A geometric invariant characterising initial data for the Kerr-Newman spacetime

We describe the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr-Newman spacetime in terms of Killing spinors. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant.


Introduction
The Kerr-Newman solution to the Einstein-Maxwell equations, describing a stationary charged rotating black hole, is one of the most interesting and well studied exact solutions in General Relativity, and yet there still remain several unresolved questions. For example, the current family of uniqueness results regarding the Kerr-Newman solution contain assumptions on the spacetime that are often considered too restrictive, such as analyticity -see e.g. [13] for a review on the subject. Also, although there has been significant progress on the linear stability of the Kerr-Newman solution, the question of non-linear stability has been far more stubborn -see e.g. [15] for a discussion on this topic.
Making progress on these unresolved questions concerning electrovacuum blackholes provides the motivation for finding characterisations of the Kerr-Newman spacetime. Different methods for characterising the exact solution can be tailored to emphasise specific properties, and so address each of these unresolved properties directly. One such characterisation is expressed in terms of Killing spinors, closely related to Killing-Yano tensors, which represent hidden symmetries of the spacetime. These symmetries cannot be expressed in terms of isometries of the spacetime. It has been shown in [14] that an asymptotically flat electrovacuum spacetime admitting a Killing spinor which satisfies a certain alignment condition with the Maxwell field must be isometric to the Kerr-Newman spacetime -see Theorem 1 below.
Once the motivation for a characterisation of the Kerr-Newman spacetime in terms of Killing spinors has been established, it is useful to investigate how the existence of such a spinor can be expressed in terms of initial data. The initial value problem in General Relativity has played a crucial role in the systematic analysis of the properties of generic solutions to the Einstein field equations -see e.g. [16,21,22]. It also provides the framework necessary for numerical simulations of spacetimes to be performed -see e.g. [1,8].
Representing symmetries of a spacetime in terms of conditions on an initial hypersurface is not a new idea; the Killing initial data (KID) equations -see e.g. [10]-are conditions on a spacelike Cauchy surface S which guarantee the existence of a Killing vector in the resulting development of the initial data. Thus, isometries of the whole spacetime can be encoded at the level of initial data. The resulting conditions form a system of overdetermined equations, so do not necessarily admit a solution for an arbitrary initial data set. In fact, it has been shown that the KID equations are non-generic, in the sense that generic solutions of the vacuum constraint Einstein equations do not possess any global or local spacetime Killing vectors -see [11]. An analogous construction can, in principle, be performed for Killing spinors. This analysis has been performed for the vacuum case giving explicitly the conditions relating the Killing spinor candidate and the Weyl curvature of the spacetime -see [17] and also [4]. These conditions are, like the KID equations, an overdetermined system and so do not necessarily admit a solution for an arbitrary initial surface. However, in [3,4] it has been shown that given an asymptotically Euclidean hypersurface it is always possible to construct a Killing spinor candidate which, whenever there exists a Killing spinor in the development, coincides with the restriction of the Killing spinor to the initial hypersurface. This approximate Killing spinor is obtained by solving a linear second order elliptic equation which is the Euler-Lagrange equation of a certain functional over S. The approximate Killing spinor can be used to construct a geometric invariant which in some way parametrises the deviation of the initial data set from Kerr initial data. Variants of the basic construction in [4] have been given in [5,6].
The purpose of this article is to extend the analysis of [4] to the electrovacuum case. In doing so, we rely on the characterisation of the Kerr-Newman spacetime given in [14] which, in turn, builds upon the characterisation provided in [18] for the vacuum case and [24] for the electrovacuum case. As a result of our analysis we find that the Killing spinor initial data equations remain largely unchanged, with extra conditions ensuring that the electromagnetic content of the spacetime inherits the symmetry of the Killing spinor. These electrovacuum Killing spinor equations, together with an appropriate approximate Killing spinor, are used to construct an invariant expressed in terms of suitable integrals over the hypersurface S whose vanishing characterises in a necessary and sufficient manner initial data for the Kerr-Newman spacetime. Our main result, in this respect, is given in Theorem 6. Overview of the article Section 2 provides a brief overview of the theory of Killing spinors in electrovacuum spacetimes. Section 3 discusses the evolution equations governing the propagation of the Killing spinor equation in an electrovacuum spacetime. The main conclusion from this analysis is that the resulting system is homogeneous in a certain set of zero-quantities. The trivial data for these equations gives rise to the conditions implying the existence of a Killing spinor in the development of some initial hypersurface. In Section 4 a space-spinor formalism is used to reexpress these conditions in terms of quantities intrinsic to the initial hypersurface. In addition, in this section the interdependence between the various conditions is analysed and a minimal set of Killing spinor data equations is obtained. Section 5 introduces the notion of approximate Killing spinors for electrovacuum initial data sets and discusses som basic ellipticity properties of the associated approximate Killing spinor equation. Section 6 discusses the solvability of the approximate Killing spinor equation in a class of asymptotically Euclidean manifolds. Finally, Section 7 brings together the analyses in the various section to construct a geometric invariant characterising initial data for the Kerr-Newman spacetime. The main result of this article is given in Theorem 6.

Notation and conventions
Let (M, g, F ) denote an electrovacuum spacetime -i.e. a solution to the Einstein-Maxwell field equations. The signature of the metric in this article will be (+, −, −, −), to be consistent with most of the existing literature using spinors. We use the spinorial conventions of [19]. The lowercase Latin letters a, b, c, . . . are used as abstract spacetime tensor indices while the uppercase letters A, B, C, . . . will serve as abstract spinor indices. The Greek letters µ, ν, λ, . . . will be used as spacetime coordinate indices while α, β, γ, . . . will serve as spatial coordinate indices. Finally A, B, C, . . . will be used as spinorial frame indices.
The conventions for the spinorial curvature are set via the expressions We systematically use of the following expression for the (once contracted) second derivative of a spinor:

Killing spinors in electrovacuum spacetimes
In this section we provide a systematic exposition of the properties of Killing spinors in an electrovacuum spacetime.

The Einstein-Maxwell equations
Using standard spinorial notation, the Einstein-Maxwell equations are given by In particular, from the Maxwell equation (3b) it follows that The Bianchi identity is given by Given an electrovacuum spacetime, applying the derivative ∇ A ′ C to the Maxwell equation in the form ∇ A A ′ φ AB = 0 one obtains, after some standard manipulations, the following wave equation for the Maxwell spinor:

Killing spinors
A Killing spinor κ AB = κ (AB) in an electrovacuum spacetime (M, g, F ) is a solution to the Killing spinor equation In the sequel, a prominent role will played by the integrability conditions implied by the Killing spinor equation. More precisely, one has the following: Lemma 1. Let (M, g, F ) denote an electrovacuum spacetime endowed with a Killing spinor κ AB . Then κ AB satisfies the integrability conditions: Proof. The integrability conditions follow from applying the derivative ∇ D A ′ to the Killing spinor equation (6), then using the identity (2) together with the box commutators (1) and finally decomposing the resulting expression into its irreducible terms -the only non-trivial trace yields equation (7b) while the completely symmetric part gives equation (7a).

