hep-th/0305247

A Charged Rotating Black Ring

Henriette Elvang

Department of Physics, UCSB, Santa Barbara, CA 93106

February 17, 2021

We construct a supergravity solution describing a charged rotating black ring with horizon in a five dimensional asymptotically flat spacetime. In the neutral limit the solution is the rotating black ring recently found by Emparan and Reall. We determine the exact value of the lower bound on , where is the angular momentum and the mass; the black ring saturating this bound has maximum entropy for the given mass. The charged black ring is characterized by mass , angular momentum , and electric charge , and it also carries local fundamental string charge. The electric charge distributed uniformly along the ring helps support the ring against its gravitational self-attraction, so that can be made arbitrarily small while remains finite. The charged black ring has an extremal limit in which the horizon coincides with the singularity.

## 1 Introduction

Recently, Emparan and Reall [1] found an exact vacuum solution describing a rotating black ring in a five dimensional asymptotically flat spacetime. The black ring solution is the first explicit example of non-uniqueness in higher dimensional gravity in the sense that the asymptotically determined quantities do not uniquely specify the solution: in five dimensions there exist asymptotically flat vacuum solutions with the same mass and angular momentum, but with distinct horizon topologies — one is the rotating black hole with horizon and the other is the rotating black ring with horizon. We extend this non-uniqueness result to charged solutions of low energy heterotic string theory.

In five dimensions, the Myers-Perry black hole [2] is characterized by the mass and two independent angular momenta, and . Taking and , the dimensionless ratio constructed from and has an upper bound,

Emparan and Reall showed that for the black ring with mass and angular momentum the dimensionless ratio has a lower bound,

and they found [1]. In this paper we show that the exact value is . The solution with is the black ring that maximizes the entropy for the given mass. For

The interpretation of the lower bound on is that for a given mass it takes a certain angular momentum to balance the gravitational self-attraction of the ring. An electric charge distributed uniformly around the black ring would help support the ring so that the ratio could be made arbitrarily small. We find that this is indeed possible.

Applying the solution generating techniques of Hassan and Sen [3] to the rotating black ring, we find a solution describing a charged rotating black ring. This is a solution of the low energy limit of heterotic string theory (heterotic supergravity) and besides carrying a U(1) electric charge , the black ring also carries local fundamental string charge. The charged black ring can be viewed as the field of a rotating excited loop of fundamental string with electric charged added. We find that the ratio can be made arbitrarily small, while the dimensionless ratio of charge to mass approaches a constant. The charge and mass satisfy , independent of the angular momentum. We also compute the magnetic moment and gyromagnetic ratio. The charged rotating black ring has an extremal limit for which the ring is extremally charged, . For the extremal solution, the horizon coincides with the singularity.

It is unknown if supersymmetric black rings exist. In [4] all supersymmetric solutions of minimal supergravity in five dimensions are constructed. Furthermore, as a first uniqueness result, it is argued in [5] that the only supersymmetric, asymptotically flat black hole solutions in this theory are the Breckenridge, Myers, Peet, and Vafa (BMPV) [6] black holes, which are characterized by their mass and angular momentum. However, we have here a black ring solution not to minimal supergravity, but to heterotic supergravity with five dimensions compactified. Hence the uniqueness result of [5] does not exclude the possibility of a supersymmetric black ring for which the extremal limit of the black ring found in this paper may be a candidate.

The matter of uniqueness is interesting in its own right, but is also important for the string theory calculations of the entropy of supersymmetric or nearly supersymmetric black holes [7]. For these derivations it is assumed that the black hole solutions are specified uniquely by their asymptotic charges.

The solutions presented in this paper are *not* uniquely
specified by their asymptotic charges. For a certain range of
parameters there are charged rotating black rings and spherical black
holes (obtained by applying the Hassan-Sen transformation to a
Myers-Perry black hole with a single nonzero angular momentum) with
the same asymptotic charges. The extremal limit of the charged black
ring hasvanishing horizon area.

