r = as v 0. This result means that, while the radius of m √m2 a2 and w = , is a stationary null hyper- the tw→o-sphereS is finite, θo is actually infinite (in these sur−face – it−s lightlike gene∞ratorshave zero rate of expan- coordinates). sion. Moreover,thesegeneratorsareshearfreesinceRay- Usingtheinvariantdefinitionofthemassinaspherical chaudhuri’s equation implies that shear produces con- geometry traction. This is not a generic situation; a linear perturbation analysis shows that there is usually a flux of 2m(xα) q2 r2eλθθ backscattered radiation crossing the CH, along with the 1 + :=gαβr r = (2.28) − r r2 ,α ,β − 2 blueshifted influx parallel to it [4]. Based on this obser- e vation and using the spherical case as a guide, we will we know that θ+θ+ 0 as v 0, which means that show that the CH is the locus of a scalar curvature sin- → → θo =0. Substituting (2.20) into (2.17) and using θo =0 gularity. The leading divergence may be associated with e propagatingmodesofthegravitationalfield,manifesting e r−2 θ− = 2(u u0)βH(u u0). e(2.29) itselfinthe formofadivergingWeylcurvatureofPetrov − − − type-N. The radius oef the CH is then r−2 =ro2−(u−u0)2βH(u−u0), (2.30) Formalism − andfinallyθ isgivenby(2.19). ThesquareoftheWeyl We employ the dual null formalism of Hayward [11] curvature on the CH is which is based on a 2+2 null decomposition of spacetime [18]. In terms of coordinates u and v which label 4 1 the intersectingnullhypersurfacesthatfoliatethespace- ν+ = θ+ (3.10) time, the line element can be written as −2 ds2 = 2e−λdudv+(s sa)du2+2s dxadu+h dxadxb it is possible to obtainthe solution to the Einstein equa- − a a ab tions on S+. Each of the equations (A7)-(A15) can be (3.2) integrated(intheordertheyappear)providedoneknows k on this hypersurface, since ab where the lower-case latin indices range over 1,2 . There remains freedom to rescale u and v, and to{tran}s- σ+σab =(kab∂ k+)(kcd∂ k+). (3.11) ab + + v bc + v da form the coordinates xa so that sa vanishes on a chosen v =constantsurface,wetakethistobetheCHlocatedat We cannot evolve generic initial data from the event v =0(see Fig.2). h is the 2-metric, sa ashift 2-vector horizonofanon-spherical,rotatingblackholesowemake ab and λ a scaling function on S(u,v), the spatial 2-surface anansatz for this conformal2-metric. It is inferred from whichistheintersectionofthenullsurfaceslabelledbyu the work of Ori [8] that the conformal metric may be andv. The2-metriccanbedecomposedintoaconformal written as factor Ω and a conformal 2-metric k so that ab k+ =ko (xa)+K+(ln v ,xa) (3.12) Ω=√h, k =Ω−1h . (3.3) ab ab ab | | ab ab where det(ko )=1 and asymptotically ab Thus k has unit determinant. Each of the quantities definedahbere depend on all four variables u,v,xa. Ka+b ≃Fab(xc)(−ln|v|)−n+... . (3.13) ThevacuumEinsteinequations(A7)-(A23)arewritten as v 0. The functions (xc) are well behaved for in a first order