Title:

Aspects of Lorentz violating theories of gravity

Lorentz symmetry is arguably the most fundamental symmetry of physics, at least in its modern conception. On the other hand, some of the issues that plague the currently accepted theory of gravitation could be solved by breaking such symmetry. The theory proposed by Petr Horava in 2009 brings forward exactly this aspect. The theory, dubbed Horava gravity, is a UV complete theory of gravity that is also renormalisable. It represents therefore a good candidate for a quantum theory of gravity. There are some issues though, which typically arise in any theory which explicitely violates Lorentz symmetry. In this thesis we will be concerned with two of these issues, in particular the matter problem and the existence of black holes. The first issue mentioned arises every time we try to couple matter to a Lorentz violating theory of gravity. Indeed, in the matter sector Lorentz symmetry is extremely well constrained, and therefore we need to find a way to avoid the percolation of Lorentz violations to the matter sector through higher order operators. One possible solution based on the separation of scales was proposed in the last few years (Pospelov et al.,2010). While studying the proposed mechanism though, the authors uncovered a naturalness problem in the vector sector of the theory. The solutions they proposed relies on the use of some higher derivative terms that are not normally present in the ``traditional'' Horava theory. It is unclear then what impact this type of terms can have on the whole theory. In our work we precisely addressed this question. We analysed the perturbations around Minkowski of the most generic theory extended to these type of terms, both from the point of view of the stability of the theory and of the renormalisability. What we found is that the theory retains its renormalisability, but some instabilities occur in the scalar sector. More work is hence required in order to understand whether such instabilities could be tamed, or if the mixed derivatives should be abandoned in favour of some alternative solution, not presently available. The second theme we concentrated on is that of the existence of black holes. The definition of black hole in general relativity rely strongly on the causal structure dictated by Lorentz symmetry. As soon as Lorentz symmetry is broken it is therefore unclear whether black holes will still exist. Surprisingly enough black holes have been shown to exist in Lorentz breaking theories, but a rigorous definition was still to be found. In our work we developed the mathematically rigorous definitions for the causal structure of foliated spacetimes and we defined for the first time black holes in such spacetimes. We also uncovered a number of interesting properties of this objects and we developed a local characterisation that allows one to locate horizons without the knowledge of the whole structure of the complete spacetime. Finally we developed the Initial Value Problem for these types of theory in the hope that new simulations of gravitational collapse will be performed using our analysis as a starting point. The thesis is organised as follows. In the first Chapter we give an introduction on gravity and the problems with its renormalization. We also introduce some of the theories that have been proposed to solve this difficulties. In the second Chapter we start discussing Lorentz violations and we provide a proof of the powercounting renormalizability of a toy model of a Lorentz violationg scalar field theory. We also introduce the theories that we will be studying throughout the thesis. In the third Chapter we discuss the mixed derivative extension to Horava gravity and we discuss the consequences of the new terms that occur in the theory. In the fourth and fifth Chapters we introduce the causal structure of spacetimes which violate Lorentz symmetry by means of a preferred foliation, we discuss the notion of black holes and horizons and we formalise some results present in the literature adapting them to our framework. In the sixth Chapter we then discuss the Initial Value Problem for such spacetimes, with some attention to the process of gravitational collapse leading to the formation of black holes. Finally in the last Chapter we draw some conclusions and discuss some ideas for future work.
