Black Lenses in Kaluza-Klein Matter

Black Lenses in Kaluza-Klein Matter Marcus A. Khuri∗ and Jordan F. Rainone† arXiv:2212.06762v1 [gr-qc] 13 Dec 2022 Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology L(p, q). These 5-dimensional static/stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers 1 ≤ q < p, in particular conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realising the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by product, we obtain new examples of regular gravitational instantons in higher dimensions. What are the possible shapes of a black hole? Fifty years ago, Hawking [8] provided an answer to this fundamental question in spacetime dimension 4, with his horizon topology theorem. This result asserts that crosssections of the event horizon, in asymptotically flat stationary black hole spacetimes satisfying the dominant energy condition, must be topologically a 2-sphere S 2 . In 2002, Emparan-Reall [3] discovered the first regular asymptotically flat non-spherical black hole, a 5dimensional black ring with horizon topology S 1 × S 2 . Not only did this give impetus to the question above, but it also showed that the traditional black hole no hair theorem is false in higher dimensions [4]. Shortly thereafter, Galloway-Schoen [7] generalized Hawking’s theorem to higher dimensions, stating that horizon crosssections must be of positive Yamabe invariant. This condition is equivalent to the underlying topology admitting a metric of positive scalar curvature, and leads to a concise list of possible horizon topologies in five dimensions [6]. Namely, orientable horizons in this setting must be either a quotient of the 3-sphere S 3 (spherical space form), the ring S 1 × S 2 , or a finite connected sum thereof. Further restrictions are possible in the case of extreme black holes [18], in particular all but one connected sum can be ruled out. The basic question of whether each topology on the list is achieved by a black hole solution has remained unresolved. The totality of non-spherical black holes found to date consists of the ring S 1 × S 2 [3, 24], and the lens spaces L(p, 1) discovered initially by KunduriLucietti when p = 2 [19] and extended to all positive integers p by Tomizawa-Nozawa [25], see also [13]. While the ring is a vacuum solution, the lenses are solutions of minimal supergravity. Moreover, there is evidence that suggests regular asymptotically flat vacuum black lenses do not exist [20]. Symmetry yields further restrictions on topology. Indeed, the rigidity theorem [10, 22] guarantees that generically stationary black holes come with at least one rotational symmetry, and in fact almost all known solutions in 5-dimensions have U (1)2 rotational symmetry; see [13] for recent examples admitting only U (1) symmetry. In this setting of bi-axisymmetry, the list of possible horizon topologies reduces to the sphere S 3 , the ring S 1 × S 2 , and the lens spaces L(p, q) ∼ = S 3 /Zp for any choice of relatively prime integers 1 ≤ q < p. It is also possible to classify the list of possible domain of outer communication (DOC) topologies [9, 12, 14] in this regime. Namely, the compactified Cauchy surfaces within the DOC must either be the 4-sphere S 4 , a connected sum of S 2 × S 2 ’s, or in the non-spin case a connected sum of complex pro2 jective planes CP2 and CP . There have been a number of attempts to realize all the topologies in these lists, however they have all suffered from the presence of either naked singularities [1, 5], or conical singularities on the axes [12, 16, 20], when trying to implement the more complicated configurations. The purpose of this note is the following. We show that all possible horizon topologies, and DOC topologies from the classification, including all combinations of multiple horizon configurations, are realized by regular formally asymptotically flat black hole solutions. In particular, this includes the first examples of general lens space topology L(p, q), involving a topology change between the horizon and asymptotic end. These black holes can be either static or stationary, and are supported by Kaluza-Klein matter fields in that they arise from higher dimensional vacuum solutions through reduction on tori. The methods also extend to all higher dimensions, allowing for the construction of solutions, realising the full range of possible