Iosif Bena
email: iosif.bena@cea.fr

Simon F Ross
email: s.f.ross@durham.ac.uk

Nicholas P Warner
email: warner@usc.edu

Coiffured black rings

Keywords: black hole hair, supersymmetric solutions, black rings PACS numbers: 04.70.Dy, 11.30.Pb, 04.65.+e

We describe a new type of hair on supersymmetric black string and black ring solutions, which produces the largest known violation of black hole uniqueness, parameterized by an arbitrary function and hence an infinite number of continuous parameters. The new solutions can have non-trivial density profiles for the electric fields along the horizon, and yet have a geometry that is regular, although generically not infinitely differentiable, at the horizon. Both neutral and charged probes can cross the horizon without experiencing divergent forces. We also find restricted examples, parameterized by a few arbitrary continuous parameters, where the charge densities fluctuate but the metric does not and hence is completely differentiable. Our new class of solutions owes its existence to a mechanism reminiscent of the Q-ball: in the simplest examples the metric has more symmetry than the matter that supports it.

Introduction

Five-and six-dimensional supergravity theories have a surprisingly rich solution space. The construction of smooth, horizonless microstate geometries in these theories has led to a new perspective on the microstates of black holes. The simplest class of microstate geometries is supertubes [1], in which two types of branes form a bound state that has a non-trivial dipole profile of arbitrary shape. In the D1-D5 frame this dipole charge corresponds to a Kaluza-Klein monopole, and the resulting smooth geometry is parameterized by several continuous functions that depend on the embedding of the world-volume of this monopole in the spacetime [START_REF] Lunin | Metric of the multiply wound rotating string[END_REF][START_REF] Lunin | Gravity solutions for the D1-D5 system with angular momentum[END_REF]. One can imagine superposing several different types of supertubes to obtain a black ring [START_REF] Bena | One ring to rule them all ... and in the darkness bind them?[END_REF][START_REF] Elvang | Supersymmetric black rings and threecharge supertubes[END_REF][START_REF] Gauntlett | General concentric black rings[END_REF], but if one tries to do this with supertubes that have a non-trivial charge profile and obtain a BPS black ring with varying charge densities [START_REF] Bena | Black rings with varying charge density J. High Energy Phys[END_REF] one finds that the horizon becomes singular [START_REF] Horowitz | How hairy can a black ring be?[END_REF]. Thus, it appears that the horizon of a black ring cannot support the kind of non-trivial charge hair that supertubes can possess.

As a consequence of this, for almost ten years the prevailing lore has been that, while the space of black objects in five-dimensional supergravity is much richer than in four dimensions, the violations of black-hole uniqueness in five dimensions come only from the discrete choices of the black-ring dipole charges and that, not only are there no infinite-dimensional moduli spaces worth of hair (as one might have expected from supertubes), but there is not even a finite-dimensional moduli space of solutions parameterized by a finite number of continuous parameters. In this paper we show that this lore is incorrect by explicitly constructing some very hairy black-ring solution that can be coiffured so as to preserve a lot of the hair and yet have well-behaved horizons. In particular, we find families of solutions that depend upon several continuous parameters for which the metric is C ∞ across the horizon. More generally, we argue that the hair can be arranged so that the Riemann tensor remains at least continuous across the horizon while the hair still depends upon an arbitrary continuous function.

The key idea behind this construction comes from our recent discovery of a new BPS object in M-theory-the magnetube [START_REF] Bena | On the oscillation of species[END_REF]. This object has M5 and momentum charges (which are magnetic in five dimensions), as well as several electric M2 charge densities that can oscillate between positive and negative values along the M5-P common direction. The M2 charge densities can have non-trivial oscillating profiles along this direction but these density profiles can be arranged so that the total energy-momentum tensor they source does not oscillate and yet the overall density configuration has an arbitrary function's worth of freedom.

In this paper, we consider putting similar oscillating electric charges on supersymmetric black strings and black rings. We will show that, unlike the simple single-species fluctuations considered in [START_REF] Bena | Black rings with varying charge density J. High Energy Phys[END_REF][START_REF] Horowitz | How hairy can a black ring be?[END_REF], the profiles of the oscillating species can be chosen such that the first few terms in the near-horizon expansion of the metric are independent of the angle around the ring. Furthermore, in the simplest example, the metric can be made completely independent of this angle, even though the Maxwell fields fluctuate. Thus these solutions are rather reminiscent of Q-balls [START_REF] Coleman | [END_REF][START_REF] Friedberg | Scalar soliton stars and black holes[END_REF] and hence we dub them 'O-rings. ' The smoothness of the metric at the O-ring horizon is achieved by arranging that the energy-momentum tensor of the fields is either independent of, or has much weaker dependence upon, the direction in which the other fields fluctuate. The curvatures at the horizon then remain finite, and an infalling observer can survive the passage across the horizon. The electric fields will oscillate infinitely many times along the trajectory of an infalling observer, but since the fields remain finite, the force on a charged particle falling across the horizon is also finite, and given the oscillating nature of the fields the integrated force is also finite. Thus by 'coiffing' a suitably rich variety of hair, in the form of charge densities, one can arrange for regularity of the gravitational background.

Our very hairy black rings provide by far the largest known violation of black-hole uniqueness, being regular black objects with an arbitrary continuous function's worth of hair. While our solutions are supersymmetric, we do not anticipate obstructions in adapting and generalizing our results to non-supersymmetric black rings. It would be very interesting to investigate this in detail and determine the possible fluctuation directions that can exploit the mechanism underpinning O-rings.

∫

In section 2 we review the basic supergravity setting for our new class of solutions. Since horizon regularity is a 'near-ring' issue we start by examining the simplest possible solutions: black strings in five dimensions. In section 3 we review the non-oscillating strings (that have a 3 × S 1 base space) and describe, in some detail, the horizon regularity and the adapted coordinates allowing continuation across the horizon. In section 4 we find the simplest BPS oscillating strings: O-rings in 3 × S 1 and derive the constraints on the charge densities for smoothness across the horizon. Section 5 contains the solutions for O-rings in 4 : there are no additional issues of smoothness compared to the O-rings in 3 × S 1 . In section 6 we consider more general classes of O-rings in 4 and perform the computations in the more traditional separable black-ring coordinate system. The impatient reader who is familiar with the standard description of black rings and does not care about their construction as BPS solutions with a Gibbons-Hawking space, could skip directly to this section as it is relatively selfcontained. Section 7 contains some concluding comments.

The supergravity setting of the new BPS solutions

The supergravity action

As in [START_REF] Bena | On the oscillation of species[END_REF], we use N = 2 supergravity in five dimensions coupled to vector multiplets, except that here we consider more vector multiplets. We will thus follow the conventions of [START_REF] Gauntlett | General concentric black rings[END_REF][START_REF] Gutowski | General supersymmetric AdS(5) black holes J. High Energy Phys[END_REF]. The action of N = 2, five-dimensional supergravity coupled to N U (1) gauge fields is 1 ⎛ 1

I 1 I J K ⎞ S = 2κ5 -g d 5 x ⎜ ⎝ R -QIJ F μν F Jμν -QIJ ∂μX I ∂ μ X J - 24 CIJK Fμν Fρσ Aλ ϵ ¯μνρσλ ⎠ ⎟ , (2.1)
with I, J = 0 ,..., N . The extra photon lies in the gravity multiplet and so there are only N independent scalars. It is, however, convenient to parametrize them by N + 1 scalars X I , satisfying the constraint

1/6 CIJK X I X J X K = 1. (2.2)
Following standard practice, introduce

XI ≡ 1/6 CIJK X J X K . (2.
3)

The scalar kinetic term can then be written as

QIJ = 9/2 XI XJ -1/2 CIJK X K . (2.4)
The Chern-Simons structure constants are required to satisfy the constraint

C IJK C J′(LM C PQ) K′ δ JJ′ δ KK′ = 4/3 δ I (L C MPQ) . (2.5)
It is also convenient to define

C IJK = δ II ′ δ JJ′ δ KK′ C I ′J′K′ . (2.6)
Using the constraint (2.5), one can show that the inverse, Q IJ , of QIJ is given by:

Q IJ = 2 X I X J -6 C IJK XK , (2.7)
and one can show that

1/6 C IJK XI XJ XK = 1/27. (2.8)
As in [START_REF] Bena | On the oscillation of species[END_REF][START_REF] Giusto | Adding new hair to the 3-charge black ring Class[END_REF][START_REF] Vasilakis | Bubbling the newly grown black ring hair[END_REF], we will focus on theories that can be obtained from eleven-dimensional supergravity reduced on a T 6 . The Maxwell fields then descend from the tensor gauge field, C (3) , via harmonic 2-forms on T 6 and the structure constants CIJK are given by the intersection product of the homology cycles and the X I are moduli of the T 6 . The restriction to torus compactifications is made for simplicity and convenience and it will be evident that rather richer classes of O-ring solutions can be obtained from Calabi-Yau compactifications with more complicated cohomology and intersection forms.

The supersymmetry conditions

We start with the most general stationary five-dimensional metric:

ds 2 = -Z -2 (dt + k) 2 + Z ds 2 , ( 2.9)
where Z is simply a convenient warp factor. Supersymmetry implies that the metric ds 2 on the spatial base manifold, B, must be hyper-Kähler. One now defines N + 1 independent functions, ZI, by

ZI = 3 Z XI , (2.10)
and then (2.8) implies

Z = (1/6 C IJK Z Z Z ) 1/3 .
(2.11)

It is more convenient to think of the solution as parametrized by the N + 1 independent scalars, ZI; the warp factor is then determined by (2.11). Supersymmetry requires that the Maxwell potentials all have the form

A (I) = -1/2 Z -3 C IJK ZJ ZK (dt + k) + B (I) , (2.12)
where B (I) are purely magnetic components on the spatial base manifold, B. One defines the magnetic field strengths accordingly:

Θ (I ) = d B (I) . (2.13)
Having made all these definitions, the BPS equations take their canonical linear form [START_REF] Bena | One ring to rule them all ... and in the darkness bind them?[END_REF]:

Θ (I ) = ⋆4 Θ (I ) , (2.14)
 2 ZI = 1/2 CIJK ⋆4 Θ (J ) ∧ Θ (K ) (2.15)
dk + ⋆4dk = ZI Θ (I ) , (2.16) where ⋆4 is the Hodge dual in the four-dimensional base metric ds 2 , and  2 is the (fourdimensional) Laplacian in this metric.

(4)

In section 6 we will take B to be flat 4 written in the canonical black-ring coordinates. However, it is very useful for our initial analysis to take the metric on B to be a Gibbons-Hawking metric because this will enable us to move easily between the solutions on 3 × S 1 and 4 . It will also make subsequent generalizations of our analysis to black rings in Taub-NUT [START_REF] Elvang | Supersymmetric 4-D rotating black holes from 5-D black rings J. High Energy Phys[END_REF][START_REF] Gaiotto | 5D black rings and 4D black holes J. High Energy Phys[END_REF][START_REF] Bena | Black rings in Taub-NUT[END_REF] and to multi-centered solutions relatively straightforward.

