Lower bound for angular momenta of microstate geometries in five dimensions

We study the BPS solutions of the asymptotically flat, stationary microstate geometries with bi-axisymmetry and reflection symmetry in the five-dimensional ungauged minimal supergravity. We show that the angular momenta of the microstate geometry with a small number of centers (at least, five centers) have lower bounds, which are slightly smaller than those of the maximally spinning BMPV black hole. Therefore, there exists a certain narrow parameter region such that the microstate geometry with a small number of centers admits the same angular momenta as the BMPV black hole. Moreover, we investigate the dependence of the topological structure of the evanescent ergosurfaces on the magnetic fluxes through the 2-circles between two centers.

physics. From this point of view, it is an important issue to probe what extent asymptotically flat microstate geometries possess the classical features of stationary black holes with the same asymptotic structure. There are many ways to probe physical aspects of such microstate geometries.
A natural and simple way is to investigate whether these spacetimes can carry the same asymptotic charges, the mass and angular momenta, as rotating black holes [22][23][24], rotating black rings [25][26][27][28] and rotating black lenses [29][30][31][32][33] in the same theory. If not so, such spacetimes cannot be regarded as the description of these black objects. For instance, as proved mathematically in [34], there are no asymptotically static microstate geometries in higher dimensional Einstein-Maxwell theory, which implies that any static black hole cannot be described by the soliton solutions. In particular, it is well known that there exist the microstate geometries corresponding to maximallyspinning black holes and maximally-spinning black rings that have zero horizon area, which are referred to "zero-entropy microstate geometries" [37]. Moreover, using the merge of such zeroentropy microstate geometries, Refs. [35,36] constructed the first microstate geometries with the same charges as black holes and black rings which have nonzero horizon area. In general, it is, however, not known how to construct the microstate geometries that correspond to black holes and black rings with non-zero horizon area without introducing the merger of zero-entropy microstate geometries.
The main purpose of this paper is to investigate whether there exist the microstate geometries in five dimension having the same asymptotic charges (mass and angular momenta) as the black hole, without using zero-entropy microstate geometries and by merely imposing a simple symmetry. In this paper, based on the work developed by Gauntlett et al. [38] in the framework of the five-dimensional minimal ungauged supergravity, we consider asymptotically flat, stationary and bi-axisymmetric BPS microstate geometries with n centers on the z-axis of the Gibbons-Hawking space. In addition, we impose reflection symmetry, which means the invariance under the transformation z → −z, on the solution since such an assumption dramatically simplifies the constraint equations for the parameters included in the solutions, a so-called "bubble equations", and this enables us to solve the constraint equations for the parameters. It can be shown that under the symmetry assumptions, the geometry has equal angular momenta. It is of physical interest to compare the mass and angular momenta of the Breckenridge-Myers-Peet-Vafa (BMPV) solution [24] since it describes a spinning black hole with equal angular momenta in the same theory. We will show that asymptotically flat, stationary, bi-axisymmetric and reflection-symmetric microstate geometries (at least, for five centers) can have the same mass and angular momenta as the BMPV black hole.
The rest of the paper is organized as follows: In the following Sec. II, we review the BPS solutions of the microstate geometries in the five-dimensional minimal supergravity. In Sec. III, we compute the mass, angular momenta and magnetic fluxes through the bubbles, and show the existence of evanescent ergosurfaces. In Sec. IV, imposing reflection symmetry, we simplify the solution and the bubble equations and thereafter show numerically that the microstate geometries have the same angular momentum as the BMPV black hole. In Sec. V, we summarize our results and discuss possible generalizations of our analysis.

