The ‘non-Kerrness’ of domains of outer communication of black holes and exteriors of stars

Thomas Bäckdahl , Juan A. Valiente Kroon

Abstract

In this paper, we construct a geometric invariant for initial datasets for the vacuum Einstein field equations Embedded Image, such that Embedded Image is a three-dimensional manifold with an asymptotically Euclidean end and an inner boundary Embedded Image with the topology of the 2-sphere. The hypersurface Embedded Image can be thought of being in the domain of outer communication of a black hole or in the exterior of a star. The geometric invariant vanishes if and only if Embedded Image is an initial dataset for the Kerr spacetime. The construction makes use of the notion of Killing spinors and of an expression for a Killing spinor candidate, which can be constructed out of concomitants of the Weyl tensor.

1. Introduction

Let Embedded Image be an initial dataset for the vacuum Einstein field equations such that Embedded Image has two asymptotically Euclidean ends, but otherwise trivial topology.1 In Bäckdahl & Valiente Kroon (2010a), a geometric invariant for this type of initial datasets has been constructed—see also Bäckdahl & Valiente Kroon (2010b) for a detailed discussion. This invariant is a non-negative number having the property that it vanishes if and only if the initial dataset corresponds to data for the Kerr spacetime. Thus, the invariant measures the non-Kerrness of the initial data.

In view of possible applications of the non-Kerrness to the problem of the uniqueness of stationary black holes and the nonlinear stability of the Kerr spacetimes, a different type of initial hypersurface is of more interest: a three-dimensional hypersurface with the topology of the complement of an open ball in Embedded Image, Embedded Image. This type of 3-manifold can be thought of as a Cauchy hypersurface in the domain of outer communication of a black hole or the exterior of a star. In the present paper, we discuss the construction of a geometric invariant measuring the non-Kerrness of this type of initial hypersurface.

(a) Outline of the article

In §2, we provide a brief summary of the theory of non-Kerrness invariants developed in Bäckdahl & Valiente Kroon (2010a,b). This is provided for quick reference and contains the essential ingredients required in the construction of the present paper. Section 3 contains a discussion of properties of vacuum Petrov-type D spacetimes that are relevant for our discussion. In particular, it provides a formula of a Killing spinor candidate written entirely in terms of concomitants of the Weyl tensor. For a spacetime that is exactly of Petrov-type D, this expression provides a Killing spinor of the spacetime. This expression is used in the sequel to provide the boundary value of an elliptic problem. Section 4 provides a discussion of a boundary value problem for the approximate Killing spinor equation. Section 5 makes use of the solution to the boundary value problem to construct the non-Kerrness invariant. Finally, in §6 we provide some conclusions and outlook. This paper also includes two appendices. The first one provides a summary of the results on boundary value problems for elliptic systems used in our construction. The second appendix contains an improved theorem characterizing the Kerr spacetime in terms of Killing spinors. This theorem removes some technical assumptions made in Bäckdahl & Valiente Kroon (2010a,b).

(b) Notation

Throughout, Embedded Image will denote an orientable and time orientable, globally hyperbolic vacuum spacetime. It follows that the spacetime admits a spin structure (Geroch 1968, 1970). In what follows, μ,ν,… will denote abstract four-dimensional tensor indices. The metric gμν will be taken to have a signature (+,−,−,−). Let ∇μ denote the Levi–Civita connection of gμν. The triple Embedded Image will denote initial data on a hypersurface of the spacetime Embedded Image. The symmetric tensors hab and Kab will correspond, respectively, to the 3-metric and the extrinsic curvature of the 3-manifold Embedded Image. The metric hab will be taken to be negative definite. The indices a,b,… will denote abstract three-dimensional tensor indices, while i,j,… will denote three-dimensional tensor coordinate indices. Let Da denote the Levi–Civita covariant derivative of hab. Spinors will be used systematically. We follow the conventions of Penrose & Rindler (1984). In particular, A,B,… will denote abstract spinorial indices, while A,B,… will be indices with respect to a specific frame.