form in terms of the fields all xa→, and n is an integer.F(aTbhe value implied by Ori’s θ =Ω−1 Ω, θ =Ω−1 Ω (3.4) work is n 6.) This form is based on the observation Ll Ln that non-li≥near metric perturbations decay according to σ =Ω k , σ =Ω k (3.5) ab Ll ab eab Ln ab aninversepower-lawofadvancedtime[8],andtheexpec- ν = lλ, ν = nλ (3.6) tation that Hayward’s coordinate is related to external L e L 1 advanced time by ωa = eλΩkab lsbe, (3.7) 2 L v e−κow(1+...) (3.14) where indicates the Lie derivative, and ≃− L as v 0 (or w ). While this cannot be proven to l= ∂ n= ∂ sa ∂ (3.8) hold,→it seems re→aso∞nable as long as the surface S+ does ∂v ∂u − ∂xa not encounter a caustic (Ω = 0) at or before the CH – this condition is satisfied a postiori implying, at least, a are null vectors. The notation and the equations are self-consistent treatment. given in appendix A, more detail may be found in Hay- As in the spherical analysis, we give no detail of the ward [11]. − gravitationaldataonS savetopointoutthatthegrav- Our aim is to calculate the Weyl curvature scalars on theintersectingnullhypersurfacesS+ andS− bymaking itationalenergy flux crossingthe CHshouldnot diverge. Infact,onphysicalgrounds,itisexpectedthatanyradi- an ansatz for the free gravitational data (k ) on these ab − ation from the star will be exponentially redshifted near surfaces. S is taken to coincide with the CH at v = 0, P inFig.2. Furthemoreoncetheincomingradiationgets whileS+ isanoutgoingnullhypersurface,paralleltothe − scattered by the gravitational potential inside the black eventhorizon,crossingS neartoP inFig.2. Thechoice hole, it should only produce a slow contraction of the of the data is motivated by our understanding of the CH [12] spherical situation and by the non-linear perturbation Tosummarise,weassumethataportionoftheCHex- analysis of Ori [8]. istswhichiscausticfreeneartoP inFig.2. Furthermore, thegravitationalperturbationspropagatingintothehole decayaccordingtoaninversepowerlawofadvancedtime Assumptions which is related to v by (3.14). The focussing equation on S+ is [Eq. (A7)] ∂θ 1 1 = θ2 νθ σ σab . (3.9) ab ∂v −2 − − 4 Haywardnoticed that making use of coordinate freedom on S+ (rescaling v), this equation may be linearised. Thus choosing v such that 5 as v 0. Since σ σab is v−2 times a slowly varying ab functi→on (near the CH), θ+ is asymptotically C SH v = 0 θ+ ≃θ0(xa)+ 4nv2(koablnkocdvF)b2c(nF+d1a) +... (3.19) − | | whichdivergesattheCHforthesamereasonthat∂ K+ P v ab does. It is now straightforward to formally integrate the re- mainingequations,howeverourmainobjectiveistoshow = 0 +: u which quantities diverge and which remain bounded at S the CH. Observing that the integral v0 I = vv˜−1( ln v˜)−ndv˜= 1 ( ln v )−n+1 (3.20) H E Z − | | n 1 − | | − is finite as v 0 (provided n>1), it is clear that → v 1 dv˜ v˜−1K+ ( ln v )−n+1 (3.21) Z ab → n 1 Fab − | | (cid:0) (cid:1) − FIG. 