Trapping Horizons as inner boundary conditions for black hole spacetimes
J.L. Jaramillo, E. Gourgoulhon, I. Cordero-Carrion, J.M. Ibanez
aa r X i v : . [ g r- q c ] S e p Trapping Horizons as inner boundary conditions for black hole spacetimes
Jos´e Luis Jaramillo,
1, 2, ∗ Eric Gourgoulhon, † Isabel Cordero-Carri´on, ‡ and Jos´e Mar´ıa Ib´a˜nez § Instituto de Astrof´ısica de Andaluc´ıa, CSIC, Apartado Postal 3004, Granada 18080, Spain Laboratoire Univers et Th´eories (LUTH), Observatoire de Paris, CNRS,Universit´e Paris Diderot ; 5 place Jules Janssen, 92190 Meudon, France Departamento de Astronom´ıa y Astrof´ısica, Universidad de Valencia, Valencia, Spain (Dated: 18 september 2007)We present a set of inner boundary conditions for the numerical construction of dynamical blackhole space-times, when employing a 3+1 constrained evolution scheme and an excision technique.These inner boundary conditions are heuristically motivated by the dynamical trapping horizonframework and are enforced in an elliptic subsystem of the full Einstein equation. In the station-ary limit they reduce to existing isolated horizon boundary conditions. A characteristic analysiscompletes the discussion of inner boundary conditions for the radiative modes.
PACS numbers: 04.25.Dm, 04.70.Bw, 02.60.Lj
General problem.
The aim of this report is to discuss aset of inner boundary conditions (BC) for dynamical evo-lutions of black hole spacetimes using an excision tech-nique. These BCs are derived in the context of the dy-namical trapping horizon framework [1, 2, 3, 4]. In paral-lel with the recent black hole numerical studies based onfree evolution schemes, which have led to the successfulsimulations of binary black hole coalescence through themerger phase (see e.g. [5] for extensive references), a 3+1scheme for a fully-constrained evolution of Einstein equa-tion has been presented in Ref. [6]. This approach maxi-mizes the number of elliptic equations to be solved duringthe evolution, resulting in a coupled elliptic-hyperbolicPDE system [7]. Spectral methods [8] are then employedboth to solve the elliptic subsystem and to handle thespatial part of the relevant hyperbolic operators. We dealwith the black hole singularity by means of the excisiontechnique. This raises the question about the appropriatechoice of inner BCs on the excised sphere, both for theelliptic and the hyperbolic parts of the system. Regard-ing the hyperbolic equations, this inner boundary issueis intimally related to the metric type of the world-tubehypersurface generated by the time evolution of the ex-cision sphere. As observed in Ref. [9], certain choicesfor the excision surface render this excision hypersurfacepartially time-like, leading to ill-posedness if inconsistentBCs are supplied for the radiative modes. A solutionto this problem is suggested by the quasi-local approachto the evolution of black hole horizons, embodied in the dynamical trapping horizon framework (see review arti-cles [3, 4] and also Ref. [10]). This formalism moti-vates a natural geometric choice for the excision surface.The basic underlying idea goes back to Eardley’s work[11] and consists in modeling the black hole horizons by ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] S × R world-tubes sliced by apparent horizons , that sat-isfy certain additional conditions guaranteeing the phys-ical growth of the horizon area (see below). On the onehand, apparent horizons at each given 3-slice of the timeevolution provide non-ambiguous geometric choices forthe excision sphere that are guaranteed to lay inside theevent horizon, and therefore are causally disconnectedfrom the rest of the spacetime. On the other hand, dy-namical trapping horizons are space-like hypersurfacessuggesting that no conditions must be supplied at theinner boundaries for the modes propagating in the bulk.In sum, this proposal recasts Eardley’s program [11] inthe dynamical trapping horizon setting. In the followingwe describe the fully-contrained scheme, then we presentinner BCs for the elliptic part that guarantee that theexcised sphere generates a (dynamical) trapping horizon,and finally we show that the combination of a Dirac-likegauge [6] and dynamical trapping horizon inner BCs forthe elliptic part of the PDE system, actually imply thatno BCs must be prescribed for the hyperbolic fields atthe inner excised sphere. Fully-constrained evolution scheme.