In the next sections, the four-dimensional metric will therefore be of the form:

ds 2 = V -1 (dψ + ) 2 + V dy ⃗ • dy ⃗ = V -1 (dψ + ) 2 + V ( dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 )),
(2.17)

V 3 4 2 (4)
I + with  ⃗ × ⃗ =  ⃗ V . (2.18)
The magnetic potentials that solve the supersymmetry conditions are:

B I = K I (dψ + ) + ξ ⃗ I • dy ⃗ + α I (2.19)
with  2 K I = 0,  ⃗ × ξ ⃗ I = - ⃗ K I , and dα I = 0. The closed form, α I , is usually set to zero, but we will see later that we need to take it to be non-zero in order to have regular gauge potentials at the horizon. The functions ZI are

ZI = CIJK K J K K + LI (2.20)
with  2 LI = 0. We will assume that K I and V are independent of ψ, and depend only on the coordinates y ⃗ on the base. In a suitable gauge, the last supersymmetry equation is satisfied by taking k = μdψ + ω, with and

1 6 V 2 CIJK K I K J K K + K I L M 2 V
(2.21)

N+1 I I
where and

D × ω⃗ + V ∂ψω⃗ = V DM -M DV + ∑ ( K DLI -LI DK ) . (2.22) I =1 D ⃗ ≡  ⃗ -⃗ ∂ψ , (2.23)
 2 M = 0 .
(2.24)

Adding species on the T 4

From the eleven-dimensional perspective, we are going to add extra Maxwell fields coming from three-form potentials with two legs on the T 4 defined by (x 5 , x 6 , x 7 , x 8 ) but leave the fields on the other T 2 , defined by (x 9 , x 10 ) unchanged and supporting only one Maxwell field, which we have labelled as A (0) . Thus the only non-zero components of the intersection product CIJK are

C0JK = C ‸ = C ‸ , (2.25) JK KJ
and one has [START_REF] Bena | On the oscillation of species[END_REF]: (2.11) now implies that the space-time metric warp factor, Z, is given by

⎛ ⎜ 0 1 0 0 ⎞ ⎟ C ‸ = ⎜ 1 0 0 0 ⎟ , (2.26) IJ ⎜ ⎝ 0 0 -1 0 ⎟ ⎠ 0 0 0 -1 which satisfies C ^IJ C ^KL δ JK = δIL. Observe that
1 2 V μ = 1 1 r r r = I 2.3.1. IJ ⎛ 1 2 2 ⎞ Z 3 = 2 Z0 ( C ‸ ZI ZJ ) = Z0 ⎜ ⎝ Z1 Z2 - 2 ( Z 3 + Z 4 ) ⎠ ⎟ .
(2.27)

It is also convenient to define the quadratic combination

P ≡ C ‸ IJ Z Z = ⎛ ⎜ Z Z - 1 Z 2 + Z 2 ⎞ ⎟ .
(2.28)

I J ⎝ 1 2 2 ( 3 4 ) ⎠
Since we are considering black objects, we will work entirely in the five-dimensional theory; we do not need to uplift to the higher-dimensional description.

String with no oscillation

We will first discuss the simplest solutions: black strings with oscillating species, so that we may examine horizon smoothness without the algebra becoming unnecessarily complicated. We therefore take the base metric to be simply a flat metric on 3 × S 1 , which means we set V ≡ 1 and ≡ 0 in the metric (2.17). The string is then aligned along the ψ direction. A useful starting point is to review the horizon smoothness of the non-oscillating electrically and magnetically charged black string solution [START_REF] Bena | Splitting hairs of the three charge black hole[END_REF], which is independent of ψ and is given by the harmonic functions:

K I p , r L = 1 + q (3.1)
for I = 0, 1, 2, and K I = 0, LI = 0 for I = 3, 4. The functions in the metric are then and

Z = 1 + q p 2 , (3.2) r 2 μ = p 3 + 3pq + 3p , (3.3) r 3 2r 2 r
choosing M = 3p so that we can simply solve (2.22) by taking ω = 0. The scalars are all constants, X I 2r = 1 for I = 0, 1, 2, X I = 0 for I = 3, 4. The metric and gauge fields are clearly finite for r ≠ 0. The solution has a coordinate singularity at r = 0, and we want to see how this corresponds to an event horizon. As r → 0, the metric is asymptotic to 2 ds 2 ≈ -r 4 dt 2 -2r dtdψ + 3 (q 2 -4p 2 )dψ 2 + p dr 2 + p 2 (dθ 2 + sin 2 θdϕ 2 ) , (3.4)

p 4 p 4p 2 r 2
which is singular in the dr direction and degenerate in the dt direction. To see that this is simply a coordinate singularity one can use a simple generalization of the Eddington-Finkelstein coordinate system:

t = v + a + b log (r p), ψ = ψˆ + c log (r p). (3.5) + 3 q 2 -4p 2 ψψ ψψ ψψ g (0) p r t tψ tψ tψ tψ tψ ψψ rr 4p
Expand the metric into power series in r:

g ψ = r ( g (0) + rg (1) + r 2 g (2)
) , g ψψ = g (0) + rg (1) + r 2 g (2) ,

g = 1 ( g (0) + rg (1) + r 2 g (2)
) . To the first few orders in r, the components of interest in the new coordinate system are

g′ ψ ˆ = -2 ⎛ ⎜ a - b ⎞ ⎟ g + 2 c g r ⎝ r 2 r ⎠ tψ r ψψ = 1 ( 2cg (0) -2ag (0) ) + ( 2cg (1) -2ag (1) + 2bg (0) ) + … (3.7) and ⎛ ⎜ a r ψψ tψ ψψ tψ tψ b ⎞ 2 ⎛ a b ⎞ c c 2 g′rr = ⎝ r 2 - r ⎠ ⎟ g tt -2 ⎝ ⎜ r 2 - r ⎠ ⎟ r g tψ + r 2 g ψψ + g rr = 1 ( -2acg (0) + c 2 g (0) + g (0) ) + 1 ( -2acg (1) + 2bcg (0) + c 2 g (1) + g (1) ) r 2 tψ ψψ rr r tψ tψ ψψ rr + ( -a 2 p -4 -2acg (2) + 2bcg (1) + c 2 g (2) + g (2) ) + … . (3.8)
There is a potential singularity in g′rψˆ at order r -1 and cancelling this fixes a in terms of c:

g (0) a = ψψ c = - tψ 3 c (q 2 -4p 2 ). (3.9)
Then there are potential singularities in g′rr at both order r -2 and r -1 , which fix the other two parameters c, b,

c = 2p 2 , b = (q 3 -3p 2 q) . ( 3

.10)

With these choices, the metric near r = 0 reduces to ds 2 ≈ -2c dvdr + f dψ ˆ 2 + 2 hdrdψ ˆ + jdr 2 + p 2 (dθ 2 + sin 2 θdϕ 2 ), (3.11) where f = g (0) , h = 2cg (1) -2ag (1) + 2bg (0) , ψψ ψψ tψ tψ j =a 2 p -4 -2acg (2) + 2bcg (1) + c 2 g (2) + g (2) . (3.12)

tψ tψ ψψ rr
The determinant of the metric is g ≈fc 2 p 2 sin 2 θ. The metric is thus completely regular and non-degenerate near r = 0. We should also consider the regularity of the matter fields. For the vector fields,

A 3 = A 4 = 0 while A 0 = A 1 = A 2 = A = Z -1 (dt + μdψ ) - p dψ + p cos θdϕ + α . ( 3.13)
(Note that the vector field, A here, should not be confused with in (2.17).) All the components are finite at r = 0 in the original coordinates, but we need to consider the r r

∫ 4 I r 0 I
behaviour in the regular coordinates:

A = Z -1 ⎛ ⎜ dv - a dr - b dr ⎞ ⎟ + ⎛ ⎜ μZ -1 - p ⎞ ⎟ ⎛ ⎜ dψ ˆ + c dr ⎞ ⎟ + p cos θdϕ + α. (3.14) ⎝ r 2 r ⎠ ⎝ r ⎠ ⎝ r ⎠ Now Z ∼ r -2
, so the first part is finite, but a constant term in Aψ could produce a divergence in Ar in the new coordinates. Indeed μZ -1 -p ≈ q . But this can be cancelled by

r 2p q 5
taking advantage of the freedom to introduce a closed form, setting α = -2 p dψ . has:

Then one

⎛ a b ⎞ ⎛ p q ⎞ ⎛ c ⎞ A = Z -1 ⎜ dv -dr -dr ⎟ + ⎜ μZ -1 -- ⎟ ⎜ dψˆ + dr ⎟ + p cos θdϕ , (3.15) ⎝ r 2 r ⎠ ⎝ r 2p ⎠ ⎝ r ⎠
which is finite at the horizon, r = 0, in the new coordinates. Thus, this solution describes a regular supersymmetric black string with electric and magnetic charges.

Oscillating black string

We now add oscillations to the black string solution, promoting the LI, which control the electric charges, to functions of ψ. We will not change the magnetic charge functions, so

K 0 = K 1 = K 2 = p ,
but we now take

K 3 = K 4 = 0 , (4.1)
L = 1 + q , L = 1 + q FI (ψ , r) (4.2) r for I = 1, 2, 3, 4. The LI are required to be solutions of  2 LI = (q + ρ (ψ ))δ 3 (x ⃗ ).
Since we are working on a flat base, it is easy to write an explicit integral form for the solutions:

Δψ F (ψ , r) = r ρI (ψ ′)dψ ′ . (4.3) I 0 π (r 2 + (ψ -ψ ′) 2 )
It is useful to note that if ρI (ψ ) = Re [ρI (z)], where ρI (z) has no poles in the upper-half plane, then the result of doing this integral can be written as FI (r, ψ ) = Re [ρI (ψ + ir)] [START_REF] Horowitz | How hairy can a black ring be?[END_REF]. We will subsequently assume that the charge distributions are purely oscillating, so the integral of ρI over the circle vanishes; this part carries no net charge. This implies ∫ dψFI = 0 as well.

Our goal is to suppress as much of the ψ-dependence from the metric as possible and to that end we will choose the densities such that the fluctuations cancel in the source term of (2.16) and thus the angular momentum vector is ψ-independent. This is achieved by choosing the opposite oscillating densities:

ρ2 (ψ ) = -ρ1 (ψ ) , (4.4)
which implies that F2 = -F1, exactly as in the magnetube solutions in [START_REF] Bena | On the oscillation of species[END_REF].The functions ZI are 2

Z = 1 + q + p , Z = 1 + q ± F 1 + p 2 , Z = F 3,4 , (4.5) 0 r r 2 1,2 r r r 2 3,4
r 5 Note this is not a gauge transformation, as ψ is a periodic coordinate, so this addition is changing the holonomy around the circle. Thus the solution that is regular at the horizon has some non-trivial holonomy at infinity.

