# Black Holes' New Horizons (16w5008)

Arriving in Oaxaca, Mexico Sunday, May 15 and departing Friday May 20, 2016

## Objectives

Black hole theory is one of the most interesting and intriguing topics of the modern science. It is in the focus of interests of astrophysics as well as of theoretical and mathematical physics. Black holes are solutions of the Einstein equations, which possess a number of remarkable properties. Recently the status of black holes has dramatically changed. There exist more and more observational evidence that stellar mass and supermassive black holes do exist in our Universe. Theoretical and mathematical interest in black holes has greatly increased after understanding that black holes allow one to test new fundamental ideas such as string theory and extra dimensions. Study of higher-dimensional black holes requires contemporary results from topology, differential and Riemannian geometry and the global theory of partial differential equations. Recent wide discussion of the information puzzle in black holes (e.g., concerning a possible firewall at the horizon) has attracted new attention to longstanding problems of the foundations of quantum theory and quantum gravity. The study of the motion of massive particles and photons, and of the propagation of fields, in the spacetime of higher-dimensional black holes provides one with new mathematically interesting results: for example, a new wide class of completely integrable dynamical systems, a new broad generalization of spheroidal functions etc. The study of black-hole formation in the processes of the coalescence of binary relativistic objects and in the collisions of highly relativistic objects in our and higher dimensions require the development of numerical methods for integrating highly non-linear partial differential equations. Developing of the corresponding mathematical tools for the study of these and other problems of black-hole theory has now become highly important. To be more concrete let us illustrate this by a few examples.

• The black-hole information puzzle. By an adopted definition, a black hole is a spacetime region from which information-carrying signals cannot escape to infinity. This means that the global causal structure of the spacetime in the presence of a black hole is non-trivial. Quantum effects result in the evaporation of the black hole. If it disappears in this process and the information captured by the black hole does not return to the external space, one would effectively deal with the breaking down of the unitarity property of the theory. This is a longstanding problem; however recently it attracted a lot of attention. String theory strongly advocates a point of view that information is not fundamentally lost in black hole formation and evaporation, but this view is not yet universally accepted, and in fact there are many unknown issues connected with how information may get back out. One recent very hot topic is the argument for firewalls at the horizon of a sufficiently old black hole, with about one hundred papers on this question in the past year alone. A firewall is a very unpalatable conclusion, but many researchers do not see a plausible way to get around it, so it has become a key paradox in our present incomplete understanding of black holes. In his recent publication (January 2014) Hawking expressed his opinion that the adopted mathematical definition of the black hole must be modified in order to include a possibility of unitarity restoration. This as well as the firewall proposal has generated a lot of discussion.

• Black hole entropy and the AdS/CFT correspondence. In the presence of a bulk AdS black hole, its Hawking temperature is connected with thermality of the corresponding conformal field theory at the boundary. Construction of solutions for such bulk higher dimensional black holes and studying their properties is an important problem. The AdS/CFT correspondence has been recently applied for studying physics in the vicinity of extreme rotating black holes and to the problem of the microscopic origin of black hole entropy.

• The subject of higher-dimensional gravity has become very hot and has attracted a lot of attention of theoreticians and mathematicians. Black holes are important elements in this study playing a role of probes of extra dimensions. It has been demonstrated that there exists a large variety of black hole solutions in higher-dimensional Einstein theories of gravity. These solutions differ by the topology of their event horizons. Explicit examples were obtained in 5 dimensions. There are indications that similar and more complicated black hole solutions exist in 6 and higher dimensions. An important problem, which requires developing appropriate mathematical tools, is to find such solutions. Another open problem is the dynamical stability of higher-dimensional black holes.

• In models with large extra dimensions the latter are usually assumed to be compact. As a result, the variety of `black objects’ is wider and includes different types of black strings, branes and so on. An interesting problem is to study `different phases' in a space of higher-dimensional black-hole solutions in a spacetime with compactified extra dimensions and transitions between these phases. This work began with the discovery by Gregory and Laflamme in 1993 of an instability of black strings, but many details of this process remains unclear until now.

• It has been known for a long time that geodesic equations in the Kerr metric are completely integrable. Recently it was discovered that the complete integrability is a characteristic common property of all stationary higher-dimensional black holes with horizons having spherical topology. Geodesic motion in these spacetimes are new physically interesting cases of completely integrable systems. There exist a number of interesting mathematical problems, such as the construction of Lax pairs for such systems and the application of the Kolmogorov-Arnold-Moser method for studying general properties of these dynamical systems.

• It was shown that the same hidden symmetries, which are responsible for the complete integrability of geodesic equations, also imply complete separation of variables in the physically interesting field equations. Ordinary differential equations obtained by such a separation are second-order linear equations with polynomial coefficients. The power of these polynomials grows with the increasing number of the spacetime dimensions. The Sturm-Liouville problem for `angular' eigen-modes and general properties of the `radial equations' are open problems, with interesting physical applications.

• Entanglement entropy and minimal surfaces in AdS spaces.

• Black-hole numerics.

These problems of black hole theory lie in the mainstream of modern theoretical and mathematical physics. They are widely discussed now. Their study requires contemporary mathematical tools. It happens that theoretical, mathematical and computational methods which are used for their study have many common features. This is one of the reasons why these topics were selected for the proposed workshop. Our goal is to understand better those properties of the black objects which are common for four and higher dimensions, and those which are different. In particular we shall focus on the transitions between different black objects in higher dimensions, trying to understand the main features of such transitions in which the topology of the horizon changes. There are indications that these processes may be similar to the critical phase transitions in thermodynamics and the critical collapse phenomena in gravity. One may also expect that during these transitions naked singularities can be formed, so that they can provide an interesting and important counter-example to the famous Penrose conjecture on cosmic censorship. The above list of selected subjects, proposed for discussion at the workshop, represents some of the `hottest’ problems of black holes at the present time.