topologies for the horizon as well as the DOC, that are compatible with multi-axisymmetry in which the orbit space is 2-dimensional. Furthermore, as a by product, we obtain new examples of regular gravitational instantons in higher dimensions. The basic strategy consists of the following steps. Given the desired DOC Mn+3 for a (n + 3)-dimensional static/stationary spacetime admitting U (1)n symmetry with n ≥ 1, we show how to encode its topology in 2 a higher dimensional DOC M̃n+3+k having a relatively simple topological structure. On this higher dimensional spacetime manifold, we solve the static/stationary vacuum Einstein equations with U (1)n+k symmetry, and take advantage of the simple topology to balance any conical singularities. A dimensional reduction, or quotient procedure, is then carried out in order to obtain a regular solution with Kaluza-Klein matter on the original topology Mn+3 . Due to global hyperbolicity, the topology of the spacetimes considered here will always be of the form Mn+3 = R × M n+2 . The time slice M n+2 is assumed to admit an effective action by the torus T n = U (1)n , and hence the quotient map M n+2 → M n+2 /T n exhibits M n+2 as a T n -bundle over a 2-dimensional base space with any degenerate fibers occurring on the boundary. In particular, while fibers over interior points are n-dimensional, fibers over boundary points can be (n − 1) or (n − 2)dimensional. Those points where the fiber is (n − 1)dimensional are called axis rods, and the points with an (n − 2)-dimensional fiber are discrete and called corners. Consistency with topological censorship demands that the base space M n+2 /T n is homeomorphic to a half plane R2+ [11]. The entire topology of M n+2 may be recorded in the boundary ∂R2+ of this half-plane. This is achieved by dividing it into disjoint intervals separated by corners or horizon rods (assumed to be finite) where the fibers do not degenerate. Associated to each axis rod interval Γi ⊂ ∂R2+ is a vector vi ∈ Zn referred to as the rod structure, which determines the 1-dimensional isotropy subgroup R/Z · vi ⊂ Rn /Zn ∼ = T n for the action of T n on points that lie over Γi . We then have M n+2 ∼ = (R2+ × T n )/ ∼, (1) where the equivalence relation ∼ is given by (p, φ ) ∼ (p, φ +λvi ) with p ∈ Γi , λ ∈ R/Z, and φ ∈ T n . Together, the rod structures form a (n-dimensional) rod data set D = {(v1 , Γ1 ), . . . , (vk , Γk )} which enshrines the topology of the DOC. Rod data sets may be chosen arbitrarily except for an admissibility condition when n ≥ 2 that guarantees the total space is a manifold, namely if rod structures vi , vi+1 arise from neighboring rods separated by a corner then the 2nd determinant divisor det2 (vi , vi+1 ) = gcd{|Qj1 j2 |} is 1, where Qj1 j2 is the determinant of the 2 × 2 minor obtained from the matrix defined by the column vectors vi , vi+1 by picking rows j1 and j2 . We will assume without loss of generality that each rod structure vi is primitive, in the sense that its components are relatively prime gcd(vi1 , . . . , vin ) = 1. Given a topology Mn+3 that we wish to realize as the DOC for a static/stationary solution of the Einstein equations, and which is characterized by a n-dimensional admissible rod data set D, the first goal is to encode this into a higher dimensional rod data set having a simpler structure. To this end, we define a new (n + k)dimensional rod data set D̃ = {(ṽ1 , Γ1 ), . . . , (ṽk , Γk )} by ṽi = v̄i + en+i for i = 1, . . . , k, where ej is an element of the standard basis for Zn+k having 1 in position j and zeros elsewhere, and v̄i = (vi , 0) ∈ Zn+k . Note that each vector ṽi is primitive since the same is true for vi , and similarly since det2 (ṽi , ṽj ) divides det2 (vi , vj ) the data set D̃ inherits the admissibility property from D. In particular, the analogous quotient M̃ n+k+2 as in (1) defined with respect to D̃ yields an (n + k + 2)-dimensional manifold admitting an effective action by U (1)n+k , which will serve as a DOC time slice for a static/stationary spacetime. We claim that topologically the new higher dimensional manifold is relatively simple, in that it is the product of a torus with a connected sum consisting of products of spheres, and can be described by a rod structure having only standard basis elements. To see this, we note that changing coordinates on the torus fibers T n+k ∼ = Rn+k /Zn+k