A space spinor formalism will be used throughout. A very brief introduction to this formalism is given in what follows—see Sommers (1980), Bäckdahl & Valiente Kroon (2010b) for more detailed expositions. Let τAA denote the spinorial counterpart of the normal τμ to the surface Embedded Image, with normalization τAAτAA=2. Given the spacetime solder forms Embedded Image and Embedded Image satisfying Embedded Imagethe relation τAAτBA=ϵAB allows us to introduce the spatial solder forms Embedded ImageOne has that Embedded ImageAny spatial tensor has a space-spinor counterpart. For example, if Tμν is a spatial tensor (i.e. τμTμν=0 and τνTμν=0), then its space-spinor counterpart is given by Embedded Image.

Let ∇AA denote the spinorial counterpart of the spacetime connection ∇μ. Besides the connection ∇AA, two other spinorial connections will be used: DAB, the spinorial counterpart of the Levi–Civita covariant derivative Da and ∇AB, the Sen covariant derivative of Embedded Image. The Sen connection is defined by Embedded Image.

2. Killing spinors and non-Kerrness

In this section, we provide a brief account of the theory of non-Kerrness developed in Bäckdahl & Valiente Kroon (2010a,b).

(a) Killing spinors and Killing spinor initial data

The starting point of the construction in Bäckdahl & Valiente Kroon (2010a,b) is the space-spinor decomposition of the Killing spinor equation Embedded Image2.1where κAB=κ(AB) and the spinorial conventions of Penrose & Rindler (1984) are being used.

Important for our purposes is the idea of how to encode that the development of an initial dataset Embedded Image admits a solution to the Killing spinor equation (2.1). This question can be addressed by means of the space-spinor formalism discussed in the previous section.

The space-spinor decomposition of equation (2.1) renders a set of three conditions intrinsic to the hypersurface Embedded Image: Embedded Image2.2a Embedded Image2.2b Embedded Image2.2cwhere we have written Embedded Imageand ∇AB denotes the spinorial version of the Sen connection associated with the pair (hab,Kab) of intrinsic metric and extrinsic curvature. It can be expressed in terms of the spinorial counterpart, DAB of the Levi–Civita connection of the 3-metric hab, and the spinorial version, KABCD=K(AB)(CD)=KCDAB, of the second fundamental form Kab. For example, given a valence 1 spinor πA one has that Embedded Imagewith the obvious generalizations to higher valence spinors. In equations (2.2b) and (2.2c), the spinor ΨABCD denotes the restriction to the hypersurface Embedded Image of the self-dual Weyl spinor. Crucially, the spinor ΨABCD can be written entirely in terms of initial data quantities via the relations Embedded Imagewith Embedded Imageand Embedded Imageand where ΩABCDK(ABCD), KKPQPQ. Furthermore, the spinor rABCD is the Ricci tensor, rab, of the 3-metric hab.

The key property of equations (2.2a)–(2.2c) is contained in the following result proved in Bäckdahl & Valiente Kroon (2010b)—see also García-Parrado & Valiente Kroon (2008).

Proposition 2.1

Let equations (2.2a)–(2.2c) be satisfied for a symmetric spinor Embedded Image on an open set Embedded Image. Then the Killing spinor equation (2.1) has a solution, κAB, on the future domain of dependence Embedded Image.

(b) Approximate Killing spinors

The spatial Killing spinor equation (2.2a) can be regarded as a (complex) generalization of the conformal Killing vector equation. It will play a special role in our considerations. As in the case of the conformal Killing equation, equation (2.2a) is clearly overdetermined. However, one can construct a generalization of the equation which under suitable circumstances can always be expected to have a solution. One can do this by composing the operator in equation (2.2a) with its formal adjoint—see Bäckdahl & Valiente Kroon (2010a). This procedure renders the equation Embedded Image2.3which will be called the approximate Killing spinor equation. One has the following result proved in Bäckdahl & Valiente Kroon (2010b).

Lemma 2.2

The operator L defined by the left-hand side of equation (2.3) is a formally self-adjoint elliptic operator.

In Bäckdahl & Valiente Kroon (2010a,b) it has been shown that if Embedded Image has the same topology as Cauchy slices of the Kerr spacetime, and if the pair (hab,Kab) is suitably asymptotically Euclidean, then there exists a certain asymptotic behaviour at infinity for the spinor κAB for which the approximate Killing spinor equation always admits a solution.

