A stationary and biaxisymmetric four-soliton solution in five dimensions
Shinya Tomizawa Takashi Mishima













^1 Department of Liberal Arts, Tokyo University of Technology, 5-23-22, Nishikamata, Otaku, Tokyo, 144-8535, Japan, 

^2 Laboratory of Physics, College of Science and Technology, Nihon University,
Narashinodai, Funabashi, Chiba 274-8501, Japan


 
Using the inverse scattering method to solve the five-dimensional vacuum Einstein equations, we construct an asymptotically flat  four-soliton solution as a stationary and bi-axisymmetric solution.   
We impose certain boundary conditions on this solution so that it includes a rotating black hole whose horizon-cross section is topologically a lens space of L(2,1).
The solution has nine parameters but three only is physically independent due to the constraint equations. 
The remaining  degrees of freedom correspond to the mass and two independent angular momenta of the black hole. 
We analyze a few simple cases in detail, in particular, the static case with two zero-angular momenta and the stationary case with a single non-zero angular momentum.


04.50.+h  04.70.Bw


A stationary and biaxisymmetric four-soliton solution in five dimensions

    Shinya Tomizawa^1[tomizawasny@stf.teu.ac.jp] and Takashi Mishima^2[tmishima@phys.ge.cst.nihon-u.ac.jp]
    December 30, 2023
==========================================================================================================








§ INTRODUCTION





The studies on higher-dimensional black hole solutions to Einstein's equations have played roles in the microscopic derivation of Bekenstein-Hawking entropy <cit.>, and the realistic black hole production at an accelerator in the scenario of large extra dimensions <cit.>.  
Despite two decades of research and development in techniques of solution-generation, our understanding of higher dimensional black holes is still not enough. 
 The topology theorem for a stationary black hole generalized to five dimensions <cit.> states that the topology of the spatial cross section of the event horizon must be either  a sphere S^3, a ring S^1× S^2 or lens spaces L(p,q) (p,q : coprime integers), if the spacetime is asymptotically flat and admits two commuting axial Killing vector fields which also commutes a stationary timelike Killing vector. 
 As for the first two topologies, the exact solutions to vacuum Einstein's equations <cit.> have already been found. 
In contrast, a regular vacuum black hole solution with the horizon of lens space topology has been difficult to find in spite of a few trials,   since the resultant solutions always suffer from naked singularities. 
 
 


The inverse scattering method (ISM) is known as one of the most useful tools to obtian exact solutions of Einstein equations with D-2 Killing isometries (D: spacetime dimension). 
In this method, new solutions with the same isometries can be systematically obtained by the soliton transformation from a certain known simple solution, which is often called seed. 
In general, the direct application of the original method formulated by Belinski and Zakharov <cit.> to higher dimensions yields singular solutions but  
Pomeransky modified the ISM so that it can generate regular solutions even in higher dimensions <cit.>. 
Remarkably, combined with the rod structure <cit.>, it has achieved a great success so far, concerned with, in particular, five-dimensional vacuum black hole solutions. 
The first example of the generation of black hole solutions by the modified ISM is the re-derivation of the five-dimensional Myers-Perry black hole solution <cit.>.  
Thereafter, the S^2-rotating black ring  was  re-derived <cit.> by the ISM from the Minkowski seed (this solution was first derived in Ref. <cit.> independently), 
but it turned out that the generation of the S^1 rotating black ring  has a more delicate problem on how to choose the seed, since an facile choice of the seed always results in the generation of a singular solution. 
The suitable seed to derive the black ring with the S^1-rotation was first considered in <cit.>. 
Subsequently, the regular black ring solution with both of S^1 and S^2 rotations was constructed by Pomerasnky and Sen'kov <cit.>.
 


Using the ISM, a few authors attempted to construct asymptotically flat black lens solutions to the five-dimensional vacuum Einstein equations. 
First, Evslin <cit.> attempted to construct a static black lens with the lens space topology of L(n^2+1,1) but  found that curvature singularities cannot be eliminated, whereas both conical and orbifold singularities can be removed.  
Subsequently, Chen and Teo <cit.> constructed a black lens solution with the horizon topology of L(n,1)=S^3/ℤ_n by the ISM but observed that it must have either conical singularities or naked curvature singularities. 
Thus, the major obstacle in constructing a black lens solution is always suffering from naked singularities. 
However, the sudden breakthrough in this line have come from supersymmetric solutions. 
Based on the well-known framework of the construction for supersymmetric solutions in the bosonic sector of five-dimensional minimal supergravity developed by Gauntlett et al. <cit.>, 
Kundhuri and Lucietti <cit.> succeeded in the derivation of the first regular exact solution of an asymptotically flat black lens with the horizon topology of L(2,1)=S^3/ℤ_2. 
This solution was subsequently generalized to the more general supersymmetric black lens with the horizon topology  L(n,1)=S^2/ℤ_n (n≥ 3) in the same theory <cit.>.  









The supersymmetric solutions provides us various useful information on the corresponding vacuum solutions when we do not yet know them. 
In particular,  the discovery of the supersymmetric black lens solutions <cit.> gives us a nice  guideline for understanding  the rod structure of unfound regular vacuum black lens solutions.  
From now,  for simplicity, we consider the vacuum solution with the horizon of the special lens space topology L(2,1).
Figure 1 shows the rod diagram of the Kunduri-Lucietti's supersymmetric black lens in <cit.>, where the horizon rod is drawn as a point because the supersymmetric black hole has a degenerate horizon. 
On the contrary, Chen and Teo in <cit.>  considered the black lens with the rod structure displayed in FIG.2, though  this solution has conical singularities on z∈[z_3,z_4][More precisely, this has two branches: one has conical singularities only on the axis, whereas the other has curvature singularities on the surface surrounding the point z=z_4. Here we consider the former case.].
 The main difference between these rod diagrams except the shapes of the horizon rods lines in the signatures of the 3rd component of the rod vectors (0,2,±1) on [z_3,z_4]. 
 In this paper, we consider to construct the vacuum solution of a black lens with the combined rod diagram displayed in FIG.4 where the rod vector on [z_3,z_4] in FIG.2 is replaced with one in FIG.1.   
 