The paper is organized as follows. We consider the neutral rotating black ring in section 2: in section 2.1 we review the neutral black ring of [1] and in section 2.2 we derive the exact lower bound of for the black ring. We review the Hassan-Sen solution generating technique in section 3. The Hassan-Sen technique gives the transformed solutions implicitly. Starting from quite general solutions we offer in appendix A.1 and appendix A.2 explicit expressions for the transformed solutions. In section 4, we apply the Hassan-Sen transformation to the rotating black ring to obtain the charged ring solution (section 4.1). We investigate the physical properties (section 4.2) and study an extremal limit of the charged black ring (section 4.3). We compare the local behavior of the charged black ring to the local behavior of a charged black string obtained by applying the Hassan-Sen transformation to a boosted black string (section 4.4). Somewhat unrelated to the black rings we discuss in section 5 solutions for charged black strings and their extremal limits. We summarize and discuss the results in section 6.

## 2 The Black Ring

Emparan and Reall found vacuum solutions describing black rings with horizons in a five dimensional asymptotically flat spacetime. The static black ring solution [8] has conical singularities preventing the ring from collapsing, but these conical singularities can be avoided if the ring is rotating fast enough to provide a force to balance the ring under its own gravitational attraction [1]. We review the rotating black ring in section 2.1, and in section 2.2 we derive the exact lower bound on .

### 2.1 Review of the Neutral Rotating Black Ring

The metric of the black ring was obtained by a Wick rotation of a metric in [9]. The solution is characterized by a parameter and a scaling . Written in C-metric coordinates (we adopt the notation of and follow closely Ref. [1]) the metric is

where

(2.2) |

It is assumed that which guarantees that has three distinct real roots, , , and . The roots can be ordered as .

Analyzing the metric (2.1), one finds that in order to keep the signature Lorentzian and to avoid conical singularities, the coordinate ranges are required to be

(2.3) |

and the angular coordinates and must have periodicities

(2.4) |

Furthermore . If the solution (2.1) describes a black hole with horizon topology. A coordinate transformation [1] identifies it as the Myers-Perry rotating black hole [2] in five dimensions with one rotation parameter set to zero. In the following we assume . Since the orbit of then vanishes at both and , there are two distinct conditions imposed on . These conditions are solved by setting

(2.5) |

implying that . Equation (2.5) can be viewed as the tuning of the angular momentum to uphold the ring.

The limit is asymptotic infinity and it can be shown that the solution is asymptotically flat: rescale and by taking and , so that , and define . Via the coordinate transformation

(2.6) |

the asymptotic metric can be written as

(2.7) |

The Killing vector vanishes when , and since the metric is regular here the coordinate can naturally be extended past allowing to take negative values. The coordinates break down at , where blows up. However, this is just a coordinate singularity that can be removed by a change of coordinates. In fact, one finds that there is an horizon at and behind it an curvature singularity is hiding at . The region is the ergoregion. The ergosurface at is regular and has topology .

Locally, the rotating black ring is expected to look like a boosted black string, and indeed the near singularity behavior of the boosted black string matches that of the black ring (up to numerical factors and distortion, see also sections 4.2-4.4).

The physical quantities such as the ADM mass, the angular momentum, and the surface gravity are given for the black ring in [1]. Dimensionless quantities can be formed by multiplying the physical quantities by suitable powers of the mass. For the angular momentum and the horizon area we have

(2.8) | |||||

(2.9) |

We have used (2.5) to eliminate from these expressions, and we set throughout the paper. In section 4, we compute the physical quantities for the charged black ring, and the mass and angular momentum for the neutral black ring can then be obtained by taking the transformation parameter to zero (see eqs. (4.9) and (4.10)).

