Higher dimensional Kerr-Schild spacetimes
aa r X i v : . [ g r- q c ] J a n Higher dimensional Kerr-Schild spacetimes
Marcello Ortaggio ∗ , Vojtˇech Pravda † , Alena Pravdov´a ‡ Institute of Mathematics, Academy of Sciences of the Czech RepublicˇZitn´a 25, 115 67 Prague 1, Czech RepublicOctober 30, 2018
Abstract
We investigate general properties of Kerr-Schild (KS) metrics in n > k is geodetic (or, equivalently, if T ab k a k b = 0). We subsequently specialize to vacuum KS solutions, whichnaturally split into two families of non-expanding and expanding metrics.After demonstrating that non-expanding solutions are equivalent to theknown class of vacuum Kundt solutions of Weyl type N, we analyze ex-panding solutions in detail. We show that they can only be of the type IIor D, and we characterize optical properties of the multiple Weyl alignednull direction (WAND) k . In general, k has caustics corresponding tocurvature singularities. In addition, it is generically shearing. Neverthe-less, we arrive at a possible ‘weak’ n > Kerr-Schild spacetimes [1] have played an important role in four-dimensionalgeneral relativity. In particular, all vacuum KS solutions have been knownfor some time [1–3]. They are a subset of algebraically special spacetimes,and the corresponding KS null congruence is a geodetic, shearfree, re-peated principal null direction of the Weyl tensor. Notably, they includethe Kerr metric, arguably one of the physically most important knownexact solutions of Einstein’s equations in vacuum. Other well-known vac-uum solutions which can be put in the KS form are, for example, pp-waves and type N Kundt waves. In addition, the KS class can also admit(aligned) electromagnetic or matter fields, so as to include, for instance,the Kerr-Newmann metric, the Vaidya solution and pp -waves coupled to ∗ ortaggio(at)math(dot)cas(dot)cz † [email protected] ‡ [email protected] null Maxwell field or to pure radiation. (We refer the reader to [4] for acomprehensive review and for a number of original references.)In recent years, gravity in more than four spacetime dimensions hasbecome an active area of ongoing studies, mainly motivated by the in-creasing interest in string theory and extra-dimensional scenarios. In thiscontext, the KS ansatz led to the remarkable discovery of rotating vacuumblack holes in an arbitrary dimension n > k is geodetic if and only if T ab k a k b = 0(similarly as in four dimensions [4]).In Sec. 3 we specialize to the case with a geodetic KS vector k andshow that in that case the spacetime is of Weyl type II or more special.In the rest of the paper we further restrict ourselves to vacuum solu-tions. These naturally split into a non-expanding and an expanding class.In Sec. 4 we study the non-expanding class and show that it is equivalentto type N Kundt vacuum solutions. As a consequence, we also observethat n > optical constraint , i.e. a purelygeometric condition on the KS null congruence k in the flat background.This is then used to show that expanding vacuum KS spacetimes arenecessarily of Weyl type II or D. By integration of the Ricci identities wesubsequently determine the dependence of the Ricci rotation coefficients(in particular, of optical quantities) on an affine parameter r along theKS congruence. This enables us also to find the r -dependence of the KSfunction and thus to study general basic properties of KS geometries. Inparticular, we discuss a ‘weak’ extension of the Goldberg-Sachs theoremvalid for vacuum KS spacetimes, which are generically shearing. We alsodemonstrate, in the ‘generic’ case, the presence of a curvature singularityat a caustic of k . This is in particular true for all expanding non-twistingspacetimes, on which we comment as a special subset of KS solutions.We conclude in Sec. 6 with a summary and final remarks. In ap-pendix A frame components of the Riemann tensor are given for KS pacetimes with a geodetic k . Subsequent appendices B–D contain severalproofs and technical details related to the results presented in the maintext. We use the notation of [8] (see also [9]) and, in particular, we introduce anull frame m (0) = n , m (1) = ℓ , m ( i ) ( i = 2 , . . . , n − ℓ a n a = 1 and m ( i ) a m ( j ) a = δ ij ( a, b = 0 , , . . . , n − g ab = 2 ℓ ( a n b ) + δ ij m ( i ) a m ( j ) b (sum over i, j ).Directional derivatives along frame vectors will be denoted by D ≡ ℓ a ∇ a , △ ≡ n a ∇ a , δ i ≡ m ( i ) a ∇ a . The full set of the corresponding Ricci rotationcoefficients is defined in [8,9]. In particular, we will often use the definition ℓ a ; b = L cd m ( c ) a m ( d ) b (sum over c, d ). In fact, it will be convenient toadapt the frame vector ℓ to coincide with the KS null vector k . When k is geodetic, we will denote by r the corresponding affine parameter.Quantities that do not depend on r will be denoted by a subscript orsuperscript index 0 (e.g., H , s i ) , etc.).Let us also anticipate here that we will be using five types of indiceswith different ranges, namely a, b, c, . . . = 0 , , . . . , n − , i, j, k, . . . = 2 , . . . , n − ,α, β, . . . = 2 p + 2 , . . . , m + 1 , ρ, σ, . . . = m + 2 , . . . , n − , (1) µ, ν = 1 , . . . , p, with 0 ≤ p ≤ m ≤ n − , where p and m are fixed integers defined later on. Einstein’s summationconvention is employed except for indices µ, ν (for which summation willbe indicated explicitly), or when at least one of the repeated indices is inbrackets (e.g., there will be no summation over i in DA ij = − s ( i ) A ij ),unless (only in very few exceptional cases) stated otherwise. Note thatfor indices i, j, . . . we will not distinguish between subscripts and super-scripts since there is no difference between covariant and contravariantcomponents.When referring to equations presented in previous papers it will besometimes convenient to denote them by the corresponding equation num-ber followed by the reference number, e.g. eq. (11f, [9]). In this section we define KS spacetimes and we study their general prop-erties, without imposing Einstein’s equations and without requiring theKS null congruence to be geodetic.
We study an n -dimensional spacetime with a metric in the KS form g ab = η ab − H k a k b , (2) ith η ab being the Minkowski metric diag( − , , ...., H a scalar func-tion and k a a 1-form that is assumed to be null with respect to η ab , i.e. η ab k a k b = 0 ( η ab is defined as the inverse of η ab and η ab = g ac g bd η cd ).From (2) it follows for the corresponding vector that k a ≡ η ab k b = g ab k b ,so that k a is null also with respect to g ab , and that the inverse metric hasthe form g ab = η ab + 2 H k a k b . (3)One also gets for the metric determinant g = η = − . (4)Straightforwardly from the definition of the Christoffel symbols weobtainΓ abc = − ( H k a k b ) ,c − ( H k a k c ) ,b + η ad ( H k b k c ) ,d + 2 H k a k d ( H k b k c ) ,d , (5)from which Γ abc k b k c = 0 , Γ abc k a k b = 0 , Γ aab = 0 , (6)and consequently also k a ; b k b = k a,b k b , k a ; b k b = k a,b k b . (7)In particular, this implies that k a is geodetic in the flat geometry η ab iffit is geodetic in the full geometry g ab . We also haveΓ abc k a = ( H k b k c ) ; d k d , Γ abc k b = − ( H k a k c ) ; d k d , (8) k a ; c k a = k a , c k a = 0 , k a ; bc k a + k a ; b k a ; c = k a , bc k a + k a , b k a , c = 0 . (9)Since Γ aab = 0, we can express the Ricci tensor as R ab = Γ cab,c − Γ cdb Γ dac . (10)After employing the above formulas, Einstein’s equations for the projectedcomponent R ab k a k b read R ab k a k b = 2 H g ab ( k a ; c k c )( k b ; d k d ) = κ T ab k a k b . (11)From now on, when using a null frame ℓ , n , m ( i ) , we will make theconvenient choice ℓ = k , (12)adapted to the KS ansatz.Using eq. (11) and (see [8]) k a ; b k b = L k a + L i m ( i ) a (13)(sum over i ), we can formulate the following Proposition 1