does not change the topology of M̃ n+k+2 . Such a coordinate change may be described by a matrix U ∈ SL(n + k, Z) defined by U (ej ) = ej for j = 1, . . . , n and U (ṽi ) = en+i for i = 1, . . . , k so that   In V −1 U = , 0 Ik where In is the n × n identity matrix, Ik is the k × k identity matrix, and V is the n × k matrix consisting of the rod structures [v1 , . . . , vk ]. Thus, after this coordinate change the rod data set becomes D̃′ = {(en+1 , Γ1 ), . . . , (en+k , Γk )}, so that according to [21, Theorem 3.4] (see also the discussion in the proof of [15, Theorem 2 (iv)]) the topology of the compactified manifold M̃cn+k+2 is given by     k−3 k − 2 2+ℓ (2) #ℓ S × S k−ℓ × T n . ℓ+1 ℓ=1 for k ≥ 4, whereas for k = 2, 3 the topology is S 4 × T n , S 5 × T n respectively. Here the compactified manifold is obtained from M̃ n+k+2 by filling-in each horizon as well as infinity with a 4-ball cross a torus B 4 × T n+k−2 , since horizons are characterized by neighboring rod structures ei , ei+1 showing that they have topology S 3 × T n+k−2 , and similarly for the cross-section at infinity. We will now solve the Einstein equations on M̃n+k+3 = R × M̃ n+k+2 to obtain a regular static vacuum spacetime realizing this DOC topology; at the end it will be explained how to similarly obtain the rotating stationary analogues. An ansatz, studied initially by Emparan-Reall [2], will be imposed that restricts the 3 metric along the torus fibers to be given as a diagonal matrix function yielding the following form for the spacetime metric g̃ = −ρ2 e− Pn+k i=1 ui dt2 + e2α (dρ2 + dz 2 ) + n+k X eui dψ i i=1
2 , where all coefficients depend only on ρ > 0, z which parameterize the orbit space half-plane R2+ , and the Killing fields ∂ψi generate the U (1)n+k rotational isometries with 0 ≤ ψ i < 2π. In this setting the static vacuum Einstein equations reduce to finding n + k axisymmetric harmonic functions ui on R3 \ Γ, where R3 is parameterized in cylindrical coordinates (ρ, z, φ) with 0 ≤ φ < 2π and Γ denotes the z-axis. The remaining Einstein equations may be solved for α by quadrature, using harmonicity of the ui as an integrability condition i h
αρ = ρ8 (Σui,ρ )2 − (Σui,z )2 + Σ u2i,ρ − u2i,z − ρ4 Σui,ρ , i h αz = ρ4 (Σui,ρ )(Σui,z ) + Σui,ρ ui,z − ρ2 Σui,z . Observe that with the spacetime metric ansatz, axes can only exhibit rod structures of type ei , i = 1, . . . , n + k. Moreover, for an axis rod Γl having the rod structure el , we find that the corresponding logarithmic angle defect [15] is given by   1 (3) bl = lim log ρ + α − ul . ρ→0 2 The harmonic functions will be taken as potentials for a uniform charge distribution along associated axis rods. More precisely, suppose that the axis rods consist of the intervals Γ1 = (−∞, b1 ], Γi = [ai , bi ] for i = 2, . . . , k − 1, and Γk = [ak , ∞), where ai < bi ≤ ai+i < bi+1 for each i. We then set un+i = log(rai − zai ) − log(rbi − zbi ) for i = 2, . . . , k − 1, and un+1 = 2 log ρ − log(rb1 − zb1 ), un+k = log(rak − zak ), p where ra = ρ2 + (z − a)2 and za = z − a. Observe that the functions with i = 2, . . . , k − 1 are negatively valued and satisfy the following properties: un+i ∼ 2 log ρ near Γi , and un+i = (ai − bi )/r + O(r−2 ) as r → ∞. The remaining functions are set to uj = 0, j = 1, . . . , n since they are not linked to axis rods. Clearly then, these harmonic functions guarantee that the desired rod data set D̃′ is achieved through the metric g̃. The spacetime (M̃n+k+3 , g̃) has the desired topology, satisfies the static vacuum equations, and is asymptotically Kaluza-Klein. However, it may possess conical singularities along axis rods. Nevertheless, due to the diagonal matrix structure of the torus fiber metrics, any conical singularity along an axis rod Γi may be resolved by adding an appropriate constant to the associated harmonic function un+i 7→ un+i + ci , where the constant ci is chosen to ensure that the logarithmic angle defect bi = 0 in (3). This translation in the harmonic functions does not alter any of the properties listed above for the spacetime. We note that a related balancing procedure was employed by Emparan-Reall in [2] for certain examples. Furthermore, absence of conical singularities leads to full regularity of the spacetime metric, a fact which may be established analogously to [17, Section 5.1]. A similar procedure may be used to produce