A. Solutions
First, we begin with supersymmetric solutions in the five-dimensional minimal ungauged supergravity [38], whose bosonic Lagrangian consists of the Einstein-Maxwell theory with a Chern-Simons term. In this theory, the metric and the gauge potential of Maxwell field for the supersymmetric solutions take the form: Here, the four-dimensional metric ds 2 M is the metric of an arbitrary hyper-Kähler space, where we use the Gibbons-Hawking space metric [39] which is written as with r : = (x, y, z), The function H in Eq.(5) is a harmonic function with n point sources (n centers) on threedimensional Euclid space E 3 , and the 1-form χ on E 3 is determined by * dχ = dH, where the Hodge dual * is associated with E 3 . χ can be written as where the 1-formω i on E 3 , which is defined by * dω can be written asω The vectors ∂/∂t and ∂/∂ψ are commuting Killing vector fields. The Gibbons-Hawking metric (3) is preserved under the scaling transformation H → λ 2 H, χ → λ 2 χ, ψ → λψ and x i → λ −1 x i , which enables us to fix the period of the coordinate ψ as 0 ≤ ψ < 4π. These Gibbons-Hawking spaces are nontrivial U (1) fibration over a flat space E 3 and the unique class of four-dimensional hyper-Kähler metric with tri-holomorphic isometry.
The function f −1 and the 1-forms (ω, ξ) are given by where the functions K, L and M are harmonic functions on E 3 , The 1-formsω are ξ are determined by * dω = HdM − M dH and take the formsω where the 1-formω ij (i = j) on E 3 , which is determined by can be written asω In this paper, we set r i = (0, 0, z i ) (i = 1, . . . , n), by which x∂/∂y−y∂/∂x becomes another U (1) Killing vector field, and assume z i < z j for i < j (i, j = 1, . . . , n) without loss of generality. In terms of standard spherical coordinates (r, θ, φ) such that (x, y, z) = (r sin θ cos φ, r sin θ sin φ, r cos θ), the 1-formsω i andω ij are simplified as and so the 1-formω can be written aŝ where we have added the integration constant c sinceω is determined by only the derivatives in Eq. (19).

B. Boundary conditions
As the detail is reviewed in [10,40], in order that the supersymmetric solution describes the BPS microstate geometry solution of physical interest, we must impose suitable boundary conditions (i) at infinity, (ii) at the Gibbon-Hawking centers r = r i (i = 1, ..., n) and (iii) on the z-axis x = y = 0 of E 3 in the Gibbons-Hawking space. More precisely, we consider the following boundary conditions: (i) at infinity r → ∞, the spacetime is asymptotically Minkowski spacetime.
(ii) at the n centers r = r i (i = 1, ..., n) such that each harmonic function diverges, the spacetime is regular, and behaves as the coordinate singularities like the origin of the Minkowski spacetime. The spacetime admits no causal pathology such as closed timelike curve (CTCs) around these points.
(iii) on the z-axis I = {(x, y, z) | x = y = 0} of E 3 in the Gibbons-Hawking space, there appear no Dirac-Misner strings, no orbifold singularities and no conical singularities.
At infinity r → ∞, the metric functions f and H behave, respectively, as

Gibbons-Hawking centers
The metric obviously has divergence at the points r = r i (n = 1, ..., n) on the Gibbons-Hawking space. We hence impose the boundary conditions at the points r = r i (n = 1, ..., n) so that these become regular points like the origin of Minkowski spacetime: Let us choose the coordinates (x, y, z) on E 3 of the Gibbons-Hawking space so that the ith point r = r i is an origin of E 3 . Near the origin r = 0, the four harmonic functions H, K, L and M behave as, respectively, which lead to where the constants c 1(i) and c 2(i) are defined by where we have used h 2 i = 1 (h i = ±1 will be imposed below. See Eq. (65)) in the second equalities of Eqs. (49) and (50). The 1-formsω j andω kj are approximated bỹ and hence, 1-forms χ andω behave as where One therefore obtains the asymptotic behavior of the metric around the ith point as To remove the divergence of the metric, it is sufficient to impose the following conditions on the parameters (k i , l i , m i ) (i = 1, ..., n): which are equivalent to the condition for the parameters (l i , m i ), and these yield the equation Introducing the new coordinates (ρ, ψ , φ ) by we can write the metric near r = r i as Comparing the (φ , ψ )-part of the above metric (64) with the boundary condition (44), we must To ensure the five-dimensional metric with Lorentzian signature, the following inequities must be satisfied The above metric (64) is locally isometric to the flat metric, but CTCs necessarily appear near ρ 0 because the Killing vector ∂/∂ψ = ∂/∂ψ becomes timelike. To avoid the existence of CTCs around r i (i = 1, . . . , n), c 2(i) = 0 and ω 0(i) = 0 must be simultaneously satisfied at r = r i (i = 1, ..., n) but it is sufficient to impose only c 2(i) = 0, which can be written as These equations are so-called "bubble equations" in Refs. [9,41], which physically means the balance between the gravitational attraction and the repulsion by the magnetic fluxes over the 2-cycles. Moreover, let us note thatω 0(i) = 0 automatically hold for all i = 1, ..., n, if we impose (67) since from Eqs. (60) and (61),ω 0(i) can be shown to vanish, where we have used Eq. (67) for the 2nd term in the right-hand side of the first equality, and the last equality can be shown by long but simple computations.
Furthermore, the n bubble equations c 2(i) = 0 (i = 1, . . . , n) are not independent because the summation of h i c 2(i) (i = 1, . . . , n) automatically vanishes, regardless of the bubble equations, as where we have used Eqs. (40) and (42) in the second equality and the antisymmetry for i and j in the last summation. Thus, the bubble equations h i c 2(i) = 0 (i = 1, . . . , n) give (n − 1) independent constraint equations for the parameters (k i , z i ) (i = 1, . . . , n).
This means that there are no Dirac-Misner strings in the spacetime, which can be obtained as the result of the bubble equations (67) (see [10,41]).
Next, we show the absence of orbifold singularities. On the intervals I ± , the 1-form χ becomes and on the intervals I i (i = 1, . . . , n − 1), it takes the form The two-dimensional (φ, ψ)-part of the metric on the intervals I ± and I i can be written in the form Here let us use the coordinate basis vectors (∂ φ 1 , ∂ φ 2 ) with 2π periodicity, instead of (∂ φ , ∂ ψ ), where the coordinates φ 1 and φ 2 are defined by vanishes, where we have used i h j = 1 in the last equation.
3. on the interval I + , the Killing vector v + : From these, we can observe that the Killing vectors v ± , v i on the intervals satisfy with Therefore, it turns out that |det hold, which means that there exist no orbifold singularities at adjacent intervals, as proved in Ref. [42].