The null vector k in the KS metric (2) of an arbitrarydimension is geodetic if and only if the energy-momentum tensor satisfies T ab k a k b = 0 . Then, using frame indices, the condition of proposition 1 reads T = 0(or, equivalently, R = 0), and it holds iff the null frame components of T ab do not include a term proportional to n a n b . This condition is ofcourse satisfied in the case of vacuum spacetimes, also with a possiblecosmological constant, or in the presence of matter fields aligned with theKS vector k , such as an aligned Maxwell field (defined by F ab k a ∼ k b ) oraligned pure radiation (i.e., T ab ∼ k a k b ). .2 Optics of the KS vector field It is interesting to discuss how the optical properties of k a in the two ge-ometries g ab and η ab are related, i.e. to compare the Ricci rotation coeffi-cients [8, 9] L ab and ˜ L ab defined with respect to g ab and η ab , respectively.In order to do so one has to set up ‘null’ frames for both metrics. Let k , n , m ( i ) be the frame for the full spacetime metric g ab as discussed above.Then, from eq. (2) it follows that η ab = 2 k ( a [ n b ) + H k b ) ] + δ ij m ( i ) a m ( j ) b (sum over i, j ). Thus, k , ˜ n , m ( i ) is now a convenient frame for the flatmetric η ab , provided one takes˜ n a = n a + H k a . (14)Now, from eqs. (8) and (13) one gets k a ; b = k a,b − ( D H + 2 H L ) k a k b − H L i m ( i )( a k b ) (15)(sum over i ). Expanding k a ; b on the frame k , n , m ( i ) and k a,b on the frame k , ˜ n , m ( i ) in terms of L ab and ˜ L ab , respectively, and using (14) one findsthat L ab = ˜ L ab except for L = ˜ L − D H − H ˜ L and L i = ˜ L i − H ˜ L i .In particular, one has L i = ˜ L i (so that k is geodetic with respect to g ab iff it is geodetic with respect to η ab , as already observed in section 2), andthe matrix L ij ≡ k a ; b m ( i ) a m ( j ) b = k a,b m ( i ) a m ( j ) b (16)is the same with respect to both metrics. The optical scalars shear, twistand expansion [8, 9] are thus unchanged (note that when k is geodeticthis statement is independent of the particular choice (14) since L ij isthen invariant under null rotations with k fixed [9]). In the rest of the paper we will assume that the null KS vector k is geodetic and affinely parametrized (i.e., L i = 0 = L ), which seems tobe the simplest and physically most interesting case (recall proposition 1).The following definitions [8, 9] will thus be useful: S ij ≡ L ( ij ) = σ ij + θδ ij , A ij ≡ L [ ij ] ,θ ≡ n − S ii , σ ≡ σ ij σ ij , ω ≡ A ij A ij . (17)We shall refer to S ij , σ ij and A ij as the expansion , shear and twist matri-ces, respectively, and to θ , σ and ω as the corresponding scalars.Under the geodetic assumption, eq. (15) reduces to k a ; b = k a,b − ( D H ) k a k b , (18)which will be employed in the following. Using k a,b = η ac k c,b (and,from now on, denoting ( ) , d η cd by ( ) ,c , e.g. k ,ba = η bc k a,c etc.) oneeasily finds k a,b k a,c = k a ; b k a ; c , k a,b k c,a = k a ; b k c ; a , k ,ba k a,c = k ; ba k a ; c , k a , bc k b = − k a , b k b , c , k a , bc k b k c = 0 and similar identities, also to be used Cf [13] in the case n = 4. n following calculations. In addition, certain expressions will take a morecompact form if we write them in terms of the Ricci rotation coefficientsand of directional derivatives as k a,a = L ii = S ii , k a,b k a,b = L ij L ij , k a,b k b,a = L ij L ji , H ,a k a = D H , H ,ab k a k b = D H . (19)From eq. (5) we thus getΓ eab,e = ( H k a k b ) e, e − ( H k a k e ) ,be − ( H k b k e ) ,ae + 2 ˆ H D H + ( D H + H L ii ) D H ˜ k a k b , (20)Γ efb Γ fae = 2 ˆ ( D H ) − H ω ˜ k a k b , (21)The Ricci tensor then reads R ab = ( H k a k b ) e, e − ( H k a k e ) ,be − ( H k b k e ) ,ae + 2 H ˆ D H + L ii D H + 2 H ω ˜ k a k b . (22)Consequently, k is an eigenvector of the Ricci tensor, i.e. R ab k b = − [ D H + ( n − θD H + 2 H ω ] k a . (23)For certain applications it may be useful to observe that the mixedRicci components, i.e. R ab = ( H k a k b ) s, s − ( H k a k s ) ,bs − ( H k b k s ) a, s , arelinear in H [14, 15]. The Ricci scalar is thus also linear and reads R = − ˆ D H + 2( n − θD H + H ( n − n − θ + H ( ω − σ ) ˜ . (24)So far we have worked in Minkowski coordinates adapted to the flatbackground metric η ij . Note, however, that expressions for scalars, suchas frame components of tensors, do not depend on that choice. From eq. (22), one finds the non-vanishing frame components of the Riccitensor R = − [ D H + ( n − θD H + 2 H ω ] , (25) R ij = 2 H L ik L jk − D H + ( n − θ H ] S ij , (26) R = δ i ( δ i H ) + ( N ii − H L ii ) D H + (4 L j − L j − i M ji ) δ j H − L ii ∆ H + 2 H “ δ i L [1 i ] + 4 L i L [1 i ] + L i L i − L L ii + 2 L [1 j ] j M ii − A ij N ij − H ω ” , (27) R i = − δ i ( D H ) + 2 L [ i D H + 2 L ij δ j H − L jj δ i H + 2 H “ δ j A ij + A ij j M kk − A kj i M kj − L jj L i + 3 L ij L [1 j ] + L ji L (1 j ) ” . (28)Note that R = 0 = R i identically. This formula for R ab is equivalent to eq. (32.10) given in [4] for n = 4, up to rewritingpartial derivatives as covariant ones. .2 Algebraic type of the Weyl tensor The full set of the frame components of the Riemann tensor is given inappendix A for any KS geometry with a geodetic, affinely parametrizedKS vector. Thanks to R i j = R i = R ijk = 0 and to R = R i = 0,for the Weyl tensor we find immediately C i j = 0 , C i = 0 , C ijk = 0 . (29)The Weyl tensor components with boost order 2 and 1 are thus identicallyzero, which enables us to conclude Proposition 2
Kerr-Schild spacetimes (2) in arbitrary dimension with ageodetic KS vector k are of Weyl type II (or more special), with k beinga WAND of order of alignment ≥ Note that the KS null vector k must indeed be geodetic for a wideclass of matter fields, in particular in vacuum (cf proposition 1), so thatproposition 2 applies in those cases.Another interesting result follows from the observation [16] that space-times (not necessarily of the KS class) which are either static or stationarywith ‘expansion’ and ‘reflection symmetry’ can be only of the Weyl typesG, I i , D or O (see [16] for details and precise definitions). Taking the‘intersection’ of this family with the set of KS spacetimes considered inproposition 2, we can conclude that in arbitrary dimension n ≥ Proposition 3
All static spacetimes of the KS class with a geodetic k are of Weyl type D or O. All stationary spacetimes of the KS class with‘reflection symmetry’ and with a geodetic, expanding k are of Weyl typeD or O. In both cases, if the Weyl tensor is non-zero, k is a multipleWAND. As an immediate consequence, the higher-dimensional rotating blackholes of Myers and Perry (indeed obtained using the KS ansatz [5]), arenecessarily of Weyl type D in any dimension, as we anticipated in [16](this had previously been demonstrated by an explicit calculation of theWeyl tensor in [17]). This also applies to uniform black strings, sinceadding flat dimensions to a KS metric clearly preserves the KS structure.On the other hand, proposition 2 implies that five-dimensional vacuumblack rings [18] do not admit a KS representation, since they are of Weyltype I i [19]. Let us also emphasize that proposition 3 is not restrictedto vacuum KS solutions. For instance, it can also be used to concludethat static black holes with electric charge (and, possibly, a cosmologicalconstant) [20] are also of Weyl type D. Vacuum solutions must satisfy R ab = 0. We note from eqs. (25)–(28) thatthe Ricci tensor components R and R ij are simple and do not involveRicci rotation coefficients other than L ij , which characterize the opticalproperties of k . It is thus natural to start from the corresponding vacuum quations. Imposing R ij = 0 gives ( D ln H ) S ij = L ik L jk − ( n − θS ij . (30)Contracting eq. (30) with δ ij we obtain( n − θ ( D ln H ) = L ik L ik − ( n − θ = σ + ω − ( n − n − θ , (31)while its tracefree part (i.e., R ij − R kk n − δ ij = 0) is ( D ln H ) σ ij = “ σ ij − n − σ δ ij ” − “ A ij + n − ω δ ij ” + 2 σ k ( i