2. A schematic representation of the generic black as v 0, and remains bounded at the CH. Each of the → hole interior. Thenull surface S− coincides with the Cauchy equations must be treated in turn, examining the diver- horizon,CH.Theothernullsurface S+ isatu=0andinter- gent part of the source and its integral near the CH. We sects the Cauchy horizon at S before a caustic forms. Initial quickly see that Ω is bounded at the CH, and so all the values of the fields are specified on the intersection between intrinsicquantitiesonthetwodimensionalsurfacesoffo- thehypersurfacev=v0andS+. Alsoindicatedistheingoing liation remain finite all the way to the CH. To see this, gravitational wave tail, and its scattered component first integrate (A8) to get 1 v λ =λ dv˜θ+ , (3.22) Solution on S+ + o− 2Z vo then substituting in the asymptotic form for θ+ we find With this aspect in hand we can proceed to the solutions of Eqs. (A7)-(A14). Integrating (3.9) v n2 v dv˜θ+ kabkcd dv˜v˜−1( ln v˜)−2(n+1) 1 v Z ≃ 4 o o FbcFdaZ − | | θ+ =θo(xa) dv˜σ+σab (3.15) n2 − 4Zv0 ab + ≃ 4(2n+1)koabkocdFbcFda(−ln|v|)−(2n+1) (3.23) A (sub) super-script o indicates the initial value of the and hence λ constant as v 0. The gauge choice functionatv inS+ (seeFig. 2). Onthis twosurfacewe + → → 0 (3.10) implies that expect allthe fields to be wellbehaved,bounded (andin some cases non-zero) functions of xa. Ω+ =Ω e−2(λ+−λo) (3.24) o Based on the assumption (3.12) and (3.13) about k+ ab wecanestimatethebehaviourofthisintegralneartothe and so Ω and kab are both bounded at the CH. Now CH.Firstlynotethattheinverseoftheconformalmetric continuing the integration can be written ω+ =ωoe2(λ+−λo) (3.25) b b kab =kab+ǫacǫbdK+ (3.16) v + o cd +1e2λ+ dv˜e−2λ+( aσ+ 3 θ+ θ+ λ ), 2 Z △ ab− 2△b − △b + where ǫbc is the 2-dimensional, antisymmetric matrix vv0 with ǫ12 =1. The functional form of K+ implies that sb =sb +2 dv˜e−λ+ωb (3.26) ab + o Z + v0 1 ∂K+ ∂ K+ = ab (3.17) and v ab v∂ln v | | θ+ =θ e2(λ+−λo) (3.27) o will diverge since v e +ee2λ+ dv˜e−3λ+ 1(2)R+ωaω+ 1 a λ ∂Ka+b n ( ln v )−n−1 (3.18) Zvo (cid:18)−2 + a − 2△ △a + ab ∂ln v ≃ F − | | +1 aλ λ +ωa λ ωa , | | 4△ +△a + +△a +−△a +(cid:19) 6 fromwhichitisclearthatonlyfinitetermsappearinthe The integration now proceeds as it did for S+, and in − − equation for θ, which therefore cannot diverge on CH. particular we arrive at expressions for θ and σ ab theσesa+hbe≃ar(fiofnietthee)−ing21oeiλn+gh+ancuZllv0vrdav˜yse−oλr+thθeohgcodnσadbl ,to S(3(u.28=) θ− =+e2eλ2−λ−(cid:2)eZ−02uλd+u˜θ+e−(cid:3)v3λ=(cid:0)0−21(2)R+ωaωa (4.3) 0,v), is also bounded at the CH. Finally 1 a λ + 1 λ aλ ωa λ+ ωa − 2△ △a 4△a △ − △a △a v (cid:1) ν =ν + dv˜ 1σ+σab 1θ+θ+ (3.29) and e++ ee−oλ+(Z−vo12(2)Rn4+a3bωea++ω−+a −2 41△eaλ+△aλ+−ω+a△aλ+)o σa−b =eλ−h−fahe−λ+uhf+dσd+biv=0 (4.4) 1h− eλ− du˜ e−λθhdfσ 4e−2λhdf ω ω which is finite despite the presence of divergent terms − 2 fa Z db− (d b) in the