A holographic journey from fluids to black holes

Fluid dynamics and gravitational physics are deeply interconnected by means of holography, a recent paradigm stating that any theory of gravity can be equivalently described by a quantum field theory in one dimension less.
The last 15 years of research have seen many developments toward a better understanding of such a paradigm which is believed to be a key ingredient for any theory of quantum gravity. Black holes, for instance, can be related to quantum field theories at thermal equilibrium and slightly perturbed black holes are holographically dual to the fluid behaviour of such thermally equilibrated systems. Many questions remain, however, still unanswered, for example how to go beyond the simplest case of Anti-de Sitter gravity, where the best example of holography can be realised; or how to probe physics beyond the black hole horizon etc.
Motivated by a better understanding of holography and fluid dynamics, in this thesis i) we successfully construct a holographic dual of the dissipationless effective action for conformal fluids providing a first derivation of such an effective action from first principles. ii) We generalise to a certain extent the procedure of obtaining holographic fluid behaviour from gravity for cases which are not necessarily Anti-de Sitter. iii) We show the differences between holographic fluid behaviour and an older approach to black hole physics going under the name of the membrane paradigm. iv) Finally we show how this membrane paradigm can be used instead as an approximation scheme for neglecting the interior of a black hole and provide its limits of validity.