A j ) k − ( n − θσ ij . (32)Next, the equation R = 0 requires D H + ( n − θD H + 2 H ω = 0 . (33)Using eq. (31), this can also be rewritten as D H = [ − σ − ω + ( n − n − θ ] H .Note that eq. (31) involves the function H in a non-trivial way onlywhen θ = 0. It will thus be convenient to study non-expanding andexpanding solutions separately in the following sections. The remainingvacuum equations seem to be of little help in a general study, and thereis no need to write them down at this stage. The case θ = 0 turns out to be somewhat special, since from eq. (31) weimmediately get σ = 0 = ω , and thus L ij = 0 , (34)and from this and eq. (33) D H = 0 , (35)i.e., H = H + G r (recall that in our notation r denotes an affine pa-rameter along k and D H = 0 = D G ). The vacuum equations R ij = 0and R = 0 are thus identically satisfied. Note, in particular, that thesole non-expanding condition implies that we are restricted to a subset ofthe higher-dimensional Kundt class of non-expanding, non-shearing andnon-twisting vacuum solutions.Setting L ij = 0 in the Ricci components (27) and (28), the remainingvacuum equations read δ i ( δ i H ) + N ii D H + (4 L j − L j − i M ji ) δ j H + 2 H “ δ i L [1 i ] + 4 L i L [1 i ] + L i L i + 2 L [1 j ] j M ii ” = 0 , (36) δ i ( D H ) − L [ i D H = 0 . (37) Hereafter, for brevity we shall write H − D H = D ln H , where it is understood that ln H should be replaced by ln |H| whenever H < For n = 4 this reduces to ( D ln H ) σ ij = 2 σ k ( i A j ) k . .2 Equivalence with Kundt solutions of type N We have seen above that vacuum non-expanding KS solutions are Kundtspacetimes. In general, the higher-dimensional Kundt class is known to beof Weyl type II or more special, provided R = 0 = R i (thus in particularin vacuum) [9]. However, here we show that, in arbitrary dimension,vacuum KS spacetimes of the Kundt class are restricted to type N. Infact, it has already been shown in section 3 that the components of theWeyl tensor with boost weight 2 and 1 vanish. Using eqs. (34), (35) and(37) together with eqs. (A2)–(A6) we also find that all Weyl componentswith boost weight 0 and − R abcd = C abcd ).The Weyl tensor is thus of type N, q.e.d.. We have thus demonstrated that vacuum solutions of the KS class with anon-expanding KS vector are a subset of Kundt spacetimes of Weyl typeN. We now show that the converse is also true, namely that vacuum Kundtsolutions of Weyl type N are of the KS form, so that the two families ofsolutions coincide.Vacuum Kundt solutions of Weyl type N belong to the family of space-times with vanishing scalar invariants (VSI) [21, 22]. We can thus beginwith the higher-dimensional VSI metric [22, 23]d s = 2d u h d r + H ( r, u, x k )d u + W i ( r, u, x k )d x i i + δ ij d x i d x j , (38)with i, j, k = 2 , . . . , n − k a d x a = d u being the multiple WAND.Similarly as for n = 4, higher-dimensional vacuum Kundt spacetimesof Weyl type N consist of two invariantly defined subfamilies [22, 23]:Kundt waves (with L i = L i = 0) and pp -waves (for which L i = L i =0 = L ). It is convenient to discuss these subfamilies separately.The metric functions H and W i for higher-dimensional type N Kundtwaves are [22] W = − rx ,W s = x q B qs ( u ) + C s ( u ) , (39) H = r x ) + H ( u, x i ) , where s, q = 3 , . . . , n − B qs = − B sq and H must obey a field equationgiven in [22]. As shown in [22], metric (38), (39) is flat for H ( u, x i ) = H flat = 12 n − X s =3 W s + x F ( u ) + x x i F i ( u ) , (40)where F ( u ) and F i ( u ) are arbitrary functions of u . Therefore metric (38),(39) is in the KS form for an arbitrary choice of H ( u, x i ), since it canalways be rewritten as d s = d s flat + ( H − H flat )d u . similar argument can be used to show that also higher-dimensionaltype N pp -waves belong to the KS class – the corresponding metric func-tions W i and H flat entering (38) can be found in [22]. We can thus summarize the results of this section in the following(see theorem 32.6 of [4] for n = 4) Proposition 4
In arbitrary dimension n ≥ , the Kerr-Schild vacuumspacetimes with a non-expanding KS congruence k coincide with the classof vacuum Kundt solutions of Weyl type N. pp -waves for n > A comment on some differences between n = 4 and n > pp -waves is now in order. By a natural extension of the n = 4 terminology of[4], in any dimension pp -waves are defined as spacetimes (not necessarilyof the KS form) admitting a covariantly constant null vector ℓ , i.e. ℓ a ; b =0. It then follows directly from the definition of the Riemann tensor that R abcd ℓ d = 0 . (41)For four-dimensional vacuum spacetimes this is equivalent to the def-inition of the type N [4]. It is also known that, in addition to being ofWeyl type N, all four-dimensional vacuum pp -waves can be cast in theKS form [4].By contrast, in higher dimensions eq. (41) is only a necessary, but notsufficient condition for type N. It only says that the type is II (or morespecial) [19], and in fact for n > pp -waves of types III, II and D, as we now briefly discuss.Vacuum pp -wave metrics of Weyl type III can be obtained directly byspecializing results of [22] to the vacuum case. One simple five-dimensionalexample is metric (38) with W = 0 , W = h ( u ) x x , W = h ( u ) x x , (42) H = H = h ( u ) » ` ( x ) + ( x ) ´ + h ( x , x , x ) – , (43)where h ( x , x , x ) is linear in x , x , x .In addition, one can also construct ( n + n )-dimensional vacuum pp -waves of Weyl type II, e.g. by taking a direct product of a n -dimensionalvacuum pp -wave of Weyl type N or III with an Euclidean n -dimensionalRicci-flat (but non-flat) metric (with both n , n ≥
4, and n ≥ n -dimensional pp -wave is of type III). Similarly, ( n + n )-dimensional There is a typo in eq. (96) of [22]: just drop the inequality m ≤ n in the second sum. Wethank Nicos Pelavas for correspondence on this point. The proof of these statements is straightforward and we just sketch it. First, all thementioned products are really pp -waves since a covariantly constant vector field ℓ definedin the n -dimensional geometry can be trivially lifted to the ( n + n )-dimensional productgeometry, in which it will still be covariantly constant. Such products are necessarily ofWeyl type II or more special (with the lifted ℓ being a multiple WAND), as follows from thedecomposable form of the Weyl tensor of product geometries [6, 16]. They cannot be of Weyltype III or N since they inherit non-zero curvature invariants (e.g., Kretschmann) from the n -dimensional Euclidean space. They cannot be of type D since, starting from a frame adapted acuum pp -waves of Weyl type D arise if one takes a direct product of a n -dimensional flat spacetime with an Euclidean n -dimensional curved,Ricci-flat space (with n ≥ n ≥ n > pp -waves ofWeyl type II, D or III do not admit a KS form. They thus representa ‘counterexample’ to both the n = 4 results, namely they are (higher-dimensional) vacuum pp -waves that are neither of type N nor of the KSform.Note, in addition, that pp -waves of Weyl type II and D cannot belongto the VSI class of spacetimes (which is compatible only with types III,N, O [21]), and therefore they necessarily possess some non-vanishing cur-vature invariants. In table 1 we summarize the aforementioned propertiesof various types of vacuum pp -waves in four (4D) and higher dimensions(HD). Weyl type KS VSI4D N √ √
HD N √ √