integrand [which look like v−1 inverse powers 0 (cid:0) (cid:8) of ( ln v ), see Eq. (3.20)]. This ana×lysis implies that − 12△(d△b)λ+ 41△(dλ△b)λe−ω(d△b)λ+△(dωb) θg+enca−endisσ|oaf+|bthdeivsearmgeeninastiudreetahseinbltahcekshpohleer,icaanldmtohdeedloivnecre- −h−abeλ−Z udu˜e−2λ ωcωc− 12△c△cλ+ 41△cλ△(cid:9)c(cid:1)λ 0 (cid:0) σabσab isinterpretedasthe gravitationalenergycrossing ωc cλ+ cωc . S+. − △ △ (cid:1) We saw in the previous section how IV. THE CAUCHY HORIZON SINGULARITY σ+ lim ab , (4.5) v→0(cid:26)θ+ (cid:27)→∞ The solution obtained in the previous section can be − − usedtoshowthatascalarcurvaturesingularityispresent thusθ andσ aregenericallyunboundedontheCH.Of ab on the CH due to the blueshifted influx of radiation. On theremainingfields,onlyν containsdivergentquantities. S− theshiftvectorsaissettozerobyacoordinatetrans- From Eq. (A23) we see that formationxã =xã(u,xb)withoutalteringtheformofthe umleatrroicn(S3.−2)i.n(Tthheeosericgoinoardlicnoaotredsinexatisetspyrsotvemid.edEsqa. i(s3r.2e6g)- ν− ≃Z udu˜(14σa−bσ−ab− 12θ−θ−) + ... , (4.6) 0 suggests that this is true.) It is assumed that this trans- e e whichisactuallyinfiniteonCH.Imposingthefieldequa- formation was made in section III; this in no way alters tionsontheWeylscalarsonefindsthatallbutΨ contain the previous analysis. 4 − divergent quantities, and hence the CH is singular. Hayward’s scheme to integrate the equations on S It is possible to say a little more about the nature can now be completed. Making the gauge choice of the singularity by examining the rate of growth of ν− = 1θ− , (4.1) the curvature along S+. Once again, imposing the field −2 equations in Eq. (B11) gives Eq. (A16) implies tehat e Ψ+ = 1e2λmamb 2 σ +2νσ σ hmnσ . u 0 4 { Ll ab ab− am nb}|+ θ− = θ+ 1 du˜(kab∂ k−)(kcd∂ k−). (4.2) (cid:12)v=0− 4Z0 − u˜ bc − u˜ da (4.7) e e (cid:12) (cid:12) − Using σ+, etc. the asymptotic expression is Asmentionedintheintroductionθ (andallotherquan- ab tities) are formally known once k−(u,xc) is specified. − eab Ψ+ (finite) v2( ln v )n+1 −1 (4.8) Ouranalysisassumesthatkabisaslowlyvaryingfunction 0 ≃ × − | | of u which tends to a non-degenerate value as u ; (cid:8) (cid:9) →−∞ as v 0. This is the contribution from σ+, which is that is, near to P (in Fig. 2) kab should approach the → Ll ab dampedbythesmallestnumberofpowersof( ln v ). In value obtained on the CH in Kerr spacetime. Further- − | | − a similarmanner the leadingbehaviourcanbe extracted moreθ 0andthespacetimeshouldbeasymptotically → from Ψ and Ψ giving Kerr in the limit u . It is not a generic situation 1 2 e − → −∞− to have θ = 0 and/or σab = 0 on the CH, since some Ψ+ (finite) v( ln v )n+1 −1 (4.9) radiation (both from the star, and from backscattering 1 ≃ × − | | ofthe graevitationalwaveetailinsidetheblackhole[8,12]) Ψ+ (finite) (cid:8)v( ln v )n+1(cid:9)−1 (4.10) willusuallybepresent. Thisdiscussionimpliesthatthere 2 ≃ × − | | (cid:8) (cid:9) should exist a caustic free portion of the CH near to P where these leading contributions arise due to the pres- on which our analysis is valid. enceoftermsinvolvingtheshear(σ+)ofthesurfaceS+. ab 7 AlittlemoreworkshowsthatΨ+ isfiniteontheCH;no- the CH and are manifest in the square of the Weyl ten- 3 tice that eq. (3.26) combined with the choice sa 0 on sor(4.11)(whichvanishesforapuregravitationalshock). S−,andknowledgethatλ+ andω+ arefiniteonS→+ S− Thus a classical continuation is unlikely, as pointed out implies that s+ v as v 0, heance smΣ+ sn ∩0 on in[21],sincenon-classicalmatterisneededtoconfinethe S−. Finally Ψa i∼s indepen→dent of θ+ an+d σm+na+nd→there- divergent curvature to a thin layer along the CH. 4 ab fore is finite. These results imply that the square of the Toconclude,thehairysingularityinsideagenericblack Weyl tensor is dominated by Ψ , hole is dominated by the propagatinggravitationalwave 0 tail of the collapse (which decays in the exterior of the CαβγδCαβγδ 8(Ψ0Ψ4+Ψ0Ψ4)+... black hole). ≃ (finite) v2( ln v )n+1 −1+... (4.11) ≃ × − | | (cid:8) (cid:9) ACKNOWLEDGMENTS as v 0. → It is a pleasure to thank Werner Israel for comments V. DISCUSSION anddiscussionsontheinterpretationofourresults. CMC isfunded bythe EPSRCofGreatBritain,andgratefully The treatment of the non-spherical gravitational col- acknowledgesthehospitalityofIanPinkstoneandJames lapse is generally very difficult, however it seems likely Briginshaw of DAMTP, Cambridge where part of this that the external field settles down to that of a Kerr workwasperformed. PRBisgratefultoPeterHoganand black hole. Inside the black hole some progressmay also Adrian Ottewill for their kind hospitality at University be made by advancing two assumptions: 1) There exists College, Dublin where part of this work was carriedout. a caustic free portion of the CH, which is a null hypersurface. 2) There is an ingoing gravitational wave tail which gets blueshifted by an infinite amount at the CH (this is achieved here by our ansatz for k+). Indeed us- ab ing Hayward’s formalism the nature of the CH can be investigated in some detail. Following from the analysis in sections III and IV we conclude that the CH is the locus of a scalar curvature singularity,andthecurvatureisanintegrablefunctionof the advanced time. Ori [7,8] has argued that the latter property of the curvature suggests an extension of the spacetime through the singularity may be possible. It is worthcommentingalittlefurtheronthispoint. Wehave demonstrated the asymptotic values of each of the Weyl scalars,in particular it has been shownthat Ψ contains 0 the leading divergences. The algebraic classification of the Weyl curvature is obtained by solving the quartic Ψ a4+Ψ a3+Ψ a2+Ψ a+Ψ =0 (5.1) 0 1 2 3 4 for a and examining the degeneracy of its roots [19]. Basedoneqs.