Besides of the review talks of the specially invited leading experts, we are planning informal discussions and shorter talks containing the progress reports. We are also planning to invite several PhD students and PDFs working in the area of physics and mathematics of black hole.

We have experience in organizing the BIRS workshops. Valeri Frolov and Don Page have organized the 2 day BIRS workshop 05w2041 2005-05-12 and two 5-day workshops, 08w5033 2008-11-09 and 11w5099-2011-11. In addition, Valeri Frolov was a co-organizer of the BIRS 5 day workshop 06w5080-2006-07-29.

• The black-hole information puzzle. By an adopted definition, a black hole is a spacetime region from which information-carrying signals cannot escape to infinity. This means that the global causal structure of the spacetime in the presence of a black hole is non-trivial. Quantum effects result in the evaporation of the black hole. If it disappears in this process and the information captured by the black hole does not return to the external space, one would effectively deal with the breaking down of the unitarity property of the theory. This is a longstanding problem; however recently it attracted a lot of attention. String theory strongly advocates a point of view that information is not fundamentally lost in black hole formation and evaporation, but this view is not yet universally accepted, and in fact there are many unknown issues connected with how information may get back out. One recent very hot topic is the argument for firewalls at the horizon of a sufficiently old black hole, with about one hundred papers on this question in the past year alone. A firewall is a very unpalatable conclusion, but many researchers do not see a plausible way to get around it, so it has become a key paradox in our present incomplete understanding of black holes. In his recent publication (January 2014) Hawking expressed his opinion that the adopted mathematical definition of the black hole must be modified in order to include a possibility of unitarity restoration. This as well as the firewall proposal has generated a lot of discussion.

• Black hole entropy and the AdS/CFT correspondence. In the presence of a bulk AdS black hole, its Hawking temperature is connected with thermality of the corresponding conformal field theory at the boundary. Construction of solutions for such bulk higher dimensional black holes and studying their properties is an important problem. The AdS/CFT correspondence has been recently applied for studying physics in the vicinity of extreme rotating black holes and to the problem of the microscopic origin of black hole entropy.

• The subject of higher-dimensional gravity has become very hot and has attracted a lot of attention of theoreticians and mathematicians. Black holes are important elements in this study playing a role of probes of extra dimensions. It has been demonstrated that there exists a large variety of black hole solutions in higher-dimensional Einstein theories of gravity. These solutions differ by the topology of their event horizons. Explicit examples were obtained in 5 dimensions. There are indications that similar and more complicated black hole solutions exist in 6 and higher dimensions. An important problem, which requires developing appropriate mathematical tools, is to find such solutions. Another open problem is the dynamical stability of higher-dimensional black holes.

• In models with large extra dimensions the latter are usually assumed to be compact. As a result, the variety of `black objects’ is wider and includes different types of black strings, branes and so on. An interesting problem is to study `different phases' in a space of higher-dimensional black-hole solutions in a spacetime with compactified extra dimensions and transitions between these phases. This work began with the discovery by Gregory and Laflamme in 1993 of an instability of black strings, but many details of this process remains unclear until now.

• It has been known for a long time that geodesic equations in the Kerr metric are completely integrable. Recently it was discovered that the complete integrability is a characteristic common property of all stationary higher-dimensional black holes with horizons having spherical topology. Geodesic motion in these spacetimes are new physically interesting cases of completely integrable systems. There exist a number of interesting mathematical problems, such as the construction of Lax pairs for such systems and the application of the Kolmogorov-Arnold-Moser method for studying general properties of these dynamical systems.

• It was shown that the same hidden symmetries, which are responsible for the complete integrability of geodesic equations, also imply complete separation of variables in the physically interesting field equations. Ordinary differential equations obtained by such a separation are second-order linear equations with polynomial coefficients. The power of these polynomials grows with the increasing number of the spacetime dimensions. The Sturm-Liouville problem for `angular' eigen-modes and general properties of the `radial equations' are open problems, with interesting physical applications.

• Entanglement entropy and minimal surfaces in AdS spaces.

• Black-hole numerics.

These problems of black hole theory lie in the mainstream of modern theoretical and mathematical physics. They are widely discussed now. Their study requires contemporary mathematical tools. It happens that theoretical, mathematical and computational methods which are used for their study have many common features. This is one of the reasons why these topics were selected for the proposed workshop. Our goal is to understand better those properties of the black objects which are common for four and higher dimensions, and those which are different. In particular we shall focus on the transitions between different black objects in higher dimensions, trying to understand the main features of such transitions in which the topology of the horizon changes. There are indications that these processes may be similar to the critical phase transitions in thermodynamics and the critical collapse phenomena in gravity. One may also expect that during these transitions naked singularities can be formed, so that they can provide an interesting and important counter-example to the famous Penrose conjecture on cosmic censorship. The above list of selected subjects, proposed for discussion at the workshop, represents some of the `hottest’ problems of black holes at the present time.

Besides of the review talks of the specially invited leading experts, we are planning informal discussions and shorter talks containing the progress reports. We are also planning to invite several PhD students and PDFs working in the area of physics and mathematics of black hole.

We have experience in organizing the BIRS workshops. Valeri Frolov and Don Page have organized the 2 day BIRS workshop 05w2041 2005-05-12 and two 5-day workshops, 08w5033 2008-11-09 and 11w5099-2011-11. In addition, Valeri Frolov was a co-organizer of the BIRS 5 day workshop 06w5080-2006-07-29.