 
 
 
 
 




 



 
  
 
 
 
 To this end, using the Pomeransky's ISM for the five-dimensional vacuum Einstein equation, we construct a four-soliton solution by regarding the singular solution with a diagonal metric as a seed (see FIG.3 on the rod diagram), which is the same as one used for the construction of the black ring solution. 
In order to obtain a black lens solution of physical interest, we impose suitable boundary conditions at infinity, on the horizon, on a symmetry of axis as follows: 
 (i)  Infinity must be such that the spacetime is asymptotically flat. 
(ii) The horizon corresponds to a smooth null surface, whose spatial cross section has a topology of the lens space L(2,1)=S^3/ℤ_2.
 (iii) On the axis, there appear no curvature singularities, no conical singularities,  no Dirac-Misner strings, and besides, orbifold singularities at isolated points must be eliminated. There indeed exist the parameter regions where all these boundary conditions are satisfied, however, 
 it seems to be considerably hard to deal with the four-soliton itself owing to the rather complex metric and five constraints on the parameters. 
 For this reason, when we study the physical properties of the solution, we restrict ourselves to a few simple cases,  a case with a single angular momentum and a static case.
As shown later, as for the  the black lens with a single angular momentum, there exist unavoidable naked CTCs surrounding the nut z=z_4 outside the horizon, even if all of (conical, curvature, and orbifold) singularities and Dirac-Misner strings can be removed at these boundaries. 
For the static case,  there are two branches, though there necessarily exist conical singularities on [z_3,z_4] only. 
One has CTCs region around the point z=z_4, but the other does not.  





We organize the present paper as follows. 
In the following section <ref>, by using the Pomeransky's ISM, we present the four-soliton solution in five dimensions which admits three commuting Killing vectors,  stationary and bi-axially symmetric Killing vectors. 
The solution contains many parameters. 
In the general choice of these parameters, the solution cannot be necessarily asymptotically flat and regular, and the lens space topology of the horizon is not guaranteed.
Therefore, in section <ref>, we impose on the parameters the boundary conditions under which the spacetime 
is asymptotically flat, neither (curvature, conical, and orbifold) singularities nor Dirac-Misner strings exist, at least, on the axis and horizon.  
It is shown that the boundary conditions  finally reduce the number of the independent parameters to three. 
In Section <ref>, we  analyzes a few simple cases,  the case with a single angular momentum and a static case. 
In particular, for the former case, we will discuss the phase diagram and the existence of CTCs.    
 In the final section <ref>, we devote ourselves to the summary and discussion on our results.
 








§ BLACK LENS SOLUTIONS






















































First, let us start from  the construction of the seed solution.
We consider a five-dimensional, stationary and bi-axisymmetric spacetime whose has three commuting commuting Killing vectors, a stationary Killing vector ∂/∂ t, and two axisymmetric Killing vectors  ∂/∂ϕ_1, ∂/∂ϕ_2.
The diagonal metric of the five-dimensional vacuum solution whose rod diagram is given by FIG.3 can be written as

    ds^2=-μ_1/μ_3dt^2+μ_2μ_4/μ_1dϕ_1^2+ρ^2μ_3/μ_2μ_4dϕ_2^2+k^2μ_3R_11R_22R_33R_44R_24^2/μ_2μ_4R_12R_13R_14R_23R_34(dρ^2+dz^2),

where μ_i and R_ij  (i,j=1,…,4) are defined, respectively,  by

    μ_i=√(ρ^2+(z-z_i)^2)-(z-z_i),   R_ij=μ_iμ_j/ρ^2+μ_iμ_j.

z_i are constants,  and we assume z_1<z_2<z_3<z_4. 
As seen later, k is the integration constant which is determined from the requirements of the absence from conical singularities at infinity. 
Note that this metric is the exactly same as that of the seed solution used for the derivation of the rotating black ring by the ISM.
As shown later, although there exist naked curvature singularities on the rod ρ=0, z∈ [z_1,z_2] which has a negative density,  
they have disappeared under some appropriate boundary conditions after the four-soliton transformation. 
Now, we briefly explain how we have obtained the four-soliton solution, following the Pomeransky's procedure: 
 
 
 

 
 
 


  * First, let us remove trivial solitons from the four points z=z_1,z_2,z_3, and z_4 with the BZ vectors (0,1,0), (0,0,1), (1,0,0) and (0,0,1), respectively.


  * After the solitons are removed, the obtained metric  is written as

    g̃_0   =   g_0 · diag( -ρ^2/μ̃_3^2,-ρ^2/μ̃_1^2,(-ρ^2/μ̃_2^2) (-ρ^2/μ̃_4^2))
       =    diag(ρ^2/μ̃_1μ̃_3,ρ^4/μ̃_1μ̃_2μ̃_4,-ρ^4/μ̃_4μ̃_3μ̃_4),

where μ̃_i=-ρ^2/μ_i=-√(ρ^2+(z-z_i)^2)-(z-z_i).

  * Performing the formal replacement of μ̃_i→μ̃_i-λ (i=1,…,4), ρ^2 →ρ^2-2zλ -λ^2 in the matrix g̃_0 (λ is a so-called spectrum parameter), we can obtain the generating matrix Ψ_0[λ,ρ,z] corresponding to g̃_0:

    Ψ_0[λ,ρ,z]
       =    diag(
    ρ^2-2zλ-λ^2/(μ̃_1-λ)(μ̃_3-λ),
    (ρ^2-2zλ-λ^2)^2/(μ̃_1-λ)(μ̃_2-λ)(μ̃_4-λ),
           -(ρ^2-2zλ-λ^2)^2/(μ̃_2-λ)(μ̃_3-λ)(μ̃_4-λ)).





  * Next, let us add back the nontrivial solitons at z=z_1,z_2,z_3 and z_4 with the BZ vectors m_0a^(1)=(C_1,1,0),m_0a^(2)=(0,C_2,1),m_0a^(3)=(1,0,C_3) and m_0a^(4)=(0,C_4,1), respectively, where the constants C_1,C_2,C_3 and C_4 are often called BZ parameters.

Then,  using the BZ vectors and the inverse matrix of the generating matrix whose spectrum parameter λ is substituted into μ_k , we can construct the three-dimensional vectors m_a^(k),

    m_a^(k)=m_0b^(k)[Ψ_0(μ_k,ρ,z)]_ba.

Thus, we obtain the metric of the four-soliton solution as

    g_ab=(g̃_0)_ab-∑_k,l=1^4(g_0)_acm_c^(k)(Γ^-1)_klm_d^(l)(g_0)_bd/μ_kμ_l,


    Γ_kl=m_a^(k)(g̃_0)_abm_b^(l)/ρ^2+μ_kμ_l.

 Note that in the original BZ's procedure, g_ab  does not satisfy the normalization condition det (g_ab)=-ρ^2 (in the final step one must normalize g_ab),  whereas in the Pomeransky's procedure, g_ab automatically satisfies the condition without such a normalization process. 
The two-dimensional  conformal factor f:=g_ρρ(=g_zz) for the new solution is obtained from the factor f_0 for the seed  as

    f=f_0 det Γ/ det Γ_0,

where the matrix Γ_0=(Γ_0kl) is obtained by putting  C_i=0 in Eq.(<ref>).




























































































§ BOUNDARY CONDITIONS

The four-soliton solution that we have obtained in the previous section has nine parameters (z_i,C_i,k) but 
the general choice of these parameters cannot guarantees the horizon topology of a lens space,  regularity  and even asymptotic flatness.
 In order that the four-soliton solution describes a physically interesting solution, we need impose suitable boundary conditions at infinity, on the horizon, and on a symmetry of axis : 
(i) The spacetime is asymptotically flat at infinity. 
(ii) The spacetime has a smooth horizon whose spatial topology is the lens space L(2,1)=S^3/ℤ_2.
 (iii) The spacetime has  no curvature singularities, no conical singularities,  no Dirac-Misner strings, and besides, no orbifold singularities at isolated points on the axis. 
In what follows, using the rod structure <cit.>, we  show that there indeed exist the parameter regions where all these boundary conditions are satisfied. 