Remark 1.
Observe that although every solution to the Killing spinor equation (6) satisfies the wave equation (7b), the converse is not true. In what follows, a symmetric spinor satisfying equation (7b), but not necessarily equation (6), will be called a Killing spinor candidate. This notion will play a central role in our subsequent analysis -in particular, we will be concerned with the question of the further conditions that need to be imposed on a Killing spinor candidate to be an actual Killing spinor.
A well-known property of Killing spinors in a vacuum spacetime is that the spinor is the counterpart of a (possibly complex) Killing vector ξ a . A similar property holds for electrovacuum spacetimes -however, a further condition is required on the Killing spinor.
Lemma 2. Let (M, g, F ) denote an electrovacuum spacetime endowed with a Killing spinor κ AB . Then ξ AA ′ as defined by equation (8) is the spinorial counterpart of a Killing vector ξ a if and only if Proof. The proof follows by direct substitution of the definition (8) into the derivative ∇ AA ′ ξ BB ′ . Again, using the box commutators (1) one obtains, after some manipulations that from which the result follows.
Remark 2. Condition (9) implies that the Killing spinor κ AB and the Maxwell spinor φ AB are proportional to each other -thus, in what follows we refer to (9) as the matter alignment condition.
Remark 3. In the sequel we will refer to a spinor ξ AA ′ obtained from a symmetric spinor κ AB using expression (8) (not necessarily a Killing spinor) as the Killing vector candidate associated to κ AB .

Zero-quantities
In order to investigate the the consequences of the Killing spinor equation (6) in a more systematic manner it is convenient to introduce the following zero-quantities: Observe that if H A ′ ABC = 0 then κ AB is a Killing spinor. Similarly, if S AA ′ BB ′ = 0 then ξ AA ′ is the spinor counterpart of a Killing vector, while if Θ AB = 0 then the matter alignment condition (9) holds.
The decomposition in irreducible components of ∇ AA ′ κ BC can be expressed in terms of H A ′ ABC and ξ AA ′ as Similarly, a further computation shows that for ξ AA ′ as given by equation (8) one has the decomposition where If ξ AA ′ is a real Killing vector then the spinor η AB encodes the information of the so-called Killing form.
Remark 4. From equation (12) it readily follows by contraction that independently of whether the alignment condition (9) holds or not -i.e. the Killing vector candidate ξ AA ′ defined by equation (8) is always divergence free. This observation, in turn, implies that S AA ′ AA ′ = 0, so that one has the symmetry Remark 5. The zero-quantities introduced in equations (10a)-(10c) are a helpful bookkeeping device. In particular, a calculation analogous to that of the proof of Lemma 1 shows that Thus, the integrability conditions of Lemma 1 can be written, alternatively, as In particular, observe that if κ AB is a Killing spinor candidate, then the zero quantity H A ′ ABC is divergence free.

A characterisation of Kerr-Newman in terms of spinors
The following definition will play an important role in our subsquent analysis: is an open interval and B R is a closed ball of radius R. In the local coordinates (t, x α ) defined by the diffeomorphism the components g µν and F µν of the metric g and the Faraday tensor F satisfy where C and C ′ are positive constants, r ≡ (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 , and η µν denote the components of the Minkowski metric in Cartesian coordinates.
Remark 6. It follows from condition (14c) in Definition 1 that the stationary asymptotically flat end M ∞ is endowed with a Killing vector ξ a which takes the form ∂ t -a so-called time translation. From condition (14d) one has that the electromagnetic fields inherits the symmetry of the spacetime -that is L ξ F = 0, with L ξ the Lie derivative along ξ a .
Of particular interest will be those stationary asymptotically flat ends generated by a Killing spinor : Definition 2. A stationary asymptotically flat end M ∞ ⊂ M in an electrovacuum spacetime (M, g, F ) endowed with a Killing spinor κ AB is said to be generated by a Killing spinor if the spinor ξ AA ′ ≡ ∇ B A ′ κ AB is the spinorial counterpart of the Killing vector ξ a . Remark 7. Stationary spacetimes have a natural definition of mass in terms of the Killing vector ξ a that generates the isometry -the so-called Komar mass m defined as where S r is the sphere of radius r centred at r = 0 and dS ab is the binormal vector to S r . Similarly, one can define the total electromagnetic charge of the spacetime via the integral Remark 8. In the asymptotic region the components of the metric can be written in the form where m is the Komar mass of ξ a in the end M ∞ , ǫ αβγ is the flat rank 3 totally antisymmetric tensor and S β denotes a 3-dimensional tensor with constant entries. For the components of the Faraday tensor one has that -see e.g. [23]. Thus, to leading order any stationary electrovacuum spacetime is asymptotically a Kerr-Newman spacetime.
In [14] the following result has been proved: Theorem 1. Let (M, g, F ) be a smooth electrovacuum spacetime satisfying the matter alignment condition with a stationary asymptotically flat end M ∞ generated by a Killing spinor κ AB . Let both the Komar mass associated to the Killing vector ξ AA ′ ≡ ∇ B A ′ κ AB and the total electromagnetic charge in M ∞ be non-zero. Then (M, g, F ) is locally isometric to the Kerr-Newman spacetime.
The above result can be regarded as a refinement of the characterisations of the Kerr-Newman spacetime given in [24].

The Killing spinor evolution system in electrovacuum spacetimes
In this section we systematically investigate the interrelations between the zero-quantities H A ′ ABC , S AA ′ BB ′ and Θ AB . The ultimate objective of this analysis is to obtain a system of homogeneous wave equations for the zero-quantities.

A wave equation for ξ AA ′
Given a Killing spinor candidate κ AB , the wave equation (7b) naturally implies a wave equation for the Killing vector candidate ξ AA ′ . We first notice the following alternative expression for the field S AA ′ BB ′ : Lemma 3. Let κ AB denote a symmetric spinor field in an electrovacuum (M, g, F ). Then, one has that Proof. To obtain the identity one starts by substituting the expression ξ AA ′ = ∇ Q A ′ κ QA into the definition of S AA ′ BB ′ , equation (10b). One then commutes covariant derivatives using the commutators (1) and makes use of the decompositions of ∇ AA ′ κ BC , ∇ AA ′ ξ BB ′ and S AA ′ BB ′ given by equations (11), (12) and (13), respectively, to simplify.

Remark 9.
Observe that in the above result the spinor κ AB is not assumed to be a Killing spinor candidate.
The latter is used, in turn, to obtain the main result of this section: Lemma 4. Let κ AB denote a Killing spinor candidate in an electrovacuum spacetime (M, g, F ).
Then the Killing vector candidate ξ AA ′ ≡ ∇ Q A ′ κ AQ satisfies the wave equation Proof. One makes use of the definition of S AA ′ BB ′ and the identity (15) to write The above expression can be simplified using the Maxwell equations. Moreover, commuting covariant derivatives in the terms Finally, using that ξ AA ′ is a Killing vector candidate (see Remark 4) and that ∇ AA ′ H A ′ ABC = 0 (see Remark 5) the result follows.
Remark 10. Important for the subsequent discussion is that the wave equation (17) takes, in tensorial terms, the form where J a is defined in spinorial terms by In terms of the zero-quantity ζ AA ′ to be introduced in equation (19) one has Thus, J AA ′ is an homogeneous expression of zero-quantities and does not involve their derivatives.