### 2.2 An Exact Result for the Black Ring

Plotting the dimensionless ratio of angular momentum to mass as function of , Emparan and Reall found that has a global minimum,

In this section we determine the exact value of to be . The black ring with is the solution that maximizes the dimensionless measure of entropy, .

The quantities and are given in (2.8) and (2.9) in terms of the roots, , , and , of the cubic equation

(2.10) |

We assume that in order for the equation to have three distinct real roots. For , the roots and coincide. Using standard methods for obtaining the roots of a cubic equation, we find

where

(2.11) |

We can now write the dimensionless ratio of angular momentum to mass in terms of as

(2.12) |

The global minimum of

(2.13) |

Evaluating (2.12) at we find the minimum value to be

giving the simple result

(2.14) |

for the black ring. It is peculiar that the value — a number produced by extremizing a function which depends solely of the roots of the cubic equation (2.10) — cancels exactly the factor which comes from the normalization of the mass and the angular momentum. It would be interesting to understand if there is any significance to this cancellation.

In terms of , the dimensionless ratio of the horizon area and the mass is

In the given range, this function has a global maximum for given in (2.13), and we find

In conclusion, the black ring with is the black ring with minimum angular momentum and maximum entropy for the mass given. As increases, decreases, for fixed mass (see also Fig. 3 of [1]).

We find that the value of for which

## 3 Review of Hassan-Sen Transformations

In the early 90s it was shown [10, 11, 12, 13, 14] that in any string theory the space of classical solutions that are independent of of the spacetime coordinates has an O()O() symmetry (or O()O() symmetry if the dimensions are all spatial), where the first factor acts on the left-movers and the second factor acts on the right-movers.

Hassan and Sen [3] showed that in heterotic string theory the group of transformations can be extended so that the group acting on the right-movers includes a subset of the 16 internal coordinates. If the signature of the coordinates is Lorentzian and the background gauge fields are neutral under of the U(1) generators of the gauge group, the group of transformations is O()O(). These transformations can be used to generate new inequivalent classical solutions from known classical solutions.

The symmetry can be realized explicitly for the low energy effective action, but is valid to all orders in . Hassan and Sen [3] applied the transformations to a magnetic 6-brane solution in ten dimensions to generate a new solution of heterotic supergravity carrying independent electric and magnetic charges as well as antisymmetric tensor field charge. Also, starting from the neutral Kerr black hole in four dimensions, Sen found a charged rotating black hole solution with nontrivial dilaton, magnetic fields, and antisymmetric tensor field [16]. Many other solutions have been generated using these transformations.

We shall be interested in classical solutions in spacetime dimensions, hence of the ten dimensions for the heterotic string have been compactified; massless excitations from the compactification and higher derivative terms are not included in the effective action, which is given by

(3.1) |

We consider only U(1) gauge fields, and just a single U(1) component of the gauge fields has been included in the action (3.1). The antisymmetric 3-form field includes the U(1) Chern-Simons term,

(3.2) |

Throughout the metric refers to the string frame metric. In dimensions the Einstein metric is related to the string metric by .

We apply the Hassan-Sen transformations to classical solutions that are independent of the time-direction and (at least) one spatial direction . The transformations of interest to us involve only the -part of the metric, which we denote by , , the 01-part of the antisymmetric tensor and the gauge fields . Given such a solution , where and are block diagonal, ie. for all , the transformed solution is computed as follows. Define a matrix as

(3.3) |

and a matrix ,

(3.4) |

where . In addition to the above assumptions we assume for convenience that for the original solution. We have and and the symmetry on the space of solutions is O(1,1)O(2,1). Writing as

we can choose to be on the form

(3.5) |

where parametrizes boosts that mixes the -direction with the internal coordinate, and parametrizes boosts in -space. The O(1,1)-transformations are Lorentz boosts of the solution in the 01-plane and we choose to be the identity matrix.