(4.8)-(4.10)there is a four-folddegeneracy as v 0 implying that the gravitational field is asymp- → totically type N, with repeated principal null direction k = n+am+a∗m¯ +aa∗l (where a 0 as v 0). → → It is worth noting that gravitational shock waves are of this algebraic type (and are the only curvatures which may be confined to a thin skin without a surface layer of matter being present [20]); therefore the intuitive pic- ture of the singularity which emerges is that of a gravitational shock propagating along the CH. These results mightbetakentosupportargumentsfortheexistenceof a classical continuation of the geometry beyond the CH singularity. However, this is a highly speculative point and it should be noticed that the Weyl curvature contains contributions which are not present in an exactly type N spacetime. Indeed these contributions diverge at 8 APPENDIX A: NOTATION In this appendix we summarise the dual null formalism of Hayward [11]. The reader is referred to his articles for more technical details. This approach, based on a 2+2 decomposition, assumes that the spacetime can be foliated locally by compact orientable 2-surfaces S. Therefore given a Lorentzian manifold (with signature +++) we M − assume that there exists a smooth embedding φ : S [0,U) [0,V) with the induced metric, h , on S ab × × → M being spatial. Latin indices range over 1,2 . In terms of a basis which is Lie-propagated along ∂/∂u and ∂/∂v the { } spacetimeline elementmaybewrittenasinEq.(3.2). TheLie derivativealongavectorξ isdenoted . The natural ξ L covariantderivative of the metric h is , and the Ricci scalar is denoted by (2)R. Tensor valued quantities on the ab a 2-surfaceS aredenotedXa... (sometim△eswewillincludeasuperscript(2)toemphasisethe2-dimensionalcovariant b... nature). Introducing the null vectors lµ =(1,0,0,0), nµ =(0,1, sa). (A1) − we define Σ = h , Σ = h . (A2) ab l ab ab n ab L L To write the Einstein field equations in a first order formeHaywardintroduces the extrinsic fields: θ = 1habΣ , θ = 1habΣ , (A3) 2 ab 2 ab σab =Σab θhab, σaeb=Σab eθhab, (A4) − − ν = λ, ν = λ, (A5) Ll e Len e ω = 1eλh sb . (A6) a 2 abLl e These quantities are readily interpreted geometrically; (θ,θ) are the expansions in the null directions normal to the foliation S. The tensors (σ ,σ ) are the shears, which are traceless by definition, the vector ω is the twist. The ab ab a e two functions (ν,ν) are called the inaffinities and measure the non-affine quality of the coordinates u and v. Then the Einstein equations and conetracted Bianchi identities are: the l equations: e θ = 1θ2 νθ 1σ σab (A7) Ll −2 − − 4 ab λ =ν (A8) l L Ω =θΩ (A9) l L k =Ω−1σ (A10) l ab ab L ω = θω + 1 bσ + 1 (ν θ) 1θ λ (A11) Ll a − a 2△ ba 2△a − − 2 △a sa =2e−λωa (A12) l L θ = θθ+e−λ( 1(2)R+ω ωa 1 a λ+ 1 λ aλ+ωa λ ωa) (A13) Ll − −2 a − 2△ △a 4△a △ △a −△a Llσeab =σacehcdσdb+ 21θσab− 21θσab e +2ee−λ(ωaωb−e 21△(ae△b)λ+ 14△(aλ△b)λ+ω(a△b)λ−△(aωb)) e−λh (ω ωc 1 c λ+ 1 λ cλ+ωc