In the setting of thestandard 3+1 decomposition of a spacetime ( M , g ) byspatial slices (Σ t ), Ref. [6] proposes a particular initial-boundary problem for the spacetime evolution from aninitial Cauchy slice. Let us denote by n the unit time-like normal vector to Σ t , the spatial 3-metric by γ , i.e. γ = g + n ⊗ n , and define the extrinsic curvature of Σ t as K = − L n γ . The evolution vector t ≡ ∂ t is de-composed in terms of the lapse function N and the shiftvector β , as t = N n + β . In addition, we introduce afidutial flat metric f , satisfying L t f = ∂ t f ij = 0. Nowwe proceed by performing a conformal decomposition ofthe 3+1 fields: γ = Ψ ˜ γ , K = Ψ ˜ A + K γ , where K = γ ij K ij , the representative ˜ γ of the conformal classof the 3-metric is chosen to be unimodular, i.e. satisfiesdet(˜ γ ) = det( f ), and the traceless part ˜ A of K is writ-ten as ˜ A ij = N (cid:16) ˜ D i β j + ˜ D j β i − ˜ D k β k ˜ γ ij + ∂ t ˜ γ ij (cid:17) , ˜ D being the Levi-Civita connection associated with ˜ γ . In asecond step, a coordinate choice must be adopted. Fol-lowing the prescriptions in [6], namely maximal slicing and Dirac gauge , we set K = 0 , D k ˜ γ ki = 0 , (1)where D is the Levi-Civita connection associated withthe flat metric f (see Ref. [12] for a discussion and re-lation to other coordinate choices). Conditions (1) fixthe coordinates up to boundary terms. The Dirac gaugecondition will play a key role in the following, whereasmaximal slicing can be relaxed to an arbitrary K vanish-ing asymptotically near spacelike infinity. Inserting theconformal decomposition and gauges (1) into Einsteinequation results in a coupled elliptic-hyperbolic system[6]. The elliptic part can be written as˜ D k ˜ D k Ψ − ˜ R S Ψ [Ψ , N, β , ˜ γ ]˜ D k ˜ D k β i + 13 ˜ D i ˜ D k β k + ˜ R i k β k = S iβ [Ψ , N, β , ˜ γ ] (2)˜ D k ˜ D k N + 2 ˜ D k ln Ψ ˜ D k N = S N [ N, Ψ , β , ˜ γ ] , where the first equation on Ψ follows from the Hamilto-nian constraint, and the equation for the shift β resultsfrom the simultaneous imposition of the preservation ofthe Dirac gauge in time, i.e. ∂ t ( D k ˜ γ ki )=0, together withthe momentum constraint. The Dirac gauge ensures theelliptic character of this equation. Finally the third equa-tion follows from ∂ t K = 0. S Ψ , S β and S N represent non-linear sources given in Ref. [6]. Note the similarity withthe extended conformal thin sandwich elliptic system [13]for the construction of initial data. In the present con-text, Eqs. (2) are meant to be solved along the wholeevolution, not only on an initial slice. Regarding the evo-lution part, we solve for the deviation h of the conformalmetric from the flat fidutial one f , i.e. h = ˜ γ − f . Wechoose a second-order form for the evolution equations,that can be formally written as ∂ h ij ∂t − N Ψ ˜ γ kl D k D l h ij − L β ∂h ij ∂t + L β L β h ij = S ijh (3)where the nonlinear sources S ijh [ N, Ψ , β , ˜ γ ] do not con-tain second derivatives of h . Eqs. (2) and (3) are solvedin Ref. [6] inside a spacetime region bounded by an outertimelike tube at large spatial distances. We focus hereonly on the inner BC problem. On a first stage, dy-namical trapping horizon considerations will provide in-ner BCs for the conformal factor Ψ, the shift β and thelapse N . In a second step we will analyse the hyperbol-icity of the subsystem (3) and, most importantly in thepresent context, we will evaluate its characteristics fieldsand speeds to assess if inner BCs must be provided at allfor h .As mentioned above, we do not discuss here the impor-tant outer BC problem. In this sense, a very interestingalternative has been recently presented by Moncrief etal. [14]. They propose a (conformal) 3+1 constrainedscheme, which differs crucially from [6] in one point: the chosen slicing, involving constant mean curvature slices,extends up to future null infinity I + , a natural boundaryfor physical outgoing radiation conditions. This strategypermits to bypass the boundary problem at the outertimelike border. The feature of [14] we highlight in thecontext of the present work is the shared adoption ofan inner excision approach to the black hole singularityproblem. An alternative geometric choice for the innersurface is proposed in [14], namely the use of minimalsurfaces. However, our proposal of rather employing ap-parent horizons instead, straightforwardly translates alsointo their scheme. Inner BCs for the elliptic part: dynamical trapping hori-zons.