HD III X √ HD II (D) X XTable 1: Properties of various types of vacuum pp -waves. Non-expanding vacuum solutions of the KS class have been fully classifiedaccording to the discussion of the previous section. Let us now considersolutions with an expanding k , i.e. θ = 0. For θ = 0, from (31) one can write D ln H as D ln H = L ik L ik ( n − θ − ( n − θ. (44)Substituting into (30) one gets L ik L jk = L lk L lk ( n − θ S ij . (45) to the canonical form of the (type N or III) Weyl tensor of the n -dimensional spacetime, thecomponents of negative boost weight turn out to be unchanged under null rotations, and thuscannot be set to zero. The only possible Weyl type is thus indeed II. In fact, in view of our previous comments a similar table applies to all Kundt solutions,in which case type III appears also in 4D [4], again VSI but not KS. nterestingly, this equation is independent of the KS function H and itis thus a purely geometric condition on the KS null congruence k in theMinkowskian ‘background’ η ab (recall the discussion of subsection 2.2).We will thus refer to it as the optical constraint . Its consequences on theform of L ij will be studied in detail in appendix C and D, and they willbe employed in the following sections.Using (17), eq. (45) can also be rewritten in terms of the shear andtwist matrices as( n − θ h“ σ ij − n − σ δ ij ” − “ A ij + n − ω δ ij ” + 2 σ k ( i A j ) k i = ˆ σ + ω − ( n − θ ˜ σ ij . (46)This traceless equation is in fact equivalent to (32), after using (44) (sincethe trace of eq. (45) is obviously an identity). Contracting (46) with σ ij one finds the scalar constraint σ ˆ σ + ω − ( n − θ ˜ − ( n − θ ˆ σ ij σ ij − σ ij A ji ˜ = 0 . (47)So far we have discussed consequences of the vacuum equation R ij = 0.Next, one has to make sure that eq. (33) (i.e., R = 0) is now compatiblewith (44). In fact, taking the D -derivative of (44) and using the scalarSachs equations of [9] and eq. (47), one exactly recovers eq. (33). This isthus automatically satisfied, provided the preceding equations hold. Ofcourse, when looking for an explicit solution one should also solve theremaining Einstein equations, namely R = 0 and R i = 0. These are,however, too involved unless one makes some further assumptions (suchas the presence of symmetries, etc.). In any case, they are not needed inthe following discussion. From the general result of proposition 2, we already know that the Weyltensor of any KS metric with a geodetic k is of type II or more special.Here we show that, in fact, the types III and N are not possible for vacuumexpanding solutions.By reductio ad absurdum , let us thus assume that all components ofthe Riemann (Weyl) tensor with boost weigh zero, i.e. (A2)–(A4), vanish. This constraint is of course identically satisfied in the trivial case σ = 0 = ω (i.e., L ij = S ij = θδ ij ) which includes, e.g., the Schwarzschild-Tangherlini solution [20]. Less trivialexamples are provided by the KS congruence of static black strings, for which σ = 0 and ω = 0,or of rotating black strings and Myers-Perry black holes (which were indeed constructed in [5]using the KS anstatz), both having σ = 0 = ω . (To explicitly verify these statements one mayuse the corresponding optical quantities calculated in any dimension in [8, 16].)On the other hand, a simple example of a geodetic null congruence violating the opticalconstraint is provided by the vector field k = p f ( φ ) ∂ t + f ( φ ) ∂ ρ + ∂ z in the flat geometryd s = − d t + d ρ + ρ d φ + d z + δ AB d y A d y B (with A, B = 1 , . . . , n − f = 0, and violates the optical constraint if f ,φ = 0,in which case it is also twisting. In odd dimensions, it implies that one cannot have σ ij = 0 if A ij = 0 (this was alreadyknown in a more general context [9]). n particular, we can multiply R i j by L lj . Using the optical constraint(45), the condition R i j L lj = 0 gives S il D ln H = − A ki S kl , (48)where we have dropped an overall factor L jk L jk = σ + ω + ( n − θ >
0. Using as a ‘spatial’ basis an eigenframe of S ij , we have S ij = diag( s (2) , s (3) , s (4) , . . . ). At least one eigenvalue must be non-zero,say s (2) = 0. Then from (48) with i = l = 2 we get s (2) D ln H = − s (2) A , (49)which obviously can never be satisfied since A = 0 (and since the case D H = 0 is ‘forbidden’, cf appendix B). This contradiction completes theproof of Proposition 5
In arbitrary dimension n ≥ , Kerr-Schild vacuum space-times with an expanding KS congruence k are of algebraic type II or D. For a similar result for n = 4 cf [4, 24]. r -dependence As detailed in appendices C and D, the optical constraint (45) and theSachs equations [9] imply that there exists an appropriate frame, satisfying i M j = 0, such that the matrix L ij takes the block diagonal form L ij = L (1) . .. L ( p ) ˜ L . (50)The first p blocks are 2 × L is a ( n − − p ) × ( n − − p )-dimensional diagonal matrix. They are given by L ( µ ) = „ s (2 µ ) A µ, µ +1 − A µ, µ +1 s (2 µ ) « ( µ = 1 , . . . , p ) ,s (2 µ ) = rr + ( a µ ) ) , A µ, µ +1 = a µ ) r + ( a µ ) ) , (51)˜ L = 1 r diag(1 , . . . , | {z } ( m − p ) , , . . . , | {z } ( n − − m ) ) , (52)with 0 ≤ p ≤ m ≤ n −
2. The integer m denotes the rank of L ij , so that L ij is non-degenerate when m = n − ccordingly, the optical scalars are given by( n − θ = 2 p X µ =1 rr + ( a µ ) ) + m − pr , (53) ω = 2 p X µ =1 a µ ) r + ( a µ ) ) ! , (54) σ = 2 p X µ =1 rr + ( a µ ) ) − θ ! + ( m − p ) „ r − θ « + ( n − − m ) θ . (55)Note, in particular, that expansion θ indicates the presence of a causticat r = 0, except when 2 p = m (with m even). Note also that shear isgenerically non-zero (the special case σ = 0 will be discussed below). Thetwist is zero if and only if p = 0 (or, equivalently, all a µ ) vanish).One can similarly fix the r -dependence of all Ricci rotation coefficients(at least with the additional ‘gauge’ condition N i = 0). Since this is notneeded in the following discussion, corresponding results are relegated toappendix D.Knowing the form of L ij enables us to solve the vacuum equation (44),where now L ik L ik = ( n − θr − . The r -dependence of H is thus givenby H = H r m − p − p Y µ =1 r + ( a µ ) ) . (56)This includes in particular the solution of (44) in the case when its rhsvanishes, i.e. D H = 0, which happens for m = 1 (implying p = 0). Thisis, however, incompatible with the Bianchi identities [8] as explained inappendix B. Hence, in the following we shall restrict to2 ≤ m ≤ n − . (57)For example, the r -dependence in the special case of Myers-Perry so-lutions is obtained by setting m = n − p = 1, andstatic black holes to p = 0 (see also subsection 5.7).In general, the asymptotic behavior of H for r → ∞ is given by H = H r m − + O ( r − m − ) , (58)i.e., the function H behaves as a Newtonian potential in ( m + 1) spacedimensions with r as a radial coordinate. Let us now use the above results to comment on a partial extension tohigher dimensions (but restricted to KS solution) of the Goldberg-Sachs heorem. In four dimensions, this is a well-known theorem stating that ina vacuum (non-flat) spacetime, a null congruence is geodetic and shearfreeif and only if it is a multiple principal null direction of the Weyl tensor[4, 11, 12].For n = 4, the matrix L ij associated with a generic null vector ℓ [8]is 2 × L ij = − „ ( ρ + ¯ ρ ) + ( σ + ¯ σ ) − i ( ρ − ¯ ρ ) + i ( σ − ¯ σ ) i ( ρ − ¯ ρ ) + i ( σ − ¯ σ ) ( ρ + ¯ ρ ) − ( σ + ¯ σ ) « . (59)In vacuum, when ℓ is a multiple principal null direction of the Weyltensor the Goldberg-Sachs theorem implies σ = 0, so that L ij reduces to L ij = − „ ρ + ¯ ρ − i ( ρ − ¯ ρ ) i ( ρ − ¯ ρ ) ρ + ¯ ρ « . (60)We have observed in previous sections that for vacuum KS spacetimesthe KS vector ℓ = k is indeed a (geodetic) multiple WAND for any n ≥ n = 4, eqs. (50)–(52) must be of theshearfree form (60). It is easy to verify that this is indeed the case, sincefor n = 4 we necessarily have m = 2 (cf (57)) and therefore L ij eithercoincides with one of the blocks L ( µ ) (for p = 1, i.e. ρ − ¯ ρ = 0) oris proportional to the two-dimensional identity matrix (for p = 0, i.e. ρ − ¯ ρ = 0).However, it has been pointed out [5, 8, 9, 16, 26] that the Goldberg-Sachs theorem cannot be extended to higher dimensions in the most di-rect formulation ‘multiple WANDs are geodetic and shearfree in vacuum’.Indeed, we have already noticed that k is generically shearing. Neverthe-less, for any n > restricted to KS spacetimes :in a suitable