A wave equation for H A ′ ABC
Lemma 5. Let κ AB denote a Killing spinor candidate in an electrovacuum spacetime (M, g, F ). Then the zero-quantity H A ′ ABC satisfies the wave equation Proof. We consider, again, the identity (15) in the form Applying the derivative ∇ D B ′ to the above expression one readily finds that Using the identity (2) and the box commutators (1) one obtains, after using the Maxwell equations to simplify, the desired equation.
Remark 11. Observe that the righthand side of the wave equation (18) is an homogeneous expression in the zero-quantity H A ′ ABC and the first order derivatives of Θ AB and S AA ′ BB ′ .

A wave equation for Θ AB
In order to compute a wave equation for the zero-quantity associated to the matter alignment condition it is convenient to introduce a further zero-quantity: Clearly, if the matter alignment condition (9) is satisfied, then ζ AA ′ = 0. The reason for introducing this further field will become clear in the sequel. Using the above definition one obtains the following: Lemma 6. Let κ AB denote a symmetric spinor field in an electrovacuum spacetime (M, g, F ).
Then, one has that Proof. The wave equation follows from applying the derivative ∇ B A ′ to the definition of ζ AA ′ and using the identity (2) together with the box commutators (1).

Remark 12.
A direct computation using the definitions of Θ AB and ζ AA ′ together with the expression for the irreducible decomposition of ∇ AA ′ κ BC given by equation (11) and the Maxwell equations gives that Remark 13. It follows directly from equation (20) that Alternatively, this property can be verified through a direct computation using the identity (21).
As the right hand side of equation (20) is an homogeneous expression in Θ AB and a first order derivative of ζ AA ′ , one needs to construct a wave equation for ζ AA ′ . The required expression follows from an involved computation -as it can be seen from the proof of the following lemma: Lemma 7. Let κ AB denote a symmetric spinor field in an electrovacuum (M, g, F ). Then, one has that Proof. Consider the identity (21) and apply the derivative ∇ A B ′ to obtain Some further simplifications yield To obtain the required wave equation we apply ∇ D B ′ to the above expression and make use of the decomposition (2) on the terms Finally, substitution of the wave equations for φ AB and H A ′ ABCD , equations (5) and (18) yields the required expression homogeneous in zero-quantities.

A wave equation for S AA ′ BB ′
The discussion of the wave equation for the spinorial field S AA ′ BB ′ is best carried out in tensorial notation. Accordingly, let S ab denote the tensorial counterpart of the (not necessarily Hermitian) spinor S AA ′ BB ′ . Key to this computation is the wave equation for the Killing vector candidate ξ a , equation (17).
Let κ AB denote a Killing spinor candidate in an electrovacuum spacetime (M, g, F ).
Then the zero-quantity S ab satisfies the wave equation where L ξ T ab denotes the Lie derivative of the energy-momentum of the Faraday tensor.
Proof. The required expression follows from applying = ∇ a ∇ a to commuting covariant derivatives, using the wave equation (17), the Einstein equation and the relation A straightforward computation computation shows that the Lie derivative of the electromagnetic energy-momentum tensor can be expressed in terms of the Lie derivative of the Faraday tensor and the zero-quantity S ab as Furthermore, the Lie derivative of the Faraday tensor can be expressed in terms of the Lie derivative of the Maxwell spinor as where the Lie derivative of the Maxwell spinor is defined by -see Section 6.6 in [20]. This expression can be written in terms of zero quantities by using the wave equations for the Killing and Maxwell spinors, the Maxwell equations and the identity along with the wave equations for the Killing and Maxwell spinors and the Maxwell equations, equations (7b) and (17), so as to obtain From the previous discussion it follows that: Lemma 9. Let κ AB denote a Killing spinor candidate in an electrovacuum spacetime (M, g, F ).
Then the Lie derivative L ξ T ab can be expressed as an homogeneous expression in the zeroquantities and their first order derivatives.

Remark 14.
In the context of the present discussion the object L ξ φ AB , as defined in (24), must be regarded as a convenient shorthand for a complicated expression. It is only consistent with the usual notion of Lie derivative of tensor fields if ξ AA ′ is the spinorial counterpart of a conformal Killing vector ξ a -see [20], Section 6.6, for further discussion on this point.

Summary
We summarise the discussion of the present section in the following: Let κ AB denote a Killing spinor candidate in an electrovacuum spacetime (M, g, F ).
Then the zero-quantities satisfy a system of wave equations, consisting of equations (18), (20), (22) and (23), which is homogeneous on the above zero-quantities and their first order derivatives.
A direct consequence of the above and the uniqueness of solutions to homogeneous wave equations is the following: Theorem 2. Let κ AB denote a Killing spinor candidate in an electrovacuum spacetime (M, g, F ) and let S denote a Cauchy hypersurface of (M, g, F ). The spinor κ AB is an actual Killing spinor if and only if on S one has that Proof. The initial data for the homogeneous system of wave equations for the fields H A ′ ABC , Θ AB , ζ AA ′ and S AA ′ BB ′ given by equations (18), (20), (22) and (23) consists of the values of these fields and their normal derivatives at the Cauchy surface S. Because of the homogeneity of the equations, the unique solution to these equations with vanishing initial data is given by Thus, if this is the case, the spinor κ AB satisfies the Killing equation on M and, accordingly, it is a Killing spinor. Conversely, given a Killing spinor κ AB over M, its restriction to S satisfies the conditions (25a)-(25d).

Remark 15.
As the spinorial zero-fields H A ′ ABC , Θ AB , ζ AA ′ and S AA ′ BB ′ can be expressed in terms of the spinor κ AB , it follows that the conditions (25a)-(25d) are, in fact, conditions on κ AB , and its (spacetime) covariant derivatives up to third order. In the next section it will be shown how these conditions can be expressed in terms of objects intrinsic to the hypersurface S.

The Killing spinor data equations
The purpose of this section is to show how the conditions (25a)-(25d) of Theorem 2 can be reexpressed as conditions which are intrinsic to the hypersurface S. To this end we make use of the space-spinor formalism outlined in [4] with some minor notational changes.

The space-spinor formalism
In what follows assume that the spacetime (M, g) obtained as the development of Cauchy initial data (S, h, K) can be covered by a congruence of timelike curves with tangent vector τ a satisfying the normalisation condition τ a τ a = 2 -the reason for normalisation will be clarified in the following -see equation (28). Associated to the vector τ a one has the projector projecting tensors into the distribution τ ⊥ of hyperplanes orthogonal to τ a .
Remark 16. The congruence of curves needs not to be hypersurface orthogonal -however, for convenience it will be assumed that the vector field τ a is orthogonal to the Cauchy hypersurface S.
Now, let τ AA ′ denote the spinorial counterpart of the vector τ a -by definition one has that In the following we restrict the attention to spin-dyads such that It follows then that consistent with the normalisation condition (26). As a consequence of this relation, the spinor τ AA ′ can be used to introduce a formalism in which all primed indices in spinors and spinorial equations are replaced by unprimed indices by suitable contractions with τ A A ′ .
Remark 17. The set of transformations on the dyad {o A , ι A } preserving the expansion (27) is given by the group SU (2, C).

