Studies of Black Hole Horizons

Abstract

This thesis has focused entirely on classical and thermodynamical aspects of black hole physics. We have developed four different projects involving different kinds of black holes.

1 BLACK BRANES IN A BOX Neutral black branes with extended horizons are dynamically unstable to long wavelength perturbations along their horizons; this instability is known as the Gregory-Laflamme instability. In some regimes, the dynamics of black branes can be captured by an effective hydrodynamic description. We have studied the effective hydrodynamics of neutral black branes inside a cylindrical cavity to investigate their dynamic and thermodynamic instabilities. We have used the size of the box as a control parameter for stability (smaller cavities increase rigidity and contribute to the stability of the solutions); we have ob¬served that both instabilities disappear at the same critical value of the cavity radius. We have discussed the Correlated Stability Conjecture, which relates thermodynamic and dynamic instabilities in these objects and we have argued that its correct interpretation is given by the Correlated Hydrodynamic Stability (CHS). The CHS relates the presence of unstable hydrodynamic modes to the local thermodynamic instability; this is transparent in our approach. In the effective fluid description we have computed the specific quantities that characterize the fluid. Finally we have studied the system close to the critical point at which the instability disappears and we have obtained that the wave number that marks the onset of the instability vanishes with a critical behaviour ruled by a critical exponent of 1/2.

2 BLACK STRING FLOW We have constructed an event horizon describing a heat flow, that remains constant in time, between to asymptotic regions at constant temperature. This horizon is the smooth interpolation between the horizon of a black string and a planar acceleration horizon. This was the first exact description of a flowing horizon connecting a stringlike horizon with a planar one (this can also be an infinitely big spherical black hole); the construction is valid for any number of dimensions greater than four. We obtained the horizon generators as well as the exact geometry and we showed that this horizon resembles that of flowing funnels. We computed a surface gravity that approaches on one end, the black string's surface gravity, and on the other, the infinite black hole's surface gravity which is 0. We also computed the expansion associated to the horizon generators and it vanishes in both asymptotic regions; thus reflecting the property that the black string flow horizon interpolates between two asymptotic horizons, each of which is asymptotically in equilibrium at different temperature. This construction shows that stationary black holes with non-killing horizons are possible with non-AdS asymptotics.

3 BUMPY BLACK HOLES We have constructed numerically three new families of stationary black holes with a single angular momentum. These black holes have spherical topology but they differ from the Myers Perry solution (higher dimensional generalisation of Kerr solution) in that the radius of the sphere transverse to rotation varies non-monotonically with the polar angle. We have seen that half of these solutions connect, in the space of solutions, the Myers Perry family with other families featuring non-spherical topology such as the black ring, the black saturn, etc. We found strong evidence for the presence of cones in the horizons of solutions close to the topological transition in solution space. The other half of the solutions spread widely in the rotation plane and develop a singularity along their equator. These probably do not connect to other stationary black hole branches. We have also studied stability properties of all branches.

4 BLACK HOLE MERGER We have described in an exact analytic way the event horizon of a black hole merger in the extreme mass ratio (EMR) limit; we have done it for four and five dimensions. Curiously numerical computation in which the ratio of the masses is large are difficult and not very well studied. We hope our exact result can serve as check/guide for future results in the area. We constructed the event horizon of this dynamical process by computing its null generators. We extracted a number of parameters that characterise the merger. We identified the line of caustics, the critical radius at which both horizons touch, the big horizon relaxation timescale among other things. We showed that our hypersurface describes all possible mergers, in the EMR limit, for which the small black hole is non-rotating. Finally we analysed the instants shortly before and after the pinch-on and found evidence for critical behaviour in the forming of the cusp and in the initial growth of the throat.

Esta tesis está enmarcada en el campo de los agujeros negros. En ella se han realizado cuatro proyectos que involucran diferentes tipos de agujeros negros.

1 BRANAS NEGRAS Hemos estudiado el sistema de una brana dentro de una cavidad cilíndrica con condiciones de Dirichlet para investigar la relación entre las inestabilidades dinámicas y termodinámicas presentes en la brana. Hemos empleado las técnicas de la teoría efectiva de worldvolumes para tiranas; en esta teoría la descripción del sistema (en algunas condiciones) se da a través de variables y ecuaciones hidrodinámicas. Hemos estudiado el cambio de la inestabilidad de Gregory-Laflamme al variar el radio de la caja. Hemos identificado el radio crítico que estabiliza las soluciones y el comportamiento crítico de la inestabilidad en ese punto.

2 AGUJEROS NEGROS QUE FLUYEN Hemos construido un horizonte de sucesos que describe un flujo de calor, constante en el tiempo, entre dos regiones asintóticas a temperatura constante. Este horizonte es la interpolación entre el horizonte de una cuerda negra y un horizonte planar. La cuerda negra tiene cierta temperatura y el horizonte planar, en este caso, está a temperatura cero. La construcción se ha hecho en espacio asintóticamente plano, mostrando así que una constante cosmológica negativa no es estrictamente necesaria para la existencia de agujeros negros estacionarios con que no son de Killing.

3 AGUJEROS NEGROS BUMPY Hemos construido numéricamente tres familias nuevas de agujeros negros estacionarios con un solo momento angular en seis dimensiones. Estos agujeros tienen topología esférica pero el radio de la esfera transversa a la rotación varía de manera no monótona a lo largo del ángulo polar. La mitad de estas soluciones conectan a la familia de Myers Perry con otras de topología no esférica como el anillo negro o el saturno negro, etc.

La otra mitad, se extienden mucho en el plano de rotación y acaban por tener una singularidad localizada en el ecuador.

4 FUSIÓN DE AGUJEROS NEGROS

Para el caso en que las masas de dos agujeros negros difieran mucho una de la otra hemos mostrado que una descripción analítica del proceso de fusión de dos agujeros negros es posible. Hemos obtenido los rayos de luz que generan el horizonte de sucesos de una colisión de agujeros negros en el límite de razón de masas extremas. Extraemos propiedades importantes y damos una caracterización muy completa del proceso de fusión.