C. Gauge freedom
As discussed in Ref. [43], the supersymmetric solutions have a gauge freedom, which means that for the linear transformation for the harmonic functions H, K, L and M , for a certain m (m = 1, . . . , n) because the coefficient of 1/r m in K changes k m → k m +λh m .
Moreover, using the shift of z → z+const., one can set for a certain m (m = 1, . . . , n).

III. PHYSICAL PROPERTIES
Under the appropriate boundary conditions mentioned in the previous section, let us investigate some physical properties of the solutions.

A. Conserved quantities
To begin with, we consider conserved quantities of the microstate geometries. From the boundary conditions at infinity (40)-(43), the ADM mass and two ADM angular momenta can be computed as where Q is the electric charge, which saturates the BPS bound [44].
Each interval I i (i = 1, ..., n − 1), which is introduced in Sec. II B 3, denotes the bubble which is topologically a two-dimensional sphere since the ψ-fiber of the Gibbons-Hawking space (3) collapses to zero at the centers z = z i and z = z i+1 , and so along the interval, the fiber sweeps out twodimensional sphere. Since the Maxwell gauge field A µ is obviously smooth on the bubbles, the magnetic fluxes through I i (i = 1, ..., n − 1) can be defined as which are computed as

B. Evanescent ergosurface
The existence of ergoregions gives rise to strong instability due to a superradiant-triggered mechanism in spite of the existence of the horizon [45,46]. It was demonstrated that a certain class of non-supersymmetric microstate geometries with ergoregion in type IIB supergravity are unstable, which is a general feature of horizonless geometries with ergoregion [47]. The BPS microstate geometries does not admit the presence of ergoregions but evanescent ergosurfaces [10,48], which are defined as timelike hypersurfaces such that a stationary Killing vector field becomes null there and timelike everywhere except there. Reference [49] proved that on such surfaces, massless particles with zero energy (E = 0) relative to infinity move along stable trapped null geodesics. Since this stably trapping leads to a classical non-linear instability of the spacetime [45,49,50], it is of physical importance to investigate the existence of evanescent ergosurfaces, which exist at f = 0 which corresponds to For simplicity, let us consider the microstate geometries with reflection symmetry z m = −z n−m+1 and k m = k n−m+1 (m = 1, . . . n). For the microstate geometries with three centers (n = 3) and (h 1 , h 2 , h 3 ) = (1, −1, 1), they intersect the z-axis at the points It turns out from simple computations that F 3 (z) = 0 has no root on I ± and a single root I i (i = 1, 2). As seen FIG. 1, the evanescent ergosurfaces on the timeslice t =const. is the closed surface surrounding the center r 2 = (0, 0, 0), where we have introduced the radial coordinate by ρ = For the microstate geometry with five centers (n = 5) and (h 1 , h 2 , h 3 , h 4 , h 5 ) = (1, −1, 1, −1, 1), they intersect the z-axis at the points z satisfying F 5 (z) = 0, where F 5 (z) is written as The roots of the equation to the five centers that are located at r i (i = 1, . . . , 5) on the z-axis, and the red curves denote the evanescent ergosurfaces, whose shapes depend on k 1 and k 2 .