The Sen connection
The space-spinor counterpart of the spinorial covariant derivative ∇ AA ′ is defined as The derivative operator ∇ AB can be decomposed in irreducible terms as where The operator P is the directional derivative of ∇ AA ′ in the direction of τ AA ′ while D AB corresponds to the so-called Sen connection of the covariant derivative ∇ AA ′ implied by τ AA ′ .

The acceleration and the extrinsic curvature
Of particular relevance in the subsequent discussion is the decomposition of the covariant derivative of the spinor τ BB ′ , namely ∇ AA ′ τ BB ′ . A calculation readily shows that the content of this derivative is encoded in the spinors corresponding, respectively, to the spinorial counterparts of the acceleration and the Weingarten tensor, expressed in tensorial terms as It can be readily verified that In the sequel it will be convenient to express K ABCD in terms of its irreducible components. To this end define so that one can define If the vector field τ a is hypersurface orthogonal, then one has that Ω AB = 0, and thus the Weingarten tensor satisfies the symmetry K ab = K (ab) so that it can be regarded as the extrinsic curvature of the leaves of a foliation of the spacetime (M, g). If this is the case, in addition to the second symmetry in (31) one has that In particular, K ABCD restricted to the hypersurface S satisfies the above symmetry and one has Ω AB = 0 -cfr. Remark 16. In what follows denote by D AB = D (AB) the spinorial counterpart of the Levi-Civita connection of the metric h on S. The Sen connection D AB and the Levi-Civita connection D AB are related to each other through the spinor K ABCD . For example, for a valence 1 spinor π A one has that with the obvious generalisations for higher order spinors.

Hermitian conjugation
Given a spinor π A , its Hermitian conjugate is defined as This operation can be extended in the obvious way to higher valence pairwise symmetric spinors. The operation of Hermitian conjugation allows to introduce a notion of reality. Given spinors ν AB = ν (AB) and ξ ABCD = ξ (AB)(CD) , we say that they are real if and only if If the spinors are real then it can be shown that there exist real spatial 3-dimensional tensors ν i and ξ ij such that ν AB and ξ ABCD are their spinorial counterparts. We also note that independently of whether ν AB and ξ ABCD are real or not. Finally, it is observed that while the Levi-Civita covariant derivative D AB is real in the sense that the Sen connection D AB is not. More precisely, one has that

Commutators
The main analysis of this section will require a systematic use of the commutators of the the covariant derivatives P and D AB . In order to discuss these in a convenient manner it is convenient to define the Hermitian conjugate of the Penrose box operator In terms of AB and AB , the commutators of P and D AB read Remark 18. Observe that on the hypersurface S the commutator (32b) involves only objects intrinsic to S. Notice, also, that the Sen connection D AB has torsion. Namely, for a scalar φ one has that

Basic decompositions
The purpose of this section is to provide a systematic discussion of the irreducible decompositions of the various spinorial fields and equations that will be required in the subsequent analysis.

Space-spinor decomposition of the Killing spinor and Maxwell equations
For reference, we provide a brief discussion of the space-spinor decomposition of the Killing equation, equation (6), and the Maxwell equation, equation (3b).
Contracting the Killing spinor equation (6) in the form where ∇ AB is the differential operator defined in equation (29). Using the decomposition (29) one further obtains Taking, respectively, the trace and the totally symmetric part of the above expression one readily obtains the equations Equation (33a) can be naturally interpreted as an evolution equation for the spinor κ AB while equation (33b) plays the role of a constraint.
A similar calculation applied to the Maxwell equation, equation (3b), in the form ∇ A A ′ φ AC = 0 yields the equations Again, equation (34a) is an evolution equation for the Maxwell spinor φ AB while (34b) is the spinorial version of the electromagnetic Gauss constraint.
Remark 19. The operation of Hermitian conjugation can be used to define, respectively, the electric and magnetic parts of the Maxwell spinor: It can be readily verified that Thus, E AB and B AB are the spinorial counterparts of 3-dimensional tensors E i and B i -the electric and magnetic parts of the Faraday tensor with respect to the normal to the hypersurface S.

The decomposition of the components of the curvature
Crucial for our subsequent discussion will be the fact that the restriction of the Weyl spinor Ψ ABCD to an hypersurface S can be expressed in terms of quantities intrinsic to the hypersurface. In analogy to the case of the Maxwell spinor φ AB , the Hermitian conjugation operation can be used to decompose the Weyl spinor Ψ ABCD into its electric and magnetic parts with respect to the normal to S as The electrovacuum Bianchi identity (4) implies on S the constraint Finally, using the Gauss-Codazzi and Codazzi-Mainardi equations one finds that where r ABCD is the spinorial counterpart of the Ricci tensor of the intrinsic metric of the hypersurface S.

Decomposition of the spatial derivatives of the Killing spinor candidate
Given a spinor κ AB defined on the Cauchy hypersurface S, it will prove convenient to define: These spinors correspond to the irreducible components of the Sen derivative of κ AB , as follows: Using the commutation relation for the Sen derivatives, equation (32b), we can also calculate the derivatives of ξ and ξ AB . The irreducible components of D AB ξ CD are given on S -where Ω AB = 0-by where we have also used the Hermitian conjugate of the Maxwell spinor, defined by Note that in (36b), the term D AB ξ appears -there is no independent equation for the Sen derivative of ξ.
We can also calculate the second order derivatives of ξ. Again, on the hypersurface S these take the form: Remark 20. It is of interest to remark that equation (37b) is just the statement that the Sen connection has torsion -cf. Remark 18.
An important and direct consequence of the above expressions is the following: Proof. The proof of the above result follows from direct inspection of equations (36a)-(36c) and (37a)-(37c).

Remark 21.
We observe that the above result is strictly not true if ξ ABCD = D (AB κ CD) = 0.

The decomposition of the Killing spinor data equations
In this section we provide a systematic discussion of the decomposition of the Killing initial data conditions in Theorem 2. The main purpose of this decomposition is to untangle the interrelations between the various conditions and to obtain a minimal set of equations which is intrinsic to the Cauchy hypersurface S.
For the ease of the discussion we make explicit the assumptions we assume to hold throughout this section: Assumption 1. Given a Cauchy hypersurface S of an electrovacuum spacetime (M, g), we assume that the hypothesis and conclusions of Theorem 2 hold.
Also, to ease the calculations, without loss of generality we assume: Assumption 2. The spinor τ AA ′ which on S is normal to S is extended off the initial hypersurface in such a way that it is the spinorial counterpart of the tangent vector to a congruence of g-geodesics. Accordingly one has that K AB = 0 -that is, the acceleration vanishes.

Decomposing H A ′ ABC = 0
Splitting the expression τ D A ′ H A ′ ABC into irreducible parts, and using the definitions (35a)-(35c) gives that the condition H A ′ ABC = 0 is equivalent to Equation (38a) is a condition intrinsic to the hypersurface while (38b) is extrinsic -i.e. it involves derivatives in the direction normal to S.

