<p>The topological censorship theorem suggests that higher-dimensional black holes can possess a domain of outer communication (DOC) of nontrivial topology. In this paper, we seek for a black hole coexisting with two bubbles adjacent to the horizon in five-dimensional minimal supergravity, under the assumptions of stationarity and biaxisymmetry. For simplicity, we also assume that the spacetime is symmetric under the exchange of the two axisymmetric Killing vectors. To find the solution, we combine the inverse scattering method and the Harrison transformation, and we present the conditions for the absence of conical, orbifold, and Dirac-Misner string singularities, respectively. As the result, we find that the black hole with topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>3</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>2</mn></msup><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></math> can be supported by two bubbles if we admit the conical singularities (deficits).</p>

Japan

Ryotaku Suzuki

Shinya Tomizawa

Exploring nonsupersymmetric black holes with multiple bubbles in five-dimensional minimal supergravity

<p>We present a new non-BPS solution describing an asymptotically flat, stationary, biaxisymmetric capped black hole in the bosonic sector of five-dimensional minimal supergravity. This solution describes a spherical black hole, while the exterior region of the horizon exhibits a nontrivial topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">[</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>4</mn></msup><mo>#</mo><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="double-struck">P</mi></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mo>\</mo><msup><mrow><mi mathvariant="double-struck">B</mi></mrow><mn>4</mn></msup></mrow></math> on a time slice. This solution extends our previously constructed three-parameter solution to a more general four-parameter solution. To derive this solution, we utilize a combination of the Ehlers and Harrison transformations and then impose appropriate boundary conditions on the solution’s parameters. It can be shown that the resultant solution is free from curvature, conical, Dirac-Misner string and orbifold singularities, as well as closed timelike curves on and outside the horizon. Characterized by four independent conserved charges—mass, two angular momenta, and electric charge—this solution reveals two distinct branches: a small bubble branch and a large bubble branch, distinguished by nonconserved local quantities such as magnetic flux or magnetic potential. This shows the nonuniqueness for spherical black holes, even among capped black holes. For equivalent sets of conserved charges, we find that the large/small bubble branch can have larger/smaller entropy than the Cvetič-Youm black hole.</p>

Nonuniqueness of capped black holes: Large and small bubbles

We present a new exact solution for a general non-BPS black ring in the bosonic sector of five-dimensional minimal supergravity. This obtained solution carries four independent conserved charges: the mass, two angular momenta, an electric charge, and an additional dipole charge related to other charges. By employing the Ehlers-Harrison transformation, we derive this solution by transforming a five-dimensional vacuum solution into a charged solution in the theory. Previously, our work produced a vacuum doubly rotating black ring solution possessing a Dirac-Misner string singularity by using the Ehlers transformation. In this study, we use the singular black ring as the seed for the Harrison transformation. The resultant solution is regular, free from curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. We show that within a specific parameter range, the black ring presents two branches for the same mass, two angular momenta and electric charge but these are distinguished by a dipole charge, which exhibits discontinuous nonuniqueness. Furthermore, this newly obtained black ring seamlessly connects to various physically significant solutions, such as the Pomeransky-Sen’kov black ring, the extremal black ring, the supersymmetric black ring, and the charged singly spinning black ring.

New black ring with all independent conserved charges in five-dimensional minimal supergravity

Using the Ehlers transformation, we derive an exact solution for a doubly rotating black ring in five-dimensional vacuum Einstein theory. It is well known that the vacuum Einstein theory with three commuting Killing vector fields can be reduced to a nonlinear sigma model with <math><mi>S</mi><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>R</mi><mo>)</mo></math> target space symmetry. As shown previously by Giusto and Saxena, the <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math> subgroup in the <math><mi>S</mi><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>R</mi><mo>)</mo></math> can generate a rotating solution from a static solution while preserving asymptotic flatness. This so-called Ehlers transformation actually transforms the five-dimensional Schwarzschild black hole into the five-dimensional Myers-Perry black hole. However, unlike the case with the black hole, applying this method directly to the static black ring or the Emparan-Reall black ring, does not yield a regular rotating black ring due to the emergence of a Dirac-Misner string singularity. To solve this undesirable issue, we use a singular vacuum solution of a rotating black ring/lens that already possesses a Dirac-Misner string singularity as the seed solution for the Ehlers transformation. The resulting solution is regular, indicating the absence of curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. We show that this solution obtained by the Ehlers transformation coincides precisely with the Pomeransky-Sen’kov solution. We expect that applying this method to other theories may lead to the finding of new exact solutions, such as solutions for black lenses and capped black holes, as well as black ring configurations.