IV. MICROSTATE GEOMETRIES WITH REFLECTION SYMMETRY
In Sec. II, we have considered the stationary and bi-axisymmetric microstate geometries with n centers on the z-axis of the Gibbons-Hawking space which satisfy the bubble equations (67). The nasty constraint equations (for the parameters included in the solutions) make it difficult for us to understand the physical properties. In this section, in addition to such symmetry assumptions, we impose a further reflection symmetry on the solutions: which means the invariance of the solutions under the transformation z → −z. This additional assumption extremely simplifies the bubble equations so that one can solve them and express z i (1, . . . , n) in terms of k i (i = 1, . . . , n), at least, for small n. In particular, it is easy to show from Eq. (90) that the angular momentum J φ always vanishes under the additional symmetry assumption. In this section, for simplicity, let us consider only two cases of n = 3 and n = 5.

A. Three-center solution
First, let us consider the solution with three centers (n = 3) and (h 1 , h 2 , h 3 ) = (1, −1, 1) that describes the simplest asymptotically flat, stationary and bi-axisymmetric microstate geometry, which has the four parameters (k 1 , k 3 , z 1 , z 3 ), where we have set k 2 = 0 and z 2 = 0 from the two gauge conditions (86) and (87). Moreover, under the assumption of the reflection symmetry the bubble equations (67) are simply written as which imply It is obvious that in the former case h i c 1(i) = 0 (i = 1, 2, 3), and so the inequalities (66) cannot be satisfied. In the meanwhile, in the latter case, the inequalities (66) can be automatically satisfied because h i c 1(i) (i = 1, 2, 3) can be directly computed as Therefore, for arbitrary nonzero k 1 , this describes a regular and causal solution of an asymptotically flat, stationary microstate geometry with the bi-axisymmetry and reflection symmetry. This solution was previously analyzed in Ref. [10].
The z-axis of E 3 in the Gibbons-Hawking space consists of the four intervals: Thus the rod structure of this three-center microstate geometry is displayed in Fig. 3.
and the magnetic fluxes in Eq. (92) are written as

B. Five-center solution
Next, let us consider the stationary, bi-axisymmetric microstate geometry with five centers (n = 5), which has the four parameters (k 1 , k 2 , z 1 , z 2 ) under the reflection-symmetric conditions where we note that Eqs. (107)-(109) are not independent due to the constraint equation 5 i=1 h i c 2(1) = 2h 1 c 2(1) + 2h 2 c 2(2) + h 3 c 2(3) = 0. Therefore, this solution has only two independent parameters. If we regard a and b as the functions of k 1 and k 2 from Eqs. (107), (109), this solution is a two-parameter family for (k 1 , k 2 ). Furthermore, the parameters k 1 and k 2 must satisfy the inequalities (66), which are reduced to together with the inequalities In the below, we assume k 1 = 0 and k 2 = 0 because from Eqs. (107) and (109), the case k 1 = 0 leads to where only the solution with the positive sign can satisfy (110)-(113) and has j 2 = 25/24, and from Eqs. (107) -(109), the case k 2 = 0 yields (a, b) = (k 2 1 /3, 0), which cannot satisfy one of the inequalities (113). In what follows, we remove both cases of k 1 = 0 and k 2 = 0.
As shown in Fig.4, these inequalities are equivalent with vanishes, and 6. on I + , the Killing vector v + = −∂ φ 2 vanishes, Thus, it turns out that this five-center microstate geometry has the rod structure displayed in and the magnetic fluxes in Eq. (92) are written as