Decomposing
If H A ′ ABC = 0 on S, it readily follows that D EF H A ′ ABC = 0 on S. Thus, in order investigate the consequences of the second condition in (25a) it is only necessary to consider the transverse derivative PH A ′ ABC . It follows that and so as H A ′ ABC | S = 0, the irreducible parts of τ D A ′ PH A ′ ABC = 0 are given by Taking equation (39a) and commuting the D AB and P derivatives, and using equations (38a) and (38b), gives Substituting for the derivative of ξ AB using (36c), and using equations (38a) and (38b) again, gives To reexpress condition (39b), we use the following result which is obtained by commuting the D AB and P derivatives: Recall that the Killing spinor candidate κ AB satisfies the homogeneous wave equation (7b). We can use the space-spinor decomposition to split the wave operator into Sen and normal derivative operators. The result is: Applying conditions (38a) and (38b) to the right hand side of the latter, evaluating at S (where Ω AB = 0) and setting K AB = 0 gives Then, using equations (41) and (36b), as well as (38a) and (38b) as needed, it can be shown that which is exactly the condition we needed. Thus, we have shown that the condition (39b) is purely a consequence of the evolution equation for the Killing spinor candidate, along with the conditions arising from H A ′ ABC | S = 0.

Decomposing Θ AB = 0
As Θ AB has no unprimed indices, it is already in a space-spinor compatible form -we have the condition:

Decomposing ∇ EE
If Θ AB | S = 0, one only needs to consider the normalderivative PΘ AB . Using the evolution equation for the spinor φ AB implied by Maxwell equations, equation (34a), along with (38b) in the condition PΘ AB = 0 gives the spatially intrinsic condition In summary, assuming (38b) holds, then:

Decomposing
Our point of departure to decompose the condition S AA ′ BB ′ | S = 0 is the relation linking S AA ′ BB ′ to Θ AB and the derivative of H A ′ ABC given by equation (15). Splitting the derivative of H A ′ ABC into normal and tangential parts gives We already have conditions ensuring that Θ AB | S = H A ′ ABC | S = PH A ′ ABC | S = 0, and so as a consequence we automatically have that S AA ′ BB ′ | S = 0.

Decomposing
Again as S AA ′ BB ′ | S = 0, one only needs to consider the normal derivative PS AA ′ BB ′ . Taking the normal derivative of equation (45) and using that one has a Gaussian gauge gives on S that The first and second terms on the right hand side are zero as a consequence of conditions (43) and (44). The last term can be also shown to be zero by commuting the derivatives and using (38a), (38b) and (40). This leaves Eliminating the primed indices by multiplying by factors of τ AA ′ gives Thus, if this condition is satisfied on S, then we have that PS AA ′ BB ′ | S = 0. In the following we investigate further the consequences of this condition. As in a Gaussian gauge Pτ AA ′ = 0 it readily follows that, in fact, one has Splitting into irreducible parts, one obtains two necessary conditions: Let us first consider condition (47a). We can commute the Sen derivative with one of the normal derivatives to obtain Now, we can use our previous conditions on S to eliminate terms. For example, the second term in the bracket is zero from conditions (43) and (44). The fifth, sixth and seventh terms vanish from (38a) and (40). We can also use (38b) and (42) to replace the last term -alternatively, one can commute the derivatives, use the substitution and then commute back; the result is the same. From this substitution one obtains a factor D (AB ξ CD) inside the normal derivative, which can be replaced using (36c) -this equation is valid on the whole spacetime rather than just the hypersurface, so one is allowed to take normal derivatives of it. Proceeding as above, condition (47a) can be reduced to Now, splitting the covariant derivatives in the Bianchi identity (4) into normal and tangential components gives the following space-spinor version: One can use the latter expression to further reduce condition (48) to This is an intrinsic condition on S.
In order to obtain insight into condition (47b) we make use, again, of the wave equation (7b) for the spinor κ AB . Taking a normal derivative of this equation one obtains Splitting the spacetime derivatives into normal and tangential parts and rearranging gives As before, we can use our previous conditions to eliminate terms. The fourth and eight terms on the right hand side vanish due to (38a) and (40). Also, we can use equation (36b) to replace the the seventh term -this is because the relation (36b) holds on the whole spacetime, and so one can take normal derivatives of it freely. These steps give Alternatively, consider the second derivative of ξ AB , given by applying a normal derivative to equation (41) -note that equation (41) applies on the whole spacetime), so one can take the normal derivative. This yields As before, we can use the conditions (38a), (38b), (40) and (42), and the identity (36b) to reduce this to By comparing terms, we find that which is exactly the second condition (47b). So, no further conditions are needed to be prescribed on the hypersurface -this condition arises naturally from the evolution equation for the Killing spinor.

Decomposing
Again, if ζ AA ′ | S = 0 then one one only needs to consider the transverse derivative Pζ AA ′ . By definition one has that where the last equation has been obtained by commuting the Sen and normal derivatives, and using (44). Therefore one only needs to show that Now, recalling the wave equation for Θ AB , equation (20), one readily notices that the right hand side vanishes on S as a consequence of (38a), (38b) and (40), so that one is left with Finally, expanding the left hand side one finds that on S where the last line follows by commuting the derivatives where appropriate and using conditions (43) and (44). Finally, as τ CC ′ τ CC ′ = 2 by definition, we get that P 2 Θ AB = 0 as a consequence of the evolution equation for Θ AB .