+

Class. Quantum Grav. 31 (2014) 165015

I Bena et al 9 ⎝ + ⎟ 2r = = = and ⎛ 1 ⎞ ⎛ q p 2 ⎞ Z 3 = Z0 ⎜ ⎝ Z1Z2 -( Z 2 + Z 2 ) ⎠ ⎟ = ⎜ 1 + + ⎟ 2 3 4 ⎝ r r 2 ⎠ ⎛ ⎞ 2 ⎜ ⎛ F 2 + 1 ( F 2 + F 2 ) ⎟ ⎞ ⎞ ⎟ × ⎜ ⎜ 1 + q ⎜ ⎝ r p 2 ⎝ 1 2 3 r 2 ⎠ - r 2 ⎠ ⎟ . (4.6) ⎟ ⎠
We can again take

μ = p 3 + 3pq + 3p , (4.7) r 3 2r 2 r
choosing M = 3p . The choice (4.4) implies that there are no ψ-dependent sources in the equation (2.22) for ω, so with this choice for M this equation is again satisfied by taking ω = 0. Note that ψ-dependence in the metric only enters through Z. In fact, it is possible to find particular solutions with non-trivial ψ-dependence in the fields but in which there is no ψ-dependence at all in the metric. For example, if one takes then

ρ1 (ψ ) = -ρ2 (ψ ) = Q cos (kψ ), ρ3 = ρ4 = Q sin (kψ ), (4.8
)

F = Q cos kψ e -kr , F = F = Q sin kψ e -kr ⇒ F 2 + 1 F 2 + F 2 = Q 2 e -2kr , (4.9) 1 3 4 1 2 ( 3 4 )
and Z and hence the metric will be completely independent of ψ. The non-trivial ψdependence would appear just in the gauge field and scalars. Thus ∂ is a Killing vector but is ∂ψ not a symmetry of the complete solution. It is in this sense that our solutions are analogous to Q-balls [START_REF] Coleman | [END_REF][START_REF] Friedberg | Scalar soliton stars and black holes[END_REF]. More generally, if the source functions are some arbitrary periodic harmonic functions, then the F I will decay exponentially at large r and will be non-singular near r = 0, where F I (r, ψ ) = Re [ρI (ψ + ir)] ≈ ρI (ψ ) + (r). We can therefore write the combination appearing in Z in some power series expansion about r = 0,

F 2 + 1 ( F 2 + F 2 ) = F (0) (ψ ) + F (1) (ψ ) r + F (2) (ψ ) r 2 + … (4.10) 1 2 3 4
where

F (0) (ψ ) = ρ 2 + 1 (ρ 2 + ρ 2 ).
⎜ ⎛ 1 2 ⎞ ⎟ 1 3 X 0 = ⎝ Z1Z2 - 2 (Z3 + Z 4 ) ⎠ 2 3 , 0 X 1 = Z0 Z2 , Z 2 X 2 Z0 Z1 , Z 2 X 3 Z0 Z3 , Z 2 X 4 Z 0 Z 4 . (4.11) Z 2
Since Z0 is unchanged, X 0 clearly remains finite everywhere and, assuming

F 2 + 1 (F 2 + F 2 ) 1 2 3 4
is exponentially decaying, we can choose q big enough so that Z is finite everywhere so all the

2 4 Z 3q 2 -12p 2 -4F (0) p 3q 2 -12p 2 -4F (0) r 2 g (0)
p ψψ 4p scalars are regular for r ≠ 0. As we approach r = 0, X I → 1 for I = 0, 1, 2 and X I → 0 for I = 3, 4.

The metric.

As before, the metric has an apparent singularity and degeneracy at r = 0. As r → 0, we have:

ds 2 ≈ - r 4 dt 2 - 2r dtdψ + 1 (3q 2 -12p 2 -4F (0) )dψ 2 p 4 p 4p 2 + p 2
dr 2 + p 2 (dθ 2 + sin 2 θdϕ 2 ).

(4.12)

One wants to make the same coordinate transformation (3.5) to attempt to make the metric regular at r = 0. It was shown in [START_REF] Horowitz | How hairy can a black ring be?[END_REF] that in the previous attempt to build hairy black rings [START_REF] Bena | One ring to rule them all ... and in the darkness bind them?[END_REF][START_REF] Bena | Black rings with varying charge density J. High Energy Phys[END_REF] this coordinate transformation was obstructed, as the leading term in g ψψ was ψ dependent, so we cannot set a = cg (0) /g (0) as in (3.9) for constant a, c. Similarly here there is ψψ tψ generically ψ dependence in g ψψ ; but we can choose charge densities such that the sum of the squares

F (0) (ψ ) = ρ 2 + 1 (ρ 2 + ρ 2 ) is a constant. This condition of constant amplitude for 1 2 3 4
the fluctuating densities is familiar from our study of magnetubes [START_REF] Bena | On the oscillation of species[END_REF].

Imposing this constraint, we can eliminate the potential singularity in g′rψ setting in (3.7) by

g (0) a = ψψ c = - tψ 1 (3q 2 -12p 2 -4F (0) )c . (4.13)
The leading singularity in g′rr in (3.8) can also be eliminated by setting

c = 2p 2 . ( 4.14)
But there is a further obstruction in eliminating the sub-leading singularity. We want to set

b = q 3 -3p 2 q -F (0) q -F (1) p 2 , ( 4.15)
but again F (1) is generically a function of ψ, so we could not satisfy this for constant b. Thus, we need to choose the ρI (ψ ) such that both F (0) and F (1) are constants to have a coordinate transformation that will eliminate the singularities in the metric at r → 0. We have three free functions ρ1, ρ3, ρ4, so there should be an arbitrary function's worth of freedom even after satisfying this constraint. We have also seen explicitly above that there are non-trivial solutions with constant F (0) and F (1) . When we restrict to charge densities such that F (0) and F (1) are constants, the metric near r = 0 in the new coordinates takes the same general form,

ds 2 ≈ - 2c dvdr + f dψ ˆ 2 + 2 hdrdψ ˆ + j (ψ )dr 2 + p 2 (dθ 2 + sin 2 θdϕ 2 ), (4.16)
where f , h and j (ψ ) are still given by the same general expressions (3.12), but now the grr component j is a function of ψ, as ψ = ψˆ + c log (r /p), so j g (2) involves F (2) (ψ ). In the near-horizon region, finitely many times as is some periodic function which oscillates in we approach the horizon. Thus, while the components of the metric are all finite at the horizon in this new coordinate system, they are not smooth functions of the coordinates there. Remarkably, an explicit calculation of the components of the Riemann tensor reveals that all components are finite as r → 0. If we took the approximate metric (4.16) to be the exact solution, the only non-zero components of the Riemann tensor are R ψˆ ψˆ = -1 ∂ 2 j, Rθϕθϕ = p 2 sin 2 θ, (4.17) r r 2 ψˆ which remain finite as r → 0. Keeping sub-leading terms in the expansion in r, radial derivatives of j will appear, but they are multiplied by positive powers of r, so that we don't get any divergences. Geodesics remain at finite values of the coordinates as r → 0 in (4.16), so the finiteness of the Riemann tensor components implies that the geodesics can be extended beyond r = 0, so that this is a regular horizon for the oscillating black string solutions. Thus, the solution will have a metric which is regular at r = 0 if we choose the sources such that F (0) and F (1) are constants. There should be a free function's worth of solutions which satisfy these constraints, so we find black string solutions with a function's worth of hair.

4.1.3. The vector fields. For the vector fields, we have

⎛ a b ⎞ ⎛ c ⎞ A I = Z -1 X I ⎜ dv -dr -dr ⎟ + (Z -1 X I μ -K I ) ⎜ dψˆ + dr ⎟ + p cos θdϕ + α I , (4.18) ⎝ r 2 r ⎠ ⎝ r ⎠
with dα I = 0. Again, these components are all finite in the original coordinate system, but we need to check carefully that they are finite in the regular coordinate system as r → 0. For I = 0,

Z -1 X 0 μ -K 0 ≈ q , ( 4.19)
2p so we take α 0 = -q dψ as before. For I = 1, 2,

Z -1 X I μ -K I ≈ q ∓ ρ 1 (ψ ) . ( 4

.20) 2p p

Taking α I = -q dψ will eliminate the first term as before. Since we take the charge distributions to be purely oscillating, the new term is actually pure gauge, so it can be removed by taking α I = -q dψ + dβ I (ψ ) β I (ψ ). Similarly for I = 3, 4, for some suitably chosen periodic functions,

Z -1 X I μ ≈ - ρ I (ψ ) , (4.21)
which can also be eliminated by a gauge transformation. All the vector potentials are then finite as r → 0.

There is a subtlety here similar to the one we saw above in the metric. The Av and Aψˆ components vanish at the horizon after we do this gauge transformation, so the surviving components at the horizon are Aϕ = p cos θ and Ar. The finite contribution to Ar from the first term in (4.18) is independent of ψ, but the contribution from the second term is

A ∼ c ⎛ ⎜ ∂rFI (ψ , r) r=0 - qρ I (ψ ) ⎞ ⎟. (4.22) r ⎝ p 2p 3 ⎠
This is bounded, but as ψ = ψˆ + c log (r /p), it will oscillate infinitely many times along infalling geodesics which approach the horizon at finite ψˆ . Since it is only the radial

r 2 + R 2 -2rR cos θ 1 r + R -r1
component of the gauge field that exhibits this behaviour, it does not lead to a divergent field strength; the non-zero components of the field strength at the horizon are

F ψ ˆr = ∂ ψ ˆ A r , Fθϕ = p sin θ. (4.23)
Thus, the electric field is bounded but oscillates infinitely many times along infalling geodesics approaching r = 0. Thus, the response of a charged test particle remains bounded, and the horizon at r = 0 remains regular 6 .

Oscillating black rings

We now extend the discussion to the most interesting example, an oscillating black ring. As we remarked earlier, since the smooth continuation of a solution across the horizon is a local issue we expect to encounter only the same issues and constraints as we did for the black string, albeit with a slightly higher level of complexity.

The O-ring solution

To bend our black string solution into a black ring, we want to take the four-dimensional base space (2.17) to be flat 4 . This is achieved by introducing a single GH centre at r1 = 0 and setting V = 1/r1. Note that it is rather straightforward to extend our construction to oscillating rings in Taub-NUT or more complicated multi-center solutions, although we will not do it here for the sake of simplicity. We will choose coordinates so that the black ring's event horizon is still at r = 0, and take the centre r1 = 0 to be at r = R, θ = 0. That means

r 2 = r 2 + R 2 -2rR cos θ.
(

We have

V = 1 r1 = 1 , A = r 1 + r cos θ -R dϕ. ( 5

.2) r1

We take the harmonic functions K I , LI to have the same structure as in the black string solution, given in (4.1), (4.2). The LI satisfy  2 LI = (q + ρ (ψ ))δ 3 (x ⃗ ), which now implies where G is the Page Green function for the GH base [START_REF] Page | Green's functions for gravitational multi-instantons[END_REF][START_REF] Bena | An infinite-dimensional family of black-hole microstate geometries[END_REF]. For the single-centred base, this Green function is simply

G = 1 sinh U , (5.4) 16π 2 r cosh U -cos ⎛ ⎜ ψ -ψ ′ ⎞ ⎟ ⎝ 2 ⎠
where 2U = ln r 1 + R + r . Note that the θ dependence in r1 implies that FI is a function of θ. As for the black string, we can do the integral in (5.3) for appropriate sources by contour integration, to get r ⎡ ⎤

FI (r, θ, ψ ) = Re ⎣ ⎡ ρI (ψ + iU) ⎤ ⎦ ≈ ρI (ψ ) + Re ⎢ ⎣ i∂zρI (z) z=ψ ⎥ ⎦ + (r 2 )
(5.5) 6 We can see that derivatives of the field strength will diverge at the horizon, so the field is not smooth there, but it is sufficiently differentiable to admit a physically meaningful extension through the horizon.