If one wants to extend the construction discussed in the previous paragraphs to a 3-manifold on, say, the domain of outer communication of a black hole or the exterior of a star so that Embedded Image, then in addition to prescribing the asymptotic behaviour of the spinor κAB at infinity, one also has to prescribe the behaviour at the inner boundary Embedded Image. One wants to prescribe this information in such a way that κAB has the right Killing behaviour at the boundary whenever all of the Killing spinor data equations (2.2a)–(2.2c) are satisfied. In this paper, we discuss how this can be done, and as a result we construct the non-Kerrness for 3-manifolds with topology Embedded Image. These 3-manifolds can be interpreted as slices in the domain of outer communication of a black hole or slices in the exterior of a star. It is expected that this construction will be of use in the reformulation of problems involving the Kerr spacetime: the uniqueness of stationary black holes, the construction of an interior for the Kerr solution, and possibly also the evolution of nonlinear perturbations of the Kerr spacetime.

3. Petrov-type D spacetimes

In order to analyse what is the right initial data to be prescribed on the boundary Embedded Image of our initial 3-manifold Embedded Image, we will look at some properties of vacuum spacetimes of Petrov-type D.

(a) The canonical form for type D

Let ΨABCD denote the Weyl spinor of a vacuum spacetime Embedded Image. We shall consider the following invariants of ΨABCD Embedded Imageand Embedded ImageThe Petrov type of the spacetime is determined as a solution of the eigenvalue problem Embedded Imagee.g. Stephani et al. (2003). The eigenvalues λ satisfy the equation Embedded Image3.1Let λ1, λ2 and λ3 denote the roots of the above polynomial. The invariants Embedded Image and Embedded Image can be expressed in terms of the eigenvalues by Embedded Image3.2aand Embedded Image3.2b

In what follows we assume λ1,λ2,λ3≠0. The Petrov-type D is characterized by the condition λ1=λ2. Using expressions (3.2a) and (3.2b), one has that the remaining root satisfies the equation Embedded Image3.3Combining equations (3.1) and (3.3) one finds that Embedded Image

For a Petrov-type D spacetime, there exist spinors (the principal spinors) αA, βA satisfying the normalization αAβA=1 such that Embedded Image3.4It will be convenient to define the spinor υABα(AβB). Observe that because of our normalization conditions one has that Embedded Image. Using the spinor υAB one obtains the following alternative expression for ΨABCD: Embedded Image3.5The hABCD term is chosen to compensate for the traces of υABυCD so the right-hand side is tracefree—and thus completely symmetric. The formula can be verified by making an irreducible decomposition or by substituting Embedded Imagein equation (3.5).

The expression (3.5) can be used to obtain a formula for the spinor υAB in terms of the Weyl spinor ΨABCD. Let ζAB denote a non-vanishing symmetric spinor. Contracting (3.5) with an arbitrary spinor ζAB one obtains Embedded Image3.6aand Embedded Image3.6bUsing equation (3.6a) to solve for υAB and equation (3.6b) to solve for υPQζPQ one obtains the following formula for υAB in terms of ΨABCD and the arbitrary spinor ζAB Embedded Image3.7with Embedded Image3.8In the last formulae it is assumed that ζAB is chosen such that Embedded Image

(b) The Killing spinor of a Petrov-type D spacetime

Let κAB be a solution to the Killing spinor equation (2.1). An important property of a Killing spinor is that Embedded Image3.9satisfies the (spinorial version of the) Killing vector equation Embedded ImageIn general, the Killing vector ξAA given by formula (3.9) is complex—that is, it encodes the information of two real Killing vectors. This property is closely related to the fact that all vacuum-type D spacetimes admit, at least, a pair of commuting Killing vectors—e.g. Kinnersley (1969). Vacuum spacetimes of Petrov-type D for which ξAA is real are called generalized Kerr-NUT (Newman–Unti–Tamburino) spacetimes.

Every vacuum spacetime of Petrov-type D has a Killing spinor—see Penrose & Rindler (1986) and references therein. Indeed, in the notation of the previous section, one has that Embedded Image3.10satisfies equation (2.1). Using formula (3.7), one obtains the following result.

Proposition 3.1

Let Embedded Image be a vacuum spacetime. If on Embedded Image the spacetime is of Petrov-type D and ζAB is a symmetric spinor satisfying Embedded Imagethen Embedded Image3.11with Ξ given by equation (3.8) is a Killing spinor on Embedded Image. The formula (3.11) is independent of the choice of ζAB.