 §.§ rod diagram


To ensure that the spacetime is asymptotically flat,  the two semi-infinite rods (-∞,z_1] and [z_4,∞) must have the rod vectors (0,0,1) and (0,1,0), respectively.  At a glance, this seems not to be satisfied for the obtained metric, but we can confirm that it is automatically satisfied under the global rotation g→ A^TgA, where A is a 3×3 matrix that satisfies the condition det A=1 and is written as

    A   =   ζ(
        [                                                                                       ζ^-1                C_3 (-C_2 z_12 z_34-C_4 z_14 z_23)/C_1 C_3 z_1 (C_2 z_34+C_4 z_23)+z_3 z_24                      2 C_1 z_1 z_3 (C_2-C_4) z_13/C_1 C_3 z_1 (C_2 z_34+C_4 z_23)+z_3 z_24;                                                                                          0                                                                                          1 -C_1 C_2 C_3 C_4 z_1 z_24+z_3 (C_2 z_21+C_4 z_14)/C_1 C_3 z_1 (C_2 z_34+C_4 z_23)+z_3 z_24;                                                                                          0 -C_1 C_2 C_3 C_4 z_1 z_24+z_3 (C_2 z_21+C_4 z_14)/C_1 C_3 z_1 (C_2 z_34+C_4 z_23)+z_3 z_24                                                                                          1 ]),

where z_ij:=z_i-z_j, and ζ, which is determined from det A=1, can be written as

    ζ   =   -[C_1 C_3 z_1 (C_2 z_43+C_4 z_32)+z_3 z_42]
       ×   [ z_3(C_2z_21-C_4z_41-z_42)-C_1C_3z_1(C_2C_4z_42+C_2z_43+C_4z_32) ]^-1/2
       ×   [ z_3(C_2z_21-C_4z_41+z_42)-C_1C_3z_1(C_2C_4z_42-C_2z_43-C_4z_32) ]^-1/2.










  * The semi-infinite rod z∈(-∞,z_1] has the rod vector 

    v_1   =   ( 2z_1z_3z_31C_1(C_2-C_4)/z_1C_1C_3( z_32C_4+z_43C_2 )+z_3z_42,-z_1z_42C_1C_2C_3C_4+z_3(z_21C_2-z_41C_4)/z_1C_1C_3( z_32C_4+z_43C_2 )+z_3z_42,1).

After the global rotation, v_1' =A^-1v_1 becomes proportional to (0,0,1). 
Moreover, the condition for conical singularities not to exist on the rod is given by

    m_1:=lim_ρ→ 0√(ρ^2f/g'_ij(v'_1)^i(v'_1)^j)=1,

which determine k as

    k^2   =   -z_3^2 z_42^2/ z_3(C_2z_21-C_4z_41-z_42)-C_1C_3z_1(C_2C_4z_42+C_2z_43+C_4z_32) 
       ×   1/z_3(C_2z_21-C_4z_41+z_42)-C_1C_3z_1(C_2C_4z_42-C_2z_43-C_4z_32) .


  * The semi-infinite rod [z_4,∞) has the rod vector 

    v_4   =   (C_3 (C_2 z_21 z_43+C_4 z_41 z_32)/C_1 C_3 z_1 (C_2 z_43+C_4 z_32)+z_3 z_42,1,-C_1 C_2 C_3 C_4 z_1 z_42+z_3 (C_2 z_21-C_4 z_41)/C_1 C_3 z_1 (C_2 z_43+C_4 z_32)+z_3 z_42).

After the global rotation, v_4' =A^-1v_4 becomes proportional to (0,1,0). 
The condition (<ref>) automatically guarantees the absence from  the conical singularities on the axis rod. 
In what follows, we define (ϕ'_1,ϕ'_2) by ∂/∂ϕ'_1:=v_4' and  ∂/∂ϕ'_2:=v_1'.   









  * The  finite rod [z_1,z_2] has the rod vector 

    v_12   =   (2 z_3 (C_2-C_4) z_21 z_31 z_41/C_3 z_21 z_41 (C_2 z_43+C_4 z_32)+2 C_1 z_1 z_3 z_31 z_42,
          C_2 z_21 (2 C_1 z_1 z_3 z_31-C_3 C_4 z_41 z_42)-2 C_1 C_4 z_1 z_3 z_31 z_41/C_3 z_21 z_41 (C_2 z_43+C_4 z_32)+2 C_1 z_1 z_3 z_31 z_42,1) .

To eliminate the naked curvature singularities on [z_1,z_2] which indeed exists for the seed solution, 
we require that the finite rod [z_1,z_2] should be parallel to the the semi-infinite rod (-∞,z_1]. 
It can be easily shown that this can be accomplished if we impose the condition 

    C_1=±√(z_21z_41/2z_1^2z_31).

It turns out that after the global rotation, v_12' =A^-1v_12 becomes proportional to (0,0,1).



  * The timelike finite rod [z_2,z_3] has the following rod vector 

    v_23   =   (1,
    -2 C_1 z_1z_3 z_31  (z_32+C_2 C_4 z_41)+C_3 C_4 z_32 z_42 z_41/-2z_3z_31z_41(C_2C_4z_21+z_32),
          2 C_1 C_2 z_1 z_3z_31 z_42 +C_3 z_41 z_32 (C_2 C_4z_21-z_43)/-2z_3z_31z_41(C_2C_4z_21+z_32)).