basis, the matrix L ij (associated with the geodetic, multipleWAND k ) consists of × blocks that are ‘shearfree’ , i.e. reflecting thefour-dimensional shearfree condition (60) in various orthogonal 2-planes. The only ‘exceptions’ to this of course arise in odd spacetime dimensions(there will be an isolated one-dimensional block) and in the degeneratecase det L = 0, in which one may add an arbitrary number of zeros alongthe diagonal (most simply, by just taking a direct product with flat dimen-sions). This ‘generalized Goldberg-Sachs condition’ can also be expressedin a basis-independent form simply as (dropping the matrix indices)[ S, A ] = 0 , A = S − F S, (61)where S = ( L + L T ) / A = ( L − L T ) / L , and F can be fixed as in (C1) by taking thetrace. The first equation in (61) is clearly equivalent to the condition Note that, exceptionally in this subsection only, the complex shear σ does not coincidewith the real shear scalar defined in section 3 and used throughout the paper (although theyare simply related), but it is the usual Newman-Penrose spin coefficient. Let us recall that we have been focusing on the expanding case θ = 0, since θ = 0 ⇒ L ij = 0 for vacuum KS spacetimes (see section 4). L, L T ] = 0, i.e. L is a (real) normal matrix. The second equation in (61)is just a consequence of the first one when n = 4.Let us also emphasize that the above ‘weak Goldberg-Sachs theorem’has, in fact, been proven in more generality (i.e., without the KS assump-tion) for higher-dimensional vacuum spacetimes of Weyl type III and N:the WANDs of such spacetimes must be geodetic, and the associated ma-trix L ij is indeed of the form (50), with only one non-zero 2 × For a possible complete extension of the Goldberg-Sachs theorem, oneshould study the remaining possibilities, i.e. n > L ij of the form (50) implies that the algebraic type is II or more special (thisis already known in the simple Kundt case L ij = 0 [9]). Let us recallthat the ‘geodetic part’ of the n > n > The r -dependence of the matrix L ij determines explicitly also the r -dependence of the Weyl components with boost weight 0 (cf appendixA). In order to express these components compactly, it is convenientto introduce a ( n − × ( n −
2) matrix Φ ij ≡ C i j along with itssymmetric and antisymmetric parts Φ Sij ≡ Φ ( ij ) and Φ Aij ≡ Φ [ ij ] . Allboost order zero components of the Weyl tensor are then determinedin terms of the ( n − n − / Aij andthe ( n − n − ( n − /
12 independent components of C ijkl , sinceΦ Sij = − C ikjk , C ij = 2Φ Aij , C = Φ ≡ Φ ii [8, 16].It follows from appendix A that, using the adapted frame of ap-pendix C, the matrix Φ ij inherits the block structure of L ij , with theonly non-zero components and trace given byΦ µ, µ = Φ µ +1 , µ +1 = − H A µ, µ +1 − s (2 µ ) D H , (62)Φ µ, µ +1 = Φ A µ, µ +1 = − D ( H A µ, µ +1 ) , (63)Φ αβ = − r δ αβ D H , Φ = D H = − ( n − θD H − H ω . (64)Up to index permutations, the non-zero C ijkl components are C µ, µ +1 , µ, µ +1 = 2 H (3 A µ, µ +1 − s µ ) ) , (65) C µ, µ +1 , ν, ν +1 = 2 C µ, ν, µ +1 , ν +1 = − C µ, ν +1 , µ +1 , ν = 4 H A µ, µ +1 A ν, ν +1 , (66) More precisely, this has been proven for all n > n > n = 5 solutions of type III. For twisting type IIIsolutions with n > µ, ν, µ, ν = C µ, ν +1 , µ, ν +1 = − H s (2 µ ) s (2 ν ) , (67) C ( α )( i )( α )( i ) = − H s ( i ) r − , (68)where ν = µ .Let us observe that all the above components fall off at r → ∞ as1 /r m +1 or faster (in the non-twisting case, i.e. when all A µ, µ +1 vanish,only 1 /r m +1 terms are present, cf also [10]).Since (as shown above) expanding KS spacetime can be only of Weyltype II or D in vacuum, Weyl components of boost weight 0 fully determinethe Kretschmann scalar, i.e. R abcd R abcd = 4( R ) + R ijkl R ijkl + 8 R j i R i j − R ij R ij . (69)In vacuum R abcd = C abcd , and we can use the above compact notation toreexpress R abcd R abcd = 4Φ + C ijkl C ijkl + 8Φ Sij Φ Sij − Aij Φ Aij . (70)This will be useful soon in the discussion of singularities.To conclude, let us observe that Weyl components of boost weight − − D , and we cannot studytheir r -dependence at this general level. Type D KS spacetimes are a special subclass of general KS metrics. Bydefinition of type D, only boost weight 0 components of the Weyl tensor(as given above) are non-zero in an adapted frame, i.e. they admit asecond multiple WAND not parallel to the KS (geodetic) vector k . Herewe show that this second WAND is also geodetic (as already known in thespecial case of Myers-Perry black holes [17], see also [16]).First, note that, from (63), one has Φ Aij = 0 ⇔ ω = 0 (cf eqs. (51)and (56)). In addition, since Φ Sij = 0 (cf (64) and appendix B), for ω = 0 (i.e., p = 0) it is easy to see from (64) that Φ ( i )( i ) = − Φ for allvalues of i . Thus, for type D KS spacetimes proposition 6 of [16] impliesthat the second multiple WAND is indeed geodetic (since one can alwaysalign the frame vector n to it, and this is compatible with the assumedcondition i M j = 0). This is a general consequence of the Goldberg-Sachstheorem in four dimensions, but it is a non-trivial result for n > The form (56) of the KS function suggests there may be singularities at r = 0, except in the cases 2 p = m ( m even) and 2 p = m − m odd). Inorder to invariantly identify possible curvature singularities, let us analyzethe behavior of the Kretschmann scalar (70).Eq. (70) clearly consists of a sum of squares, except for the last term,which is negative. For our purposes, it suffices to focus on the first andthe last terms determined by (63) and (64) with (51) and (56). .6.1 ‘Generic’ case ( p = m , p = m − ) Excluding for now the special cases 2 p = m and 2 p = m −
1, from (56)one easily finds that for r ∼ H ∼ r − ( m − p − , D H ∼ r − ( m − p ) , D H ∼ r − ( m − p +1) . (71)Inserting this into (63) and the second of eqs. (64), it is clear thatthe first term in (70) will dominate over the last term near r ∼
0, andtherefore the Kretschmann scalar will diverge at r = 0, thus confirming thepresence of a curvature singularity. Note that this ‘generic’ case includes,in particular, all non-twisting solutions, for which p = 0 (and Φ Aij = 0).As an explicit example, n > m = n − p = 1) fall in this subclass, and in appropriatecoordinates [5] one has 2 H = − µr − n / ( r + a cos θ ) ( a is the angularmomentum parameter). See [5] for a detailed discussion, including otherpossible examples with more than one rotation parameter. p = m and p = m − Let us now comment on the cases 2 p = m ( m even) and 2 p = m − m odd), for which H = r H Q m/ µ =1 [ r +( a µ ) ) ] − and H = H Q ( m − / µ =1 [ r +( a µ ) ) ] − , respectively. With these assumptions H and its D -derivativesare clearly non-singular at r = 0 (since p denotes the number of non-vanishing terms a µ ) in (56)). However, in general, a µ ) may be functionsof spacetime coordinates different from r . Given a specific KS solution,there may thus exist ‘special points’ where some of the a µ ) vanish. Letus say, e.g., that q (with q ≥