∫ 2R r r r + ⎟ ⎜ 1 + + ⎟ 0
The functions ZI are now given by:

Z = 1 + q p 2 , r 2 V Z 1,2 = 1 + q ± F 1 + p 2 , r 2 V Z 3,4 F 3,4 ,
(5.6) r and so the warp factor appearing in the metric is:

Z 3 = ⎛ ⎜ 1 2 2 ⎞ ⎟ Z0 ⎝ Z1Z2 - 2 ( Z 3 + Z 4 ) ⎠ ⎛ ⎛ ⎜ 2 ⎜ ⎛ F 2 + 1 ( F 2 + F 2 ) ⎟ ⎞ ⎞ ⎟ = q p 2 ⎞ ⎜ ⎛ q p 2 ⎞ ⎝ 1 2 3 4 ⎠ ⎟ r r 2 V ⎠ ⎜ ⎝ ⎝ r r 2 V ⎠ - . (5.7) r 2
⎟ ⎠ We also have

p 3 r 3 V 2 3pq 2r 2 V 3p 2rV + M. (5.8)
As before, this part of the solution does not involve the oscillation. We therefore choose M to have the same value as for the non-oscillating black ring,

M = 3p R -r (5.9) 5.1.1.
r The one-form is then also the same as for the non-oscillating ring,

ω = - 3pR ( r -R cos θ + r1 cos θ) + 3p A . (5.10) ϕ 2 R r1 2 ϕ
Again, dependence on the oscillation enters the metric only through Z.

As before, we can find a simple solution in which the metric is completely independent of ψ, by choosing ρ1 (ψ ) = -ρ2 (ψ ) = Q cos (kψ ), so that (5.5) gives ρ3 = ρ4 = Q sin (kψ ), (5.11)

F = Q cos kψ e -kU , F = F = Q sin kψ e -kU ⇒ F 2 + 1 F 2 + F 2 = Q 2 e -2kU .
(5.12)

1 3 4 1 2 ( 3 4 )
Thus ∂ is a Killing vector. Note that the warp factor is still a function of θ, unlike the black-∂ψ string solution. The form of U implies that the oscillation now has power-law decay at large r, as expected for multipole moments in an asymptotically-flat space. More generally, if the source functions were arbitrary, the F I would have power-law decay at large r, and one could use equation (5.5) to see that the expansion around r = 0 becomes

F 2 + 1 ( F 2 + F 2 ) = F (0) (ψ ) + F (1) (ψ )r + F (2) (θ, ψ )r 2 + … (5.13) 1 2 3 4
A key point is that the θ dependence enters only at quadratic order; near r = 0, U ≈ r /2R, so the linear term in the expansion of FI is still θ-independent, as indicated in (5.5).

Regularity

The regularity analysis near r = 0 now proceeds in much the same manner as it did for the black string.

+ 1 = + μ = + + 2 r 2 g (0) p ψψ ≈ ≈ 4p 5.2.1.
The metric and scalars. The scalars take the same form (4.11), and as we approach r = 0, X I → 1 for I = 0, 1, 2 and X I → 0 for I = 3, 4. For the metric, we note that as r → 0,

V 1 , R A r 2
2R 2 sin 2 θdϕ, ω ≈ -3p r sin 2 θdϕ, (5.14) so the cross terms involving ϕ become negligible near r = 0. The metric near r = 0 is then

ds 2 ≈ - r 4 dt 2 - 2r dtdψ + 1 (3q 2 -12p 2 R -4F (0) )dψ 2 p 4 R 2 p 4p 2 + p 2
dr 2 + p 2 (dθ 2 + sin 2 θdϕ 2 ).

(5.15)

If we choose charge densities such that the sum of the squares

F (0) (ψ ) = ρ 2 + 1 (ρ 2 + ρ 2 ) is 1 2 3 4
a constant, we can eliminate the potential singularity in g′rψ in (3.7) by setting (5.16) and the leading singularity in g′rr in (3.8) can also be eliminated choosing c appropriately. We also need to choose sources such that F (1) (ψ ) is also a constant, so that we can eliminate the sub-leading singularity in g′rr by choosing b appropriately.

g (0) a = ψ ψ c = - tψ c (3q 2 -12p 2 R -4F (0) ),
When we restrict to charge densities such that F (0) and F (1) are constants, the metric near r = 0 in the new coordinates takes the same general form, ds 2 ≈ -2c dvdr + f dψ ˆ 2 + 2 hdrdψ ˆ + j (θ, ψ )dr 2 + p 2 (dθ 2 + sin 2 θdϕ 2 ), (5.17) where again f , h and j (θ, ψ ) are given as in (3.12), but now j is a function of both θ and ψ, as g (2) contains F (2) (θ, ψ ). As before, in the near-horizon region, ψ = ψˆ + c log (r /p), so j is some periodic function that oscillates infinitely many times as we approach the horizon. Thus, while the components of the metric are all finite at the horizon in this new coordinate system, they are not smooth functions of the coordinates there.

Remarkably, it is still true that the components of the Riemann tensor in the approximate metric (4.16) are finite as r → 0 even after including the θ dependence. The leading metric now has the Riemann tensor

R ψˆ ψˆ = - 1 ∂ 2 j, R ψˆ θ = - 1 ∂θ∂ψˆj, r r 2 ψˆ r r 2 R = - 1 ∂ 2 j, R = - 1 ∂ j sin θ cos θ, rθrθ 2 θ rϕrϕ 2 θ Rθϕθϕ = p 2 sin 2 θ, (5.18)
which is finite as r → 0. Including sub-leading terms in the metric, radial derivatives of j again appear multiplied by positive powers of r, so we do not get any divergences in the Riemann tensor. Geodesics remain at finite values of the coordinates as r → 0 in (4.16), so the finiteness of the Riemann tensor components implies that the geodesics can be extended beyond r = 0, so that this is a regular horizon for the oscillating black ring solutions. Thus, the solution will have a metric that is regular at r = 0 if we choose the sources such that F (0) and F (1) are constants. Since these are in general functions only of ψ, this imposes two restrictions on our three free source functions, so there should be a free function's worth of solutions that satisfy these constraints. Thus we find black ring solutions with a function's worth of hair. It is essential that the regularity only requires constancy of F (0) and F (1) ;

V 2p 2p p ⎞
requiring constancy of F (2) , or even higher order terms, which are functions of both θ and ψ in general, would overconstrain the sources.

The vector fields.

The analysis for the vector fields is essentially identical to that for black strings. The fields are:

⎛ a b ⎞ ⎛ K I ⎞ ⎛ c ⎞ A I = Z -1 X I ⎜ dv - dr -dr ⎟ + ⎜ Z -1 X I μ - ⎟ ⎜ dψˆ + dr ⎟ ⎝ r 2 r ⎠ ⎝ V ⎠ ⎝ r ⎠ + p cos θdϕ + α I . ( 5.19)
The components in the new coordinates are made finite by a suitable choice of exact part. For I = 0, as r → 0

Z -1 X 0 μ - K 0 ≈ q ,
(5.20) 2p so we take α 0 = -q dψ as before. For I = 1, 2,

Z -1 X I μ - K I ≈ q ∓ ρ 1 (ψ ) , (5.21) V 2p p
and the divergence of the vector field can be cancelled by a gauge transformation and taking α I = -q dψ + dβ I for a suitable choice of β I (ψ ). Similarly for I = 3, 4,

Z -1 X I μ ≈ - ρ I (ψ ) , (5.22)
which can be eliminated by a gauge transformation involving the choice of β I (ψ ). All the vector components are then finite as r → 0. As for black strings, there is a finite Ar component that is a function of ψ = ψˆ + c log (r /p), 7 which gives an electric field that is bounded but oscillates infinitely many times along infalling geodesics. Thus, the response of a charged test particle remains bounded, and the horizon at r = 0 remains regular.

The solution in canonical ring coordinates

We can easily recast the solutions above in the, perhaps more familiar, black-ring bipolar coordinates. This will facilitate the comparison with the singular black ring solution that only has two oscillating charge densities [START_REF] Bena | Black rings with varying charge density J. High Energy Phys[END_REF], and to this end we also modestly generalize the result above by using distinct magnetic dipole moments.

The oscillating solutions

One can write the 4 spatial base metric of (2.9) in the usual spherical bipolar coordinates:

ds 2 = R 2 ⎛ ⎜ dy 2 + (y 2 -1)dψ 2 + dx 2 + (1 -x 2 ) dϕ 2 ⎟. (6.1) (x -y) 2 ⎝ y 2 -1 1 -x 2 ⎠ with -1 ⩽ x ⩽ 1, -∞ < y ⩽ -1.
The ring is located at corresponds to y = -1.

y =while spatial infinity ψ

In these coordinates the magnetic flux sources are very simple:

Θ ( j ) = 2 q j (dx ∧ dϕ -dy ∧ dψ ) , j = 0, 1, 2 (6.2)
and the electrostatic potentials that solve 2.15 have the form [START_REF] Bena | Black rings with varying charge density J. High Energy Phys[END_REF]:

Zi = 1 + Q i (x -y) -4q j q k (x 2 -y 2 ) + π (x -y)Λ (y, ψ ) , R R2 R i {i, j, k} = {0, 1, 2} (6.3)
Z = 2 π (x -y)Λ (y, ψ ) , i = 3, 4, (6.4)
i R i
where, for later convenience, we have introduced factors of harmonic, which means that they satisfy:

into Λ3 and Λ4. The Λi are (y 2 -1) ∂y (( y 2 -1) ∂y Λi ) + ∂ 2 Λi = 0 , (

The Fourier analysis is elementary and the smooth solutions that fall off at infinity have expansion:

∞ ⎛ y + 1 ⎞ n 2 Λi (y, ψ ) ≡ ∑ ⎜ ⎟ ( a i cos (n ψ ) + b i sin (n ψ ) ) . (6.6) n=1 ⎝ y -1 ⎠ n n
Indeed, if one takes y =coth ξ then this may be written as the real part of an analytic function of z where z ≡ e -ξ+iψ . Note that we focus on solutions where the additional potentials, Z3 and Z4, only have a fluctuating piece. Moreover, the source in the third BPS equation (2.16) only involves the sum 2 Λ ≡ ∑ q i Λi . The solution to the third BPS equation (2.16) with generic fluctuations was given in [START_REF] Bena | Black rings with varying charge density J. High Energy Phys[END_REF], however, our purpose here is to minimize, or at least suppress, the ψ-dependence of the metric. We will therefore consider charge densities that fluctuate while keeping Λ ≡ 0. We also want to try to remove the ψ-dependence from Z, which is given by (2.27) and therefore consider a solution where the charge corresponding to Λ0 does not fluctuate and thus Λ0 ≡ 0 as well 8 . Thus we have Λ0 ≡ 0 , q 1 Λ1 + q 2 Λ2 ≡ 0 . (

With these choices, the angular momentum vector, k, is precisely what it was for the nonoscillating black ring:

k = k1 dψ + k2 dϕ , (6.9)
where k1 = (y 2 -1) ( 1/3C (x + y) + 1/2B) -A (y + 1) , (6.10) k2 = (x 2 -1) ( 1/3C (x + y) + 1/2B) , (6.11)

π 2 a⃗ 2 ⎡ 2 2 i i ⎤ i i 2 and
A ≡ 2(q + q + q ), B ≡ 2 (Q q + Q q + Q q ), C ≡ -24q 0 q 1 q 2 . (6.12)

0 1 2 R 0 0 1 1 2 2 R 2
Thus the ψ-dependence has almost been eliminated from the five-dimensional metric. Indeed, the only such dependence comes from the warp factors (2.27) via (2.28):

P = ⎛ ⎜ Z Z - 1 Z 2 + Z 2 ⎞ ⎟ . (6.13) ⎝ 1 2 2 ( 3 4 ) ⎠
The fluctuating part, P ^, of P is given by:

P ‸ = π (x -y) ( Λ + Λ ) + π (x -y) 2 ( Q Λ + Q Λ ) R 1 2 R 2 2 1 1 2 + π 2 (x -y) 2 Λ Λ -Λ 2 + Λ 2 , (6.14) R 2 ( 1 2 ( 3 4 ))
where we have used (6.8).

