Degenerating Black Saturns

We investigate the possibility of constructing degenerate Black Saturns in the family of solutions of Elvang-Figueras. We demonstrate that such solutions suffer from naked singularities.


Introduction
In [1] Elvang and Figueras have presented a family of axisymmetric black hole solutions to vacuum 4+1-dimensional Einstein equations. Due to the specific topology of the event horizon: R × (S 1 × S 2 ) ∪ S 3 ) it has been named Black Saturn. It can be regarded as a spherical Myers-Perry black hole [2] surrounded by a black ring [3,4]. The configuration is kept in balance by the angular momenta.
The Black Saturn metrics are of great significance since they provide an example of well-behaved stationary black hole space-times with disconnected Killing horizon. This shows a sharp contrast between solutions to Einstein equations in 4+1 and 3+1-dimensions since, as proven recently [5][6][7][8], in the latter case analytic stationary two black hole space-times are nakedly singular.
A thorough analysis (see [1,10,11]) has shown that, under the assumption of parameters a i being pairwise distinct, the metrics in the Black Saturn family describe asymptotically flat, stably causal black hole space-times with smooth domains of outer communications. To guarantee the above listed desired properties of a well-behaved black hole space-time one needs to tune the parameters c 1 , c 2 , k and q in terms of a i 's.
The purpose of this work is to investigate the possibility of obtaining a well-behaved metric from the Black Saturn family in the case of coalescence of some of the parameters a i . Such coalescence corresponds to the "pole fusion effect" in the inverse scattering method, which may lead to extremal black-hole solutions (see [9,Chapter 8.3]). There are however various ways (paths in the parameter space) in which one can obtain a coalescence of two or more a i 's. The result, will a priori depend on the chosen limiting procedure as demonstrated in [12] 1 . For instance, in [1, Section A.1] it has been shown that to obtain the limiting case of a Myers-Perry black hole from the balanced Black Saturn configuration one needs to take first a 5 ր a 4 and then a 1 ր a 4 . In general, the assumption that the coalescence is to be considered after the fine tuning of parameters c 1 , c 2 , k, q already imposes restrictions on the limiting procedure, since, for example, a 1 ր a 5 causes c 1 to diverge (see [1, (3.7)] or [10, (2.3)]). Let us note that the parameters c 1 and c 2 may a priori assume infinite values. Indeed, the line element (A.1) has a well-defined limit for c 1 → ±∞ and/or c 2 → ±∞, which moreover commutes with every coalescence considered in this paper. However, these cases need separate analysis of possible balance conditions.
When the balance conditions are imposed on the Black Saturn solution, the areas of the horizons of the two disconnected components [1, (3.26, 3.27)] tend to zero in the limits a 3 ր a 2 and a 5 ր a 4 respectively. This suggests that the possible degenerate solutions are nakedly-singular. However, there is no a priori reason for the procedures of coalescence of parameters and imposition of the balance conditions to commute, so this observation does not exclude the possibility of obtaining well-behaved extremal solutions via some other limiting procedure. Let us note, that the same coalescence that leads to the vanishing of the horizon area of the black ring component implies the divergence of its temperature [1, (3.28)]. This suggests that the limiting procedures adopted in [1] are not the right ones, as one should expect T = 0 for an extremal black-hole solution, since the temperature is proportional to surface gravity.
The strategy we adopt in this paper is to consider the limits a i → a j at the level of metric functions of the full Black Saturn solution and then investigate whether the balance conditions can be fulfilled by a fine tuning of parameters c 1 , c 2 , k and q. To make the paper self-contained we present in the Appendix the Black Saturn metric of [1] in generalised Weyl coordinates. For the details of construction and properties we refer the reader to [1] and [10,11].