Eliminating redundant conditions
The discussion of the previous subsections can be summarised in the following: Theorem 3. Let κ AB denote a Killing spinor candidate on an electrovacuum spacetime (M, g, F ). If κ AB satisfies on a Cauchy hypersurface S the intrinsic conditions and its normal derivative at S is given by then κ AB is, in fact, a Killing spinor.
Remark 23. We note that Using this fact, one can show that (50d) and (50e) can be more simply expressed as a condition on the proportionality between the Killing spinor κ AB and the Maxwell spinor φ AB .
In order to simplify the conditions in Theorem 3 and to analyse their various interrelations we proceed by looking at the different algebraic types that the Killing spinor can have. First, we consider the algebraically general case: Lemma 11. Assume that a symmetric spinor κ AB satisfies the conditions on an open subset U ⊂ S. Then, there exists a spin basis {o A , ι A } with o A ι A = 1 such that the spinors κ AB and φ AB can be expanded as Furthermore, if Q ≡ ϕe 2κ is a constant on U, then conditions (50d) and (50e) are satisfied on U.
Proof. The first part of the lemma follows directly from κ AB κ AB = 0, and the fact that κ (A C φ B)C = 0 implies that φ AB ∝ κ AB . The condition Ψ F (ABC κ D) F = 0 also allows us to expand the Weyl spinor in the same basis: To show the redundancy of (50d) and (50e), we first decompose the equation D (AB κ CD) = 0 into irreducible components. To simplify the notation, we borrow the D, ∆, δ symbols from the Newman-Penrose formalism to represent directional derivatives: The components of D (AB κ CD) = 0 then become: Using these, one can show that Now, using the electromagnetic Gauss constraint, equation (34b), together with the expansion for φ AB one obtains that using the basis expansion for φ AB one obtains δϕ + 2ϕδκ = 0 (53) on S. Now, the spacetime Bianchi identity (4) implies the contraint on S. To find the basis expansion of the Hermitian conjugate φ AB , note that: where k a ≡ o AōA ′ . As τ a is timelike and k a is null, this scalar product is non-zero, and the pair {o A , o A } forms a basis. We expand the spinor ι A in this basis as Contracting this with o A , we find 1/α = o A o A ≥ 0, and so α ≥ 0. Performing a Lorentz transformation on the basis {o A , ι A } parametrised by the complex function one has that This transformation preserves the value of o A ι A and the symmetrised product o (A ι B) , and thus, it preserves the form of the basis expansions of κ AB and φ AB . Moreover, one has that So, by choosing |λ| 2 = 1/α andβ = βλ 2 , and dropping the tildes, we get Using the above expressions we can find the basis expansion ofφ AB . Namely, one has that: Now, using the basis expansion for the Weyl spinor, contracting with combinations of o A and ι A and using the relations given in (52a) and (53), the components of (54) become Exploiting the conditions (52a), the expansions of the Maxwell and the Bianchi constraints it can be shown that condition (50e) can be decomposed into the following non-trivial irreducible parts: Assuming ϕ = 0, these conditions along with the Maxwell constraint (53) are equivalent to the following basis-independent expression, also independent of the value of β: The latter can be written as D AB ϕe 2κ = D AB Q = 0.
Therefore, under the hypotheses of the present lemma, equation (50e) is equivalent to the requirement of Q being constant in a domain U ⊂ S. In a similar way, substituting the above relations in equation (50d) and splitting into irreducible parts gives the following set of equivalent conditions: e κ (Dϕ + 2φDκ) = 0, e κ (∆ϕ + 2ϕ∆κ) = 0, e κ (δϕ + 2ϕδκ) = 0.
As e κ is non-zero, this set of conditions is again equivalent to the constancy of Q in U ⊂ S.
Next, we consider the case when the Killing spinor is algebraically special: Lemma 12. Assume the symmetric spinor κ AB satisfies the conditions on an open subset U ⊂ S. Then, there exists a normalised spin basis {o A , ι A } such that the spinors κ AB and φ AB can be expanded as Furthermore, the equations (50d) and (50e) are satisfied on U ⊂ S.
Proof. The first part of the lemma follows directly from the hypothesis κ AB κ AB = 0, κ AB κ AB = 0, and the fact that κ (A C φ B)C = 0 implies φ AB ∝ κ AB . The condition Ψ F (ABC κ D) F = 0 also allows us to expand the Weyl spinor in the same basis as In this basis, the components of the equation D (AB κ CD) = 0 become Using these relations one can show that The Maxwell constraint, equation (34b), on S is equivalent to on S, as a consequence of the previous relations, is equivalent to the following condition: Then, by substituting all the relevant basis expansions into (50d) and (50e), and splitting the equations into irreducible parts, one finds that both conditions are automatically satisfied as a result of the above relations.
We round up the discussion of this section with the following electrovacuum analogue of Theorem in [6]: Lemma 13. Assume that one has a symmetric spinor κ AB satisfying the conditions on the Cauchy hypersurface S and that the complex function Then the domain of dependence, D + (S), of the initial data set (S, g, K, F ) will admit a Killing spinor.
Proof. Let U 1 be the set of all points in S where κ AB κ AB = 0 and U 2 be the set of all points in S where κ AB κ AB = 0. The scalar functions κ AB κ AB : S → C and κ AB κ AB : S → R are continuous. Therefore, U 1 and U 2 are open sets. Now, let V 1 and V 2 denote, respectively, the interiors of S \ U 1 and V 1 \ U 2 . On the open set V 1 ∩ U 2 , we have that κ AB κ AB = 0 and κ AB κ AB = 0. Hence, by Lemma 12, the conditions (50d) and (50e) are satisfied on V 1 ∩ U 2 . Similarly, by Lemma 11, conditions (50d) and (50e) are satisfied on U 1 . On the open set V 2 , we have that κ AB = 0 and therefore (50d) and (50e) are trivially satisfied on V 2 . Using the above sets, the 3-manifold S can be split as By hypothesis, all terms in conditions (50d) and (50e) are continuous, and the conditions themselves are satisfied on the open sets U 1 , V 2 and V 1 ∩ U 2 . By continuity, the conditions are also satisfied on the boundaries ∂U 1 and ∂V 2 . Therefore, (50d) and (50e) are satisfied on int S, and by continuity this extends to the whole of S.

Summary
We can summarise the discussion of the present section calculations in the following theorem: Theorem 4. Let (S, h, K, F ) be an initial data set for the Einstein-Maxwell field equations where S is a Cauchy hypersurface. If the conditions are satisfied on S, then the development of the initial data set will admit a Killing spinor in the domain of dependence of S. The Killing spinor is obtained by evolving (7b) with initial data satisfying the above conditions and

The approximate Killing spinor equation
In the previous section we have identified the conditions that need to be satisfied by an initial data set for the Einstein-Maxwell equations so that its development is endowed with a Killing spinor -see Theorem 4. Together with the characterisation of the Kerr-Newman spacetime given by Theorem 1, the latter provide a way of characterising initial data for the Kerr-Newman spacetime.
The key equation in this characterisation is the spatial Killing spinor equation As it will be seen in the following, this equation is overdetermined and thus, admits no solution for a generic initial data set (S, h, K, F ). Following the discussion of Section 5 in [4], in this section we show how to construct a elliptic equation for a spinor κ AB over S which can always be solved and which provides, in some sense, a best fit to a spatial Killing spinor. This approximate Killing spinor will be used, in turn, to measure the deviation of the electrovacuum initial data set under consideration from initial data for the Kerr-Newman spacetime.

Basic identities
In the present section we provide a brief discussion of the basic ellipticity properties of the spatial Killing equation. In what follows, let S (AB) (S) and S (ABCD) (S) denote, respectively, the space of totally symmetric valence 2 and 4 spinor fields over the 3-manifold S. Given µ AB , ν AB ∈ S (AB) (S), ζ ABCD , χ ABCD ∈ S (ABCD) (S) one can use the Hermitian structure induced on S by τ AA ′ to define an inner product in S (AB) (S) and S (ABCD) (S), respectively, via where dµ denotes volume form of the 3-metric h.
In order to evaluate the above condition one makes use of the identity (integration by parts) (57) with U ⊂ S and where dS denotes the area element of ∂U, n AB is the spinorial counterpart of its outward pointing normal and ζ ABCD is a totally symmetric spinorial field. Now, observing that it follows then from the identity (57) that

Ellipticity of the approximate Killing spinor equation
The key observation concerning the approximate Killing spinor operator is given in the following: The operator L is a formally self-adjoint elliptic operator.
Proof. It is sufficient to look at the principal part of the operator L given by The symbol for this operator is given by where the argument ξ AB satisfies ξ AB = ξ (AB) and ξ AB = −ξ AB -i.e. ξ is a real symmetric spinor. Also, define the inner product , on the space of symmetric valence-2 spinors by The operator L is elliptic if the map is an isomorphism when |ξ| 2 ≡ ξ, ξ = 0. As the above mapping is linear and between vector spaces of the same dimension, one only needs to verify injectivity -in other words, that if ξ AB ξ (AB κ CD) = 0, then κ AB = 0. To show this, first expand the symmetrisation in the symbol to obtain where we have used the reality condition ξ AB = −ξ AB . Note also that the Jacobi identity implies that which reduces the above equation to 3κ CD |ξ| 2 + ξ CD ξ, κ = 0.
Contracting this with κ CD , and using the conjugate symmetry of the inner product, we obtain Both of these terms are positive, and so the equality can only hold if each term vanishes individually. Taking the first of these, one sees that when |ξ| 2 = 0, we must have |κ| 2 = 0. This is equivalent to κ AB = 0, completing the proof of injectivity and establishing the ellipticity of L.