An elementary example

Take q 1 = q 2 and Q1 = Q2, then Λ2 = -Λ1 and

P ‸ = - π 2 (x -y) 2 ( Λ 2 + Λ 2 + Λ 2 ) , (6.15) R 2 1 3 4
We can take the fluctuations to be a single Fourier mode:

⎛ y + 1 ⎞ n 2 Λi (y, ψ ) = ⎜ ⎟ ( a i cos (n ψ ) + b i sin (n ψ ) ) , (6.16)
and then one has

⎝ y -1 ⎠ n n P ‸ = - π 2 (x -y) 2 ( y + 1 ) n ∑ ⎢ (( a ) + ( b ) 2R 2 y -1 ⎣ n n ) i∈{1,3,4} + ( a i ) 2 -( b i ) 2
cos (2nψ ) + 2 a b sin (2nψ ) . (6.17)

( n n ) n n ⎥ ⎦
Consider the a i and b i to be vectors, a⃗ , b ⃗ ∈ 3 , then the fluctuations in the metric are

n n completely absent if |a⃗ | 2 = |b ⃗ | 2 a⃗ • b ⃗ = 0 . ( 6
.18) That is, the coefficients are orthogonal vectors of the same length in 3 and then:

‸ ⎛ y + 1 ⎞ n P = - (x -y) ⎜ ⎟ . (6.19) R 2 ⎝ y -1 ⎠
This is simply a modest extension of the example given in (5.12). Here we have a fourparameter family of solutions: a⃗ can be freely chosen and then b ⃗ is a vector of the same length in the plane orthogonal to a⃗ . Thus there is really a three parameter family of hair because only a⃗ | appears in the metric.

i i 1 2 1 2 1 1 1 1 2 3 3 4 4 1 1 2 1 1 1 2 1 0 1 2 0 1 2 0 0 1 1 2 2 1
There is a simple way to parametrize the solutions of (6.19) by setting ). We therefore see that the solution space for fixed P ^ is S 3 / 2 = SO (3) where the 2 action is simply (ζ1, ζ2 ) → -(ζ1, ζ2 ).

a⃗ + ib ⃗ = ( i ( ζ 2 + ζ 2 ) ζ 2 -ζ 2 2 ζ1 ζ2
In this example, ∂ is a Killing vector of the metric but not a symmetry of the solution.

∂ψ These solutions resemble therefore Q-balls, in that the oscillations cancel out of the energymomentum tensor and the metric, but the charges rotate around the ring.

Horizon regularity in general

We can check horizon regularity exactly as we did earlier. One first makes a change of variable u = -1/y and then tries to continue smoothly through u = 0. We will need the expansion, to first order, of the electric field terms, Λi, in the neighborhood of the horizon (y = -):

Λi (y, ψ ) = Λ (0) (ψ ) + Λ (1) (ψ )y -1 + .... (6.21) To remove the singularities one must also allow u-dependent shifts in the angular and time coordinates:

t = v + a + b log u ψ = φ + c log u ϕ = φ + c log u, (6.22) u 1 1 2 2
where a, b, c1 and c2 are constants to be determined. It is convenient to define two quantities:

Δ ≡ 4π 2 q ⎛ ⎜ q ( Λ (0) ) 2 + q Λ (0) + Λ (0) ⎞ ⎟ + 4π q (q Q -q Q )Λ (0) , (6.23) 1 1 ⎝ 1 1 2 ( ( 3 ) ( 4 ) ) ⎠ 1 1 1 2 2 1
Δ2 ≡ 4π 2 q ( q Λ (0) Λ (1) + q ( Λ (0) Λ (1) + Λ (0) Λ (1) ))

+ 2π q (q Q1 -q Q2 ) Λ (1) -2π R q ( q -q )Λ (0) . (6.24)

These quantities are in principle ψ-dependent, but as we will see, smoothness across the horizon requires that these quantities are in fact independent of ψ.

One then expands the metric about u = 0. To remove the u -1 terms proportional to du dφ2 one must set c2 = -c1. The u -1 terms proportional to du dφ1 can be removed by setting a = 8 q 0 q 1 q 2 Ω 2 c , (

where

R 2 1 Ω ≡ R ⎣ ⎡ 2( q q Q0 Q1 + q q Q0 Q2 + q q Q1Q2 ) 8 q0 q1q2 0 1 0 2 1 2 -16 q q q (q + q + q ) -(q 2 Q 2 + q 2 Q 2 + q 2 Q 2 ) -Δ ⎤ ⎦ 1 2 . ( 6

.26)

To remove the u -2 du 2 terms one must set c1 = 1 . (6.27) Ω The u -1 du 2 terms have pieces that are linear in x but these cancel as a consequence of (6.25) and (6.27). This is the equivalent in the (x, y) coordinates of the θ-independence of F (0) and

i i i 0 1 2 0 1 2 n n
F (1) in (5.13). The remaining terms involving u -1 du 2 can be removed by setting:

b = - 1 ⎡ ⎢ 1 32q q q (q + q + q ) -4R ( q 2 Q + q 2 Q + q 2 Q + RQ Q Q Ω ⎢ ⎣ 16 q 0 q 1 q 2 ( 0 1 2 0 1 2 0 0 1 1 2 2 ) 0 1 2) - Rq 0 Q0 Δ1 + 1 Δ ⎤ ⎥ . ( 6
.28) 16q 2 q 2 q 2 8q q q 2 ⎥ ⎦ With this change of variables, the metric continues smoothly across u = 0. (If the Δi depended on ψ then the change of variables (6.22) would have introduced new singular dψ terms coming from the derivatives of Δi.) We have not verified that the other fields and the Riemann tensor are finite across the horizon in these coordinates, but this is guaranteed by the analysis in section 5.

Note that if we choose q 1 = q 2 and Q1 = Q2 then we should recover the same solution as in section 5. Indeed, requiring that Δ1 and Δ2 be constant reduces to requiring the ψ-independence of

∑ i∈{1,3,4} (Λ (0) (ψ )) 2 and ∑ i∈{1,3,4} Λ (0) (ψ ) Λ (1) (ψ ) . (6.29)
This is equivalent to requiring that the following are ψ-independent:

∑ ( Λ (y, ψ )) 2 and lim ⎡ ⎢ y 2 ∂ ∑ ( Λ (y, ψ )) 2 ⎤ ⎥. (6.30) i i∈{1,3,4} y=-∞ y→-∞ ⎢ ⎣ ∂y i∈{1,3,4} ⎥ ⎦
This is precisely the same condition as requiring F (0) and F (1) in (5.13) be independent of ψ. Given these two functional constraints, we can ask how big will be the family of hair that our solutions can accommodate. We started with four charge density functions constrained by (6.8), and thus three unconstrained functions parameterized by the arbitrary Fourier coefficients a i and b i , i ∈ {1, 3, 4} in (6.17). Imposing the two extra functional constraints we found should then leave, in principle, one function's-worth of hair on the surface of the black ring. Although we did not find a simple way to prove that this will always happen, it seems very plausible particularly because we can find a simple explicit example (see section 6.2) and thus show that the set of solutions is certainly non-empty. We therefore believe our analysis provides good evidence that there is a lot of new oscillatory hair intrinsic to black Orings and thus to black rings in M-theory.

Conclusions

We have found some very simple BPS black-ring solutions that exhibit whole new varieties of hair through charge oscillations. The existence of these O-ring solutions requires that we go beyond the usual three-charge, or 'STU,' supergravity and couple more vector multiplets. It is only in this way that we can sufficiently expand the phase space of solutions to have enough freedom to coiffure the hair so as to arrange smoothness across the horizon.

We have not attempted to provide a complete classification of these new BPS O-ring solutions. There are obviously more general possibilities, even within the supergravity models that we have considered here. The first step was to make sure that all the fluctuations cancel in the source term of (2.16) so that the angular momentum vector remained ψ-independent and thus the only ψ-dependence in the metric then comes through the warp factor, Z. With more non-zero independent magnetic dipoles there are manifestly many ways to arrange this in the i source of (2.16). For simplicity, we also suppressed fluctuations in Z0. Moreover, generic Calabi-Yau manifolds give rise to many more vector fields and more complicated intersection matrices. There is therefore a very large phase space that could be explored by O-rings in string theory.

In terms of counting solutions we have argued that, even within the limited class considered here, there appears to be a whole function's worth of hair, essentially because we have three charge density functions and two functional constraints coming from requiring smoothness at the horizon. While this is not a formal proof and some subtlety might yet appear in a more careful analysis, our construction strongly suggests that such large families of very hairy solutions exist. They would give by far the largest known violation of black hole uniqueness: an arbitrary function of one variable is parameterized by an infinite number of arbitrary Fourier modes, and hence our solution would depend on an infinite number of continuous parameters.

Even if somehow these functional constraints are too strong and do not give a coiffure parameterized by an infinite number of continuous parameters, we have explicitly exhibited a three-parameter family of hair on BPS O-rings. This in itself provides a substantial enlargement of the hairiness of black objects.

One rather surprising aspect of our work is that the curvatures and gauge fields are continuous across the horizon. The functional constraints on the charge densities manifestly arrange that the metric is continuous and non-degenerate across the horizon. However in the coordinates that continue across the horizon, the warp factor, Z, depends upon oscillations whose frequency diverges as one approaches the horizon. Of course, as a function of the original coordinates, the behavior of Z is completely well-behaved and it is only through transforming to the infalling coordinates that one sees such infinite frequencies. This is presumably because such an infalling observer rotates around the ring infinitely often before crossing the horizon and so sees the charge oscillations with diverging frequency. The surprise is that in spite of this behavior, the curvature and field strengths are not divergent in crossing the horizon and test particles do not encounter infinite fields. The higher level of smoothness is evidently related to the fact that the oscillations of the metric are limited entirely to the five-dimensional warp factor and that its leading two orders of oscillation been eliminated by the charge density constraints. However, the finiteness of the curvatures and gauge fields is still surprising and we hope to investigate this further.

It seems very plausible that the ideas employed here could be extended to extremal non-BPS 9 and even non-extremal black rings [START_REF] Elvang | Non-supersymmetric black rings as thermally excited supertubes[END_REF]. Indeed, one of the inspirations for this work was the idea that one might hope to find non-BPS microstate geometries that oscillate in the manner described here and would ultimately be electrically neutral [START_REF] Mathur | Oscillating supertubes and neutral rotating black hole microstates[END_REF]. One of the other inspirations for this work was the Q-ball solutions [START_REF] Coleman | [END_REF][START_REF] Friedberg | Scalar soliton stars and black holes[END_REF], which are time-dependent and thus certainly not BPS. One might hope to find time-dependent O-rings and even timedependent microstate geometries that oscillate in the same manner. The problem with such time-dependent solutions within string theory is that such charge densities will generically couple to the Maxwell fields and thus emit electromagnetic radiation. It is, or course, possible that one might find a way to do this sufficiently coherently so as to suppress the radiation within a suitably complicated supergravity phase space.