After the global rotation, v_23' =A^-1v_23 becomes proportional to (1,Ω_1,Ω_2), where Ω_1 and Ω_2 are the angular velocities along ∂_ϕ_1' and ∂_ϕ_2', respectively, and are given by


    Ω_1   =   Ω̃_1/D_1,
    Ω_2   =   Ω̃_2/D_2,

where

    Ω̃_1   =    -4 C_1 z_1 z_3^3 z_31^2 z_42+C_3^3 C_4^2 z_21  z_32z_41^2 (C_4 z_32+C_2 z_43)
          +2 C_1 C_3^2 C_4 z_1z_3 z_31 z_41  [C_2 (z_32-z_21) z_43+C_4 z_32 (-z_21+z_32+z_42)]
          +2 C_3z_3^2 z_31 z_41  [-C_2 z_21 z_43+C_4 (z_32^2+z_42^2-z_43^2+z_1z_42+z_2z_21-z_3z_41)],


    Ω̃_2   =   -4 C_1 z_1 z_3^3z_31^2 z_42  [-C_2^2 C_4 z_21 z_41+C_4 z_41 z_32
    +C_2 (C_4^2 z_14^2+z_1z_32+z_2z_43-z_4z_42)]
          +2 C_3 z_3^2z_31 z_41[C_2^3 C_4 z_21^2 z_41 z_43+C_2 C_4 z_21 z_32 
    (-C_4^2 z_41^2-z_21z_43+z_32z_41)
          -z_32{z_42^2 z_43+C_4^2 z_41 (z_21z_32-z_42^2)}
          -C_2^2 z_21{- z_43 (z_21z_32+2z_42^2)-C_4^2 z_41 (z_21z_32-z_41z_43-z_42^2)}]
          -C_3^3 z_21 z_41 z_32[C_4^2 z_32^2 z_41 z_43-C_2^3 C_4 z_21 z_41 z_43^2-C_2 C_4 z_41 z_32{-2 z_43^2+C_4^2 (z_21z_32+z_42^2)}
          +C_2^2  z_41z_43{ z_43^2-C_4^2 (z_21z_32+z_42^2)}]
          -2 C_1 C_3^2 z_1z_3  z_31 z_41 z_42[-C_4^3 z_41 (z_32^2+C_2^2 z_21 z_23)+2 C_4 (z_32^2-C_2^2 z_21 z_32) z_43
          -C_2 C_4^2 {-C_2^2 z_21 z_41 z_43+z_32  z_42(z_41+z_43)}+C_2 z_43 (2 z_32 z_43-C_2^2 z_21 z_43)],


    ζ D_1   =   ζ D_2(z_42(C_2^2z_21+z_32))^-1
       =   -2 C_1 C_3^3 C_4^2 z_1 z_31 z_41 z_32 (C_4 z_41 z_32+C_2 z_21 z_43)
          +2 C_3^2 C_4 z_3 z_31 z_41 [-C_4 z_41 z_32^2+C_2 z_21 (C_4^2 z_41 z_42+z_32^2-z_42^2)] 
          -4  z_3^3z_31^2 z_41 (-C_4^2 z_41+C_2 C_4 z_21+z_42) 
          -4 C_1 C_3 z_1 z_3^2z_31^2  [C_2 z_42 z_43-C_4^3 z_41 z_41+C_4 z_41 z_42-C_2 C_4^2 z_41 (-z_21+z_42)] .




  * The finite rod [z_3,z_4] has the rod vector

    v_34   =   (-C_3 z_41 z_43 (C_2 C_4 z_21+z_32)/C_2 (C_1 C_3 z_1 z_42 z_43-C_4 z_3 z_21 z_41)+z_3 z_41 z_43,
          -C_1 C_3 z_1 z_34 (C_2 C_4 z_41+z_32)+C_4 z_3 z_41 z_42/C_2 (C_1 C_3 z_1 z_42 z_43-C_4 z_3 z_21 z_41)+z_3 z_41 z_43,1).

After the global rotation, the rod vector v_34'=A^-1v_34 is proportional to (0,2,-1) if the constants (C_i,z_i) satisfy  

    C_4   =   -C_1C_2z_1(2z_3^2z_31+C_3^2z_32z_43)+C_2^2C_3z_3z_21z_42+C_3z_3z_32z_42/C_1z_1(C_3z_32^2+C_2^2C_3^2z_21z_42-2z_3^2z_31),

and 

    2z_42(-C_4^2 z_41+z_43) (C_1 C_2 C_3 z_1+z_3) (C_1C_2C_3z_43z_1+z_41z_3)
          -C_1^2 C_3^2 z_1^2 z_34{C_2^2 C_4 (z_1 z_43-z_2 z_42+z_32 z_4)-C_2 z_23(C_4^2 z_14+z_34)-C_4 z_23^2}
          +C_1 C_3 z_1 z_24 z_3 (C_2^2 z_12+z_23) (C_4^2 z_14+z_34)+z_14 z_3^2 {-C_2^2 C_4 z_12^2+C_2 z_12 (C_4^2 z_14+z_34)
          +C_4 (z_1 z_43-z_2 z_42+z_32 z_4)=0.

In fact, under these conditions, v'_34=(v'_34t,v'_34ϕ_1',-1) is written as

    v'_34t   =   [-2 C_1^2 C_2 C_3 z_1^2 z_31 z_3 (C_2-C_4)-C_1 z_1 {C_3^2 (C_2^2 C_4 z_21 z_42+C_2 z_32 z_43+C_4 z_32^2)
       -   2 z_3^2z_31 (C_2-C_4) }+C_3 z_3 (C_2^2 z_21^2-C_2 C_4 z_21 z_41-z_32 z_42)]ζ
       ×   [(C_1 C_2 C_3 z_1+z_3) {-C_1 C_3 z_1 (C_2 z_43+C_4z_32)-z_3 z_42}]^-1
       =   0,

and 

    v'_34 ϕ_1'   =   [C_1^2 C_3^2 z_1^2 z_34{C_2^2 C_4 (z_1 z_43-z_2 z_42+z_32 z_4)-C_2 z_32(C_4^2 z_41+z_43)-C_4 z_32^2}
       +   C_1 C_3 z_1 z_42 z_3 (C_2^2 z_21+z_32) (C_4^2 z_41+z_43)
       +   z_41 z_3^2 {-C_2^2 C_4 z_21^2+C_2 z_21 (C_4^2 z_41+z_43)+C_4 (z_1 z_43-z_2 z_42+z_32 z_4)}]
       ×    [z_42(-C_4^2 z_41+z_43) (C_1 C_2 C_3 z_1+z_3) (C_1C_2C_3z_43z_1+z_41z_3)]^-1
       =   2.




Moreover, the conical singularities are free on z∈[z_3,z_4] if the constants satisfy 

    m_34:=lim_ρ→ 0√(ρ^2f/g'_ij(v'_34)^i(v'_34)^j)=1,

which gives

    m_34^2   =   z_42^2 (C_4^2 z_41-z_43)^2 (C_1 C_2 C_3 z_1+z_3)^2(C_1C_2C_3z_43z_1+z_41z_3)^2 /z_41 z_43[ z_3(C_2z_21-C_4z_41-z_42)-C_1C_3z_1(C_2C_4z_42+C_2z_43+C_4z_32) ]^2
       ×   1/[ z_3(C_2z_21-C_4z_41+z_42)-C_1C_3z_1(C_2C_4z_42-C_2z_43-C_4z_32) ]^2
       =   1.







