Resumen de Black Holes: New Perspectives from Higher Dimensions
Nadal Haddad
However successful classical general relativity is, it faces non-normalizability problems at the quantum mechanical level. Nevertheless, it predicts the existence of black holes, which are thermal objects that are expected to still exist in a satisfactory theory of quantum gravity. This is one of the main reasons why black holes are important. String theory is considered, today, to be the most successful candidate for a quantum theory of gravity. Its mathematical consistency requires spacetime to have ten dimensions, and thus, it motivates the study of general relativity, and in particular, black holes, in higher dimensions. This is the main focus of our thesis. Moreover, the advent of the AdS/CFT correspondence, relating a theory of gravity to a quantum field theory in one less dimension have raised even more the interest in higher dimensional gravity. Compared to four-dimensional general relativity, which has been largely studied and investigated in the past years, general relativity in higher dimensions has many interesting and new features. Uniqueness is one property of four dimensional general relativity that does not continue to hold in higher dimensions. Another related property is that in four dimensions black holes can have horizons of a spherical topology only, while in higher dimensions black objects can have other topologies as well. Black rings are a manifestation of this. The last two new properties lead to topological phase transitions in the phase space of higher dimensional black holes, which is one of the topics we consider and touch upon in this thesis. The existence of extended black objects in higher dimensions, called black “p”-branes, is an important and fundamental issue. Those objects were found to be dynamically unstable. Therefore, the following of this instability has become an important problem. What is the end state of it?. This instability is known as the Gregory-Laamme instability and it is another topic we investigate in our thesis and which we relate to a type of fluid-gravity correspondence in at spacetime. The existence of black “p”-branes is attributed to the fact that in higher dimensional general relativity there are black holes that can rotate as fast as one wishes. That is, there is no analogue of the four-dimensional Kerr bound which prevents black holes from spinning very fast. The fast rotation, on the other hand, which is possible in higher dimensions has the effect that it makes the horizon of the black hole flattens along the rotation plane, and hence, in the limit it approaches the black “p”-brane geometry. In chapter [3] we give a short introduction to the blackfolds approach to higher dimensional black holes. We discuss the reasons that make new approximate methods available in higher dimensions. We introduce the basic objects, or the building blocks, of the approach, which are the boosted black “p”-branes, and explain how the blackfolds approach can be used to construct from them new stationary solutions to the gravitational field equations with new and novel topologies and how it can be used to construct dynamical solutions as well. In chapter [4] we give a brief introduction to the topic of topology change in general relativity. We discuss the Kaluza-Klein black hole phases, the homogeneous black string phase, the inhomogeneous black string phase and the localized black hole phase. We explain the mechanism under which a topological phase transition appears as one moves in the phase space from a black string to a localized black hole. In chapter [5] we managed to formulate a type of fluid-gravity correspondence in at space. We found a map between a fluctuating boosed black “p”-brane and a relativistic viscous fluid living at spatial infinity. That is, we constructed a solution to the vacuum Einstein equations of a dynamical black “p”-brane up to first order in derivatives. The dual viscous fluid is characterized by two parameters (transport coefficients) in the stress tensor - the shear and bulk viscosities. Using the gravitational solution we succeeded to compute those coefficients. Furthermore, by using the effective description of the black “p”-brane as a viscous fluid in a lower number of dimensions we studied the Gregory-Laamme instability and we found a striking agreement with the numerical results, already obtained, on this problem (which were obtained by analyzing the linearized gravitational equations). In more details, by including the damping effect of the viscosity in the unstable sound waves, we obtained a remarkably good and simple approximation to the dispersion relation of the Gregory-Laamme modes, whose accuracy increases with the number of transverse dimensions. We proposed an exact limiting form as the number of dimensions tends to infinity. Hence, instead of attacking the complicated (linearized) gravitational equations we propose instead to attack the relativistic Navier-Stokes equations of the effective (dual) fluid, as they are much simpler than the former. In chapter [6] we tackled and made progress in a different problem - topology change in higher dimensional general relativity. Before the advent of our work there had been no analytical (but only numerical) complete example of a topological phase transition in general relativity. The black string / black hole phase transition in Kaluza-Klein spaces was analyzed and studied only numerically. Neverthelss, some local models had been obtained analytically. It was argued that the phase transition passes through (or is mediated by) a singular critical configuration; a self-similar double cone geometry. This geometry was found only locally, though. In our work we have provided for the first time a complete analytical example of a topological phase transition. We took the rotating black hole inside de-Sitter space (the Kerr-de-Sitter black hole), and we showed that if one follows a trajectory in the phase space of solutions along which the black hole rotation is increased then it will pancake along the rotation plane, and finally it will touch the de-Sitter horizon along that plane. A merger transition is obtained if one continues to move along the same phase space trajectory, leading to a merged phase with a single connected horizon. In our example, the critical configuration (in which the two horizons just meet) is mediated by a self-similar double cone, thus confirming previous propositions of the mechanism under which topological transitions work. We also described local models for the critical geometries that control many transitions in the phase space of higher-dimensional black holes, such as the pinch-down of a topologically spherical black hole to a black ring or to a black p-sphere, or the merger between black holes and black rings in black Saturns or di-rings in D >/= 6. It is worth mentioning, in addition, that the cones we have found are general ones (compared to previously known ones), in the sense that they describe two horizons that intersect each other at different temperatures, not only ones that intersect each other with the same temperature. However, in the case the two horizons intersect at different temperatures, there can be no merger transition afterwards, and so the trajectory of solutions in the phase space ends there.