1 .

 1 The scalars. The scalars are given by:

4 I 4π FI

 44π (r, θ, ψ ) = r 0 G (r, θ, ψ , ψ ′)ρI (ψ ′)dψ ′ (5.3)

  ) . ζ2 ) ∈ 2 . This automatically satisfies (6.18) and |a⃗ | 2 + |b ⃗ | 2 = 2(|ζ1| 2 + |ζ2 | 2

	(6.20)
	for any (ζ1,

This comes from the linear term in FI, so it is a function just of ψ, and not of θ.

There may be more general solutions but our purpose here is to exhibit some simple families and leave the classification issue for later work.

The near-horizon geometry of extremal non-BPS black rings[START_REF] Bena | Non-BPS black rings and black holes in taub-NUT J. High Energy Phys[END_REF] appears to be more complicated than that of BPS rings[START_REF] Bossard | Octonionic black holes[END_REF], but since the underlying equations are still linear it may be possible to ensure smoothness by suitably choosing the oscillating charge distributions.

Acknowledgements

NPW is grateful to the IPhT, CEA-Saclay and to the Institut des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette, for hospitality while this work was initiated. NPW would also like to thank the Simons Foundation for their support through a Simons Fellowship in Theoretical Physics. We are all grateful to the Centro de Ciencias de Benasque for hospitality at the 'Gravity-New perspectives from strings and higher dimensions' workshop. The work of IB was supported in part by the ERC Starting Independent Researcher Grant 240210-String-QCD-BH, by the John Templeton Foundation Grant 48222: 'String Theory and the Anthropic Universe' and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 to the Foundational Questions Institute. The work of NPW was supported in part by the DOE grant DE-FG03-84ER-40168. The work of SFR was supported in part by the STFC.


Anton Gfrerrer
email: gfrerrer@tugraz.at

Paul J Zsombor-Murray

Quadrics of Revolution on Given Points

Keywords: quadric of revolution, right cone, cone of revolution, right cylinder, cylinder of revolution, special quadric, surface specification, finite given point set, repeated eigenvalues

In general, 4 points define a 3 parameter set of axisymmetric quadrics while 5 and 6 given points reduce these 3 degrees of freedom to 2 and 1, respectively. Similarly, 7 supporting points confine members of the set to a finite number. By imposing 2 constraints on the quadric coefficient matrix the 5 points are sufficient to find the axis direction of up to 6 right cylinders. Imposing only 1 constraint allows 6 points to support up to 12 right cones. Without either constraint, that implies a singular coefficient matrix or singular conic submatrix, up to 4 quadrics of revolution, possibly of mixed species, can contain 7 points. Formal arguments and proofs are presented to substantiate these observations. Algorithms are developed and applied to exhibit cases with 6 right cylinders, 12 right cones and 4 quadrics of revolution, at least 3 of which are of different type. Spheres, being uniquely defined on 4 points, are specifically excluded from consideration. The cases of 12 cones and 4 quadrics of revolution are believed to be original revelations. Methods to fit quadrics of revolution to more than 7 points are suggested.

Introduction

Real time precision camera aided inspection in industrial production, [START_REF] Koller | Optical Servoing for Industrial Surface Machining[END_REF][START_REF] O'leary | Zsombor-Murray: Direct and Least Squares Fitting of Coupled Geometric Objects for Metric Vision[END_REF][START_REF] O'leary | Instrumentation and Analysis-Methods for the Measurement of Profiles using Light Sectioning[END_REF][START_REF] Pernkopf | Image Acquisition Techniques for Automatic Visual Inspection of Metallic Surfaces[END_REF][START_REF] Tratnig | Camera Calibration, Data Segmentation, and Fitting Approaches for a Visual Edge Inspection System[END_REF], by fitting pixels in digital camera images to conic curves is a process that continues to evolve as users gain confidence in achieving economic benefit. The next challenge is to reliably and accurately fit quadric surfaces via a similar technique. Previous work in this direction seems to have been confined to fitting cylinders of revolution. Results to date have established that, in general, up to six such cylinders may be disposed on five arbitrarily selected points. Some relevant references are listed in sequence of their appearance: [START_REF]Ein geometrisches Problem der metrischen Getriebesynthese[END_REF][START_REF]Konstruktion der Drehzylinder durch vier Punkte einer Ebene[END_REF][START_REF] Devillers | Trebuchet: Circular Cylinders by Four or Five Points in Space[END_REF][START_REF] Beder | Direct Solutions for Computing Cylinders from Minimal Sets of 3D Points[END_REF][START_REF] Lichtblau | Cylinders Through Five Points[END_REF][START_REF] Zsombor-Murray | El Fashny: A Cylinder of Revolution on Five Points[END_REF], from earliest to most recent. Remarkably, there is little backward cross citation, with the exception of Schaal, Strobel (see below) and Röschel [START_REF] Röschel | Drehflächen zweiter Ordnung durch einen Kegelschnitt[END_REF] who were aware of early contributions, of Laguerre [START_REF] Laguerre | Sur la courbe enveloppée par les axes des coniques qui passent par quatre points donnés et sur les axes des surfaces de révolution du second ordre qui passent par cinq points donnés. Sur les lignes spiriques[END_REF], Narasinga [START_REF] Rao | Quadrics of Revolution through a Given Conic[END_REF] and Narasinga & Srinivasachari [START_REF] Rao | Quadrics of Revolution through a Pair of Skew Lines[END_REF], but who were not trying to find a finite number of quadric surfaces on minimal point set specifications. Furthermore there is no evidence of a unified approach to this sort of problem. The quadric of revolution appears to be a good starting point in this regard. Herein the original approach introduced to define four on seven given points will be extended to include the cylinder problem, to show how it fits into the general analytical procedure, and to expose for the first time how twelve cones of revolution may be placed on six points.

The purpose here is to clarify the way attributes of quadrics of revolution can be used to set up systems of equations whereby these surfaces can be determined on fewer than the nine given points required to uniquely define a general quadric. In the case of cylinders of revolution it has been known for some time, [START_REF] Devillers | Trebuchet: Circular Cylinders by Four or Five Points in Space[END_REF][START_REF] Beder | Direct Solutions for Computing Cylinders from Minimal Sets of 3D Points[END_REF][START_REF] Lichtblau | Cylinders Through Five Points[END_REF], and more recently demonstrated, [START_REF] Zsombor-Murray | El Fashny: A Cylinder of Revolution on Five Points[END_REF], that in general as many as six such cylinders may be located on five given points. Furthermore it is common knowledge that four arbitrarily located points will support a sphere. By examining the structure of the conic section of a quadric of revolution, where it intersects the absolute plane, one sees, via cunning reparametrization, that the equivalent of two conditions or constraint equations are implied by the specification that a pair of identical eigenvalues be imposed on a 3 × 3 symmetric matrix that represents the coefficients of this conic section of a quadric of revolution. Essentially the six unknown coefficients in the matrix are reduced to four by the parametrization. So a discrete number of axisymmetric quadrics will in general contain seven given points. Imposing the condition that the overall 4 × 4 coefficient matrix determinant be zero supplies one additional constraint allowing a number of cones of revolution to be defined on six points. Adding that the conic section coefficient matrix determinant vanishes as well obtains the cylinders of revolution on five points, but via a unified method that begins with the general quadric of revolution, rather than with a special vector construction [START_REF] Zsombor-Murray | El Fashny: A Cylinder of Revolution on Five Points[END_REF], used to find the cylinders' axial directions, that can be traced to early work by Schaal, [START_REF]Ein geometrisches Problem der metrischen Getriebesynthese[END_REF][START_REF]Konstruktion der Drehzylinder durch vier Punkte einer Ebene[END_REF], and Strobel, [START_REF] Strobel | Über die Drehkegel durch vier Punkte, Teil I. Sitzungsber[END_REF][START_REF] Strobel | Die Drehkegel durch vier Punkte[END_REF][START_REF] Strobel | Über die Drehkegel durch vier Punkte, Teil II. Sitzungsber[END_REF].

Quadrics of revolution

A quadric Q in 3-space always has an equation of the form

x ⊤ A x + 2c ⊤ x + c 0 = 0 ( 1
)
where A is a real symmetric 3

× 3 matrix, c = [c 1 , c 2 , c 3 ] ⊤ ∈ R 3 , c 0 ∈ R and x = [x, y, z] ⊤
denotes the Cartesian position vector of any point on the surface. Eq. 1 can be written compactly as follows:

[1, x ⊤ ] Q 1 x = 0 (2)
where

Q = c 0 c ⊤ c A . (3)
The following well known facts, drawn from the theory of quadrics will be used:

Q is a quadric of revolution, but not a sphere1 , if and only if the symmetric matrix A has exactly two distinct eigenvalues λ 1 and λ 2 where λ 1 has multiplicity 1, λ 2 has multiplicity 2 and λ 2 = 0. The 1-dimensional2 eigenspace belonging to λ 1 represents the (direction of the) axis of Q whereas the 2-dimensional eigenspace belonging to λ 2 represents the planes perpendicular to this axis -the planes which intersect Q on circles. Proposition 1. Let A be a symmetric 3 × 3 matrix with eigenvalues λ 1 , λ 2 = λ 3 (two of them are identical); then A can be written as

A = (λ 1 -λ 2 ) a a ⊤ + λ 2 I 3 (4)
where

a =   a 1 a 2 a 3   , a = 1, I 3 =   1 0 0 0 1 0 0 0 1   . (5)
Proof. Since A is symmetric there exists a matrix3

R = [a, b, c] =   a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3   with A = R   λ 1 0 0 0 λ 2 0 0 0 λ 2   R ⊤ ,
where R is orthogonal, i.e., R R ⊤ = I 3 . Hence,

A = [a, b, c] •   λ 2 + λ 1 -λ 2 0 0 0 λ 2 0 0 0 λ 2   •   a ⊤ b ⊤ c ⊤   = [a, b, c] •   λ 1 -λ 2 0 0 0 0 0 0 0 0   •   a ⊤ b ⊤ c ⊤   + λ 2 R R ⊤ = (λ 1 -λ 2 ) a a ⊤ + λ 2 I 3 .
Proposition 2. Let Q be a quadric of revolution different from a sphere or a pair of parallel planes, then Q has an equation of the form (2) where

A =   a 2 1 + b a 1 a 2 a 1 a 3 a 1 a 2 a 2 2 + b a 2 a 3 a 1 a 3 a 2 a 3 a 2 3 + b   (6)
and b = 0.

Proof. Because Q is a quadric of revolution matrix A has exactly two different eigenvalues λ 1 and λ 2 where λ 1 has multiplicity 1, λ 2 has multiplicity 2 and λ 2 = 0. According to Proposition 1 matrix A can be written in the form (4). Since λ 1 = λ 2 one can divide the quadric equation by λ 1λ 2 which after having put b = λ 2 λ 1λ 2 yields matrix A according to Eq. 6.

Remark 1. The following items are emphasized: a) Vector a is a direction vector of the quadric of revolution's axis. The condition that a is normalized is optional because the matrix Q can be multiplied by an arbitrary factor = 0.

b) If b = 0 parabolic cylinders or pairs of parallel planes are obtained instead of quadrics of revolution. 4 Their intersection with the plane at infinity is the absolute line of the plane a 1 x + a 2 y + a 3 z = 0.