That expression (3.10) is independent of the choice of ζAB can be verified by writing Embedded Imagewhere {αA,βA} is the dyad given by equation (3.4). Substituting the latter into equation (3.11) one readily obtains (3.10).

Observation 3.2

Formula (3.11) can be evaluated for any vacuum spacetime Embedded Image. In general, of course, it will not give a solution to the Killing spinor equation (2.1). The resulting spinor κAB will depend upon the choice of ζAB. We make the following definition:

Definition 3.3

Let Embedded Image be a vacuum spacetime. Consider Embedded Image and on Embedded Image a symmetric spinor ζAB satisfying Embedded ImageThe symmetric spinor given by Embedded Image3.12with Embedded Imagewill be called the ζAB-Killing spinor candidate on Embedded Image.

Remark 3.4

Although the choice of ζAB is essentially arbitrary, as it will be seen, in many applications there is a natural choice.

Remark 3.5

The choice of branch cut for the square root of Ξ can be chosen to be {−reiθ:r>0}, where θ is the argument of Embedded Image.

4. A boundary value problem for the approximate Killing spinor equation

In this section, we formulate a boundary value problem for the approximate Killing spinor equation (2.3) on a 3-manifold Embedded Image. As discussed in §1, this type of 3-manifold can be thought of as a Cauchy hypersurface in the domain of outer communication of a black hole or the exterior of a star. For simplicity of the presentation, it will be assumed that the initial data Embedded Image satisfy in its asymptotic region the behaviour Embedded Image4.1aand Embedded Image4.1bwith r=((x1)2+(x2)2+(x3)2)1/2, and (x1,x2,x3) are asymptotically Cartesian coordinates and m denotes the Arnowitt–Deser–Misner mass. Our present discussion could be extended at the expense of more technical details to include the case of boosted initial datasets (e.g. Bäckdahl & Valiente Kroon (2010b)). Here, and in what follows, the fall off conditions of the various fields will be expressed in terms of weighted Sobolev spaces Embedded Image, where s is a non-negative integer and β is a real number. Here, we use the conventions for these spaces given in Bartnik (1986)—see also Bäckdahl & Valiente Kroon (2010b). We say that Embedded Image if Embedded Image for all s. Thus, the functions in Embedded Image are smooth over Embedded Image and have a fall off at infinity such that ∂lη=o(rβ−|l|). We will often write Embedded Image for Embedded Image at the asymptotic end.

Following the ideas of Bäckdahl & Valiente Kroon (2010a,b), we shall look for solutions to the approximate Killing spinor equation (2.3) which expressed in terms of an asymptotically Cartesian frame and coordinates has an asymptotic behaviour and is given by Embedded Image4.2with Embedded Image

(a) Behaviour at the inner boundary

The ideas of §3 will be used to prescribe the value of the spinor κAB on the boundary Embedded Image. In general, one could choose to prescribe any value for κAB on the boundary Embedded Image as long as this choice coincides with the correct value in the exact Kerr case. However, one would like to have a coordinate independent choice. This requires constructing the candidate from geometrical objects. Equation (3.12) gives a good choice with these properties provided its normalization is adjusted so it coincides with the correct value in the exact Kerr case. Note that equation (3.12) contains an arbitrary non-vanishing spinor ζAB. In principle, one could choose any values for ζAB as long as the conditions in definition 3.2 are satisfied. If the choice of ζAB is to depend only on the geometry of the problem, then the spinorial counterpart, nAB, of the normal to the hypersurface Embedded Image, na, is the only natural choice. By convention, na is assumed to point outside Embedded Image (outward pointing). Note that because of the use of a negative definite 3-metric one has that nPQnPQ=−1. Hence, the condition ζAB≠0 is trivially satisfied if one uses ζAB=nAB.

Remark 4.1

It is, in principle, natural to suspect that there are other choices of Dirichlet boundary conditions that allow us to define a geometric invariant of the type we are interested in. However, we are not aware of any other explicit choice besides the one discussed here.

