{'fraction_non_alphanumeric': 0.035089815589277894, 'fraction_numerical': 0.00892181463336918, 'mean_word_length': 4.222129783693844, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Multipole moments for black objects in five dimensions

In higher dimensions than four, conventional uniqueness theorem in asymptotically flat space-times does not hold, i.e., black objects can not be classified only by the mass, angular momentum and charge. In this paper, we define multipole moments for black objects and show that Myers-Perry black hole and black ring can be distinguished by quadrupole moments. This consideration gives us a new insight for the uniqueness theorem for black objects in higher dimensions.

III, as an exercise, we shall consider the multipole moments of the static black objects and discuss the classification of them. In Sec. IV, we will compute the multipole moments for stationary black objects. Then we show that black ring can be distinguished from the Myers-Perry solutions by the "reduced" quadrupole moments which are well defined in center-of-mass gauge. In Sec. V, we summarize our result and discuss the possibility of uniqueness theorem using the multipole moments. In appendix A, we write down the field equations and see why our definition of multipole moments is appropriate. In appendix B, we compute the multipole moments for the black ring solutions with two angular momenta. In appendix C, for comparison, we compute the multipole moments for black objects with multiple horizons solutions like black Saturn [7] and orthogonal black di-ring solutions [10].

II. DEFINITION OF MULTIPOLE MOMENTS
In this section, following Refs. [30,38], we shall define the multipole moments in five dimensional stationary spacetimes. At first, we describe the definition of the asymptotically flatness based on the conformal completion method briefly. Then we will give a definition of the multipole moments. The definition of the multipole moments associated with angular momentum is not given by a simple extension from four to five dimensions. We also address the gauge dependence which comes from the gauge freedom of the conformal transformation.

A. Asymptotic flatness
For stationary space-times, we introduce the notion of asymptotic flatness at spatial infinity based on conformal completion method [39,40]. The metric of stationary space-times can be written aŝ where ξ = ∂/∂t is the timelike Killing vector, λ =ĝ ab ξ a ξ b andĥ ab is the metric on t = const. hypersurfaces. Since the multipole moments will be defined on t = const. hypersurfaces, we can focus on only the metric on t = const. hypersurfaces. Let us consider the conformal transformation as If there is a function Ω which satisfies the conditions Ω=0 , D a Ω=0 , D a D b Ω= 2h ab , t = const. hypersurface is called asymptotically flat space and the point Ω = 0 is identified as the spatial infinity Λ.
Here, D a is the covariant derivative with respect to the metric h ab and= stands for the evaluation on Λ. At the spatial infinity, h ab becomes the flat metric and we use the coordinate as follows ds 2 = h ab dx a dx b= dρ 2 + ρ 2 (dθ 2 + sin 2 θdφ 2 + cos 2 θdψ 2 ).
The above conditions on Ω imply the asymptotic behavior of Ω as Ω ∼ 1/r 2 ∼ ρ 2 near Λ. In this paper, we consider only the solutions of the vacuum Einstein equationsR (5) ab = 0 for simplicity.

B. Multipole moments
At first, we define mass 2 s -pole moments P a1a2···as as where O[T ab··· ] denotes the totally symmetric trace free part of the tensor T ab··· . In four dimensions, the mass multipole moments in stationary space-times are defined from the scalar potential P which consists of the lapse function λ and the twist potential σ satisfying a conformal invariant equation [31]. As seen soon, the corresponding twist potential is represented by a vector σ a , not a scalar, in five dimensions. Hence, we use only the lapse functions λ for the definition in five dimensions, and this does not matter. In fact, we can check that the twist potential does not contribute to the multipole moments in four and five dimensions. The role of σ will be important only in the proof of the smoothness of the multipole moments in four dimensions [31,33,34]. Note that the relation between the Arnowitt-Deser-Misner (ADM) mass and the mass monopole is shown in Ref. [17] M ADM = 3π 4 P.
For the definition of angular multipole moments, we assume the presence of the two commuting axisymmetric Killing vectors m = ∂/∂φ and l = ∂/∂ψ. In this case, there are many known exact solutions. However, note that this assumption might be rather strong in some senses. This is because the existence of only single axisymmetric Killing vector is guaranteed from the stationarity [41,42]. Although it is interesting to consider the multipole moments of single rotational symmetric cases too [43], this is beyond the scope of our current paper. Now, we introduce the tensorσ ab asσ ab =ε abcde ξ c∇d ξ e .
σ a satisfies the four dimensional Maxwell-type equation (see appendix A). Therefore, under the conformal transformation of Eq. (2), we can takeσ ab as conformal invariant quantity, i.e.,σ ab = σ ab andσ a = σ a . From this twist potential σ a , we can construct two scalar potentials as Then we can define the angular multipole moments in the same way as the mass multipole moments discussed above. That is, the angular 2 s -pole moments J φ a1a2···as and J ψ a1a2···as are defined recursively as and Here we have a remark: if one considers the cases having single rotational symmetry, σ a cannot be written only by scalar potential in a natural way. This means that it is not easy to construct the angular multipole moments in single rotational symmetric cases. The resolution to this difficulty is left for future study.