The approximate Killing spinor equation in asymptotically Euclidean manifolds
The purpose of this section is to discuss the solvability of the approximate Killing spinor equation, equation (58), in asymptotically Euclidean manifolds. As a result of this analysis one concludes that for this type of initial data sets for the Einstein-Maxwell equations it is always possible to construct an approximate Killing spinor.

Weighted Sobolev norms
The discussion of asymptotic boundary conditions for the approximate Killing equation makes use of weighted Sobolev norms and spaces. In this section we introduce the necessary terminology and conventions to follow the discussion. The required properties of these objects for the present analysis are discussed in detail in Section 6.2 of [4] to which the reader is directed for further reference.
Given u a scalar function over S and δ ∈ R, let u δ denote the weighted L 2 Sobolev norm of u. All throughout we make use of of Bartnik's conventions for the weights -see [7]-so that, in particular u −3/2 is the standard L 2 norm of u. Similarly, let H s δ with s a non-negative index denote the weighted Sobolev space of functions for which the norm u s,δ ≡ 0≤|α|≤s D α u δ−|α| is finite where α = (α 1 , α 2 , α 3 ) is a multiindex and |α| ≡ α 1 + α 2 + α 3 . We say that u ∈ H ∞ δ if u ∈ H s δ for all s. We will say that a spinor or a tensor belongs to a function space if its norm does -so, for instance ζ AB ∈ H s δ is a shorthand for (ζ ABζ AB + ζ A Aζ B B ) 1/2 ∈ H s δ . A property of the weighted Sobolev spaces that will be used repeatedly is the following: if u ∈ H ∞ δ then u is smooth (i.e. C ∞ over S) and has a fall off at infinity such that D α u = o(r δ−|α| ) 1 . In a slight abuse of notation, if u ∈ H ∞ δ then we will often say that u = o ∞ (r δ ) at a given asymptotic end.

Asymptotically Euclidean manifolds
We begin by spelling out our assumptions on the class of Einstein-Maxwell initial data sets to be considered in the remainder of this article. The Einstein-Maxwell constraint equations are given by where D i denotes the Levi-Civita connection of the 3-metric h, r is the associated Ricci scalar, K ij is the extrinsic curvature, K ≡ K i i , ρ is the energy-density of the electromagnetic field, j i is the associated Poynting vector and E i and B i denote the electric and magnetic parts of the Faraday tensor with respect to the unit normal of S. Assumption 3. In the remainder of this article it will be assumed that one has initial data (h, K, E, B) for the Einstein-Maxwell equations which is asymptotically Reissner-Nordström in the sense that in each asymptotic end of S there exist asymptotically Cartesian coordinates (x α ) and two constants m, q for which Remark 25. The asymptotic conditions spelled in Assumption 3 ensure that the total electric charge of the initial data is non-vanishing. In particular, it contains standard initial data for the Kerr-Newman spacetime in, say, Boyer-Lindquist coordinates as an example. More generally, the assumptions are consistent with the notion of stationary asymptotically flat end provided in Definition 1.
Remark 26. The above class of initial data is not the most general one could consider. In particular, conditions (59a)-(59d) exclude boosted initial data. In order to do so one would require that The Einstein-Maxwell constraint equations would then require one to modify the leading behaviour of the the 3-metric h αβ . The required modifications for this extension of the present analysis are discussed in [4].

Asymptotic behaviour of the approximate Killing spinor
In this section we discuss the asymptotic behaviour of solutions to the spatial Killing spinor equation on asymptotically Euclidean manifolds of the type described in Assumption 3. To this end, we first consider the behaviour of the Killing spinor in the Kerr-Newman spacetime.
In a second stage we impose the same asymptotics solutions to the approximate Killing spinor equation on slices of a more general spacetime. In what follows, we concentrate our discussion to an asymptotic end.

Asymptotic behaviour in the exact Kerr-Newman spacetime
For the exact Kerr-Newman spacetime with mass m, angular momentum a and charge q it is possible to introduce a NP frame {l a , n a , m a ,m a } with associated spin dyad {o A , ι A } such that the spinors κ AB , φ AB and Ψ ABCD admit the expansion where r denotes the standard Boyer-Lindquist radial coordinate -see [2] for more details. A further computation shows that the spinorial counterpart, ξ AA ′ , of the Killing vector ξ a takes the form where the NP spin connection coefficients µ, π, τ and ρ satisfy the conditions µκ = µκ,τκ = κπ,ρκ = κρ which ensure that ξ AA ′ is a Hermitian spinor -i.e. ξ AA ′ =ξ AA ′ . Despite the conciseness of the above expressions, the basis of principal spinors given by {o A , ι A } is not well adapted to the discussion of asymptotics on a stationary end of the Kerr-Newman spacetime.
From the point of view of asymptotics, a better representation of the Kerr-Newman spacetime is obtained using a NP frame {l ′a , n ′a , m ′a ,m ′a } with associated spin dyad {o ′A , ι ′A } such that where the vector τ a is the tensorial counterpart of the spinor τ AA ′ . It follows from the above that Notice, in particular, that from the above expression it follows that ι ′ 2ξ AA ′ , one can use the expressions (60) and (61) to compute the leading terms of the Lorentz transformation relating the NP frames {l a , n a , m a ,m a } and {l ′a , n ′a , m ′a ,m ′a }. The details of this tedious computation will not be presented here -just the main result.
In what follows it will be convenient to denote the spinors of the basis {o ′A , ι ′A } in the form Moreover, let κ AB ≡ ǫ A A ǫ B B κ AB denote the components of κ AB with respect to the basis {ǫ A A }. It can then be shown that for Kerr-Newman initial data satisfying the asymptotic conditions (59a)-(59d) one can choose asymptotically Cartesian coordinates (x α ) = (x 1 , x 2 , x 3 ) and orthonormal frames on the asymptotic ends such that with From the the above expressions one finds that on the asymptotic ends Moreover, for any electrovacuum initial data set satisfying the conditions (59a)-(59d) a spinor of the form (62) satisfies

Asymptotic behaviour for non-Kerr data
Not unsurprisingly, given electrovacuum initial data satisfying the conditions (59a)-(59d), it is always possible to find a spinor κ AB satisfying the expansion (62) in the asymptotic region. More precisely, one has: Lemma 15. For any asymptotic end of an electrovacuum initial data set satisfying (59a)-(59d) there exists a spinor κ AB such that Proof. The proof follows the same structure of Theorem 17 in [4] where the vacuum case is considered.
Remark 27. The spinors obtained from the previous lemma can be cut-off so that they are zero outside the asymptotic end. One can then add them to yield a real spinorκ AB on the whole of S such that D (ABκCD) ∈ H ∞ −3/2 and asymptotic behaviour given by (62) at each end.
In the analysis of the solvability of the approximate Killing spinor equation it is crucial that there exist no nontrivial spatial Killing spinor that goes to zero at infinity. More precisely, one has the following: Lemma 16. Let ν AB ∈ H ∞ −1/2 be a solution to D (AB ν CD) = 0 on an electrovacuum initial data set satisfying the asymptotic conditions (59a)-(59d). Then ν AB = 0 on S.
Proof. From Lemma 10 one can write D AB D CD D EF κ GH as a linear combination of lower order derivatives, with smooth coefficients. Direct inspection shows that the coefficients in this linear combination have the decay conditions to make use of Theorem 20 from [4] with m = 2. It then follows that ν AB must vanish on S.