It will be convenient to define the following set Embedded Imagewith Embedded Image

We shall make the following technical assumption on the initial dataset Embedded Image:

Assumption 4.2

The initial dataset Embedded Image is such that Ξ is a smooth function over Embedded Image satisfying

  1. Embedded Image;

  2. Embedded Image does not encircle the point z=0.

As a consequence of assumption 4.2 one can choose a cut of the square root function on the complex plane such that Ξ−1/2(p) is smooth for all Embedded Image.

Remark 4.3

The conditions in assumption 4.2 are satisfied by standard Kerr data (in Boyer–Lindquist coordinates) at the horizon. Furthermore, by construction if the data Embedded Image are data for the Kerr spacetime, then the boundary data given by the Killing spinor candidate formula in definition 3.2 give the right boundary behaviour for the restriction of its Killing spinor to Embedded Image.

Remark 4.4

In order to match the asymptotic behaviour of the nAB-Killing spinor candidate given by definition 3.3 with that given by equation (4.2) one needs to incorporate a normalization factor to equation (3.12). To this end, it is noticed that the decay conditions (4.1a) and (4.1b) imply the asymptotic expansions Embedded ImageNow, if the 2-surface Embedded Image is sent to infinity in such a way that Embedded Imageso that Embedded Image becomes more and more like a 2-sphere, one finds that Embedded ImageHence, the leading term of the nAB-Killing spinor candidate shown in equation (3.12) is given by Embedded ImageThus, in order to have a Killing spinor candidate whose asymptotic behaviour agrees with that of equation (4.2), one needs to consider the normalized expression Embedded Image4.3

(b) Existence of solutions to the approximate Killing spinor equation

Following the strategy put forward in Bäckdahl & Valiente Kroon (2010a,b), we provide an Ansatz for a solution to the approximate Killing spinor equation (2.3) that encodes the desired behaviour at infinity. To this end, let Embedded Image4.4where ϕR is a smooth cut-off function such that for R>0 large enough Embedded Imageand Embedded Image

One then has the following result:

Theorem 4.5

Let Embedded Image be an initial dataset for the Einstein vacuum field equations such that Embedded Image is a manifold with a smooth boundary Embedded Image satisfying assumption 4.1. Assume that (hab,Kab) satisfy the asymptotic conditions (4.1a) and (4.1b) with m≠0. Then, there exists a unique smooth solution, κAB, to the approximate Killing equation (2.3) with behaviour at the asymptotic end of the form (4.2) and with boundary value at Embedded Image given by the nAB-Killing spinor candidate Embedded Image of equation (4.3).

Proof.

Following the procedure described in Bäckdahl & Valiente Kroon (2010a,b), we consider the Ansatz Embedded Image4.5The substitution of the latter into equation (2.3) renders the following equation for the spinor θAB: Embedded Image4.6In view that Embedded Image vanishes outside the asymptotic region, then the value of θAB at Embedded Image coincides with that of κAB. That is, we set Embedded Image4.7By construction it follows that Embedded Imageso that Embedded ImageThe operator associated with the Dirichlet elliptic boundary value problem (4.6) and (4.7) is given by (L,B), where B denotes the Dirichlet boundary operator on Embedded Image. As discussed in Bäckdahl & Valiente Kroon (2010a,b), under assumptions (4.1a) and (4.1b), the operator L is asymptotically homogeneous—see appendix A for a concise summary of the ideas and results of the theory elliptic systems being used here. Now, elliptic boundary value problems with Dirichlet boundary conditions satisfy the Lopatinski–Shapiro compatibility conditions—see Wloka et al. (1995). Consequently, the operator (L,B) is L-elliptic and the map Embedded Imageis Fredholm—see theorem A.1 of appendix A. The rest of the proof is an application of the Fredholm alternative. Using proposition A.2 with δ=−1/2, one concludes that equation (4.6) has a unique solution if FAB is orthogonal to all Embedded Image in the Kernel of L*=L with νAB=0 on Embedded Image. If LνAB=0, then an integration by parts shows that Embedded Imagewhere Embedded Image denotes the sphere at infinity. The boundary integral over Embedded Image vanishes because of νAB∈Ker(L,B), so that νAB=0 on Embedded Image. As Embedded Image by assumption, it follows that Embedded Image and furthermore that Embedded Image. An integral over a finite sphere will then be of type o(1). Thus, the integral over Embedded Image vanishes. Hence, one concludes that Embedded ImageUsing the same methods as in Bäckdahl & Valiente Kroon (2010b, proposition 21) one finds that there are no non-trivial solutions to the spatial Killing spinor equation that go to 0 at infinity. Thus, there are no restrictions on FAB and equation (4.6) has a unique solution as desired. Owing to elliptic regularity, any Embedded Image solution to equation (4.6) is in fact a Embedded Image solution —cf. lemma A.3. Thus, θAB is smooth. □