C. Unphysical gauge dependence
Before computing the multipole moments for known black objects, we should comment on the gauge dependence of the multipole moments defined above. There are gauge freedoms in the conformal completion of Eq. (2) as where Under this gauge transformations, multipole moments are transformed as P a1...as → P a1...as − s 2 O P a1...as−1 D as ω (16) in the linear order of ω 1 . D a ω represents the 1/r-order part in ω (See Eq. (15)) and corresponds to the choice of the origin of the coordinate in the physical coordinatex a [39]. In the definition of Thorne's multipole moments [35], there is such gauge freedom, which is just a translation. Thus, the freedom of the order of D a ω can be fixed by gauge conditions just like a center-of-mass gauge. Higher order parts O(1/r 2 ) in Eq. (15), which can be written as D a D b ω, D a D b D c ω, · · · , do not contribute to the transformation of multipole moments in the linear order. On the other hand, in non-linear order of ω, the changes of multipole moments under transformations of Eq. (14) depend on not only D a ω, but also higher order terms. For example, the octupole moments are transformed as where g is a numerical constant. Hence the value of the octupole or higher-pole moments depend on the choice of O(1/r 2 ) parts in ω, while monopole, dipole and quadrupole moments depend only on D a ω. Higher multipole moments than quadrupole, that is, octupole and 2 4 -pole moments, have gauge ambiguities from the conformal transformation even in center-of-mass gauge. Since there is no tractable way to fix the gauge freedom of O(1/r 2 ) parts in ω, we will focus on the computation of only monopole, dipole and quadrupole moments for known black objects solutions in the center-of-mass gauge.

D. Modes
In the practical calculations of the multipole moments, we must specify the concrete coordinate and modes. In this paper, we use the the coordinate in Eq. (4). Then it is easy to see that the quadrupole moments have nine modes as cos 2θ, sin 2 θ sin 2φ, sin 2 θ cos 2φ, cos 2 θ sin 2ψ, cos 2 θ cos 2ψ (18) cos θ sin θ sin φ sin ψ, cos θ sin θ sin φ cos ψ, cos θ sin θ cos φ sin ψ, cos θ sin θ cos φ cos ψ.
In two rotational symmetric cases, non-vanishing quadrupole moment mode is only one mode of cos 2θ in centerof-mass gauge. In this paper, then, we define the coefficients of this mode in P ρρ as mass quadrupole moments Q 2 .

III. STATIC CASES
In this section, as a first step, we compute the multipole moments for known static solutions. Then we will discuss the classification of space-times with single horizon. Following the uniqueness theorem [13], regular static black object solutions of the vacuum Einstein equation are completely classified by its mass, that is, the mass monopole. Other solutions like static black ring have conical singularities and they are not regular solutions. However, by computing the multipole moments of these non-regular solutions, we can study the dependence of the multipole moments on the topology of horizon. Since all angular multipole moments vanish in static cases, we will consider only mass multipole moments. This section will be helpful to study the multipole moments for stationary cases in which we are interested more.
The metric of static and two rotational symmetric space-times in five dimensions can be written in the Weyl coordinate [2] where U t + U φ + U ψ = log R. The solutions are represented by the rod structure which is composed of the zero points of g tt , g φφ and g ψψ and stand for the positions of event horizons and rotational axis. For computation of the multipole moments, it is better to use new coordinate given by 1 Transformations of J φ a 1 ...as and J ψ a 1 ...as are same as Eq. (16) 2 If we define the quadrupole moment as the coefficient in other components of P ab , the difference will be only sign.
and take the conformal factor as Ω = 1/r 2 = ρ 2 . The coordinate t, r, θ, φ and ψ here are same as those in Eq. (4). The Weyl coordinate has the gauge freedom z → z + constant and this gauge freedom corresponds to r → r(1 + O(1/r 2 )) in the coordinate of Eq. (4). As mentioned in the previous section, the monopole, dipole and quadrupole moments we compute in the following are independent on this O(1/r 2 ) order gauge transformations. This implies that those moments should be written by the difference a i − a j as seen soon later.