Analysis
Since we are interested only in the solutions with two disconnected components of the event horizon (compare with the rod structure [1, Figure 1]), we shall assume the strict inequality a 4 < a 3 in the ordering (1.1). We have thus 3 possible two-fold coalescences to be considered in the next subsections. Moreover, there are 3 three-fold and 1 four-fold limit that need to be investigated. When more then two a i parameters coalesce, one can consider various different paths in the parameter space that lead to the same coalescence. Fortunately, if the limiting procedure is performed at the level of the metric, the ordering of the limits does not play a role (compare [1, Section A.1]). This is because a i ր a j implies µ i ր µ j (A.2) and all of the metric functions (see Appendix) are smooth as functions of µ i 's.
In each of the subsections we consider a particular coalescence of the a i 's parameters while keeping the other distinct. The reason for that is that the behaviour of the metric functions on the axis (ρ = 0) should be studied separately in each region of the axis a i ≤ z ≤ a j (see [10,Section 5.4]). This means that each coalescence needs a separate procedure of investigation of the metric functions on the axis.
The detailed analysis of the regularity, asymptotic flatness and causality of the seven limiting cases of the Black Saturn solution is straightforward, but lengthy -one essentially follows the strategy adopted in [10]. However, since our analysis shows that in neither of the investigated limits can one tune the parameters to obtain a balanced configuration we shall only present the part of reasoning that leads to this conclusion.

a 1 ր a 5
Let us note first, that if one takes the limit a 1 ր a 5 , then the resulting metric does not depend on the parameter c 1 anymore. Indeed, µ 1 = µ 5 implies , thus the parameter c 1 completely drops out of the line element. According to [1, p. 7] this configuration would describe a static black ring around an S 3 black hole, which are kept apart by a conically singular membrane. Indeed, one can detect the conical singularity by investigating the periodicity of the variable ϕ (compare [10, Section 4]). To avoid conical singularity at zeros of the Killing vector ∂ ϕ one needs the ratio lim ρ→0 ρ 2 g ρρ g ϕϕ to be constant on the set {z < a 1 } ∪ {a 4 < z < a 3 }, which is an axis of rotation for ∂ ϕ . By investigating the leading behaviour in ρ of the metric functions g ϕϕ and g ρρ in the relevant region of the space-time we obtain lim ρ→0 Hence, to avoid conical singularities one would need to have which is equivalent to The first case is excluded, whereas the second one would require as a 4 < a 3 by assumption. The latter however imply that either a 3 < a 1 or a 3 > a 2 , which contradicts the ordering (1.1). This means that the conical singularity on the axis cannot be avoided.

a 5 ր a 4
Let us now investigate the coalescence a 5 ր a 4 . We shall start with the analysis of the Killing vector field ∂ t on the set {ρ = 0, z ≤ a 1 }. A Mathematica calculation shows that g tt is a rational function with the denominator given by which vanishes as z approaches a 1 from below. On the other hand, its numerator has the following limit as z ր a 1 , Hence, we have now two possibilities of tuning the parameters to avoid a naked singularity at ρ = 0, z = a 1 : Keeping them in mind, we shall investigate the behaviour of the Killing vector field ∂ t on the set {ρ = 0, a 4 ≤ z ≤ a 3 }. The function g tt on this domain is a rational function with the denominator vanishing at z = a 4 . On the other hand, the numerator of g tt at ρ = 0, z = a 4 reads Thus, there is only one possibility to avoid a naked singularity at z = a 4 : set c 1 = c 2 . Combining the results obtained so far we end up with the following possible fine tunings: The choice c 1 = c 2 = 0 would bring us back to the seed solution [1], which is nakedly singular, so we are forced to set c 1 = c 2 = ± 2(a 3 − a 1 ).
Let us now analyse the behaviour of the Killing vector field ∂ ψ on the set {ρ = 0, a 1 ≤ z ≤ a 4 }. A Mathematica calculation shows that g ψψ is a rational function with the denominator given by The singularity at z = a 1 is cancelled by the tuning (2.3) since the numerator of g ψψ at z = a 1 reads On the other hand, the denominator of g ψψ is singular at z = a 4 and the numerator has the following limit for z ր a 4 , which does not vanish. This means that the naked singularity at ρ = 0, z = a 4 persists regardless of the fine tuning of parameters.
We have so far dealt with the situation of the parameters c 1 and c 2 assuming finite values. Let us now turn to the case c 1 → ±∞. In this instance g tt , being the norm of the Killing vector ∂ t , is given in the region {ρ = 0, z ≤ a 1 } by the following formula, This expression diverges as z ր a 1 and the singularity cannot be cancelled by any fine-tuning of the free parameters. For c 2 → ±∞ we obtain that g tt on the set {ρ = 0, a 1 ≤ z ≤ a 4 } is a rational function with the denominator, 2(a 4 − a 1 ) 2 (a 2 − z)(a 4 − z)(z − a 1 ), vanishing at z = a 4 . On the other hand, its numerator has the following limit for z ր a 4 , 2(a 4 − a 1 ) 2 (a 2 − a 4 ) 2 (a 3 − a 4 ).
We conclude, that in this configuration there is a naked singularity at ρ = 0, z = a 4 that cannot be avoided.