Quasi-local approaches to black hole horizons aimat modeling the boundary of a black hole region as world-tubes of apparent horizons ( S t ). At each point of a givenspacelike closed surface S t we can define (up to totalrescaling) two null vectors ℓ and k , satisfying k · ℓ = − S t . Denoting by q themetric on S t induced by the ambient metric g and by ǫ S the associated area element, we can define the expan-sion θ ( v ) and shear σ ( v ) along any vector v normal to S t by L v ǫ S = θ ( v ) ǫ S and 2 σ ( v ) = L v q − θ ( v ) q . Thesurface S t is trapped [15] if light rays emitted from itlocally converge: θ ( k ) ≤ θ ( ℓ ) ≤
0. In the lim-iting case in which one of the expansions vanishes, S t is called a marginally trapped surface (MTS). Since wewill deal with asymptotically flat 3-slices, we can unam-biguosly define an outgoing null normal, say ℓ , as theone pointing towards spacelike infinity. Then, condi-tion θ ( ℓ ) = 0 defines a marginally outer trapped sur-face (MOTS) [16]. In contrast with MTSs, MOTSs im-pose nothing on θ ( k ) . Apparent horizons are outermostMOTSs. In this context, quasi-local dynamical trappinghorizons H are S × R hypersurfaces sliced by MOTSs( S t ) and satisfying θ ( k ) <
0. Actually, slices ( S t ) are in-deed MTSs but, motivated by inner BCs below, we wishto stress the underlying MOTS structure. Following Hay-ward [1], H is a future outer trapping horizon (FOTH) if,in addition, L k θ ( ℓ ) < H is a trapped region. FOTHs can beeither null o spacelike hypersurfaces, the former repre-senting stationary situations and the latter dynamicalones. Alternatively, dynamical horizons (DH) introducedby Ashtekar and Krishnan [2] substitute the condition on L k θ ( ℓ ) by the requirement of H to be spacelike, station-arity being represented by (null)isolated horizons (IH).Both in FOTHs and DHs, condition θ ( k ) < S t )of H will always lay within a spatial surface Σ t of thechosen 3 + 1 slicing. Denoting by s the unit spacelikenormal vector to S t laying in Σ t and pointing towardsspacelike infinity, we can perform a 2+1 decompositionon the horizon. In particular, the metric q induced on S t can be written as q = γ − s ⊗ s and the shift can be decom-posed in its normal and tangential part as: β = β ⊥ s − V ,with β ⊥ = β · s and V · s = 0.A most important result in this context is the foliationuniqueness theorem by Ashtekar and Galloway [18] stat-ing that, for a given DH H , there exists a unique folia-tion ( S t ) by MTS’s. Using this, we can define a canoni-cal vector h as the vector tangent to H , normal to each S t and that Lie-drags each MTS S t of H into anotherone S t + δt . It constitutes a natural evolution vector on H and can be decomposed as h = N n + b s , where thenormalization N follows from requiring S t ∈ Σ t and isfixed up to a factor only depending on t . Defining a pa-rameter C as (half) the square norm of h with respectto g , i.e. C := h · h / b − N , it follows from theabove-commented metric type of FOTH’s that C ≥ b − N > b − N = 0 in the equilibrium (null) IH case; accordingly,we normalize the null vector ℓ as the limit of h in thestationary case: ℓ = N ( n + s ) [19, 20].Our criteria for setting BCs for Eqs. (2) are: a) to en-force the excision world-tube H to be sliced by MOTS,and b) to recover IH BCs [19, 20, 21, 22, 23] at the equi-librium limit C = 0. Motivated by this second point,but ultimately justified by the inner boundary analysisof Eqs. (3), we choose a coordinate system adapted to H by demanding t to be tangent to H . This implies β ⊥ = b ,and we have h = t + V , β ⊥ − N ≥ . (4) i) Geometric conditions for H . The first two BCs areprovided by 1) the geometric definition of S t as a MOTS: θ ( ℓ ) = 0, and 2) the Lie-dragging of MOTS into MOTSby h inside H ( trapping horizon condition): L h θ ( ℓ ) = 0.The first one yields4˜ s · ˜ D lnΨ + ˜ D · ˜ s + Ψ − K (˜ s , ˜ s ) − Ψ K = 0 , (5)where tildes refer to the conformal metric ˜ γ ; in partic-ular, ˜ s = Ψ s . The second geometric condition followsfrom the projection onto S t of one component of Einsteinequation and results in the elliptic equation [11] (cid:2) − ∆ − L · D + A (cid:3) ( β ⊥ − N ) = B ( β ⊥ + N ) , (6)where L i ≡ K kl s k q l i , A ≡
12 2 R − ˜ D · L − L · L − π T ( ˆ ℓ , ˆ k ), B ≡ σ (ˆ ℓ ) ij σ (ˆ ℓ ) ij + 4 π T ( ˆ ℓ , ˆ ℓ ), T is the stress-energy ten-sor, ˆ ℓ = n + s , ˆ k = ( n − s ) /