 §.§ Summary

The physical requirements of asymptotic flatness, regularity on the rods and horizon topology of the lens space L(2,1) impose the conditions (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) on the parameters. 
In combination with the gauge degree of freedom z→ z+α, these reduce the independent parameters from nine to three. These correspond to physical degree of freedom, mass and two angular momenta.
The ADM mass and two ADM angular momenta are given by, respectively, 

    M=3π m/8D,   J_1=π j_1/4D,


    J_2=π/42C_3(z_32+C_2^2z_21)(2z_3^2z_31+C_3^2z_32z_43)z_42z_32ζ_2/-C_3^2z_32^2+2z_3^2z_31-C_2^2C_3^2z_21z_42,


where 

    m   =   2z_31z_42[C_2^2z_21{2z_3^2z_31+C_3^2(C_4^2z_41z_42-z_43^2)  }+C_3^2C_4^2z_32^2z_41+2z_31z_3^2(z_42-C_4^2z_41)   ],
    
    j_1   =    2z_42[2C_2^2C_3C_4z_21z_31z_41z_3^2(z_42^2+z_32^2-2C_4^2z_41z_42)
          +C_3C_4z_32z_41{  C_3^2C_4^2z_32^3z_41+2z_31z_3^2(-C_4^2(z_42+z_32)+z_42^2) }+C_2C_3^3C_4^2z_41z_43
          +2C_2^3C_3z_21^2z_31z_3^2{2C_4^2z_41z_42-z_43(z_42+z_43)}  -2C_2C_3z_21z_31 z_3^2{  -C_4^2(z_32^2+z_42^2)+z_42^2z_43}
          +C_2^2C_3^3C_4z_21z_32z_41z_43(C_4^2z_32+z_43)+C_2^3C_3^3z_21^2z_43^2(C_4^2z_32+z_43)]
          +2C_1z_42[-4C_2^2z_1z_3^3z_21z_31^2(C_2C_4z_21-2C_4^2z_41+z_42+z_43)
          -4C_2C_4z_1z_3^3z_31^2( C_4^2z_41^2+z_21z_32-z_41z_43 ) 
           -2z_1z_3z_31{  C_3^2C_4^2z_32^3z_41+2z_3^2z_31(-C_4^2z_41(z_42+z_32))+z_42^2         }
           -2C_2^3C_3^2C_4z_1z_3z_21z_31{  C_4^2z_41z_42^2+z_43(z_32z_21-z_42^2)           }
          +2C_2^2C_3^2z_1z_3z_21z_31{  -C_4^2z_41(z_32^2+z_43^2)+z_43^2        }
          +2C_2C_3^2C_4z_1z_3z_31z_32{  -z_43(2z_42^2-z_1z_43+z_2z_41-z_3z_21+z_4z_32)
           -C_4^2z_41(z_42^2+z_41z_43)     }] ,
    
    D   =   C_3^2z_21z_41{ (C_2z_43+C_3z_32)^2-C_2^2C_4^2z_42^2  }   -2z_31z_3^2 { (C_2z_21+C_4z_41)^2-z_42^2  }
          -4C_1C_3z_31z_42{ C_2C_4(-C_2z_21+C_4z_41)-C_4z_32-C_2z_43} .
































§ LIMITS TO SIMPLE SOLUTION




 §.§ A black lens with a single angular momentum
 
For simplicity, we analyze a rotating black lens with only a single angular momentum, where note that  “a single angular momentum" does not mean  “a single angular velocity".  
In fact, the solution with a single angular momentum has two non-vanishing angular velocities.
One of such solutions can be obtained by taking the three-soliton limit of C_3=0, and one finds that the conditions (<ref>), (<ref>) and (<ref>) reduce, respectively, to 

    C_4= C_2, 
          z_21C_2^3+2z_41C_2^2+z_32C_2-2z_43=0, 
          z_41(z_41C_2^2-z_43)^2/(1-C_2^2)^2z_43z_42^2=1.


It should be note that this solution is different from the Chen-Teo's solution with a single angular momentum in Ref.<cit.>, since how to add back non-trivial solitons differs in what follows.
They removed a trivial soliton with (0,0,1) from z=z_2 and then added the non-trivial soliton with (C_2,0,1) at  z=z_2, whereas after we remove the trivial soliton with (0,0,1) from z=z_2, we add back the non-trivial soliton with  (0,C_2,1) at z=z_2.




  §.§.§ C-metric representation

For the present purpose, it is more convenient to use a so-called C-metric coordinates (x,y) rather than the canonical coordinates (ρ,z). The relation between these coordinates is given by

    ρ   =   2κ^2√(-G(x)G(y))/(x-y)^2,
    
    z   =   κ^2(1-xy){2+μ(x+y)}/(x-y)^2,

where

    G(ξ):=(1-ξ^2)(1+μξ),

and  μ and κ are constants, which satisfy the inequalities

    0≤μ<1,  κ>0.

To fix the gauge freedom z→ z+α, let the turning points (z_1,z_2,z_3,z_4) be

    z_1=cκ^2,   z_2=-μκ^2,   z_3=μκ^2,   z_4=κ^2.

Then, we can write (μ_2,μ_3,μ_4) in the simple forms without the square root as

    μ_2   =   -2κ^2(1-x)(1+y)(1+μ y)/(x-y)^2,
    μ_3   =   -2κ^2(1-x)(1+y)(1+μ x )/(x-y)^2,
    μ_4   =   -2κ^2(1-y^2)(1+μ x )/(x-y)^2.

Before performing the global rotation mentioned previously, the metric of the rotating black lens with the horizon topology of L(2,1)  takes the following form


    ds^2   =   -H(x,y)/H(y,x)[ dt+ω_1dϕ_1+ω_2ϕ_2]^2
          +F(x,y)/(x-y)^2H(x,y)dϕ_1^2-F(y,x)/(x-y)^2H(x,y)dϕ_2^2+2J(x,y)/(x-y)^2H(x,y)dϕ_1dϕ_2
          +κ^2H(y,x)/4(1-C_2^2)(1-μ)(x-y)^2(dx^2/G(x)-dy^2/G(y)),

where 

    ω_1   =   2κ ^2  C_1c(c-μ)  (1+y)/H(x,y)[-C_2^4(1-c) (c+μ )  (1+x)^2
          +C_2^2 { -2(1-μ)(2-μ)-μ^2x(y(1-x)+1+x)+μ(2x^2+xy+x-y)
          +c(5-4μ-μ x^2(1+y)+(1-μ)x(3-y)+y)
    }
    +4(1-μ)^4
    ], 
    ω_2   =   -2 κ ^2  C_1 C_2c  (c-μ )(1+y)/H(x,y)[C_2^2{μ^2(xy+1)(1-x)+μ(x^2-xy+5x+y-2)+4
          +c((1-2μ+μ y)x^2+((1-μ)y-3μ-1)x-y+μ-4  )}  -2(1-μ)(μ(x^2+2x-1)+2)
    ],

and the functions H(x,y), F(x,y), and J(x,y) are written, respectively,  as

    H(x,y)   :=   C_2^4 [-c^3 (x+1)^2 (1+y)-c^2 (x+1)^2 (1+y) (μ(1+y)-1)
          -c μ  (x+1)^2 (1+y) ((μ -1) y-1)+μ ^2 (x+1)^2 y (1+y)]
       +   C_2^2 [c^2 (1+y) {4 μ +x (μ  ((x-1) y+x+3)+y-3)-y-5}
          +c {-13 μ +(μ -1) μ  x^2 (1+y)^2+2 x (μ  (μ +y (2 μ +3 μ  y-y+2)-1)+2)
          +μ  (5 μ +y (μ  (y+2)+3 y-2))+8 y+12}
          -μ ^2 {(x (x+3)+4) y^2+2 x (x+5) y+(x-1) x-4 y+8}
          -μ ^3 (x-1) (y-1)^2-4 μ  (x+2 y-3)-8]
          -4 c (μ -1)^2 (1+y)+4 (μ -1)^2 (μ  (y-1)+2),