a 3 ր a 2
Let us now consider the coalescence a 3 ր a 2 .
To rule out smooth non-trivial solutions it is sufficient to investigate the behaviour of the Killing vector field ∂ t in the region {ρ = 0, a 4 ≤ z ≤ a 2 }. With the help of Mathematica we obtain that g tt is a rational function with the denominator given by 2(a 2 − a 1 ) 2 (z − a 1 )(a 2 − z)(a 5 − z), which vanishes as z approaches a 2 from below. On the other hand, its numerator has the following limit as z ր a 2 , This means, that one should impose the condition c 2 = 0 to avoid a naked singularity at z = a 2 . But setting a 3 = a 2 and c 2 = 0 completely removes the S 3 black hole component [1, Section A.2] and we are left with a single ψ-spinning black ring.
In the case c 1 → ±∞ the function g tt in the region {ρ = 0, z ≤ a 1 } reads, .
Thus, a naked singularity pops out at z = a 1 . For c 2 → ±∞, g tt in the region {ρ = 0, a 5 ≤ z ≤ a 4 } turns out to also be given by, , now leading to a singularity at z = a 4 . Similarly to the case described in Section 2.2, for c 1 , c 2 → ±∞ we have g tt = − µ 2 2 µ 1 µ 4 , which becomes singular on the axis {ρ = 0} in the whole region a 1 ≤ z ≤ a 2 .

a 1 ր a 5 ր a 4
According to [1,Section A.1] in this limit the Black Saturn metric reduces to a Myers-Perry black hole with a single angular momentum, hence no further analysis is needed. Let us stress however, that to obtain this result independently of the order of the limits one needs to compute the limits at the level of the metric functions -before the imposition of the balance conditions. 1 ր a 5 , a 3 ր a 2 Let us first investigate the behaviour of the Killing vector field ∂ t on the set {ρ = 0, a 4 ≤ z ≤ a 2 }. Again with the help of Mathematica we obtain the following formula for g tt function in this region