Remark 4.6

It is worth mentioning that similar methods can be used to obtain solutions to the approximate Killing spinor equation on annular domains of the form Embedded Image, where R2>R1. Again, one would use the Killing spinor candidate of definition 3.2 to provide boundary value data on the two components of Embedded Image. This type of construction is of potential relevance in the nonlinear stability of the Kerr spacetime and in the numerical evaluation of the non-Kerrness.

5. The geometric invariant

In this section, we show how the approximate Killing spinor κAB obtained from theorem 4.5 can be used to construct an invariant measuring the non-Kerrness of the 3-manifold with boundary Embedded Image. To this end, we recall the following lemma from Bäckdahl & Valiente Kroon (2010a):

Lemma 5.1

The approximate Killing spinor equation (2.3) is the Euler–Lagrange equation of the functional Embedded Image5.1

In what follows, it will be assumed that κAB is the solution to equation (2.3) given by theorem 4.4. Furthermore, let Embedded Image5.2aand Embedded Image5.2bThe geometric invariant is then defined by Embedded Image5.3

Remark 5.2

It can be verified that I is coordinate independent. Furthermore, if the initial dataset satisfies the decay conditions (4.1a) and (4.1b), then I is finite.

The desired characterization of Kerr data on 3-manifolds Embedded Image with boundary and one asymptotic end is given by the following theorem.

Theorem 5.3

Let Embedded Image be an initial dataset for the Einstein vacuum field equations such that Embedded Image is a manifold with boundary Embedded Image satisfying Assumption 4.1. Furthermore, assume that Embedded Image has only one asymptotic end and that the asymptotic conditions (4.1a) and (4.1b) are satisfied with m≠0. Let I be the invariant defined by equations (5.1), (5.2a), (5.2b) and (5.3), where κAB is given as the only solution to equation (2.3) with asymptotic behaviour given by (4.4) and with boundary value at Embedded Image given by the nAB-Killings spinor candidate κ′AB of equation (4.3), where nAB is the outward pointing normal to Embedded Image. The invariant I vanishes if and only if Embedded Image is an initial dataset for the Kerr spacetime.

The proof of this result is analogous to the one given in Bäckdahl & Valiente Kroon (2010a,b) and will be omitted.

Remark 5.4

In the previous theorem, for an initial dataset for the Kerr spacetime it will be understood that Embedded Image (the union of the past and future domains of dependence of Embedded Image) is isometric to a portion of the Kerr spacetime. In order to make stronger assertions about Embedded Image, one needs to provide more information about Embedded Image. For example, if it can be asserted that Embedded Image coincides with the intersection of the past and future components of a non-expanding horizon, then theorem 5.3 will give that Embedded Image is the domain of outer communication of the Kerr spacetime.

6. Conclusions and outlook

Theorem 5.3 and the methods developed in the present articles are expected to be of relevance in several outstanding problems concerning the Kerr spacetime: a proof of the uniqueness of stationary black holes that does not assume analyticity of the horizon, and whether the Kerr solution can describe the exterior of a rotating star. The boundary value problem discussed in the present paper might also play a role in applications of Killing spinor methods to the nonlinear stability of the Kerr spacetime and in the evaluation of the non-Kerrness in slices of numerically computed black hole spacetimes. The motivation for some of these claims is briefly discussed in the next paragraphs.

For the problem of the uniqueness of stationary black holes, as mentioned in the remark after theorem 5.3, one would like to consider slices in the domain of outer communication of a stationary black hole that intersects the intersection of the two components of the non-expanding horizon. One then would have to analyse the consequences that the existence of this type of boundary have on the Killing spinor candidate constructed out of the normal to Embedded Image—the Weyl tensor is known to be of type D on non-expanding horizons (Ionescu & Klainerman 2009). The main challenge in this approach is to find a convenient way of relating the a priori assumption about stationarity made in the problem of uniqueness of black holes with the Killing vector initial data candidate (ξ, ξAB) provided by the solution, κAB, to the approximate Killing spinor equation (2.3).