    F(x,y)/κ^2   :=   -2C_2^4[ (1-c)(1+x)^3(c+μ)(c+μ x) (1-y^2)(1+μ y) ]                      
       +   C_2[ c^2 {(-5 + x (-3 + y) - y) (1 + y) + (1 + y) (4 + x (3 + x + (x-1) y)) μ}
          +c {4 (3 + x + 2 y) + (-13 - 2 x (y-1)^2 - 2 y + 3 y^2 - x^2 (1 + y)^2) μ
          + (5 + 2 y + y^2 + x^2 (1 + y)^2 + x (2 + 4 y + 6 y^2)) μ^2}
          - (x-1) (y-1)^2 μ^3+(-8 - (x-1) x + 4 y - 2 x (5 + x) y - (4 + x (3 + x)) y^2) μ^2
          -4 (-3 + x + 2 y) μ-8]
          +8 (-1 + y^2) (1-μ)^2 (2 - c (1 + x) + (x-1) μ) (1 + y μ),


    J(x,y)/κ^2   =    -2C_2^3 (-1 + c) (1 + x) (x - y) (1 + y) (c + μ) [-4 + {2 + x^2 (-1 + y)
           - 5 y - y^2 + x (-5 + y^2)}μ + {-1 + y + x^2 (-1 + y) y - x (-1 + 4 y + y^2)}μ^2 ] 
       +   4C_2(x - y) (1 - μ) [2 (c (1 + x) (1 + y) - 2 (x + y))
           -  {2 + 4 x^2 - 2 y + 4 y^2 + c (1 + x) (1 + x (-3 + y) - 3 y) (1 + y)+ 2 x (-1 + 5 y)}μ
          +  {-1 - 2 y + 3 y^2 + x (-2 + 4 y - 6 y^2) - x^2 (-3 + 6 y + y^2)
          + c (1 + x) (1 + y) (1 - y + x (-1 + 3 y))}μ^2 
          -(1-x) (-1 + x (-1 + y) - y) (1 - y) μ^3],



From Eq. (<ref>), z_1 can be written in terms of the other parameters as

    z_1=z_2C_2^3+2z_4C_2^2+z_32C_2-2z_43/C_2^3+2C_2^2,

so that the substitution into Eq. (<ref>) gives 

    z_42C_2^3-z_32C_2+2z_43/z_43(C_2+2)^2=1.

 It can be show that Eqs. (<ref>) and (<ref>) have three different roots for (C_2,z_1), which can be expressed in terms of only (μ,κ) as

    z_1   =   κ^2μ,
    
    C_2   =   -1,

and

    z_1   =   κ^2μ(-9+7μ±2√(9-8μ^2))/9(1-μ),
    
    C_2   =   z_42+5z_43±√(z_2^2+z_3^2+34z_2z_3+36z_42z_43)/2z_32=3-2μ±√(9-8μ^2)/2.

Because of our assumption z_1<z_2=-κ^2μ, we must take 

    z_1   =   κ^2μ(-9+7μ-2√(9-8μ^2))/9(1-μ), 
    
    C_2   =   3-2μ-√(9-8μ^2)/2μ.

















  §.§.§ Asymptotic charges




The mass and two angular momenta are given, respectively,  by
 
    M=3π/4z_31,   J_1=-π√(z_21z_31z_41(1-C_2^2)/2),   J_2=0.

The dimensionless angular momenta and horizon area are given by  

    j:=√(27π/32)J_1/(GM)^3/2=√(z_21z_41(1-C_2^2)/z_31^2),   j_2:=√(27π/32)J_2/(GM)^3/2=0,
            a_h:= √(27/16^2π)A_h/(GM)^3/2=2√(2)|z_21C_2^2+z_32|√(z_32z_41)/z_31z_42√((1-C_2^2)).






  §.§.§ Phase diagram

From FIG.1, we see that the dimensionless angular momentum j is a monotonically increasing function of μ and  j→1/2√(2) at μ→ 0, and j→1 at μ→ 1. 
Figure 2  illustrates the relation between j and the dimensionless horizon area a_h, where 
the black and blue curves correspond to the Myers-Perry (MP) black hole with a single angular momentum and the Emparan-Reall (ER) black ring, respectively, and the red curve corresponds to the black lens  with a single augular momentum.
The ER (thin) black ring has no upper bound for j, whereas the black lens has the upper bound j=1 (at μ=1).
The MP black hole has a zero lower bound for the angular momentum j, whereas  the black lens has the non-zero lower bound j=1/2√(2) (at μ=0)  as the MP black hole does.
It turns out from these graphs that the dimensionless horizon area of the black lens is always larger than that of MP black hole (within the range of 1/2√(2)<j<1) and a_h vanishes at j→ 1 (μ→ 1), which corresponds to the limit to the singular extreme MP black hole, and take the finite  value 2√(3) at j→ 1/2√(2) (μ→ 0), which is a singular solution without a horizon. 
Moreover, it can be seen that there indeed exists the parameter region such that a_h of the black lens can exceed to those of the other three solutions, the black hole and thin/fat  black rings.   





















  §.§.§ CTCs and curvature singularities

We wish to require absence of CTCs  in the domain of outer communication.
The necessary and sufficient conditions to ensure that CTCs do not exist in the domain of outer communication is such that the following two-dimensional matrix always becomes nonnegative in the region:

    g_2=(
        [ g_ϕ_1'ϕ_1' g_ϕ_1'ϕ_2'; g_ϕ_1'ϕ_2' g_ϕ_2'ϕ_2' ]),

namely, CTCs do not exist if and only if 

    det g_2≥ 0,   Tr g_2≥ 0.