Due to Proposition 2 every quadric of revolution has an equation of the form (2) where the 3 × 3 submatrix A of Q can be parameterized according to Eq. 6. Conversely, every symmetric 4 × 4 matrix with A, Eq. 6 as its 3 × 3 lower right minor represents a quadric of revolution if b = 0.

It is easy to verify that the determinants of the matrices A and Q can be written as follows:

det A = b 2 • δ, det Q = b • ∆ where δ = b + a 2 1 + a 2 2 + a 2 3 , (7)
∆ = b c 0 δ -c 2 1 (a 2 2 + a 2 3 + b) -c 2 2 (a 2 1 + a 2 3 + b) -c 2 3 (a 2 1 + a 2 2 + b) + 2(a 1 a 2 c 1 c 2 + a 1 a 3 c 1 c 3 + a 2 a 3 c 2 c 3 ). ( 8
)
One can classify the quadrics of revolution where b = 0 by means of δ and ∆ as shown in Table 1.

Table 1: Quadrics of revolution; classification ∆ = 0, δ = 0 : ellipsoids or hyperboloids of revolution ∆ = 0, δ = 0 : paraboloids of revolution ∆ = 0, δ = 0 : cones of revolution ∆ = 0, δ = 0 : cylinders of revolution

Problem formulation

The aim is to find quadrics of revolution through n given points P i , i = 1, . . . , n. This means to find a solution to the equation system

[1, p ix , p iy , p iz ] •     c 0 c 1 c 2 c 3 c 1 a 2 1 + b a 1 a 2 a 1 a 3 c 2 a 1 a 2 a 2 2 + b a 2 a 3 c 3 a 1 a 3 a 2 a 3 a 2 3 + b     •     1 p ix p iy p iz     = 0, i = 1, . . . , n (9)
in the 8 variables c 0 , c 1 , c 2 , c 3 , b, a 1 , a 2 , a 3 , where p ix , p iy , p iz denote the Cartesian coordinates of any given point P i . This system can also be written in the form of the fundamental equation system

P n,10 •                 c 0 c 1 c 2 c 3 a 2 1 + b a 2 2 + b a 2 3 + b a 2 a 3 a 1 a 3 a 1 a 2                 =    0 . . . 0    (10)
where c) no solution if n > 7. The case where n = 4 is satisfied by a 3 parameter set is dealt with in Section 4. Section 5 addresses problems pertaining to 5 given points. Section 6 focuses on finding right cylinders on 5 points. Section 7 treats the case n = 6. An algorithm to determine the right cones on 6 points is presented in Section 8. Finally, how to find the general quadrics of revolution on n = 7 points is exposed in Section 9.

P n,10 =     1
To avoid digression too far into the maze of special or pathological cases it is assumed that the n given points are in general position defined as follows.

Definition 1. Let P 1 , . . . , P n be points in the Euclidean 3-space then we say that P 1 , . . . , P n are in general position if in case of n = 4 the four points are not coplanar and in case of n ≥ 5 the n points are mutually distinct and neither coplanar nor cospherical.

Quadrics of revolution on four non coplanar points

Let P 1 , P 2 , P 3 , P 4 be non coplanar points then we rewrite Eq. 10 as

M 4,4 •     c 0 c 1 c 2 c 3     = -N 4,6 •         a 2 1 + b a 2 2 + b a 2 3 + b a 2 a 3 a 1 a 3 a 1 a 2         . ( 12
)
Since P 1 , P 2 , P 3 , P 4 are non coplanar the matrix

M 4,4 =     1 2p 1x 2p 1y 2p 1z 1 2p 2x 2p 2y 2p 2z 1 2p 3x 2p 3y 2p 3z 1 2p 4x 2p 4y 2p 4z    
is regular and thus Eq. 12 is equivalent to

    c 0 c 1 c 2 c 3     = -M -1 4,4 • N 4,6 •         a 2 1 + b a 2 2 + b a 2 3 + b a 2 a 3 a 1 a 3 a 1 a 2         . ( 13
)
So we have the following Result 1. If P 1 , P 2 , P 3 , P 4 are four non coplanar points, then for any choice of an axis direction [a 1 , a 2 , a 3 ] ⊤ and any b = 0 one obtains exactly one quadric of revolution on P 1 , P 2 , P 3 , P 4 .

Remark 2. Note the following two items: a) Choosing a fixed axis direction [a 1 , a 2 , a 3 ] ⊤ and varying b produces a pencil of quadrics of revolution on P 1 , P 2 , P 3 , P 4 . This case is described in detail by Strobel [16,. b) The unique sphere determined by P 1 , P 2 , P 3 , P 4 is obtained by setting a 1 = a 2 = a 3 = 0 and choosing an arbitrary value b = 0 in Eq. 13.

Quadrics of revolution on five given points in general position

Lemma 1. Let P 1 , . . . , P 5 be five points in general position and let M

5,5 and M denote the sub-matrices of P 5,10 established by the first four and one of the fifth, sixth and seventh columns, respectively, then at least one of these matrices is regular.

Proof. Suppose that all three matrices are singular; then there exist vectors

[α 0 , α 1 , α 2 , α 3 , α 4 ] ⊤ = [0, 0, 0, 0, 0] ⊤ , [β 0 , β 1 , β 2 , β 3 , β 4 ] ⊤ = [0, 0, 0, 0, 0] ⊤ , [γ 0 , γ 1 , γ 2 , γ 3 , γ 4 ] ⊤ = [0, 0, 0, 0, 0] ⊤ , so that α 0 + α 1 p ix + α 2 p iy + α 3 p iz + α 4 p 2 ix = 0, β 0 + β 1 p ix + β 2 p iy + β 3 p iz + β 4 p 2 iy = 0, γ 0 + γ 1 p ix + γ 2 p iy + γ 3 p iz + γ 4 p 2 iz = 0    (14)
for i = 1, 2, 3, 4, 5.

If one of the values α 4 , β 4 or γ 4 is zero then the five points lie in a plane contradicting the assumption of being not coplanar. Hence α 4 , β 4 , γ 4 = 0 and we may multiply the three equations of Eq. 14 by α -1 4 , β -1

4
and γ -1 4 , respectively, and then add them to show that P 1 , . . . , P 5 belong to a sphere which again contradicts our assumption of the points being in general position.

Given five points P 1 , . . . , P 5 in general position, then according to the Lemma 1 above, one of the matrices M 5,5 , is invertible. As a consequence the fundamental equation system Eq. 10 can be rewritten in the form

      c 0 c 1 c 2 c 3 a 2 1 + b       = -M -1 5,5 • N 5,5 •       a 2 2 + b a 2 3 + b a 2 a 3 a 1 a 3 a 1 a 2       . ( 15
)
The last line of these five equations has the structure

kb + q(a 1 , a 2 , a 3 ) = 0 ( 16
)
where k is a real non-zero5 number and q(a 1 , a 2 , a 3 ) is a non-trivial6 quadratic homogeneous polynomial in a 1 , a 2 , a 3 . In conclusion we can determine b uniquely from Eq. 16 for every prescribed axis direction [a 1 , a 2 , a 3 ] ⊤ and then c 0 , c 1 , c 2 , c 3 via Eq. 15. So we have the following Result 2. If P 1 , . . . , P 5 are points in general position then for any choice of an axis direction [a 1 , a 2 , a 3 ] ⊤ one gets exactly one quadric of revolution containing the five points. Only the directions which null the polynomial q(a 1 , a 2 , a 3 ) (Eq. 16) yield parabolic cylinders or pairs of parallel planes instead of quadrics of revolution.

Remark 3. Since we have assumed that the points are in general position we excluded the cases where they lie on a common sphere or in a common plane. In case of P 1 , . . . , P 5 lying in a common plane ε each quadric passing through the five points has to intersect ε along the second order curve determined by the points. This case was studied in detail by Narasinga [START_REF] Rao | Quadrics of Revolution through a Given Conic[END_REF] and Röschel [START_REF] Röschel | Drehflächen zweiter Ordnung durch einen Kegelschnitt[END_REF]. Remark 4. Laguerre [START_REF] Laguerre | Sur la courbe enveloppée par les axes des coniques qui passent par quatre points donnés et sur les axes des surfaces de révolution du second ordre qui passent par cinq points donnés. Sur les lignes spiriques[END_REF] proved the following theorem for a generic quintuple P 1 , P 2 , P 3 , P 4 , P 5 of points: The axis of each quadric of revolution passing through the five points is an asymptote of a twisted cubic containing the centers M k of the five circumspheres of P i 1 , P i 2 , P i 3 , P i 4 , (i j ∈ {1, 2, 3, 4, 5} mutually distinct).

Conversely, any point on the plane of infinity determines an axis direction and together with the five sphere centers M k a twisted cubic c, in general. The tangent of c at M k , i.e., asymptote of c, is the axis of a quadric of revolution on the five given points P 1 , P 2 , P 3 , P 4 , P 5 .

Right cylinders on five given points in general position

Cylinders of revolution are characterized by the conditions ∆ = δ = 0 where δ and ∆ are determined according to Eq. 7 and Eq. 8. If δ = 0 we have

b = -a 2 1 -a 2 2 -a 2 3 ( 17
)
and after substitution into the condition ∆ = 0:

a 1 c 1 + a 2 c 2 + a 3 c 3 = 0 (18)
Hence, the right cylinders among all quadrics of revolution can be extracted by requiring that the two Eqs. 17 and 18 are fulfilled.

Determining the right cylinders on five given points P 1 , . . . , P 5 in general position is accomplished as follows. Substituting the condition expressed by Eq. 17 into Eq. 15 yields

      c 0 c 1 c 2 c 3 -a 2 2 -a 2 3       = -M -1 5,5 • N 5,5 •       -a 2 1 -a 2 3 -a 2 1 -a 2 2 a 2 a 3 a 1 a 3 a 1 a 2       . ( 19
)
The last of these five equations has the form

-a 2 2 -a 2 3 + k 1 (a 2 1 + a 2 3 ) + k 2 (a 2 1 + a 2 2 ) + k 3 a 2 a 3 + k 4 a 1 a 3 + k 5 a 1 a 2 = 0. ( 20
)
The left hand side of this equation is a quadratic homogeneous polynomial in a 1 , a 2 , a 3 which again is non-trivial as one can easily check. Hence, Eq. 20 represents a second order curve c ∞ on the plane at infinity. Real solutions can occur only in the case where c ∞ is a pair of real lines or a real double line or a unipartite conic. In the first two cases one can always find linear homogeneous parametrizations

a i (s, t) = α i0 s + α i1 t, i = 1, 2, 3 (21)
and in the case where c ∞ being a regular conic with real points a quadratic homogeneous parametrization

a i (s, t) = α i0 s 2 + α i1 st + α i2 t 2 , i = 1, 2, 3 (22)
of c ∞ . The parameters are s and t while α ij are constants. By substitution of this parametrization into the first four lines of Eq. 19 one either gets a quadratic parametrization

c i (s, t) = γ i0 s 2 + γ i1 st + γ i2 t 2 , i = 0, 1, 2, 3 (23)
or a quartic parametrization

c i (s, t) = 4 j=0 γ ij s 4-j t j , i = 0, 1, 2, 3 (24)
of the unknowns c i . After substituting the (linear or quadratic) parametrization of a i and the (quadratic or quartic) parametrization of c i into the condition Eq. 18 one obtains a homogeneous polynomial in s, t whose degree is ≤ 6. The real zeroes s * , t * of this polynomial yield possible values for a 1 , a 2 , a 3 , c 0 , c 1 , c 2 , c 3 with the help of Eq. 21 or Eq. 22 and Eq. 23 or Eq. 24. This uniquely determines the equation of the corresponding solution cylinder.