Solving the approximate Killing spinor equation
In the reminder of this section we will consider solutions to the approximate Killing spinor equation of the form: withκ AB the spinor discussed in Remark 27. For this Ansatz one has the following: Proof. The proof is analogous to that of Theorem 25 in [4] and is presented for completeness as this is the main result of this article. It will now be shown that a spinor ν AB satisfying the above must be trivial. Using the identity (57) with ζ ABCD = D (AB ν CD) and assuming that L(ν CD ) = 0 one obtains where ∂S ∞ denotes the sphere at infinity. Now, using that by assumption ν AB ∈ H 2 −1/2 , it follows that D (AB ν CD) ∈ H ∞ −3/2 and that The integration of the latter over a finite sphere is of type o(1). Accordingly, the integral over the sphere at infinity ∂S ∞ vanishes and, moreover, Thus, one concludes that D (AB ν CD) = 0 over S so that ν AB is a Killing spinor candidate. Now, Lemma 16 shows that there are no non-trivial Killing spinor candidates that go to zero at infinity. It follows from the discussion in the previous paragraph that the kernel of the approximate Killing spinor operator is trivial and that the Fredholm alternative imposes no obstruction to the existence of solutions to (65). Thus, one obtains a unique solution to the approximate Killing spinor equation with the prescribed asymptotic behaviour at infinity.

The geometric invariant
In this section we make use of the approximate Killing spinor constructed in the previous section to construct an invariant measuring the deviation of a given electrovacuum initial data set satisfying the asymptotic conditions (59a)-(59d) from initial data for the Kerr-Newman spacetime.
In the following let κ AB denote the approximate Killing spinor obtained from Theorem 5, and let where following the notation of Section 4 one has The above integrals are well-defined. More precisely, one has that: Lemma 17. Given the approximate Killing spinor κ AB obtained from Theorem 5, one has that J, I 1 , I 2 , I 3 < ∞.
We now look at the boundedness of I 2 . By construction and due to the asymptotic conditions (59a)-(59d), one can choose asymptotically Cartesian coordinates and orthonormal frames on the asymptotic ends such that the approximate Killing spinor and Maxwell spinor satisfy Therefore, and so Θ AB ∈ H ∞ −3/2 , and I 2 < ∞. Finally, to show the boundedness of I 3 , note that in the asymptotically Cartesian coordinates and orthonormal frames used above,we have κ AB κ AB 2 = 4 81 r 4 + o ∞ r −7/2 φ AB φ AB = q 2 2r 4 + o ∞ r −9/2 and so the quantity Q satisfies Taking a derivative, one obtains D AB Q 2 = o ∞ r −3/2 and therefore D AB Q 2 ∈ H ∞ −3/2 and I 3 < ∞.
The integrals J, I 1 , I 2 and I 3 are then used to define the following geometric invariant: One has the following result combine the whole analysis of this article: Theorem 6. Let (S, h, K, E, B) denote a smooth asymptotically Euclidean initial data set for the Einstein-Maxwell equations satisfying the on each of its two asymptotic ends the decay conditions (59a)-(59d) with non-vanishing mass and electromagnetic charge. Let I be the invariant defined by equation (67) where κ AB is the the unique solution to equation (58) with asymptotic behaviour at each end given by (62). The invariant I vanishes if and only if (S, h, K, E, B) is locally an initial data set for the Kerr-Newman spacetime.
Proof. The proof follows the same strategy of Theorem 28 in [4]. It follows from our assumptions that if I = 0 then the electrovacuum Killing spinor data equations (55a)-(55d) are satisfied on the whole of the hypersurface S. Thus, from Theorem 4 the development of the electrovacuum initial data (S, h, K, E, B) will have, at least on a slab a Killing spinor. Now, the idea is to make use of Theorem 1 to conclude that the development will be the Kerr-Newman spacetime. For this, one has to conclude that the spinor ξ AA ′ ≡ ∇ Q A κ BQ is Hermitian so that it corresponds to the spinorial counterpart of a real Killing vector. By assumption, it follows from the expansions (63a)-(63c) that Together, the last two expressions correspond to the Killing initial data for the imaginary part of ξ AA ′ -thus, the imaginary part of ξ AA ′ goes to zero at infinity. It is well know that for electrovacuum spacetimes there exist no non-trivial Killing vectors of this type [9,12]. Thus, ξ AA ′ is the spinorial counterpart of a real Killing vector. By construction, ξ AA ′ tends, asymptotically, to a time translation at infinity. Accordingly, the development of the electrovacuum initial data (S, h, K, E, B) contains two asymptotically stationary flat ends M ∞ and M ′ ∞ generated by the Killing spinor κ AB . As the Komar mass and the electromagnetic charge of each end is, by assumption, non-zero, one concludes from Theorem 1 that the development (M, g, F ) is locally isometric to the Kerr-Newman spacetime.

Conclusions
As a natural extension to the vacuum case described by Backdahl and Valiente Kroon [4], the formalism presented above for the electrovacuum case has similar applications and possible modifications. For example, the use of asymptotically hyperboloidal rather than asymptotically flat slices can now be analysed for the full electovacuum case, applying to the more general Kerr-Newman solution. Another interesting alternative to asymptotically flat slices would be to obtain necessary and sufficient conditions for the existence of a Killing spinor in the future development of a pair of intersecting null hypersurfaces. For instance, one could take a pair of event horizons intersecting at a bifurcation surface, and obtain a system of conditions intrinsic to the horizon that ensures the black hole interior is isometric to the Kerr-Newman solution.
A motivation for the above analysis was also to provide a way of tracking the deviation of initial data from exact Kerr-Newman data in numerical simulations. However, in order to be a useful tool, one would still have to show that the geometric invariant is suitably behaved under time evolution (such as monotonicity). As highlighted in [4], a major problem is that it is hard to find a evolution equation for κ AB such that the elliptic equations (58) is satisfied on each leaf in the foliation. If these issues can be resolved, then this formalism may be of some use in the study of non-linear perturbations of the Kerr-Newman solution and the black hole stability problem.
Finally, the ethos of this article is to show that the characterisation of black hole spacetimes using Killing spinors is still a fruitful avenue of investigation. In the future, we hope to show that this method can be used to investigate other open questions, such as the Penrose inequality and black hole uniqueness.