2, and D , ∆ and R are respectively the covariant derivative, Laplacian andRicci scalar of ( S t , q ). The non-negative character ofthe rhs term in (6), together with the lhs elliptic op-erator under the FOTH condition (closely related to thestability condition in [17]), guarantees the positivity of( β ⊥ − N ) in (4). Moreover, null-like condition β ⊥ = N [19, 20] is recovered in the stationary IH limit, for which σ (ˆ ℓ ) = 0 = T ( ˆ ℓ , ˆ ℓ ). Condition (6) provides a relation be-tween combinations ( β ⊥ − N ) and ( β ⊥ + N ): given one,the other is fully determined. ii) Gauge conditions for the tangential part of the shift. Let us express the shear tensor along h , σ ( h ) , using thecoordinate system (4) adapted to H :2 σ ( h ) ij = (cid:18) ∂q ij ∂t − ∂∂t ln √ q q ij (cid:19) + (cid:0) D i V j + D j V i − D k V k q ij (cid:1) . (7)Imposing as a coordinate choice the vanishing of the firstparenthesis in the rhs results in D i V j + D j V i − D k V k q ij = 2 σ ( h ) ij , (8)an elliptic equation whose source is determined by theevolution equation of the shear σ ( h ) (tidal equation): L h σ ( h ) ij = − q ki q lj ℓ m ℓ n W mknl − C q ki q lj k m k n W mknl − πC (cid:20) q µi q νj T µν −
12 ( q µν T µν ) q ij (cid:21) + · · · , (9)where W is the Weyl tensor. Condition (8) fixes thetangential part of the shift V up to a linear combinationthe six conformal symetries in the kernel of the ellipticoperator in the lhs. We determine this conformal sym-metry in the evolution by continuity with the conformalKilling symmetry prescribed on the initial data. In thestationary limit, where h tends to ℓ and σ ( ℓ ) = 0, thevanishing of the rhs in Eq. (8) leads to the conformalKilling condition on V and, given the rescaling proper-ties of the conformal Killing operator, IH condition for V in Refs. [19, 20, 22] is recovered. iii) Slicing condition. Combined results in Refs. [17,18] show that, for different choices of 3-foliation (Σ t ), agiven MTS S t on a given initial 3-slice evolves genericallyinto distinct DHs. However, all these DHs are ultimatelyexpected to approach the event horizon, and thereforethere is no preferred candidate on the sole basis of thedynamical trapping horizon framework. The choice ofinner BC for N , must be adopted on the basis of thewell-posedness of the elliptic-hyperbolic system and thespecific numerical needs. In practice, this issue must benumerically addressed. Having said this, Eq. (6) suggestsan alternative in this context: prescribing an inner BCfor ( β ⊥ − N ) determines ( β ⊥ + N ) algebraically. Such isthe case of the proposal in [24], where the choice of thatDH locally maximizing the area rate of change of theslice S t leads to: β ⊥ − N = − const · θ (ˆ k ) , with const > β ⊥ + N ) [resp. N ], then Eq. (6) must be solved asan elliptic equation on S t for ( β ⊥ − N ) [resp. β ⊥ ]. Inner BCs for the hyperbolic part . Assessing the free-dom in prescribing inner BCs for Eqs. (3) is a key stepin the implementation of the fully-constrained evolutionscheme. A first analysis of the general issues concerninghyperbolicity in Eqs. (3), has been carried out in Ref. [7]by writing down the evolution equations as a first-ordersystem in conservative form, i.e. ∂ t U + A i ( U ) ∂ i U = F [ U , ... ], where the evolving variable vector U is givenby U = ( h , ∂ t h , D h ) and matrices A i are straightfor-wardly determined from Eqs. (3). First, it is shown thatimposing the Dirac gauge in (1) indeed guarantees thereal character of the eigenvalues corresponding to ma-trices A i , and therefore the hyperbolicity of the evolu-tion system. Of particular relevance for the present in-ner BC discussion is the explicit determination of the(non-vanishing) characteristic speeds associated with thevector s normal to the excision surface S t , resulting in[7] λ ( s ) ± = − β ⊥ ± N (each one of multiplicity 6) . (10)Taking into account the inequality in (4), consequenceof the choice of a coordinate system adapted to the DH H by enforcing condition (6), we conclude the absenceof ingoing radiative modes into the integration domainΣ t at the excision surface. Therefore no inner BC what-soever must be prescribed for the hyperbolic part, as aconsequence of our choice of BCs for the elliptic part.This confirms our initial motivation for using space-likeexcision worldtubes in the evolution and shows the keyinterplay between elliptic and hyperbolic modes in thecoupled fully-constrained evolution system. Discussion.