A. Schwarzschild black holes
At first, we will compute the multipole moments of the Schwarzschild black hole as a trivial example. The rod structure of the Schwarzschild black holes are shown in Fig. 1. The Schwarzschild black hole is described by two parameters a Sch and b Sch in the Weyl coordinate. As there is a gauge freedom z → z + constant, however, the independent parameter is only b Sch − a Sch . After all, the multipole moments of the Schwarzschild black hole are computed as Hence, the Schwarzschild black hole has only monopole as non-trivial multipole moments, which is proportional to the ADM mass. Here note that the 2 4 -pole moment L is given by L ∝ (a 3 Sch − b 3 Sch ). As we stressed before, however, it has the unphysical gauge dependence and then the physical meaning of this L is unclear.

B. Static black ring
Next, we compute the multipole moments of static black ring solution (the rod structure is shown in Fig. 2). Static black ring solutions have two independent parameters b BR − a BR and c BR − b BR . Then, after short calculation, we can see that the multipole moments are In our definition of quadrupole moments, Q is always non-positive. In addition, only if we take the Schwarzschild limit of b BR = c BR or flat limit of a BR = b BR , the quadrupole moment vanishes.

C. Classification issue
We can distinguish the Schwarzschild black hole from black ring solutions by quadrupole moments. Then, if the single horizon is assumed, black objects are completely classified by the mass monopole and quadrupole moments, which are well defined in center-of-mass gauge.
Although the cases with multiple horizons is beyond of our current consideration, we have some comments on that. In appendix C, we computed the multipole moments for black Saturn and orthogonal black di-ring cases as examples. Static black Saturn and orthogonal di-ring solutions have three and four independent parameters respectively. On the other hand, the mass monopole and quadrupole moments can determine only two independent parameters. Then the multipole moments up to the quadrupole moments are not enough parameters to specify the space-times uniquely. That is, higher multipole moments are needed for the classification of these solutions. As mentioned in previous section, however, the higher multipole moments have the unphysical gauge ambiguities of ω. Thus, we should fix the gauge about the conformal factor Ω completely or improve the definition of the higher multipole moments.

IV. STATIONARY CASES
In static cases, we have shown that black objects with single horizon can be classified by mass monopole and quadrupole moments completely. In this section, we consider the black objects with angular momentum and single horizon. As we will see later soon, in stationary cases, rotating black objects can be classified by mass monopole, quadrupole and angular dipole moments.
Using the fact of σ φ ∼ − tan 2 θg tψ and σ ψ ∼ cot 2 θg tφ near the spatial infinity, we define the coefficient of cos θ in J φ ρ and sin θ in J ψ ρ as angular dipole moments J φ and J ψ . Here note that cos θ is l = 1 mode of scalar harmonics in the φ-rotational plane with the metric dθ 2 + sin 2 θdφ 2 and sin θ is one in the ψ-rotational plane with the metric dθ 2 + cos 2 θdψ 2 . Note that the relations between the angular dipole moments and the ADM angular momentum are given by [17]

A. Myers-Perry black holes
Let us examine the Myers-Perry solutions. The metric of the Myers-Perry black holes is given by where and Introducing the new coordinate defined by ρ = 1/r and taking the conformal factor as Ω = ρ 2 , we can compute the multipole moments of the Myers-Perry black holes. The results are Contrasted with the Schwarzschild black hole case, the rotating black holes have non-zero mass quadrupole moments, which are contributions from the rotations. To measure the deviation of other black object solutions from the Myers-Perry black holes, it is better to define the reduced mass quadrupole moments as It is chosen so that the reduced mass quadrupole moments of the Myers-Perry black holes vanishes, that is, B. Black ring with single angular momentum Next, we compute the multipole moments of black ring with single angular momentum. For the case of black ring with two angular momenta, see appendix. B. The metric is given by [4] where and the parameter range is 0 < ν ≤ λ < 1. For the regular black ring solution which has no conical singularities, the parameters λ and ν must satisfy the relation as λ = 2ν/(1 + ν 2 ). In the following, we will consider this regular black ring solution.
For the computation of the multipole moments, it is better to introduce the coordinate (ρ, θ) defined by and take conformal factor as Ω = ρ 2 . Then, the multipole moments of black ring with single angular momentum are evaluated as and the reduced quadrupole moment becomes As in static cases, the reduced quadrupole moments of black ring solutions have always non-positive value. In the appearance of naked singularity with λ = 0, 1 or in the Myers-Perry limit or flat metric limit, the reduced quadrupole moment becomes to be zero. If given the quadrupole q, ν is completely determined, that is, thin or fat is specified.