We have numerically studied the positivity for various values of μ and the normalized κ (as κ=1) in the (ρ,z)-plane. 
In FIG.7, the point (ρ,z)=(0,1) and the interval on the z-axis {(ρ,z) | ρ=0,  -n/10≤ z ≤ n/10} represent the turning point ((ρ,z)=(0,z_4)) and the Killing horizon, respectively, for μ=n/10 (n=1,4,7,9).
In each figure, the white region represents the CTC region, which always appears around the turning point (ρ,z)=(0,z_4).  
We have confirmed that regardless of the values of μ, the white region which surrounds the point (ρ,z)=(0,z_4) exists outside the horizon.  
As a consequence, it can be seen that the existence of CTCs outside the horizon seems to be unavoidable. 


Next, let us see if curvature singularities exist inside or outside the horizon. 
For this purpose, we consider where one of scalar invariants, for instance,  Kretschmann invariant  R_μνρσR^μνρσ, diverges in the (ρ,z)-plane.
One can find from a direct computation  that it diverges at the points (x,y) satisfying H(y,x)=0, which is denoted by the red curve in each figure of FIG.7.   
It can be seen from these figures that H(y(ρ,z),x(ρ,z))=0 holds just on the spherical boundary of the CTC region around the point (ρ,z)=(0,z_4). 
Therefore, it can be concluded that the black lens with a single angular momentum unavoidably has curvature singularities outside the horizon and on the spherical boundary of the CTC region.









































































 §.§ Static solution

Finally, we consider the static limit of the four-soliton solution z_1→ z_2, C_3→0, in which case it follows from Eq. (<ref>) that C_1→ 0 holds.  
In terms of the C-metric, the metric takes the following simple form

    ds^2   =   -1+μ y/1+μ xdt^2+2κ^2(1+μ x)[C_4^2(1-x^2)(1+μ y)^2+(1-μ)^2(y^2-1)]/(x-y)^2[(1-μ)^2-C_4^2(1+μ x)(1+μ y)]dϕ_1^2
          +2κ^2(1+μ x)[C_4^2(y^2-1)(1+μ x)^2-(1-μ)^2(1-x^2)]/(x-y)^2[(1-μ)^2-C_4^2(1+μ x)(1+μ y)]dϕ_2^2
          -4κ^2C_4(1-μ )(1+μ x)[x+y+μ(1+xy)]/(x-y)[(1-μ)^2-C_4^2(1+μ x)(1+μ y)]dϕ_1dϕ_2
          +2κ^2(1+μ x)[(1-μ)^2-C_4^2(1+μ x)(1+μ y)]/(1-C_4^2)(1-μ)(x-y)^2(dx^2/G(x)-dy^2/G(y)),

where it should be noted that  C_2 automatically disappears.
In the limit,  the condition (<ref>) is automatically satisfied, and  two equations (<ref>) and (<ref>) are simplified, respectively, as

    2μ C_4/(1-μ)-C_4^2(1+μ)=2,
          [1-μ-C_4^2(1+μ)]^2/(1-C_4^2)^2(1-μ^2)=1.

From Eq. (<ref>), μ can be written as

    μ=-C_4^2/C_4^2+C_4-2,

and then Eq.(<ref>) becomes 

    C_4/2C_4^2+5C_4+2=1,

which gives C_4=-1. 
This result contradicts with Eq.(<ref>). 
Therefore, this implies that when Eq. (<ref>) holds, Eq.(<ref>) cannot be satisfied. 
We can interpret physically  that the static black lens needs conical singularities on the rod [z_3,z_4] to support the horizon against gravitational attraction.
It may be of interest to investigate such a static black lens even though it has conical singularities. 
From Eq. (<ref>), C_4 can be solved as

    C_4=C_4±:=-μ±√(4-3μ^2)/2(1+μ).

Hence, the static black lens solution has two branches according to the choice of C_4.
It can be shown numerically that for C_4=C_4+, CTCs do not exist, whereas for  C_4=C_4-,  there always exists CTC region which  surrounds the point (ρ,z)=(0,z_4). 
It can be shown that after performing  the global rotation, this  solution coincides with the static limit of the Chen-Teo's solution <cit.> in which the parameters a, c, n and the angular coordinates (ψ, ϕ) are replaced, respectively,  with C_4, μ,  -2, and (-ϕ_1',ϕ_2').

















§ SUMMARY


Using the ISM for the five-dimensional vacuum Einstein equations and starting from the same seed as for the black ring construction, we have obtained the four-soliton solution,  which includes an asymptotically flat,  stationary and biaxisymmetric black hole with the horizon topology of L(2,1)=S^3/ℤ_2.  
The solution has nine parameters (C_i,z_i,k) which obey to five constraint equations imposed from the physical requirements. 
Therefore,  except for the gauge freedom  z→ z+α, the remaining degrees of freedom  reduce to three,  which physically correspond to the mass and two angular momenta. 
 The rod diagram of the obtained solution is the same as that of the Kunduri-Lucietti's supersymmetric black lens except for the horizon rod rather than that of Chen-Teo's solution. 
We wish to emphasize that the rod diagram is different from that of Chen-Teo's solution in whether the orientation of the finite rod between the horizon and the nut is (0,2,1) or (0,2,-1), where Chen and Teo considered the former, whereas we have chosen the latter. 
We have also showed that there is the parameter region such that on the axes of symmetry  there exist no curvature singularities, no conical singularities, and  no orbifold singularities. 

Since the metric of the four-soliton solution takes a considerably complicated form even in the C-metric representation,  we have analyzed, in particular, the static case and the case of a single angular momentum, which corresponds to a two-soliton solution and a three-soliton solution, respectively. 
We have discussed the phase diagram of the black lens with a single angular momentum  in comparison with those of the MP black hole with a single angular momentum and the ER black ring. 
In contrast to the ER (thin) black ring,  the angular momentum for the black lens has the upper bound, and 
in contrast to the MP black hole with a single angular momentum, it has a certain non-zero lower bound.
We have shown that there exists the parameter region such that the four different solutions, the black hole, thin/fat black ring and the black lens, exist for the same asymptotic charges (mass and an angular momentum).
Unfortunately, we have found that for the case of single angular momentum, the existence of CTCs and curvature singularities around the nut cannot be avoidable. 
The static black lens solution whose horizon topology is L(2,1)=S^3/ℤ_2 can be obtained by simply setting z_1= z_2 and C_3=0.  
We have shown that the solution always has conical singularities between the horizon and the nut, but there are two possibilities for this static solution. 
 One has an unavoidable CTC region which includes the nut, while the other does not admit  any CTCs  in the domain of outer communication. 
 


The four-soliton solution with C_3≠0, which we have obtained in Sec.<ref>, has much more complicated metric  than the three-soliton solution with C_3=0 even if we use the C-metric representation. 
This makes it difficult that we even numerically check the existence of CTCs and regularity for the far region from the rods and horizon through the whole parameter region.  
We have numerically found that for the 4-soliton solution, there also exist the parameter region where there are neither conical singularities nor curvature singularities on the axis and horizon, but it seems that CTCs inevitably appear around the nut for several special sets of the parameters.  
To see if there are CTCs for the whole parameter region is our remaining future study.