With regards to the problem of the existence of an interior solution for the Kerr spacetime, the key question to be analysed is what kind of conditions on the boundary Embedded Image needs to be prescribed to ensure that the solution to the approxi- mate Killing spinor equation (2.3) given by theorem 4.4 renders a vanishing inva- riant I. It is to be expected that these conditions will impose strong restrictions to the type of matter models describing a hypothetical interior solution.

Finally, in what concerns numerical simulations of black hole spacetimes and the non-linear stability of the Kerr solution, the key issue is the behaviour of the geometric invariant upon time evolution. If some type of monotonic behaviour along a foliation of spacetime can be established, then our invariant could be a valuable tool for the investigation of the dynamics of the gravitational field.

The ideas touched upon in the previous paragraphs will be further elaborated in future works.

Acknowledgements

We would like to thank Gastón Ávila for helpful conversations on the boundary value problem for elliptic systems. T.B. is funded by a scholarship from the Wenner-Gren foundations. J.A.V.K. was funded by an EPSRC Advanced Research fellowship. The authors thank the hospitality and financial support of the International Centre of Mathematical Sciences (ICMS) and the Centre for Analysis and Partial Differential Equations (CANPDE) of the University of Edinburgh for their hospitality during the workshop on Mathematical Relativity 1–8 September 2010, in the course of which this research was completed.

Appendix A. Elliptic results for slices in the domain of outer communication of a black hole

In this appendix, we summarize the results on the theory of boundary value problems for elliptic systems that have been used in the present paper. The presentation is adapted from Lockhardt & McOwen (1985).

As in the main text, let Embedded Image denote a three-dimensional manifold with the topology of Embedded Image, where Embedded Image denotes the open ball of radius 1. Note that Embedded Image is closed. Assume Embedded Image to be Embedded Image. In what follows, let u denote an N-dimensional vector-valued function over Embedded Image. Following Cantor (1981) and Lockhart (1981), a second-order elliptic operator L acting on u will be said to be asymptotically homogeneous if it can be written in the form Embedded Imagewhere Embedded Image denotes a matrix with constant coefficients while aij, ai and a are matrix-valued functions of the coordinates such that Embedded Image

On Embedded Image we will consider the homogeneous Dirichlet operator B given by Embedded ImageThe combined operator (L,B) is said to be L-elliptic if L is elliptic on Embedded Image and (L,B) satisfies the Lopatinski–Shapiro compatibility conditions—see Wloka et al. (1995) for detailed definitions. Crucial for our purposes is that if L is elliptic and B is the Dirichlet boundary operator, then the Lopatinski–Shapiro conditions are satisfied and thus (L,B) is L-elliptic—see again Wloka et al. (1995, theorem 10.7).

The Fredholm properties for the combined operator (L,B) follow from Lockhardt & McOwen (1985, theorem 6.3)—cf. similar results in Klenk (1991); Reula (1989). Bartnik’s conventions are used for the weights of the Sobolev spaces Embedded Image—see Bartnik (1986).

Theorem A.1

Let L denote a smooth second-order asymptotically homogeneous operator on Embedded Image. Furthermore, let Embedded Image be smooth, and let B denote the Dirichlet boundary operator. Then for δ not a negative integer, s≥2 the map Embedded Imageis Fredholm.

The same arguments used in Cantor (1981, theorem 6.3) then allow us to provide the following version of the Fredholm alternative:

Proposition A.2

Let (L,B) as in theorem A.1. Given δ not a negative integer, the boundary value problem Embedded Imageand Embedded Imagehas a solution Embedded Image if Embedded Imagefor all Embedded Image such that Embedded Imageand Embedded Imagewhere L* denotes the formal adjoint of L.

Finally, we note the following lemma—cf. Lockhardt & McOwen (1985, eqn (1.13)).

Lemma A.3

Let (L,B) as in theorem A.1. Then for any Embedded Image and any s≥2, there exists a constant C such that for every Embedded Image the following inequality holds Embedded Image

In this lemma, Embedded Image denotes the local Sobolev space. That is, Embedded Image if for an arbitrary smooth function v with compact support, uvHs.