In the context of constrained schemes forexcised black hole evolutions such as Refs. [6, 14], innerBCs (5) and (6), together with the essentially free choiceof 3-slicing, characterize the inner excision hypersurface H as a world-tube sliced by a family ( S t ) of MOTS. If,in addition, the condition (8) is enforced, then IH innerBCs [19, 20, 21, 22, 23] are recovered in the stationary limit and one of our basic requirements is fulfilled. Eventhough the excision world-tube H is indeed expected tobe a DH in realistic contexts, such a character is notactually enforced since the MTS condition θ ( k ) < θ ( k ) could be explicitly enforced as a Robin condition on β ⊥ · Ψ (cf. Eq. (16) in Ref. [23]): together with Eqs. (5),(6) this would fix N , therefore providing an alternativemanner of fixing the slicing. However it is known thatthe future evolution of a DH can cease “momentarily” tosatisfy MTS and FOTH conditions, e.g. in the mergingof two black holes once the common horizon has shownup. In this situation, insisting in the prescription of anegative θ ( k ) probably leads to the ill-posedness of thewhole coupled elliptic-hyperbolic system. For this rea-son, we rather adopt the methodological choice of onlyprescribing MOTS as inner BCs. Regarding a possibleFOTH condition failure, and according with the charac-teristic analysis in [7], monitoring the sign of ( β ⊥ − N )determines if inner BCs must or must not be providedfor the radiative modes. This work represents an inter-mediate step in the ongoing program [6] addressing fully-constrained excised black hole numerical evolutions.This work has been supported by the Marie Curie con-tract MERG-CT-2006-043501 (J.L.J.), the doctoral fel-lowship AP2005-2857 from MEC (I.C-C.), and grantsAYA2004-08067-C03-01 from MEC, HF2005-0115 fromCNRS/MEC, and 06-2-134423 MATH-GR from ANR. [1] S.A. Hayward, Phys. Rev. D , 6467 (1994); Phys. Rev.Lett. , 251101 (2004); Phys. Rev. D , 104027 (2004);Phys. Rev. D , 104013 (2006).[2] A. Ashtekar and B. Krishnan, Phys. Rev. Lett. , 104030 (2003).[3] A. Ashtekar and B. Krishnan, Liv. Rev. Relat. , 10(2004); [4] I. Booth, Can. J. Phys. , 1073 (2005).[5] J.G. Baker, M. Campanelli, F. Pretorius and Y. Zlo-chower, Class. Quantum Grav. , S25-S31 (2007).[6] S. Bonazzola, E. Gourgoulhon, P. Grandcl´ement, and J.Novak, Phys. Rev. D , 104007 (2004).[7] I. Cordero-Carri´on, J.M a . Ib´a˜nez, J.L. Jaramillo, and E.Gourgoulhon, J. Phys. Conf. Series , 012046 (2007); I.Cordero-Carri´on, Master Thesis, University of Valencia(2007); I. Cordero-Carri´on et al., in preparation .[8] P. Grandcl´ement and J. Novak, Liv. Rev. Relat., submit-ted, preprint arXiv:0706.2286.[9] M.A. Scheel, H.P. Pfeiffer, L. Lindblom, L.E. Kidder, O.Rinne, S.A. Teukolsky Phys.Rev. D , 104006 (2006).[10] I. Booth and S. Fairhurst, Phys. Rev. D , 084019(2007)[11] D.M. Eardley, Phys. Rev. D , 2299 (1998).[12] E. Gourgoulhon, , preprint arXiv:gr-qc/0703035.[13] J.W. York, Phys. Rev. Lett. , 1350 (1999); H.P. Pfeiffer and J.W. York, Phys. Rev. D , 044022 (2003).[14] V. Moncrief, L. Buchman, H.P. Pfeiffer, O. Rinne, O.Sarbach, talk at From Geometry to Numerics
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