C. Comparison with BMPV black hole
Finally, we compare the BPS microstate geometries for n = 3 and n = 5 described in the previous section with the rotating BPS black hole in the five-dimensional minimal supergravity, i.e., the BMPV black hole [24], which carries mass (saturated the BPS bound) and equal angular momenta (J φ = 0). For this purpose, let us define a dimensionless angular momentum by For the BMPV black hole, the dimensionless angular momentum j has the range of where j = 0 corresponds to the extremal Reissner-Nordstrom black hole. The absence of CTCs around the horizon requires the upper bound, j = 1.
It is shown from Eqs. (102) and (103) that for n = 3, the squared angular momentum j 2 takes only the value of which is a larger value than the upper bound for the BMPV black hole.
Similarly, for n = 5, we evaluate the value of the squared angular momentum j 2 from Eqs. (116) and (117), where the ratio k 2 /k 1 lies in the range (115). As seen in Fig. 6, The squared angular momentum j 2 asymptotically approaches 25/24 at k 2 /k 1 → −∞. For k 2 /k 1 < −1, j 2 monotonically increases and diverges at k 2 /k 1 → −1, whereas for k 2 /k 1 > −1, it has the lower bound 0.841... at k 2 /k 1 → −0.206..., where Eqs. (107)-(109) cannot be satisfied. Thereafter, it increases and approaches 9/8 at k 2 /k 1 → 0, for k 2 /k 1 > 0 monotonically decreases and asymptotically approaches 25/24 at k 2 /k 1 → ∞. Thus, because the squared angular momentum does not have an upper bound but have the lower bound j 2 = 0.841..., we find that it must run the range From this analysis, we can conclude that the bi-axisymmetric and reflection-symmetric microstate geometry with five centers can have the angular momentum of the range 0.841... < j 2 < 1 as the BMPV black hole, while the microstate geometry with three centers cannot have.

V. SUMMARY AND DISCUSSIONS
In this paper, we have analyzed the solutions of the asymptotically flat, stationary, BPS microstate geometries with bi-axisymmetry in the five-dimensional minimal supergravity. Moreover, we have imposed additional reflection symmetry since this symmetry assumption extremely simplifies the expression of the solutions and enables us to solve the bubble equations. We have also computed the conserved charges, the ADM mass, two ADM angular momenta, and (n−1) magnetic fluxes through the bubbles between two centers. In particular, we have compared the mass and angular momenta for the three-center solution and the five-center solution of microstate geometries with those of the BMPV black hole. We have shown that the dimensionless angular momentum of the five-center microstate geometry does not have the upper bound but has the lower bound which is smaller than the angular momentum for the maximally spinning BMPV black hole, and hence there are the parameter region such that the microstate geometry has the same angular momentum as the BMPV black hole.
In our present analysis, we have restricted ourselves to the reflection-symmetric microstate geometries for n = 3 and n = 5, but it is not trivial whether there exist the reflection-symmetric solutions with a larger number of centers (n = 7, 9, . . .) which admit the same mass and angular momentum as the BMPV black hole or the microstate geometries for n = 3, 5. The bi-axisymmetric and reflection-symmetric microstate geometry with n centers seems to have (n + 3)/2 independent physical charges or fluxes [the mass M , the angular momentum J ψ or the (n − 1)/2 magnetic fluxes q[I i ] (i = 1, . . . , (n − 1)/2)], among which only (n − 1)/2 are independent since the number of the parameters reduces to half due to reflection symmetry. The analysis for such microstate geometries with n ≥ 7 deserves future works. Moreover, it may be interesting to compare the five-center solution dealt with in this paper with the spherical black holes having a topologically nontrivial domain of outer communication in Refs. [51,52], which can have not only same asymptotic charges as the BMPV black hole but also different ones. The solution without the reflection symmetry should be compared with the supersymmetric black ring [27] and supersymmetric black lenses [29,31,32] which does not admit the limit to equal angular momenta. This may be an interest issue as our future study. Finally, we comment that the solutions of the five-dimensional minimal supergravity can be uplifted to the solutions of both type IIB supergravity and eleven supergravity [53,54], and as discussed in Ref. [55], such solutions are relevant for the most general four-dimensional superconformal field theories (SCFTs) with holographic duals. This enables one to study some aspects of the dual strongly coupled thermal plasma with a non-zero R-charge chemical potential.
Therefore, it might be physically interesting to study the fluid-dynamics of the thermal plasma of the SCFTs corresponding to the microstate geometries.