The Hassan-Sen transformation acts on the solution to give

The -components of the new metric and the fields, and , are given implicitly by and can be extracted using (3.3) and (3.4) with , , and replaced by , , and . All other field components are unchanged by the transformation. In appendix A.1 and appendix A.2 we give the explicit transformed solution in terms of the original solutions.

#### Remark: Hassan-Sen transformations with

Let be a static solution satisfying the
above assumptions.
It is well-known [17] that when
applying a Lorentz boost with parameter
( and
)
and then T-dualizing in the -direction, one obtains
a new solution where the linear
momentum created by the boost is exchanged for an
-charge.^{2}^{2}2The fields are required to fall off appropriately
at infinity.
Boosting the solution in -space
with the same parameter gives a new solution
and then T-dualizing again, we find that the
resulting solution is exactly the Hassan-Sen transformed solution
with and .
This also holds true if the original metric has .

If identically for the original static solution then the last T-duality transformation has no effect: for the solution is invariant under T-duality. In section 5 we give an example of a self-T-dual charged black string.

## 4 The Charged Black Ring

### 4.1 The Solution

The rotating black ring solution (2.1) has three Killing vectors corresponding to the coordinates , , and . We apply the Hassan-Sen transformation with and to the -part of the black ring solution (2.1) to find a solution of the theory (3.1) with . The transformation is given explicitly in appendix A.2. The transformed solution is

with fields

(4.7) |

The functions and are given in (2.2) and

(4.8) |

The analysis of the metric with respect to signature and regularity works out exactly as for the neutral case. The coordinates and are restricted to the regions (2.3) and the coordinates and are periodic with the periods given in (2.4). We note that for and in the coordinate regions (2.3), the function in equation (4.8) is strictly positive.

The asymptotic region is at . Since at infinity, the coordinate transformation (2.6) takes the asymptotic metric to the form (2.7) after the appropriate rescalings. Thus the transformed metric is asymptotically flat.

For , the coordinate transformation given in [1] takes the solution given by (4.1) and (4.7) to the solution obtained by applying the Hassan-Sen transformation of section A.2 to the five dimensional Myers-Perry black hole with only one nonzero rotation parameter. This solution is the five dimensional analog of the charged rotating black hole in four dimensions found by Sen [16]. It can be generalized to a charged solution with two independent angular momenta by applying a Hassan-Sen transformation to the general five dimensional Myers-Perry rotating black hole.

In the following we assume that . Just as in the case of the neutral black ring, regularity requires to be given by (2.5).

The transformed solution is regular at , so defining we can extend the coordinate region to include just as for the neutral black ring. At , the metric component blows up while the fields stay finite. By a slight modification of the coordinate transformation given in [1] we obtain new coordinates for which the metric is regular at . The new coordinates — valid for — are defined as

so the -part of the metric becomes

The Killing vector vanishes at , so the region is the ergoregion. The determinant has a zero at and since we know that the metric is regular here, the constant- surface at defines the event horizon. There is no inner horizon. Both the ergosurface (defined as the constant- surface at ) and the horizon are topologically . The curvature blows up at , and the dilaton is singular there; this corresponds to a spacelike curvature singularity in the metric.

### 4.2 Physical Properties

Going to the Einstein metric, , and using the next to leading order behavior of the asymptotic metric we compute the ADM mass and the angular momentum [2]

(4.9) | |||||

(4.10) |

which reduce to the values for the neutral ring for . Also, the black ring has an asymptotic electric U(1) charge given by (charges are normalized as in [17])

The dimensionless ratio of angular momentum and mass is given by

(4.11) |

where we have used the result (2.14) for lower bound on for the neutral black ring, and the dimensionless ratio of charge to mass is

(4.12) |

We note that by taking large we can make arbitrarily small while approaches a constant. Thus the charge helps holding up the black ring allowing us to make the angular momentum arbitrarily small. This was of course not possible for the neutral black ring.

