Correspondences in higher-dimensional gravity

Abstract

In this thesis we have made progress on the study of higher dimensional gravity by focusing on the properties of black holes and branes and their dynamics. We have developed two main projects: • provide several maps between different spacetimes • determine the hydrodynamical behavior of fluids dual to some classes of black holes This work improves the current understanding of GR in spacetimes with general dimension and gives hints for holography in spacetimes different from AdS. Here we give a brief summary of the work developed underling the main results achieved. In Chapter 2, we introduce the techniques applied for studying black brane hydrodynamics. In the long-wavelength regime, black hole dynamics can be related to fluid dynamics and one can develop effective theories which capture the hydrodynamical description of such black holes. We review two of these: the fluid/gravity correspondence and the blackfold approach. We have hence learnt that black holes behave as fluids under certain circumstances. One can therefore compute the effective stress energy tensor associated to the fluid, extract the corresponding dissipative transport coefficients and possibly perform a stability analysis. In Chapter 3, we have introduced the AdS/Ricci flat correspondence, which is a relation between a class of AdS spacetimes and Einstein solutions with zero cosmological constant. Remarkably, we have developed an extension of such correspondence to spacetimes with positive cosmological constant, including scalar matter. This AdS/dS correspondence may possibly give hints to improve our understanding of holography in dS space. We have also found a new Kerr/AdS solution with hyperbolic horizon from a known Kerr/dS one through the map. The hydrodynamics of fluids using the KK dimensional reduction was studied in Chapter 4. Choosing a generic relativistic fluid, performing a boost in N internal dimensions, compactifing them and reducing on an N dimensional torus we have obtained a charged fluid with N charges. Therefore, we have investigated the variation of the transport coefficients, the shear and bulk viscosity, of the original theory and we were also able to compute the thermal conductivity. The same analysis has been applied to a particular fluid: the fluid dual to a black p-brane. We were able to compute the shear viscosity, bulk viscosity and thermal conductivity matrix for a black p-brane with N charges in the compact directions. This method is particularly interesting since it allows studying the hydrodynamics of charged objects without performing a perturbative analysis but only applying dimensional reduction techniques. Using the AdS/Ricci flat correspondence we have checked that our mapped transport coefficients coincide with the ones obtained for a known charged AdS black branes.

In Chapter 5 we have investigated the hydrodynamics properties of fundamentally charged (dilatonic) black branes and branes with Maxwell charge smeared over their worldvolume. We have determined the dissipative behavior of the effective fluids associated to those branes in terms of the transport coefficients of the effective stress energy tensor. Studying the response to small long-wavelength perturbations we have analyzed the dynamical stability of both classes of charged black branes. We have moreover modified the AdS/Ricci flat correspondence to include charged cases using a non-diagonal KK reduction.

In this thesis we have shown how higher dimensional gravity is surprisingly rich of new phenomena and bizarre features. Playing with spacetime dimension is the key to probe GR. Hopefully, we will able to improve our comprehension of this mysterious and powerful theory. Holography is an extremely useful tool available for this aim. Mapping apparently unrelated theories living in different number of dimensions has revealed various successful predictions and results but above all opens new perspective for our perception and understanding of GR.

Esta tesis se centra principalmente en el estudio de la gravedad en dimensiones superiores con un enfoque en las relaciones entre diferentes tipos de espaciotiempo y el análisis y caracterización de agujeros negros. Para este último objetivo hemos desarrollado y adaptado teorías efectivas que nos permiten estudiar la dinámica de agujeros negros en ciertos regímenes. Hemos presentado dos de ellas: la "fluid/gravity correspondence" y el metodo de "blackfold". Se puede demostrar entonces que los agujero negros admiten una descripción hidrodinámica y se puede calcular el tensor energía-impulso asociado al fluido dual al agujero negro y extraer los coeficientes de transporte al primer orden en derivadas. Hemos utilizado estas técnicas para analizar propiedades hidrodinámicas de branas negras en el caso en que las branas llevan cargas de diferentes tipos. En particular, consideramos los casos en que la brana negra está acoplada a un potencial de (p+1)-forma, que llamamos brana con carga fundamental, y brana acoplada a un campo de Maxwell. También hemos investigado las propiedades de estabilidad de estos sistemas hidrodinámicos . Otra línea de investigación es el estudio de la hidrodinámica de fluidos utilizando la reducción dimensional de Kaluza Klein. Empezamos considerando un fluido genérico y luego hemos particularizado el cálculo al fluido dual a una p-brana negra. Hemos investigado como varían los coeficientes de transporte de la teoría inicial como la "shear and bulk viscosity" y además hemos conseguido calcular la matriz de conductividad térmica. Como último proyecto hemos desarrollo mapas entre espaciotiempos diferentes. En particular hemos extendido el "AdS/Ricci-flat correspondence" para espacios de Einstein con curvatura positiva y negativa. Una vez derivado el mapa, lo hemos aplicado a espacios de Sitter (dS) y AdS y a agujeros negros de Schwarzschild-dS/AdS. Además, hemos estudiado perturbaciones en la frontera de AdS, que a través del mapa nos dan sugerencias sobre una posible construcción de holografía en espacio de dS. De hecho, la frontera de un espacio asintóticamente AdS se mapea en una brana en el centro de dS y las perturbaciones cerca de la frontera tienen como fuente un tensor energía-impulso confinado en esta brana.