 




From our results in this paper, we cannot conclude immediately that a supersymmety plays an essential role in the existence of a regular black lens without naked CTCs simply because such a black lens has been found only within a class of sympersymmetric solutions so far. 
As for the supersymmetric solutions <cit.>, the existence of magnetic fluxes rather than an electric charge seems to be essential to support the horizon of the black lens. 
Therefore, it may be also possible that a non-BPS black lens with a magnetic flux exists in five-dimensional minimal supergravity. 
Moreover, it may be an interesting issue whether a vacuum solution of the more general horizon topology L(p,q) (p,q: comprime integers) under the same symmetry assumptions, since it was shown in Ref. <cit.> that such a general black lens cannot exist, at least, within a class of  asymptotically flat, stationary and bi-axisymmetric supersymmetric solutions in five-dimensional minimal supergravity. 
These issues deserve further study.







This work was supported by Grant-in-Aid for Scientific Research (C) (No. 17K05452) from Japan Society for the Promotion of Science (S.T.).





99


  

Strominger:1996sh 
  A. Strominger and C. Vafa,
  “Microscopic origin of the Bekenstein-Hawking entropy,”
  Phys. Lett. B 379, 99 (1996)
  [hep-th/9601029].
  
  
  
    Argyres:1998qn 
  P. C. Argyres, S. Dimopoulos and J. March-Russell,
  “Black holes and submillimeter dimensions,”
  Phys. Lett. B 441, 96 (1998)
  [hep-th/9808138].
  
  


  
 Hollands:2007aj 
  S. Hollands and S. Yazadjiev,
  “Uniqueness theorem for 5-dimensional black holes with two axial Killing fields,”
  Commun. Math. Phys.  283, 749 (2008)
  [arXiv:0707.2775 [gr-qc]].
  
Hollands:2010qy 
  S. Hollands, J. Holland and A. Ishibashi,
  “Further restrictions on the topology of stationary black holes in five dimensions,”
  Annales Henri Poincare 12, 279 (2011)
   [arXiv:1002.0490 [gr-qc]].
   
   Cai:2001su 
  M. l. Cai and G. J. Galloway,
  “On the Topology and area of higher dimensional black holes,”
  Class. Quant. Grav.  18, 2707 (2001)
  [hep-th/0102149].
 

Galloway:2005mf 
  G. J. Galloway and R. Schoen,
  “A Generalization of Hawking's black hole topology theorem to higher dimensions,”
  Commun. Math. Phys.  266, 571 (2006)
  [gr-qc/0509107].


  
Tangherlini:1963bw 
  F. R. Tangherlini,
  “Schwarzschild field in n dimensions and the dimensionality of space problem,”
  Nuovo Cim.  27, 636 (1963).
 
  
  
 
Myers:1986un 
  R. C. Myers and M. J. Perry,
  “Black Holes in Higher Dimensional Space-Times,”
  Annals Phys.  172, 304 (1986).
 

Emparan:2001wn 
  R. Emparan and H. S. Reall,
  “A Rotating black ring solution in five-dimensions,”
  Phys. Rev. Lett.  88, 101101 (2002)
  [hep-th/0110260].
 


Pomeransky:2006bd 
  A. A. Pomeransky and R. A. Sen'kov,
  “Black ring with two angular momenta,”
  hep-th/0612005.
  

Evslin:2008gx
  J. Evslin,
  “Geometric Engineering 5d Black Holes with Rod Diagrams,”
  JHEP 0809 (2008) 004
  [arXiv:0806.3389 [hep-th]].
   

Chen:2008fa
Y. Chen and E. Teo, “A Rotating black lens solution in five dimensions”,  Phys. Rev. D 78, 064062 (2008).


  
Belinsky:1979mh
  V. A. Belinsky and V. E. Sakharov,
  “Stationary Gravitational Solitons with Axial Symmetry,”
  Sov. Phys. JETP 50, 1 (1979).
  
  
  
Belinski:2001ph 
  V. A. Belinski and E. Verdaguer,
  Gravitational Solitons,
  (Cambridge University Press, Cambridge, 2001).


book exact solution
  H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, 
  Exact solutions of Einstein's Field Equations, 2nd ed. 
  (Cambridge University Press, Cambridge, 2003).




 
Pomeransky:2005sj
  A. A. Pomeransky,
  “Complete integrability of higher-dimensional Einstein equations with additional symmetry, and rotating black holes,”
  Phys. Rev. D 73 (2006) 044004
  [hep-th/0507250].
 
 Harmark:2004rm
  T. Harmark,
  “Stationary and axisymmetric solutions of higher-dimensional general relativity,”
  Phys. Rev. D 70 (2004) 124002
  [hep-th/0408141].
 
  
Tomizawa:2005wv
  S. Tomizawa, Y. Morisawa and Y. Yasui,
  “Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method,”
  Phys. Rev. D 73 (2006) 064009
  [hep-th/0512252].
 





Figueras:2005zp
  P. Figueras,
  “A Black ring with a rotating 2-sphere,”
  JHEP 0507 (2005) 039
  [hep-th/0505244].
  

 
Mishima:2005id
  T. Mishima and H. Iguchi,
  “New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness,”
  Phys. Rev. D 73 (2006) 044030
  [hep-th/0504018].
   
  

Iguchi:2006rd
  H. Iguchi and T. Mishima, “Solitonic generation of five-dimensional black ring solution,”
  Phys. Rev. D 73 (2006) 121501
  [hep-th/0604050].
 

Tomizawa:2006vp
  S. Tomizawa and M. Nozawa,
  “Vacuum solutions of five-dimensional Einstein equations generated by inverse scattering method. II. Production of black ring solution,”
  Phys. Rev. D 73 (2006) 124034
  [hep-th/0604067].
  
Gauntlett:2002nw 
  J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall,
  “All supersymmetric solutions of minimal supergravity in five- dimensions,”
  Class. Quant. Grav.  20, 4587 (2003) 
  [hep-th/0209114].



Kunduri:2014kja 
 H. K. Kunduri and J. Lucietti,“Supersymmetric Black Holes with Lens-Space Topology”,  Phys. Rev. Lett.  113, no. 21, 211101 (2014).







Tomizawa:2016kjh
  S. Tomizawa and M. Nozawa,
  “Supersymmetric black lenses in five dimensions,”
  Phys. Rev. D 94 (2016)  044037
  [arXiv:1606.06643 [hep-th]].
  



Breunholder:2017ubu
  V. Breunhölder and J. Lucietti,
  “Moduli space of supersymmetric solitons and black holes in five dimensions,”
  Commun. Math. Phys. (2018)
   [arXiv:1712.07092 [hep-th]].
 








    
  