Result 3. The task of determining the right cylinders on 5 given points in general position is a sextic problem.

This confirms a result of Zsombor-Murray and El Fashny [START_REF] Zsombor-Murray | El Fashny: A Cylinder of Revolution on Five Points[END_REF]. Applying this method to finding the right cylinders on the five points

Quadrics of revolution on six given points in general position

Let now six points P 1 , . . . , P 6 in general position be given. In this section we will discuss how to determine all quadrics of revolution on these points.

Since the six points are in general position (mutually distinct and neither coplanar nor cospherical) one can re-enumerate them always in a way that the quadruple P 1 , P 2 , P 3 , P 4 is not coplanar and the points of the quintuples {P 1 , P 2 , P 3 , P 4 , P 5 } and {P 1 , P 2 , P 3 , P 4 , P 6 } are also in general position. (This can be easily checked by elementary considerations.)

Because P 1 , P 2 , P 3 , P 4 , P 5 are in general position we have

b = q 1 (a 1 , a 2 , a 3 ) (25)
where q 1 is a quadratic homogeneous and non-trivial polynomial according to the results of Section 5. Analogously, we find

b = q 2 (a 1 , a 2 , a 3 ) (26)
with another quadratic homogeneous and non-trivial polynomial q 2 by applying the results of Section 5 to the quintuple P 1 , P 2 , P 3 , P 4 , P 6 . In conclusion the axis direction numbers a 1 , a 2 , a 3 have to fulfill the homogeneous quadratic condition f (a 1 , a 2 , a 3 ) = q 2 (a 1 , a 2 , a 3 )q 1 (a 1 , a 2 , a 3 ) = 0 which represents a second order curve c ∞ on the plane at infinity. The only exception occurs if the two polynomials q 1 (a 1 , a 2 , a 3 ) and q 2 (a 1 , a 2 , a 3 ) are identical in which case the condition on a 1 , a 2 , a 3 vanishes.

As in the previous section c ∞ can be linearly parameterized as shown by Eq. 21 when c ∞ is a pair of real lines or a double line or quadratically as shown by Eq. 22 if c ∞ is a unipartite conic.

Substituting this parametrization into Eq. 25 or Eq. 26 yields a quadratic or quartic homogeneous parametrization of b in s and t:

b = b(s, t) (27)
Finally, since P 1 , P 2 , P 3 , P 4 are not coplanar Eq. 13 can be used to obtain quadratic or quartic parameterizations of c 0 , . . . , c 3 by substituting Eq. 21 or Eq. 22 and Eq. 27:

    c 0 (s, t) c 1 (s, t) c 2 (s, t) c 3 (s, t)     = -M -1 4,4 • N 4,6 •         a 2 1 (s, t) + b(s, t) a 2 2 (s, t) + b(s, t) a 2 3 (s, t) + b(s, t) a 2 (s, t) • a 3 (s, t) a 1 (s, t) • a 3 (s, t) a 1 (s, t) • a 2 (s, t)         (28)
Thus a homogeneous parametrization of the set of all quadrics of revolution on the given six points P 1 , . . . , P 6 is available.

Result 4. In general, there is a one-parametric set of quadrics of revolution on six points P 1 , . . . , P 6 in general position. The points at infinity on the axes of these quadrics lie on a second order curve c ∞ .

Right cones on six given points in general position

Based on Section 7 the quadrics of revolution on six given points in general position will usually establish a family determined by the parameterizations Eq. 21 or Eq. 22 of a 1 , a 2 , a 3 , Eq. 27 of b and Eq. 28 of c 0 , c 1 , c 2 , c 3 . In order to select the right cones from this one degree of freedom family one must substitute these parameterizations into the condition

∆ = b c 0 (b + a 2 1 + a 2 2 + a 2 3 ) -c 2 1 (a 2 2 + a 2 3 + b) -c 2 2 (a 2 1 + a 2 3 + b) -c 2 3 (a 2 1 + a 2 2 + b) + 2(a 1 a 2 c 1 c 2 + a 1 a 3 c 1 c 3 + a 2 a 3 c 2 c 3 ) = 0. ( 29
)
This yields a homogeneous bivariate polynomial in s, t of degree ≤ 12. Every real solution s * , t * of this polynomial yields a solution cone.

Result 5. The problem of finding the right cones on 6 given points in general position can be reduced to the task of determining the zeroes of a univariate polynomial of degree 12, in general. Every real solution of this polynomial corresponds to exactly one solution cone.

in general. Analogously, each quadric of revolution on P 1 , P 2 , P 3 , P 4 , P 5 , P 7 has an axis whose point at infinity generally lies on another second order curve c 2,∞ : f 2 (a 1 , a 2 , a 3 ) = 0 (31) Hence, we have Result 6. The points at infinity of the axes of the quadrics of revolution on seven given points in general position lie in general on the intersection of two second order curves c 1,∞ and c 2,∞ .

In conclusion there are at most four possible intersection points if c 1,∞ and c 2,∞ are not coincident in whole or in part; the latter in the case of distinct line pairs sharing a line.

If a * 1 , a * 2 , a * 3 are the coordinates of an axis' point at infinity satisfying Eq. 30 and Eq. 31 then the corresponding values for b and c 0 , c 1 , c 2 , c 3 can be determined uniquely by applying the method described in Section 7.

Result 7. In general there are at most four quadrics of revolution on seven given points in general position.

Conclusions and unresolved problems for future research

The following list describes some relevant topics that remain to be investigated.

• Omitted cases where points lie on a sphere might be considered in the light of Remark 4, i.e., Laguerre's work [START_REF] Laguerre | Sur la courbe enveloppée par les axes des coniques qui passent par quatre points donnés et sur les axes des surfaces de révolution du second ordre qui passent par cinq points donnés. Sur les lignes spiriques[END_REF].

• Quadrics of revolution on n = 5, 6 or 7 given coplanar points exist only if these lie on c a proper conic in that plane. This is generally the case where n = 5. It was treated by Narasinga [START_REF] Rao | Quadrics of Revolution through a Given Conic[END_REF] and Röschel [START_REF] Röschel | Drehflächen zweiter Ordnung durch einen Kegelschnitt[END_REF] but the special case where c degenerates to a line pair or a double line has not received attention.

• In Section 7 (quadrics of revolution on 6 points) the case where the two quadratic polynomials q 1 (a 1 , a 2 , a 3 ) and q 2 (a 1 , a 2 , a 3 ) are identical was excluded. How can this case be geometrically characterized?

• An infinite number of quadrics might contain 7 given points (Section 9) if the two second order curves c 1,∞ and c 2,∞ have a common component. What are the geometric characteristics of this case?

• Moreover methods to discriminate among the types of quadrics of revolution, i.e., ellipsoids, paraboloids and hyperboloids of one and two sheets, have yet to be formulated.

• Another task, pertinent to application in camera aided inspection, is to find the most suitable quadric of revolution through an overdetermined set of points {P 1 , . . . , P n }, n > 7. Using a setup with the parametrization Eq. 6 it can be shown that this problem can be reduced to a pair of homogeneous quartic equations in the axis direction numbers a 1 , a 2 , a 3 . Alternatively, one could choose various subsets of 7 points from a given set of n > 7 points and then apply the method described in Section 9 to determine a quadric of revolution. This quadric could then be taken as a starting point in, e.g., a leastsquares Gauss-Newton fit optimization procedure. Deficiencies or defects stemming from singularity of the involved matrices M k,k are easy to detect. In such a case the chosen 7-tupel of points can be easily replaced by another. After all, camera data contain a great many points.

It is believed that the unified treatment of the three types of quadrics of revolution exposed herein is original and constitutes a solid foundation upon which to build effective, precise methods to quickly fit overdetermined point sets to such surfaces. This will be invaluable in camera aided inspection of manufactured structural and machine elements of such cross section, a fruitful extension of this research that will have important industrial applications.

P 1 .

 1 . . [0.0, 0.0, 0.0] ⊤ , P 2 . . . [1.0, 0.0, 0.0] ⊤ , P 3 . . . [0.0, 1.0, 0.0] ⊤ , P 4 . . . [0.0, 0.0, 1.0] ⊤ , P 5 . . . [0.8, 0.8, 1.0] ⊤ produces a solution with the six cylinders shown in Fig. 1.

Figure 1 :

 1 Figure 1: Six right cylinders on five given points

Figure 3 :

 3 Figure 3: Four quadrics of revolution on seven given points

  1, -1.2, 2.2] ⊤ , P 2 . . . [3.5, 1.4, 2.4] ⊤ , P 3 . . . [3.3, -2.5, -0.5] ⊤ , P 4 . . . [4.2, 3.1, -0.9] ⊤ , P 5 . . . [-2.2, 4.3, 0.36] ⊤ , P 6 . . . [-1.9, 3.2, 2.5] ⊤ , P 7 . . . [-2.6, -3.8, 2.8] ⊤

Spheres are characterized by a single eigenvalue λ = 0 whose multiplicity is then 3.

In case of symmetric matrices the algebraic multiplicity of an eigenvalue is the same as the geometric multiplicity, i.e., the dimension of its corresponding eigenspace.

The first, second and third column of R contains an eigenvector of A belonging to λ 1 , λ 2 and λ 3 = λ 2 , respectively.

Of course, pairs of parallel planes can also be considered as special quadrics of revolution.

The assumption k = 0 leads to a contradiction as follows: If k = 0 then Eq. 16 is fulfilled for a 1 = a 2 = a 3 = 0 and any choice of b = 0. Substitution of these values in Eq. 15 yields unique values of c 0 , c 1 , c 2 and c 3 . Therefore the 5 points lie on a sphere and are not in the required general configuration.

The polynomial q(a 1 , a 2 , a 3 ) is non-trivial because the coefficient of a 2 1 is 1.

Acknowledgements

This research is supported by a Natural Sciences and Engineering Research Canada "Discovery" grant. Thanks are also extended to Manfred Husty from the University of Innsbruck and Johannes Wallner and Andreas Weinmann, both from TU Graz, for some fruitful discussions during the preparation of this paper.

 [START_REF] Müller | Durch vier Punkte einen Drehkegel von vorgegebener Achsenrichtung zu legen[END_REF]studied the problem of finding the right cones with prescribed axis direction on four points. The algorithm described above was applied to the following six given points:

In this case the twelve solution cones displayed in Fig. 2 were obtained.

Quadrics of revolution on seven given points in general position

If 7 points P 1 , . . . , P 7 in general position are given then it is easy to verify that one can always find sextuples of this set, say, {P 1 , P 2 , P 3 , P 4 , P 5 , P 6 } and {P 1 , P 2 , P 3 , P 4 , P 5 , P 7 } whose points are also in general position.

Then, according to Section 7 the axes' points at infinity of the quadrics of revolution on P 1 , P 2 , P 3 , P 4 , P 5 , P 6 are situated on a second order curve c 1,∞ : f 1 (a 1 , a 2 , a 3 ) = 0, (30)