1) of such terms a µ ) vanish simultaneouslythere. Then, proceeding as above one finds that at those special points,when r → H ∼ r − q +1 , D H ∼ r − q , D H ∼ r − q − (2 p = m ) , (72) H ∼ r − q , D H ∼ r − q − , D H ∼ r − q − (2 p = m − . (73)Again, the first term in (70) will dominate over the last one and theKretschmann scalar will diverge at the ‘special points’ when r → n = 4 Kerr solution ( m = 2, p = 1) andthe n = 5 Myers-Perry metric with only one spin ( m = 3, p = 1) theonly non-zero a µ ) function is determined by ( a ) = a cos θ . Hence,there is a curvature singularity at r = 0 and θ = π/ n = 5 Myers-Perry solution with two non-zero spins (again, m = 3, p = 1)contains ( a ) = a cos θ + b sin θ (in the notation of, e.g., [26]), whichnever vanishes. Possible singularities of rotating black hole spacetimes arestudied in more generality in [5]. We now briefly comment on special subfamilies of expanding KS solu-tions, characterized by a non-twisting or a non-shearing KS vector k .In relation to the first possibility, let us note that general properties of igher-dimensional non-twisting vacuum spacetimes (not necessarily KS)have been recently studied in [10], and that all such solutions are explic-itly known with the additional assumptions of non-zero expansion andvanishing shear (‘Robinson-Trautman spacetimes’) [32, 33]. The KS vector k is non-twisting if and only if L ij is a symmetric matrix,i.e. for p = 0. This clearly gives L ij = S ij = 1 r diag(1 , . . . , | {z } m , , . . . , | {z } ( n − − m ) ) , (74)so that (cf (53)–(55)) θ = 1 r mn − , ω = 0 , σ = 1 r m ( n − − m ) n − . (75)From eq. (56), the r -dependence of H becomes simply H = H r m − . (76)For Weyl components with boost weight 0 we now have the non-zerocomponents Φ αβ = Φ Sαβ = H r m +1 ( m − δ αβ , (77) C ( α )( β )( α )( β ) = − H r m +1 . (78)As mentioned above, all non-twisting KS solutions contain a curvaturesingularity at r = 0.Explicit n -dimensional examples of non-twisting KS spacetimes with m ≥ m + 2)-dimensional Schwarzschild black hole with( n − m − s = h − (2d r + K d u )d u + r dΩ m + δ AB d y A d y B i + µr m − d u , (79)where the metric in square brackets represents a flat n -dimensional space-time (with A, B = 1 , . . . , n − m −
2, and dΩ m being the line elementof a m -dimensional space of constant curvature K ), k a d x a = d u and2 H = − µr − m +1 .The above non-twisting KS solutions are, in addition, non-shearing iff m = n − m = 0 is ruled out here by the assumption θ = 0). In this case, they must belong to the family of higher-dimensionalRobinson-Trautman spacetimes [32]. In fact, one can see from the resultsof [32] that the only Robinson-Trautman solutions that are also KS aregiven by static Schwarzschild-Tangherlini black holes, i.e. by metric (79)with m = n − The n > .7.2 Non-shearing solutions ( n even) Non-shearing, non-twisting solutions belong to the already discussed KS-Robinson-Trautman class, so that we can now focus on non-shearing buttwisting solutions. Recall [9] that σ = 0 and ω = 0 can occur only foreven n (and for θ = 0).We see from (55) that σ = 0 requires m = n − L ij must benon-degenerate), so that m is also necessarily even. Then, we have toconsider two possible situations.1. If 2 p = n − k is shearfree iff s (2 µ ) = θ for any µ = 1 , , . . . , ( n − / a = a = . . . a n − ≡ a , and θ = rr + a , ω = √ n − a r + a . (80)This agrees with the general behavior found in [9] (note also that A ik A jk = ω n − δ ij , as expected from the optical constraint (45) with σ = 0).The function H reduces to H = r H ( r + a ) n − . (81)This includes, for instance, the n = 4 Kerr solution. Explicit solu-tions for n > p < n −
2, we have the additional condition θ = 1 /r . This implies ω = a = 0, which is the already-discussed non-twisting case. We have presented a systematic study of geometric properties of KS met-rics in higher dimensions. We have preliminary discussed general resultsthat do not require any further assumptions. Namely, the KS vector k isgeodetic if and only if the energy-momentum tensor satisfies T ab k a k b = 0(proposition 1). When this happens (e.g., in vacuum) the Weyl tensor isof type II (or more special) and k is a multiple WAND (proposition 2).Furthermore, we have shown that optical properties of k are the same withrespect to both the flat background and the full metric. Subsequently, ouranalysis has focused on vacuum solutions.For non-expanding metrics the most general KS solution is now known,since they are equivalent to the type N Kundt class in vacuum (propo-sition 4). Expanding solutions required a more detailed analysis. Again,they turned out to be algebraically special, but the only possible types areII and D (proposition 5). We have also shown that the choice of a possi-ble KS congruence is restricted in vacuum by an “optical constraint”. In n = 4 dimensions this requires k to be a shearfree congruence, in agree-ment with the standard Goldberg-Sachs theorem. For n > non-constant curvature horizons [32], and ‘exceptional’ solutions with ‘zero mass’ µ = 0 [32,33](see also [16]). These cannot be KS spacetimes because their Weyl components C ijkl containan r − term which is absent from (78). roven a partial and apparently weaker extension of this theorem, whichhowever naturally reduces to the familiar result when n = 4. This ex-tension has been derived here only for KS solutions but, interestingly, itagrees with previous results for general vacuum type III/N spacetimes [8].Moreover, by integration of the Ricci identities we have fixed the depen-dence of the optical matrix L ij and of the KS function H on the affineparameter r along k . This enabled us to prove the presence of a curvaturesingularity in “generic” expanding KS spacetimes. As a side remark, letus note that we were interested in the n > n = 4 case, for which we have rederived various knownresults previously scattered in several publications (many reviewed, how-ever, in [4]).In future work, it will be interesting to employ our results to possiblyfind new expanding KS solutions, and to understand to what extent theMyers-Perry metrics exhaust the expanding vacuum KS class. In addition,further investigation will admit non-zero matter fields and generalized KSsolutions, in which the background is not necessarily flat. These includeother important spacetimes such as rotating black holes in (A)dS [34, 35]and their NUT extensions [25].
Acknowledgments
The authors acknowledge support from research plan No AV0Z10190503and research grant KJB100190702. M.O. carried out part of his workat Departament de F´ısica Fonamental, Universitat de Barcelona, with apostdoctoral fellowship from Fondazione Angelo Della Riccia (Firenze).
A Frame components of the Riemann ten-sor ( k geodetic) When k is geodesic and affinely parametrized we find the following framecomponents of the Riemann tensor corresponding to the line element (2) R i j = 0 , R i = 0 , R ijk = 0 , (A1) R = D H , R ij = 2 A ji D H + 4 H S k [ j A i ] k , (A2) R i j = − L ij D H − H A ki L kj , (A3) R ijkl = 4 H ( A ij A kl + A k [ j A i ] l + S l [ i S j ] k ) , (A4) R i = − δ i ( D H ) + 2 L [ i D H + L ji δ j H + 2 H (2 L ji L [1 j ] + L j A ji ) , (A5) R ijk = 2 L [ j | i δ | k ] H + 2 A jk δ i H − H “ δ i A kj + L j L ki − L k L ji − L j A ki + L k A ji + 2 L [1 i ] A kj + A lj l M ki − A lk l M ji ” , (A6) R i j = δ ( i ( δ j ) H ) + k M ( ij ) δ k H + (2 L j − L j ) δ i H + (2 L i − L i ) δ j H + N ( ij ) D H − S ij ∆ H + 2 H “ δ ( i | L | j ) − ∆ S ij − L i L j )1 + 2 L i L j − L k ( i | N k | j ) − H L k ( i A j ) k L k k M ( ij ) − H A ik A jk − L k ( i k M j )1 − L ( i | k k M | j )1 ” . (A7)For certain calculations, it may be useful to note that, using the Ricciidentities (11k, [9]), the above component R ijk can also be transformedto the somewhat different form R ijk = 2 A jk δ i H − L ki δ j H + L ji δ k H + 2 H “ δ [ k S j ] i + 2 l M [ jk ] S il − l M i [ j S k ] l + 2 L i A jk − L j A k ] i ” . (A8)Let us finally emphasise that throughout sections 4 and 5 and in ap-pendix D we restrict to vacuum spacetimes, so that C abcd = R abcd andthe Weyl tensor is there given simply by the above expressions. B Expanding solutions with D H = 0 donot exist in vacuum In the special case when D H = 0, from the vacuum equation (33) we get ω = 0, i.e. L ij = S ij . Then eq. (30) reads S ik S jk = ( n − θS ij . (B1)Using an eigenframe of S ij , it is easy to see that the only possible solutionhas the form S ij = s (2) diag(1 , , , . . . ) , (B2)with s (2) = ( n − θ . Now, putting D H = 0, A ij = 0, L ij = S ij andeq. (B2) into eqs. (A2)–(A4), we find that all components of the Rie-mann (Weyl) tensor with non-negative boost weight vanish. The algebraictype must thus be III or N. However, by analyzing the Bianchi identitiesit has been proven in [8] that the canonical form of the expansion ma-trix S ij for type N and (non-twisting) type III spacetimes in vacuum is S ij = s (2) diag(1 , , , . . . ) (cf eqs. (50) and (C.20) of [8]). This is clearlyincompatible with (B2). Therefore, there do not exist vacuum solutionsof the KS class with θ = 0 and D H = 0. C Solving the optical constraint