Surprisingly, the ratio is independent of . In fact, we notice that the right hand side of (4.12) is always less than 1, so that for all we have

(4.13) |

with equality in the limit . Contrary to other solutions with angular momentum and charge, this bound does not involve the angular momentum.

As a one dimensional object in a five dimensional asymptotically flat spacetime, the black ring can carry local — but not global — fundamental string charge associated with the 3-form field . Using (3.2) we find that has only one nonzero component,

and it gives rise to the local fundamental string charge

where the integral is over a two sphere parametrized by and at a constant -cut around of the ring. In the limit , the dimensionless ratio diverges.

From the leading order behavior of the field at infinity we find the magnetic moment of the black ring. In spherical coordinates with radial coordinate and a polar coordinate we have for large

In analog to the normalization of the charges, we normalize the magnetic moment as where is the area of a unit three sphere. We find

The dependence cancels in so that the ratio depends only on . The gyromagnetic ratio is defined as and we find

(4.14) |

so that is independent of . We see from (4.14) that for the charged black ring, the -factor can take values between and . The same bounds have been found on the gyromagnetic ratio for the string theory solution describing a dilatonic rotating charged black hole in a four dimensional asymptotically flat spacetime [15]. It should however be noted that there is an ambiguity in the normalization of the magnetic moment; changing the normalization of changes the -factor.

The area of the event horizon is

As a function of , the dimensionless ratio is maximized for : for a given mass the neutral black ring always has higher entropy than the charged black ring, and increasing the charge while keeping the mass fixed, the horizon area decreases. This is qualitatively the same behavior as for a charged spherical black hole.

Associated with the horizon is a Killing vector field

(4.15) |

Outside the horizon where the original coordinates are valid, the Killing vector field is given by

Using the Killing field (4.15) we compute the surface gravity

(4.16) |

Near the singularity, for small , the metric takes the form

where we have ignored numerical constants and -dependence (for example, the -part of the metric is only topologically a two sphere). For the fields we find near the singularity

We shall compare this behavior with the near singularity behavior of the Hassan-Sen transformed boosted black string (see section 4.4).

### 4.3 Extremal Limit

In the limit the ’s behave as

The ratio approaches zero when , but it diverges for (see figure 1). We find an extremal limit of the charged black ring by taking the limit keeping fixed. The extremal metric is

and the fields are

We now have and , and the periodicities are . The solution is asymptotically flat. The curvature blows up at and this is a null singularity coinciding with the horizon.

The physical quantities for the extremal solution are

Note that the inequality (4.13) is saturated in the extremal limit so that . Also, for the extremal ring and the -factor is 2. By taking large we can make arbitrarily small. The horizon area shrinks to zero, however taking the limit of the surface gravity (4.16) gives a constant, . This is similar to the behavior found in [15, 18] for slowly rotating and non-rotating dilatonic charged spherical black hole solutions in string theory.

Defining and considering small we find that the near horizon/singularity behavior is

with

We compare this with the near-singularity behavior of the extremal limit of the charged boosted black string in section 4.4.

### 4.4 A Charged Boosted Black String

The local behavior of the neutral rotating black ring is like that of a boosted black string, hence we expect that the charged black ring behaves locally as a boosted black string with similar charges and fields. We check this by comparing the near singularity behavior of the charged black ring to that of a charged black string obtained from the boosted black string by the Hassan-Sen transformation of appendix A.2.

The black string metric in five dimensions is the four dimensional Schwarzschild solution times ,

(4.17) |

Applying a Lorentz boost to the solution (4.17) by taking and , we obtain the metric for the boosted black string

(4.18) | |||||

T-dualizing the metric (4.18) gives the solution for the non-extremal fundamental black string (see section 5). Now instead apply the Hassan-Sen transformation of section A.2 to (4.18). The transformed solution is given by

The solution is regular at and the coordinate singularity at can be removed by a coordinate transformation (valid for )

In these coordinates the metric becomes