New construction of a vacuum doubly rotating black ring by the Ehlers transformation

Utilizing the electric Harrison transformation developed in five-dimensional minimal supergravity, we construct an exact solution characterizing non-BPS (Bogomol’nyi-Prasad-Sommerfield) charged rotating black holes with a horizon cross section of a lens space <math><mi>L</mi><mo>(</mo><mi>n</mi><mo>;</mo><mn>1</mn><mo>)</mo></math>. Among these solutions, only the ones corresponding to <math><mi>n</mi><mo>=</mo><mn>0</mn></math> and <math><mi>n</mi><mo>=</mo><mn>1</mn></math> do not have any curvature singularities, conical singularities, Dirac-Misner string singularities, and orbifold singularities both on and outside the horizon; additionally, they are free from closed timelike curves. The solution for <math><mi>n</mi><mo>=</mo><mn>0</mn></math> corresponds to the charged dipole black ring that we constructed in the previous paper. The specific solution for <math><mi>n</mi><mo>=</mo><mn>1</mn></math>, referred to as the “capped black hole,” was introduced in our previous article. This provides the first example of a non-BPS exact solution, representing an asymptotically flat, stationary spherical black hole with a domain of outer communication (DOC) having a nontrivial topology in five-dimensional minimal supergravity. We demonstrate that the DOC on a time slice has the topology of <math><mrow><mo>[</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>#</mo><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>]</mo><mo>\</mo><msup><mrow><mi>B</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math>. Differing from the well-known Myers-Perry and Cvetič-Youm black holes describing a spherical horizon topology and a DOC with a trivial topology of <math><msup><mi>R</mi><mn>4</mn></msup><mo>\</mo><msup><mi>B</mi><mn>4</mn></msup></math> on a timeslice, the capped black hole’s horizon is capped by a disk-shaped bubble. We explicitly demonstrate that the capped black hole carries mass, two angular momenta, an electric charge, and a magnetic flux, with only three of these quantities being independent. Furthermore, we reveal that this black hole can possess identical conserved charges as the Cvetič-Youm black hole. The existence of this solution challenges black hole uniqueness beyond both the black ring and the BPS spherical black hole. Moreover, within specific parameter regions, the capped black hole can exhibit larger entropy than the Cvetič-Youm black hole.

Solution generation of a capped black hole

We present the first non-BPS (Bogomol’nyi-Prasad-Sommerfield) exact solution of an asymptotically flat, stationary spherical black hole having domain of outer communication with nontrivial topology in five-dimensional minimal supergravity. It describes a charged rotating black hole capped by a disk-shaped bubble. The existence of the “capped black hole” shows the nonuniqueness of spherical black holes.

Capped black hole in five dimensions

We investigate possible configurations for vacuum multiblack holes that maintain static equilibrium in expanding bubbles. Our analysis assumes a five-dimensional Weyl metric to describe the spacetime, facilitating the derivation of solutions based on the provided rod structure. We consider a spacetime having expanding bubbles caused by one or two acceleration horizons, and show that various configurations such as two bubbles, four bubbles devoid of horizons, a black saturn, a black di-ring, a bicycling black ring (orthogonal black di-ring), and a five-dimensional black hole binary can achieve equilibrium within expanding bubbles. Specifically, we demonstrate that equilibrium requires two acceleration horizons on both sides for the bicycling ring and the five-dimensional black hole binary. However, only one acceleration horizon is necessary for achieving equilibrium in the case of the black saturn and the black di-ring.

Static equilibrium of multi–black holes in expanding bubbles in five dimensions

We present an exact solution for a non-Bogomol’nyi-Prasad-Sommerfield (non-BPS) charged rotating black ring endowed with a dipole charge in the bosonic sector of five-dimensional minimal supergravity. Utilizing the electric Harrison transformation, we derive this solution by converting a five-dimensional vacuum solution into a charged solution within the realm of five-dimensional minimal supergravity. As the seed solution for the Harrison transformation, we use a vacuum solution of a rotating black ring possessing a Dirac-Misner string singularity. The resulting solution exhibits regularity, indicating the absence of curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. This obtained solution carries the mass, two angular momenta, an electric charge, and a dipole charge, with only three of these quantities being independent, similar to the charged rotating dipole black ring found previously by Elvang, Emparan and Figueras. However, aside from the vacuum case, these two solutions do not coincide. We discuss the difference between them in the phase space.

New construction of a charged dipole black ring by the Harrison transformation

We investigate the nonlinear dynamics of D = 2N + 3 Myers-Perry black holes with almost equal angular momenta, which have N equal spins out of possible N + 1 spins. In particular, we study the ultraspinning instability and the fate of its nonlinear evolution using the large D effective theory approach. We find that every stationary phase can be mapped to the counterpart in the singly rotating phase within the leading order effective theory. From the known results of singly rotating solutions, we obtain the phase diagram of almost equally rotating black holes. We also obtain a certain implication for the possible topology changing transition.

Dynamics of Myers-Perry black holes with almost equal angular momenta in odd dimensions

We describe the dynamical evaporation of a black hole as the classical evolution in time of a black hole in an Anti-de Sitter braneworld. A bulk black hole whose horizon intersects the brane yields the classical bulk dual of a black hole coupled to quantum conformal fields. The evaporation of this black hole happens when the bulk horizon slides off the brane, making the horizon on the brane shrink. We use a large-D effective theory of the bulk Einstein equations to solve the time evolution of these systems. With this method, we study the dual evaporation of a variety of black holes interacting with colder radiation baths. We also obtain the dual of the collapse of holographic radiation to form a black hole on the brane. Finally, we discuss the evolution of the Page curve of the radiation in our evaporation setups, with entanglement islands appearing and then shrinking during the decreasing part of the curve.