regular stationary vacuum solutions having the same rod data set, with prescribed angular momenta for each black hole, by utilizing the results of [12]; although we do not pursue this here. We now record two auxiliarly results, concerning the ability to achieve certain DOC topologies, that are consequences of the above arguments. Notice that the assumption n ≥ 1 is not required when constructing the higher dimensional static vacuum spacetime, and this leads to the following statement. Theorem 1 For each pair of integers n ≥ 0 and k ≥ 2, the compactified domain of outer communication topology M̃cn+k+2 given by (2), is realized by time slices of a regular, asymptotically Kaluza-Klein (or asymptotically flat when n = 0, k = 2), static vacuum solution with up to k − 1 horizons of topology S 3 × T n+k−2 . In fact, the construction proceeds just as well if no horizons are present. In this case, the z-axis consists entirely of axis rods. Furthermore, in this case, for some Pn+k constant c the function i=1 ui − 2 log ρ − c is harmonic on R3 \ Γ, tends to zero at infinity, and remains bounded upon approach to Γ. Therefore, a version of the maximum principle (or Weinstein Lemma [26, Lemma 8]) shows Pn+k that this function vanishes indentically, that is i=1 ui = 2 log ρ + c and hence the static potential is constant. It follows that the time slice is a complete Ricci flat Riemannian manifold, yielding new examples of higher dimensional gravitational instantons. Corollary 2 For each pair of integers n ≥ 0 and k ≥ 2, the topology M̃cn+k+2 gives rise to a gravitational instanton. More precisely, on the complement of a B 4 ×T n+k−2 this manifold admits a regular, complete, Ricci flat Riemannian metric which is asymptotically Kaluza-Klein (or asymptotically flat when n = 0, k = 2). In order to proceed with the original problem of realizing a static solution on the given topology Mn+3 with rod data set D, we will perform a dimensional reduction (or quotienting procedure) on the constructed static spacetime M̃n+k+3 having rod data set D̃. First note that 4 the static vacuum metric g̃ is expressed above with coordinates ψ i , on the torus fibers, that yield the simplistic rod data set D̃′ in terms of standard basis vectors, however we may change back to the original coordinates φi in which the rod data set is given by D̃. This is achieved with the unimodular matrix U = (Uji ) through the relation ψ i = Uji φj . It follows that in these coordinates g̃ = −f˜−1 ρ2 dt2 + f˜−1 e2σ (dρ2 + dz 2 ) + n+k X f˜ij dφi dφj , since m > 1 divides a, b, and c which are relatively prime. Therefore, the subtorus action must be free. The free subtorus action rotates the last k circles in the fibers of M̃n+k+3 which are parameterized by (φn+1 , . . . , φn+k ), while keeping the first n circles fixed. Hence, viewing the spacetime as a bundle with torus fibers, the projection map M̃n+k+3 → M̃n+k+3 /T k may be described by (p, φ1 , . . . , φn+k ) 7→ (p, φ1 , . . . , φn ), i=1 where (f˜ij ) = U T diag(eu1 , . . . , eun+k )U , f˜ = det(f˜ij ), and 2σ = 2α + log f˜. The reduction procedure will be carried out using a k-dimensional torus whose action is free (devoid of fixed points) on M̃n+k+3 . In fact, the desired subtorus action is defined by Tk ∼ = spanR {en+1 , . . . , en+k }/ spanZ {en+1 , . . . , en+k }. To confirm that this is indeed free, we will show that the circle action of R/Z · w ⊂ Rn+k /Zn+k is free for any primitive vector w ∈ spanZ {en+1 , . . . , en+k }. Proceeding by contradiction, assume that for some w the action is not free. Since fixed points can only occur at axis rods or corners, this implies that for some i ∈ {1, . . . , k − 1} there are λ, α, β ∈ R with 0 < λ ≤ 1, and z ∈ Zn+k , such that λw + z = αṽi + β ṽi+1 . If λ is irrational then utilize the transformation matrix U to obtain the equation λU w + U z = αen+i + βen+i+1 , and observe that then all components of U w and U z vanish except possibly those in the n + i, n + i + 1 positions. Writing U z as a linear combination of en+i , en+i+1 , and applying the inverse transformation then shows that w = α′ ṽi + β ′ ṽi+1 . However, this is impossible since wj = 0 for j = 1, . . . , n while vi and vi+1 are linearly independent. It follows that λ is rational, and hence so are α and β. We may now find relatively prime integers a, b, c, and 1 < d ≤ m, such that λ = d/m and d w + cz = aṽi + bṽi+1 . cm Let x ∈ Zn and y, wk ∈ Zk be such that z = (x, y) and w = (0, wk ), then this equation splits into two parts cx = avi + bvi+1 , cdwk = m(aei + bei+1 − cy). Clearly m cannot divide d, and also m cannot divide every component of wk = (wn+1 , . . . , wn+k ) since w is primitive. It follows that m must divide c, and thus c = mc′ for some integer c′ . Since the rod structures making up D satisfy the admissibility condition, we then have b = det2 (vi , avi + bvi+1 ) = m det2 (vi , c′ x). Hence m divides b. By a similar argument we can see that m divides a as well. We have now reached a contradiction where p ∈ R2+ . To show that the quotient space is indeed homeomorphic to the given topology Mn+3 , we observe that the projection map implies that the rod data set D̃ encoding the higher dimensional topology, descends down to the rod data set D for M̃n+k+3 /T k . Lastly, since the free subtorus action is by isometries, and the static vacuum total space M̃n+k+3 is regular, the same is true of the quotient Mn+3 . In particular, this solution is devoid of conical singularities. We note that as a consequence of the dimensional reduction on tori, Kaluza-Klein matter fields will be present. Indeed, the metric on M̃n+k+3 may be expressed in Kaluza-Klein format as g̃ = g + n+k X hµν (dφµ + Aµi dφi )(dφν + Aνj dφj ), µ,ν=n+1 where i, j = 1, . . . , n, hµν = f˜µν , hµν Aµi = f˜νi , and the (quotient) metric g on Mn+3 is given in WeylPapapetrou form by g = −(f h)−1ρ2 dt2 +(f h)−1 e2σ (dρ2+dz 2 )+ n X fij dφi dφj , i,j=1 with fij + hµν Aµi Aνj = f˜ij , f = det(fij ), and h = det(hµν ). The dimensionally reduced Lagrangian on Mn+3 may then be expressed [23, Section 11.4] as   p 1 −1 2 −1 2 2 L = hg R − (|Tr(h ∇h)| + Tr(h ∇h) + |F | ) , 4 ν where R is the scalar curvature of g, |F |2 = hµν F µij Fij µ µ with F = dA , and g = − det g. We now record what has been established. A globally hyperbolic spacetime of dimension n + 3 will be referred to as multi-axisymmetric, if a (noncompact) Cauchy slice admits the symmetry group U (1)n with a simply connected 2-dimensional orbit space, so that its topology is completely determined by an admissible rod data set D. Theorem 3 Any possible topology of the domain of outer communication for a multi-axisymmetric spacetime of dimension greater than or equal to 4, is realizable by a regular static solution of the Einstein equations with KaluzaKlein matter. In particular, these solutions are obtained 5 from a higher dimensional asymptotically Kaluza-Klein vacuum solution by dimensional reduction on tori. The 5-dimensional case is of particular interest. By choosing rod structures v1 = (1, 0) and vk = (0, 1) for the two semi-infinite rods Γ1 and Γk , cross-sections of the time slice M 4 near spatial infinity will be 3-spheres, and in this region the spacetime curvature will approach zero; solutions with these two properties will be referred to as formally asymptotically flat. It should be noted that various, often more specialized, notions of asymptotic flatness appear in the literature, which may not be applicable to the solutions discussed here. In particular, the 2-dimensional tori that foliate the S 3 cross-sections, which arise from the Hopf fibration, may not grow in the asymptotic end. We state the next result with a focus on the topology of black holes. Corollary 4 There exist 5-dimensional regular formally asymptotically flat static bi-axially symmetric solutions of the Einstein equations with Kaluza-Klein matter, supporting any finite configuration of nondegenerate black hole horizons of the form S 3 , S 1 × S 2 , or L(p, q) where 1 ≤ q < p with gcd(p, q) = 1. We remark that all solutions discussed may be written down explicitly in terms of the harmonic functions ui and rod structures vi . Moreover, it is possible to replace static solutions with stationary solutions in these results, thus giving rotation to the constructed black holes, which may be partially prescribed. This is accomplished by utilizing [12], instead of the harmonic function technique to obtain the relevant higher dimensional vacuum solutions. Acknowledgments. M. Khuri acknowledges the support of NSF Grant DMS-2104229. The authors would like to thank Hari Kunduri for helpful comments. ∗ † [email protected] [email protected] [1] Y. Chen, and E. Teo, Phys. Rev. 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