C. Classification issue
As shown above, the Myers-Perry black hole and black ring solutions with single angular momentum are classified by (reduced) mass quadrupole moments completely. When one specifies the ADM mass and angular momentum of black ring solutions, there are two different solutions, thin ring and fat ring solutions in a certain of parameter region. To see this, it is useful to introduce the quantity j 2 as Regular black ring solution has the two independent parameters R and ν satisfying 0 < R and 0 < ν < 1 as in Eq. (35). If M ADM and J ADM of black ring are given, we can compute the value of j 2 and determine the parameter ν. However, in the range 27/32 < j 2 < 1, it is known that there are two different solutions, that is, thin ring (1 > ν > 1/2) and fat ring solutions (ν < 1/2) shown in Fig. 3. Then, even if we assume the horizon topology of S 1 × S 2 , we cannot specify the solution only by M ADM and J ADM . This is well-known fact. Then, one wonders if one can distinguish these two solutions (thin or fat) by the reduced quadrupole moment. The answer is yes. To see this, we define q as The relation between the normalized area of the event horizon a H [4] and q is shown in Fig. 4. From Fig. 4, one can see that we can determine the parameter R and ν if the reduced quadrupole moments Q red and mass monopole P of black ring solutions are given. Thus, black ring solutions are completely specified by the mass monopole and the reduced quadrupole moments.
Here we comment on the multipole moments for black objects with multiple horizons, e.g., black Saturn [7] and orthogonal black di-ring solutions [10,11], although this is beyond the scope of our current paper. As shown in appendix C, these solutions all have non-vanishing reduced quadrupole moment. Regular black Saturn solution which has no conical singularity has four independent parameters. Hence, by tuning these parameters, black Saturn has the same (reduced) quadrupole and angular dipole moments of black ring solution. This means that these multiple horizon solutions cannot be classified only by monopole and quadrupole moments. Hence, we need the information about the higher multipole moments for the complete classification as we pointed out in static cases. For the details, see appendix C.

V. SUMMARY AND DISCUSSION
In this paper, we have defined mass and angular multipole moments in five dimensional stationary space-times. It is known that black holes and black ring with or without rotations cannot be distinguish by mass monopole (ADM mass) and angular dipole moment (ADM angular momentum). But, we could show that these black objects with the single horizon can be classified by introducing the (reduced) quadrupole moment. These moments are well defined in the center-of-mass gauge. In static cases, we could see that the mass quadrupole moments capture the existence of finite spacelike rods and the quadrupole moments detects the deviation of the topology of the event horizon from sphere. As seen in the previous section, this interpretation is valid for the reduced quadrupole moment in stationary cases.
Let us discuss the remaining works. When one wants to classify black objects with multiple horizons, we need the gauge independent definition of higher multipole moments. However, our current definition of multipole moments higher than quadrupole moments are not gauge invariant even in center-of-mass gauge. Therefore, we have to define the multipole moment carefully. Since the computation itself of the multipole moment defined here is hard task, we would guess that the improvement is also hard one. This might be done by additional some extra terms in our current definition of the multipole moments. It is also interesting to extend the definition of multipole moments to non-vacuum cases like Einstein-Maxwell system or so.
It is known that the metric is determined completely if we specify the all mass and angular multipole moments in four dimensional space-times [32][33][34]. The method of the proof of this theorem does not hold in five dimensions because the fact that the Weyl tensor for h ab trivially vanishes plays a key role of the proof in four dimensions. Therefore, it is rather non-trivial if the metric can be determined only by mass and angular multipole moments. To investigate this, it may be useful to use the rod structure. If we can show that the parameters of the rod structure can be constructed only by mass and angular multipole moments, the five dimensional black objects with single or multiple horizons in stationary and two rotational symmetric space-times will be classified only by mass and angular multipole moments. This is also our future work.