λ ωc) (A14) − ab c − 2△ △c 4△c △ △c −△c ν = 1σ σab 1θθ+e−λ( 1(2)R+3ω ωa 1 λ aλ ωa λ) (A15) Ll 4 ab − 2 −2 a − 4△a △ − △a and the n equatieons: e e θ = 1θ2 νθ 1σ σab (A16) Ln −2 − − 4 ab λ =ν (A17) Lne e ee e e Ω =θΩ (A18) Ln e k =Ω−1σ (A19) n ab e ab L ω = θω 1 bσ 1 (ν θ)+ 1θ λ (A20) Ln a − ea− 2△ ba− 2△a − 2 △a Lnθ =−θeθ+e−λ(−e12(2)R+ωaωea−e21△a△eaλ+ 14△aλ△aλ−ωa△aλ+△aωa) (A21) Lnσab=σacehcdσdb+ 21θσab− 12θσab e e e 9 +2e−λ(ω ω 1 λ+ 1 λ λ ω λ+ ω ) a b− 2△(a△b) 4△(a △b) − (a△b) △(a b) e−λh (ω ωc 1 c λ+ 1 λ cλ ωc λ+ ωc) (A22) − ab c − 2△ △c 4△c △ − △c △c ν = 1σ σab 1θθ+e−λ( 1(2)R+3ω ωa 1 λ aλ+ωa λ) (A23) Ln 4 ab − 2 −2 a − 4△a △ △a e e APPENDIX B: RIEMANN TENSOR AND WEYL SCALARS 1. Riemann Tensor Here we list all the components of the Riemann tensor for the metric (3.2): Rv = ν sa ν+ 1eλsaΣ sbν+2sa ω e−λω aλ+3e−λωaω +saΣ ωa vuv −Ll − △a 2 ab Ll a− a△ a ab + 1eλsa( Σ )sb+ 1saΣ bλ 1e−λ aλ λ 1eλsaΣ hbcΣ sd (B1) e2 Ll ab 2 ab△ − 4 △ △a − 4 ab cd Rv = 1 ν+ 1eλνsmΣ + 1ωmΣ + ω + 1eλsm Σ 1eλΣ hmnΣ sq+ 1 mλΣ (B2) vua −2△a 2 ma 2 ma Ll a 2 Ll ma− 4 am nq 4△ ma Rv = 1eλ Σ + 1eλνΣ 1eλΣ hmnΣ (B3) aub 2 Ll ab 2 ab− 4 am nb Rv = 1eλsmΣ λ+eλsm Σ eλsmΣ ω +2 ω + 1eλΣ Σ hnm (B4) vab 2 m[b△a] △[a b]m− m[a b] △[a b] 2 n[a b]m Racbd =−eλsc△[bΣd]a− 21eλscΣa[d△b]λ+ 12eλhcm(Σa[dΣb]m−Σm[dΣb]a)−eλescΣa[dωb]+(2)Rcabd (B5) Ravcd = 21eλ△[cλΣd]a+eλ△[cΣd]a+eλΣa[dωc] e e (B6) Rv = 1eλ Σ + 1eλsm( Σ Σ )+ 1 λ 1 λ λ ω avb 2 Ln ab 2 △m ab−△b am 2△a△b − 4△a △b −△b a + 1eλΣ wcs + 1eλsc λΣ 1eλΣ hmnΣ + 1ω λ+ 1ω λ ω ω 2 ab c 4 △c ab− 4 mb na 2 a△b 2 b△a − a b − 21eλωbsmΣam− 14eλ△bλΣmasm e (B7) Rv = 1eλsm Σ + ω + 1eλsm(sn )Σ 1eλΣ hmnΣ sq+ 1 ν+ 1(sm ) λ vva 2 Ln ma Ln a 2 △n ma− 2 ma nq 2△a 2 △m △a −2sm△aωm− 41eλ△aλsmΣmnsn− 21eλsm(△aΣmn)sn−e41sm△mλ△aeλ+ 14eλsmΣmasn△nλ + 1sm λω 1Σ mλ 1eλ(smΣ sn)ω + 1eλsmΣ hnqΣ + 1(smω ) λ 2 △m a− 4 am△ − 2 mn a 4 mn aq 2 m △a + 21eλ(snωn)smΣmea−(smωm)ωa+ 12ωmΣma+sm△mωa e (B8) Rauvb = 21eλΣabν+ 14eλsm△mλΣab+ 21eλLnΣab+ 12eeλsm△mΣab− 41eλ△bλsmΣam− 12eλsm△bΣam −e21eeλ(smwm)Σab−e41eλΣbmhmnΣean+ 12eλsmΣameωb e e (B9) Racvb = 21hcm(△aΣmb−△emΣab)−hecnsm(2)Remban− 12eλsecLnΣab− 21eλscsm△mΣab− 12sc△b△aλ + 41eλsec△bλsmΣaem+ 21eλscsm△bΣam+sc△bωa+ 14sc△aλ△bλ− 21scωb△aλ+ 14△aλhcmΣmb 1 + 1eλsmΣ scw eλsnΣ hcmΣ 1scω λ+scω ω 1hcmΣ ω e 2 am b− 4 an mb− 2 a△b a b− 2 mb a 1eλωms scΣ + 1eλhcmsnΣ eΣ + 1eλΣ 1hcmsnΣ +e−λwec 1e−λ cλ − 2 m ab 4 mn ab 2 ab 2 mn − 2 △ − 41eλsm△mλΣab+ 14eλscΣnbhenmΣma− 41eλsenΣa(cid:0)nhcmΣbm (cid:1) (B10) e e 2. The Weyl Scalars We choose a complex-null tetrad eλl,n,m,m such that 2m(µmν) =hµν and, l and n are givenby (3.8). In this { } tetrad the five Newman-Penrose Weyl scalars are in a Ricci flat spacetime Ψ = 1e2λ(2 Σ +2νΣ Σ hmnΣ )mamb (B11) 0 4 Ll ab ab− am bn Ψ = 1eλ( 2 ν+2ωmΣ +4 ω + mλΣ )ma (B12) 1 4 − △a am Ll a △ ma Ψ = 1(2eλ Σ +2 λ λ λ 4 ω eλΣ hmnΣ +2ω λ 2 −4 Ln ab △a△b −△a △b − △b a− mb an a△b +2ωb aλ 4ωaωb+eλ bλsmΣam+2eλsmΣamωb)mam¯eb (B13) △ − △ 10