a
We have a naked singularity at z = a 2 unless we set c 2 = 0. As argued in Section 2.3 this completely removes the S 3 black hole component. What is more, the conical singularity detected in Section 2.1 persists. Indeed, we have lim ρ→0 Hence, to guarantee the correct periodicity of ϕ we would have to set a 4 = a 1 , which is excluded by the assumptions of this section.
Since the parameter c 1 has dropped out of the line element in the coalescence considered in this Subsection, we need only to comment on the instance c 2 → ±∞. In this case, the function g tt in the region {z ≤ a 1 } behaves near the axis {ρ = 0} like This excludes the possibility of c 2 → ±∞ leading to a well-behaved spacetime.
2.6 a 5 ր a 4 , a 3 ր a 2 It is sufficient to analyse the behaviour of the Killing vector field ∂ t on the axis. In the region {ρ = 0, z ≤ a 1 } g tt is a rational function with the denominator given by As z ր a 1 its numerator reads Thus, to avoid a naked singularity at ρ = 0, z = a 1 one has to set Let us now switch to the region {ρ = 0, a 1 ≤ z ≤ a 4 }. A Mathematica calculation shows that g tt is a rational function with the denominator equal to The continuity of g tt at z = a 1 is easily verified for both choices of parameters (2.5). On the other hand, as z approaches a 4 , g tt becomes singular since its numerator at z = a 4 reads To bypass the naked singularity at ρ = 0, z = a 4 we need to set c 1 = c 2 in addition to (2.5).
Finally, in the region {ρ = 0, a 4 ≤ z ≤ a 2 } the denominator of g tt is given by Again, the continuity of g tt at z = a 4 is guaranteed by the tuning of parameters imposed so far. However, the numerator of g tt at z = a 2 reads so the only way to avoid a singularity at z = a 2 is to set c 2 = 0. Combining this with the previous results we conclude that to assure the smoothness of the Killing vector field ∂ t on the axis {ρ = 0} one needs to set c 1 = c 2 = 0. As already argued, this would bring us back to the seed solution [1], which is singular itself. It remains to check the possibility of cancelling the singularities by letting one or both of the parameters c 1 , c 2 go to ±∞. As c 1 → ±∞ we obtain that g tt in the region {ρ = 0, z ≤ a 1 } is given by the expression , For c 2 → ±∞ on the other hand, we obtain the following behaviour of g tt near the axis {ρ = 0} in the region {a 4 ≤ z ≤ a 2 }, Moreover, if we let both c 1 and c 2 tend to infinity we again obtain g tt = − µ 2 2 µ 1 µ 4 . We conclude that the Black Saturn solution with a 5 ր a 4 , a 3 ր a 2 and one or both of the c i parameters infinite is nakedly singular. 1 ր a 5 ր a 4 , a 3 ր a 2 As in the previous cases (see Section 2.1) the limit a 1 ր a 5 implies that the parameter c 1 is no longer present in the line element. Furthermore, an investigation of the behaviour of the Killing vector ∂ t on the axis forces us to impose c 2 = 0. Indeed, in the region {ρ = 0, a 1 ≤ z ≤ a 2 } the metric function g tt reads

a
so only c 2 = 0 allows to avoid a singularity at z = a 2 . But if c 1 drops out of the metric functions and c 2 vanishes we are again back at the seed solution [1], which is of no physical interest. Moreover, in the case c 2 → ±∞ we obtain g tt = µ 2 2 ρ 2 , that clearly leads to singularities on the axis.

Conclusions
We have investigated various different coalescences of parameters defining the Black Saturn solution. We have shown that either the resulting metric is nakedly singular or it reduces to a black hole with one connected component of the event horizon: a Myers-Perry black hole or Emperano-Reall black ring.
Led by the example given by Geroch in [12] one might think that there can still be a way of obtaining a meaningful coalescence limit in the Black Saturn family by employing a smart change of coordinate chart. However, as demonstrated in [12], the Killing vectors are inherited by any limit of a space-time with some parameters. Strictly speaking, this property has been demonstrated for a 3+1-dimensional case. Nevertheless, as the technique developed in [12,Appendix B] is general, the proof can be adapted in a straightforward way to a 4+1 dimensional space-time with three Killing vectors. Now, since our analysis consisted in uncovering singularities in the norms of Killing vector fields, we conclude that any coordinate transformation would either lead to the same results or not yield a proper limit space-time at all.
We have thus exhausted the possibility of constructing a smooth extremal Black Saturn configuration in the family of solutions of Elvang-Figueras. This outcome is in consent with the known properties of 4+1-dimensional black holes. Both spherical black holes [2] and black rings [13,14] require two non-vanishing angular momenta to admit smooth extremal configurations. Unfortunately, the Black Saturn solution of Elvang-Figueras has angular momentum in a single plane only and it is not clear if doubly-spinning components can at all be kept in balance [1]. Thus, the question of existence of smooth stationary axisymmetric black hole with disconnected degenerate Killing horizons in 4 + 1 dimensions remains open.