In this appendix we provide a solution of the optical constraint (45) forthe matrix L ij , in the case of an expanding KS vector, i.e. θ = 0. Weintroduce the compact notation F = L lk L lk ( n − θ = σ + ω + ( n − θ ( n − θ , (C1)so that (45) simply reads L il L jl = F S ij . (C2)In the following, it will be convenient to analyze separately the twopossible cases det L = 0 and det L = 0. .1 Non-degenerate case Let us start with the non-degenerate case det L = 0, so that there existsa matrix inverse of L ij . Let us denote by L − ij such inverse matrix (or,sometimes, its ( ij )-element – there will be no ambiguity according to thecontext). It is also convenient to define symbols for its antisymmetric andsymmetric parts, i.e. B ij = L − ij ] , C ij = L − ij ) . (C3)Now, multiplying (C2) by L − kj one gets L ik = F S ij L − kj , (C4)and by further multiplication by L − li we find δ lk = F C lk , so that L − ij = F − δ ij + B ij . (C5)Eq. (C4) thus becomes L ik = S ik + F S ij B kj . (C6)The symmetric and anti-symmetric parts of this equation give rise,respectively, to B kj S ji + B ij S jk = 0 , (C7)2 A ik = F ( B kj S ji − B ij S jk ) . (C8)The symmetric matrix S ij defines a natural frame of orthonormaleigenvectors. It is thus convenient to identify our basis vectors m ( i ) withsuch eigenvectors, so that S ij = diag( s (2) , s (3) , s (4) , . . . ). Eqs. (C7) and(C8) then read B ki ( s ( i ) − s ( k ) ) = 0 , (C9)2 A ik = F B ki ( s ( i ) + s ( k ) ) . (C10)If B ki = 0 for all values of the indices, then eqs. (C9) and (C10) areidentically satisfied and, by (C5), L ij = F δ ij , and all eigenvalues of S ij take the same value s ( i ) = F .If, instead, for some components B ¯ k ¯ i = 0, eq. (C9) implies that thecorresponding eigenvalues of S ij coincide (i.e., s (¯ i ) = s (¯ k ) ). After orderingthe basis vectors so as to have multiple eigenvalues next to each otheron the diagonal of S ij , the matrix B ki must thus be composed of (anti-symmetric) blocks with their diagonal on the main diagonal, each blockcorresponding to a repeated s (¯ i ) (e.g., B = 0 implies s (3) = s (2) andthen, if s (4) = s (2) , we have B = 0, etc.). Moreover, there can be atmost one eigenvalue of S ij with multiplicity 1, and this must equal F (sup-posing, e.g., that s (2) = s (ˆ k ) for ˆ k = 3 , . . . , n −
1, then one gets B k = 0, A priori , there is no reason to expect that this frame is parallely propagated along k , acondition that we do not need here. However, we shall see in the following appendix that suchan eigenframe is, in fact, compatible with parallel transport. o that, by (C5), L − k = 0 = L − k , L − = F − , and thus s (2) = F ; and ofcourse there cannot be distinct eigenvalues both equal to F ).Indices ¯ i , ¯ j , ¯ k, . . . take values within a given block (and thus s (¯ i ) = s (¯ k ) ), so that eq. (C10) now splits into separate equations for each blockof the matrix B ki , i.e. A ¯ i ¯ k = s (¯ i ) F B ¯ k ¯ i , (C11)and components of A ij with indices referring to different blocks necessarilyvanish (since thus does B ij ). The matrix A ij has thus the same block-structure of B ij (or simpler, if some of the s ( i ) vanish). Eq. (C11) alsoimplies that all eigenvalues s ( i ) (for all blocks) must be non-zero, since weare now considering the case det L = 0.Using (C5), (C6), and (C11), within each block we can thus write L − i ¯ j = F − δ ¯ i ¯ j + B ¯ i ¯ j , L ¯ i ¯ j = s (¯ i ) ( δ ¯ i ¯ j + F B ¯ j ¯ i ) . (C12)But since these two blocks must be inverse to each other (and since s (¯ i ) = 0takes the same value for any ¯ i within a block), we necessarily have B ¯ i ¯ k B ¯ j ¯ k = F − s (¯ i ) F s (¯ i ) δ ¯ i ¯ j . (C13)This implies that any block-matrix B ¯ i ¯ j must be even-dimensional (andthat ( F − s (¯ i ) ) s (¯ i ) > L − i ¯ j = F − δ ¯ i ¯ j and L ¯ i ¯ j = s (¯ i ) δ ¯ i ¯ j , so that s (¯ i ) = F ( ⇔ B ¯ i ¯ j = 0) . (C14)Hence, if there exist more than one block satisfying B ¯ i ¯ j = 0, to such valuesof indices there corresponds only one possible eigenvalue s (¯ i ) = F of thematrix S ij , i.e. S ij contains a diagonal block given by F diag(1 , , . . . , B ¯ i ¯ j = 0, S ¯ i ¯ j = s (¯ i ) δ ¯ i ¯ j andwe can thus finally perform a rotation so as to put this anti-symmetricmatrix in a canonical form with two-dimensional anti-symmetric blocks(constrained by (C13)) along the diagonal and zeros elsewhere.To summarize, we have shown that the matrix L − ij can always bewritten in a block-diagonal form. If its antisymmetric part B ij does notvanish identically, there is a number p ≤ ( n − / L − µ ) = F − −F − “ F− s (2 µ ) s (2 µ ) ” / F − “ F− s (2 µ ) s (2 µ ) ” / F − , (C15) µ = 1 , . . . , p, ≤ p ≤ n − .s (2 µ ) need not be all distinct and thus some two-blocks may be identical.In addition, there is a ( n − − p ) × ( n − − p )-dimensional diagonalblock ˜ L − = F − diag(1 , , . . . , . (C16) ince each block can be inverted separately, finding the explicit formof the matrix L ij is now straightforward. This consists of p blocks of theform L ( µ ) = s (2 µ ) h s (2 µ ) ( F − s (2 µ ) ) i / − h s (2 µ ) ( F − s (2 µ ) ) i / s (2 µ ) , (C17)and of one ( n − − p ) × ( n − − p )-dimensional diagonal block˜ L = F diag(1 , , . . . , | {z } ( n − − p ) ) . (C18)More explicitly, the matrix L ij thus takes the form L ij = L (1) L (2) . . . L ( p ) ˜ L . (C19) C.2 Degenerate case
Let us now study the degenerate case det L = 0 = det L T . This im-plies that there exists a non-zero vector v (living in the ‘transverse’ spacespanned by vectors m ( i ) ) such that L T v = 0. By (C2) this gives also Lv = 0. If we now choose an orthonormal basis such that, say, m ( n − corresponds to v , we can rewrite L T v = 0 = Lv as L ( n − ,i = 0 = L i, ( n − . (C20)In addition, the optical constraint (C2) now reduces to L ¯ i ¯ k L ¯ j ¯ k = F S ¯ i ¯ j ,with the barred indices ranging from 2 to ( n −
2) only. In other words,we have the same equation in a reduced space. We can then distinguishthe two cases det L ¯ i ¯ j = 0 and det L ¯ i ¯ j = 0, and similarly proceed until wefind a subspace with a non-degenerate reduced matrix L . Finally, we canproceed exactly as in the non-degenerate case. It is thus obvious that thegeneral form of L ij will have the same block diagonal form (C19), exceptthat now the diagonal block contains also ( n − − m ) zeros,˜ L = F diag(1 , . . . , | {z } ( m − p ) , , . . . , | {z } ( n − − m ) ) , (C21)where m ≡ rank( L ), and 0 ≤ p ≤ m . The non-degenerate case clearlycorresponds to m = n − Integration of the Ricci identities
The general form of the Ricci identities in higher dimensions has beengiven in [9]. Here we consider them only in the case θ = 0, and in a formsimplified by the fact that ℓ = k is geodetic and affinely parametrized.Further simplification could be achieved by using a null frame that isparallely transported along k (which is always a possible choice [9]), i.e., i M j = 0 = N i . For certain purposes, however, it is desirable to retainthe full freedom of null rotations of n , and for now we thus only impose i M j = 0 . (D1)We also employ the special form that the Riemann tensor takes incase of KS spacetimes, as given in appendix A, and use the compact nota-tion (62)–(64) (strictly speaking, this is defined for the Weyl tensor, but invacuum we can equivalently apply it to the Riemann tensor). Moreover,we keep into account consequences of the vacuum equations discussed inappendix C. D.1 Sachs equations: form of L ij Under the above assumptions, eq. (11g, [9]) becomes DL ij = − L ik L kj . (D2) D.1.1 Non-degenerate case