Appendix A: field equations
In this appendix, we describe the key ingredient behind the definition of the mass multipole moments. We first write down the vacuum Einstein equations R (5) ab = 0 for the metric of Eq. (1) aŝ whereD andR ab are the covariant derivative and Ricci tensor forĥ ab respectively, andσ ab =ǫ abcde ξ c∇d ξ e . Using the functionP = 1 − √ λ, we rewrite Eq. (A1) as whereR is the Ricci scalar ofĥ ab . We can regard Eq. (A2) as the Maxwell equations on the t = const. hypersurfaces andσ ab as the fields strength. From the conformal invariance of the Maxwell equation in four dimensions, we can suppose that the Maxwell fieldσ ab is conformal invariantσ ab = σ ab under the conformal completion of Eq. (2). Then, the conformal transformation transforms Eq. (A4) into where P = Ω −1P . We can regard Eq. (A5) as a Poisson-like equation with a certain of regular source. At spatial infinity Λ, it becomes Therefore, it is natural to define multipole moment using P and σ ab .

Appendix B: Black ring solutions with two angular momenta
In the main text, we focused on the black objects with single angular momentum mainly. This is because we wanted the argument to be compact as possible as we can. In this appendix, we compute the multipole moments for black ring with two angular momenta. The metric of black ring with two angular momenta is given by [5] where and The parameter ranges are 0 < ν < 1, 2 √ ν < λ < 1 + ν. Regular black ring solution with two angular momenta have three independent parameters ν, λ and k.
For computing the multipole moments, we introduce the new coordinate (ρ, θ) defined by and use the conformal factor of Ω = ρ 2 . Then, the multipole moments are computed as , P a = 0, (B8) Then the reduced quadrupole moment becomes As in one rotational case, black ring has a negative value for the mass quadrupole moment. Thus, in two rotational case, Myers-Perry black hole and black ring with two angular momenta can be classified by the mass quadrupole moment. for static black Saturn, and for static orthogonal black di-ring.
We can see that in the quadrupole moments of the static orthogonal black di-ring, Q 1 is quadrupole moments of black ring in the φ rotational plane, and Q 2 is the quadrupole moment of black ring in the ψ rotational plane. Hence, monopole and quadrupole moments are "linear" moments. Only by monopole and quadrupole moments, we cannot distinguish static black ring, black Saturn and black di-ring. Static black Saturn solutions have three independent parameters, for example, b BS − a BS , c BS − b BS and d BS − c BS . By tuning these parameters, static black Saturn can have the same mass monopole and quadrupole moments as static black ring's. As in static orthogonal black di-ring solutions, we can do same thing because its solution has four independent parameters. That is, there are several different solutions with same P and Q. This result suggests that higher multipole moments are needed to classify all these solutions.

Stationary cases
Here, we compute the multipole moments for the stationary solutions with multiple horizons, black Saturn [7] and orthogonal black di-ring [10]. Since the explicit form of the metric is complicated, we show only the rod structure of the solutions (See Figs. 7 and 8). The metric of stationary and two rotational symmetric solutions in five dimensions can be written by Weyl form as where x A = (t, φ, ψ). The rod structure is described by the zero point of det G AB and their direction ξ A determined from G AB ξ A = 0 at R = 0. For computing the multipole moments, it is better to introduce the new coordinate (ρ, θ) defined through the relation and we choose the conformal factor as Ω = ρ 2 . The rod structure of black Saturn solution is shown in Fig. 7. The angular velocities are given by for each event horizons, that is, for the central black hole and outer black ring. Note that the following regularity condition is imposedc where ǫ = 1(−1) forc 2 > −κ −1 2 (c 2 < −κ −1 2 ). Thus, black Saturn solutions has four independent parameters L, κ 1 , κ 2 and κ 3 .
After some length calculations, the mass multipole moments for black Saturn are computed as And the angular multipole moments are also computed as Next we consider the orthogonal black di-ring solution. The rod structure is shown in Fig. 8. The angular velocities are given by , Ω , Ω .
The labeling of (1), (2) specifies which event horizons we consider. The regularity conditions is also imposed as Note that there is the gauge freedom of z → z +const.. Thus, the orthogonal black di-ring has the four free parameters. The mass multipole moments for orthogonal black di-ring are computed as We can interpret the Q φ and Q ψ as the quadrupole moment of the black ring in the φ-and ψ-rotational planes, respectively.
The angular momentum multipole moment are As in static cases, regular black objects with multiple horizons have non-trivial quadrupole moments. These solutions with multiple horizons have independent parameters more than three. Hence, to classify these solutions, it is necessary to evaluate higher multipole moments such as mass 2 4 -pole moments.