Remark A.4

If L has smooth coefficients and Lu=0, then it follows that all the Embedded Image norms of u are bounded by the Embedded Image and the Embedded Image norms. Thus, it follows that if a solution to the boundary value problem exists and the boundary data are smooth, then the solution must be, in fact, smooth—elliptic regularity.

Appendix B. An improved characterization of the Kerr spacetime by means of Killing spinors

In Bäckdahl & Valiente Kroon (2010b) a characterization of the Kerr spacetime by means of Killing spinors was given. This characterization contains an a priori assumption on the Weyl tensor—namely, that it is nowhere of type N or D. The purpose of the present appendix is to show that these assumptions can be removed.

As in the main text, let κAB denote a totally symmetric spinor. Let Embedded ImageIf κAB is a solution to the Killing spinor equation (2.1), then ξAA satisfies the Killing equation Embedded ImageIn general, ξAA will be a complex Killing vector. The Killing form FAABB associated with ξAA is defined by Embedded ImageIn the cases where ξAA is real, we will consider the self-dual Killing form Embedded Image defined by Embedded Imagewhere F*AABB is the Hodge dual of FAABB. Owing to the symmetries of the self-dual Killing form one has that Embedded Image

The characterization of the Kerr spacetime discussed in Bäckdahl & Valiente Kroon (2010b) is, in turn, based on the following characterization proved by Mars (2000).

Theorem B.1 (Mars 1999, 2000)

Let Embedded Image be a smooth vacuum spacetime with the following properties:

  1. Embedded Image admits a real Killing vector ξAA such that the spinorial counterpart of the Killing form of ξAA satisfies Embedded ImageB1with φ a scalar;

  2. Embedded Image contains a stationary asymptotically flat 4-end, and ξAA tends to a time translation at infinity and the Komar mass of the asymptotic end is non-zero.

Then Embedded Image is locally isometric to the Kerr spacetime.

Remark B.2

A stationary asymptotically flat 4-end is an open submanifold Embedded Image diffeomorphic to Embedded Image, where Embedded Image is an open interval and Embedded Image is a closed ball of radius R such that in local coordinates (t,xi) defined by the diffeomorphism, the metric satisfies Embedded Imagewith C, α≥1 constants, ημν the Minkowski metric and Embedded ImageIn this context, the notions of Komar and ADM mass coincide.

We want to relate the notion of Killing form and that of Killing spinors. As discussed in , if ξAA is real, the commutators for a vacuum spacetime readily yield that Embedded ImageB2Now, vacuum spacetimes admitting a Killing spinor, κAB, can only be of Petrov-type D, N or O. If the spacetime is of type O at some point (so that ΨABCD=0), then theorem B.2 shows that Embedded Image, and the relation (A1) is trivially satisfied. If the spacetime is of Petrov-type N, then κAB has a repeated principal spinor that coincides with the repeated principal spinor of ΨABCD—e.g. Jeffryes (1984). Hence, again one has that Embedded Image, and theorem B.1 is satisfied trivially. For Petrov-type D spacetimes with a Killing spinor such that ξAA is real, it has already been shown in Bäckdahl & Valiente Kroon (2010b) that theorem B.1 is satisfied.

From the discussion in the previous paragraph, we obtain the following characterization of the Kerr spacetime in terms of Killing spinors.

Theorem B.3

A smooth vacuum spacetime Embedded Image is locally isometric to the Kerr spacetime if and only if the following conditions are satisfied:

  1. there exists a Killing spinor κAB such that the associated Killing vector ξAA is real;

  2. the spacetime Embedded Image has a stationary asymptotically flat 4-end with non-vanishing mass in which ξAA tends to a time translation.

As a consequence of this theorem, the a priori conditions on the Petrov type of the Weyl required in Bäckdahl & Valiente Kroon (2010b, theorem 28) can be dropped.

Footnotes

  • 1 More precisely, Embedded Image where Embedded Image denotes an open ball of radius 1 and # indicates that the boundaries of the two copies of Embedded Image are identified in the trivial way.

  • Received October 15, 2010.
  • Accepted December 15, 2010.

References

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