By methods similar to those of [12], the solution to (D2) can be writtenin terms of its inverse as [36] L − ij = rδ ij + ( L ) − ij , (D3)where ( L ) − ij represents L − ij at r = 0. For compatibility with the op-tical constraint, the ‘initial value’ ( L ) − ij must be of the canonical formobtained in appendix C. The solution for any r can then be obtainedimmediately by adding a term rδ ij to eqs. (C15) and (C16) evaluated at r = 0.Hence, the inverse matrix L − ij consists of p two-blocks and of one( n − − p ) × ( n − − p )-dimensional block of the form, respectively, L − µ ) = F − + r −F − „ F − s µ ) s µ ) « / F − „ F − s µ ) s µ ) « / F − + r , (D4)˜ L − = ( F − + r )diag(1 , , . . . , . (D5)Correspondingly, the matrix L ij (C19) consists of blocks L ( µ ) = 11 + 2 rs µ ) + r s µ ) F s µ ) (1 + r F ) h s µ ) ( F − s µ ) ) i / − h s µ ) ( F − s µ ) ) i / s µ ) (1 + r F ) , (D6)˜ L = F r F diag(1 , , . . . , . (D7)By construction, this matrix L ij satisfies the optical constraint at r = 0. However, note that the last two equations are of the same form of(C17) and (C18), respectively, provided one rewrites them using s (2 µ ) = s µ ) (1 + r F )1 + 2 rs µ ) + r s µ ) F , F = F r F . (D8)This implies that L ij automatically satisfies the optical constraint forany r . In other words, the canonical frame determined by the opticalconstraint is compatible with the condition i M j = 0 (and, in particular,with parallel transport along k if N i = 0 is also required).For practical purposes it will be convenient to simplify the above equa-tions by shifting the affine parameter as r = ˜ r − / F , and simultaneouslydefining the new quantities a µ ) = h s µ ) ( F − s µ ) ) i / / ( s µ ) F ). Thenone has (dropping tildes over r ) L ( µ ) = 1 r + ( a µ ) ) „ r a µ ) − a µ ) r « , ˜ L = 1 r diag(1 , , . . . , . (D9)This general structure of L ij agrees with the results of [16] (up toreordering the basis vectors) for the specific case of Myers-Perry blackholes. D.1.2 Degenerate case
When det L = 0, it is easy to see that the vector v (that satisfies Lv = 0)can be parallely transported along k [36]. We can therefore proceed asin the non-degenerate case to solve the Sachs equations in the ‘non-zeroblock’ of L ij , the remaining part of the matrix keeping its zero form. Wethus do not repeat the results here, just replace the last ( n − − m ) entriesof ˜ L by zeros in (D9). D.1.3 Summary
In the following, we will often need to refer to the elements of the matrices S ij and A ij , and we will need a notation that enables us to handle boththe non-degenerate and the degenerate case simultaneously. It will alsobe convenient to distinguish between indices α , β , . . . referring to the non-zero eigenvalues of ˜ L and ρ , σ , . . . to its zero eigenvalues. For furthercompactness, we also use a complex notation (with i being the imaginaryunit) and we summarize the non-zero elements of L ij as s (2 µ ) + iA µ, µ +1 = r + ia µ ) r + ( a µ ) ) , ( µ = 1 , . . . , p ) (D10) ( α ) = 1 r ( α = 2 p + 2 , . . . , m + 1) , (D11) s ( ρ ) = 0 ( ρ = m + 2 , . . . , n − . (D12) D.2 Remaining Ricci rotation coefficients
The above results for L ij enable us to find the r dependence of all Riccirotation coefficients by integrating the corresponding Ricci identities [9]that contain a derivative along k . From now on, however, in additionto (D1) we also use the null rotation freedom on n to set N i = 0 , (D13)i.e. we use a null frame that is parallely transported along k . D.2.1 Form of L i and L i Let us start from eq. (11b, [9]), which now takes the form DL i = − L j L ji . (D14)These equations decouple according to the block structure of L ij , and theresulting coefficients can be compactly written in complex form as L , µ + iL , µ +1 = ( l , µ + il , µ +1 )( s (2 µ ) + iA µ, µ +1 ) , (D15)where l , µ and l , µ +1 are real integration ‘constants’, and s (2 µ ) and A µ, µ +1 are given in (D10).The solution for coefficients L i with i corresponding to the diagonalblock of L ij is simply L α = l α r , L ρ = l ρ . (D16)Ricci equations (11e, [9]), i.e. DL i = − L ij L j , (D17)lead to similar expressions for the coefficients L i , namely L µ, + iL µ +1 , = ( l µ, + il µ +1 , )( s (2 µ ) − iA µ, µ +1 ) , (D18) L α = l α r , L ρ = l ρ . (D19) D.2.2 Form of i M jk Next, the coefficients i M jk are analogously determined by eq. (11n, [9]),which here takes the form D i M jk = − i M jl L lk . (D20)Again, the solutions can be given in natural pairs, each for each block of L ij , as distinguished by the value of the last index k , i.e. i M j, µ + i i M j, µ +1 = ( i m j, µ + i i m j, µ +1 )( s (2 µ ) + iA µ, µ +1 ) , (D21) ith real integration ‘constants’. When k corresponds to the diagonalpart of L ij we have simply i M jα = i m jα r , i M jρ = i m jρ . (D22)Because of the index symmetries of i M jk [8], we require i m jk + j m ik = 0for any i, j, k = 2 , . . . , n − D.2.3 Form of i M j Eq. (11m, [9]) here becomes D i M j = − i M jk L k − Aij . (D23)From (D10), one has DA ij = − s ( i ) A ij and thus, by (A2), Φ Aij = − D ( H A ij ).Using also previous results for i M jk we find i M j = p X µ =1 " rr + ( a µ ) ) ( i m j, µ l µ, + i m j, µ +1 l µ +1 , )+ a µ ) r + ( a µ ) ) ( i m j, µ l µ +1 , − i m j, µ +1 l µ, ) + i m jα l α r − i m jρ l ρ r + 2 A ij H + i m j , (D24)where the real ‘constants’ satisfy i m j + j m i = 0. D.2.4 Form of L One can also easily integrate (11a, [9]), here reduced to DL = − L i L i − R , (D25)to fix the r -dependence of L , i.e. L = − D H + l + p X µ =1 h ( l , µ l µ, + l , µ +1 l µ +1 , ) s (2 µ ) + ( l , µ l µ +1 , − l , µ +1 l µ, ) A µ, µ +1 i + l α l α r − l ρ l ρ r. (D26) D.2.5 Form of N ij Eq. (11j, [9]) here reads DN jk = − N jl L lk − Φ kj . (D27) ote that, for k = 2 µ, µ + 1, it is more natural to deal with a complexunknown, namely N j, µ + iN j, µ +1 . Then, using (62), (63) and (D10)–(D12), eq. (D27) can be solved obtaining N j, µ + iN j, µ +1 = ` n j, µ + in j, µ +1 ´ ( s (2 µ ) + iA µ, µ +1 )+ H ( s (2 µ ) − iA µ, µ +1 )( δ j, µ + iδ j, µ +1 ) , (D28) N jα = 1 r ( n jα + H δ jα ) , N jρ = n jρ , (D29)where n ij are real ‘constants’. D.2.6 Form of N i in an adapted parallely transportedframe So far, the frame vector n has not been specified, except for the require-ment that it be parallely transported along k . While retaining the lattercondition, we can still perform a null rotation ˆ k = k , ˆ n = n + z i m ( i ) − z k z k ℓ , ˆ m ( i ) = m ( i ) − z i ℓ , (D30)provided Dz i = 0, so as to simplify some of the Ricci rotation coefficients.In particular, if we choose z i = − l i (for i = 2 , . . . , m + 1) , z ρ = 0 , (D31)we obtain (see the transformation properties given in [9] – we drop hatsfrom the transformed coefficients) L i = 0 ( i = 2 , . . . , m + 1) , (D32)which is equivalent to l µ, = l µ +1 , = l α = 0, while the L ρ = l ρ areunchanged. (Alternatively, one can also set L i = 0, but this appears tobe less convenient in what follows.)With this choice, eq. (11f, [9]) becomes DN i = − n iρ l ρ + R i . (D33)In addition, using eq. (D14) and the commutator (22, [21]), eq. (A5)reduces to R i = − D ( δ i H ) − D ( L i H ) ( i = 2 , . . . , m + 1) , (D34) R ρ = − D ( δ ρ H ) − D ( L ρ H ) + l ρ D H . (D35)We can thus straightforwardly integrate eq. (D33) and find N i = − n iσ l σ r + δ i H + 2 L i H + n i , ( i = 2 , . . . , m + 1) (D36) N ρ = − n ρσ l σ r + δ ρ H + (2 l ρ − l ρ ) H + n ρ , (D37)where H and L i are specified by (56) and (D15), (D16). For certainapplications it may be useful to recall that the coefficients N i vanish if andonly if the frame vector n is geodetic (an affine parameter correspondingto L = 0). ote finally that, when l µ, = l µ +1 , = l α = 0, eq. (D24) simplifiesto i M j = − i m jρ l ρ r + 2 A ij H + i m j , (D38)and eq. (D26) becomes L = − D H + l − l ρ l ρ r. (D39) References [1] R. P. Kerr and A. Schild,
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