In this thesis, we examine the dynamical structure of Horava-Lifshitz gravity, and investigate its relationship with holography for anisotropic systems.
Horava-Lifshitz gravity refers to a broad class of gravitational models that incorporate anisotropy at a fundamental level. The idea behind Horava-Lifshitz gravity is to utilize ideas from the theory of dynamical critical phenomena into gravity to produce a theory of dynamical spacetime that is power-counting renormalizable, and is thus a candidate renormalizable quantum field theory of gravity.
One of the most distinctive features of Horava-Lifshitz gravity is that its group of symmetries consists not of the diffeomorphisms of spacetime, but instead of the group of diffeomorphisms that preserve a given foliation by spatial slices. As a result of having a smaller group of symmetries, HL gravity naturally has one more propagating degree of freedom than general relativity.
The extra mode presents two possible difficulties with the theory, one relating to consistency, and the second to its viability as a phenomenological model. (1) It may destabilize the theory. (2) Phenomenologically, there are severe constraints on the existence of an extra propagating graviton polarization, as well as strong experimental constraints on the value of a parameter appearing in the dispersion relation of the extra mode.
In the first part of this dissertation we show that the extra mode can be eliminated by introducing a new local symmetry which steps in and takes the place of general covariance in the anisotropic context. While the identification of the appropriate symmetry is quite subtle in the full non-linear theory, once the dust settles, the resulting theory has a spectrum which matches that of general relativity in the infrared. This goes a good way toward answering the question of how close Horava-Lifshitz gravity can come to reproducing general relativity in the infrared regime.
In the second part of the thesis we pursue the relationship between Horava-Lifshitz gravity and holographic duals for anisotropic systems. A holographic correspondence is one that posits an equivalence between a theory of gravity on a given spacetime background and a field theory living on the "boundary" of that spacetime, which resides at infinite spatial separation from the interior. It is a non-trivial problem how to define this boundary, but in the case of relativistic boundary field theories, there is a well-known definition due to Penrose of the boundary which produces the geometric structure required to make sense of the correspondence. However, the proposed dual geometries to anisotropic quantum field theories have a Penrose boundary that is incompatible with the assumed correspondence. We generalize Penrose's approach, using concepts from Horava-Lifshitz gravity, to spacetimes with anisotropic boundary conditions, thereby arriving at the concept of anisotropic conformal infinity that is compatible with the holographic correspondence in these spacetimes.
We then apply this work to understanding the structure of holography for anisotropic systems in more detail. In particular, we examine the structure of divergences of a certain theory of gravity on Lifshitz space. We find, using our construction of anisotropic conformal infinity, that the appropriate geometric structure of the boundary is that of a foliated spacetime with an anisotropic metric complex. We then perform holographic renormalization in these spacetimes, yielding a computation of the divergent part of the effective action, and find that it exhibits precisely the structure of a Horava-Lifshitz action. Moreover, we find that, for dynamical exponent z = 2, the logarithmic divergence gives rise to a conformal anomaly in 2+1 dimensions, whose general form is precisely that of conformal Horava-Lifshitz gravity with detailed balance.
