On the extra mode and inconsistency of Hořava gravity

Abstract

We address the consistency of Hořava's proposal for a theory of quantum gravity from the low-energy perspective. We uncover the additional scalar degree of freedom arising from the explicit breaking of the general covariance and study its properties.The analysis is performed both in the original formulation of the theory and in the Stückelberg picture. A peculiarity of the new mode is that it satisfies an equation of motion that is of first order in time derivatives. At linear level the mode is manifest only around spatially inhomogeneous and time-dependent backgrounds. We find two serious problems associated with this mode. First, the mode develops very fast exponential instabilities at short distances. Second, it becomes strongly coupled at an extremely low cutoff scale. We also discuss the "projectable" version of Hořava's proposal and argue that this version can be understood as a certain limit of the ghost condensate model.The theory is still problematic since the additional field generically forms caustics and, again, has a very low strong coupling scale. We clarify some subtleties that arise in the application of the Stückelberg formalism to Hořava's model due to its non-relativistic nature. 1Recently, Hořava has proposed a new approach to the theory of quantum gravity [1]. The key idea of the proposal is to equip space-time with a new structure: a foliation by spacelike surfaces. This foliation defines the splitting of the coordinates into "space" and "time" and breaks the general covariance of general relativity (GR). Then one can improve the UV behavior of the graviton propagator and ultimately make the theory power-counting renormalizable by adding to the GR action terms with higher spatial derivatives. At the same time the action in the ADM formalism contains only first order time derivatives, which allows to circumvent the problems with the ghosts appearing in covariant higher order gravity theories [2]. The higher derivative terms naively become irrelevant in the infrared and it was argued in [1] that the theory reduces to GR at large distances.However, the consistency of the above proposal is far from being clear. The main concern comes from the fact that the introduction of a preferred foliation explicitly breaks the gauge group of GR down to the group of space-time diffeomorphisms preserving this foliation. As already pointed out in [1] this breaking is expected to introduce extra degrees of freedom compared to GR. The new degrees of freedom can persist down to the infrared and lead to various pathologies (instabilities, strong coupling) that may invalidate the theory. An illustration of this phenomenon is provided by theories of massive gravity where special care is needed to make the additional degrees of freedom well-behaved [3,4,5].

Non-relativistic holography from Hořava gravity

“…The simplest potential term involving up to two derivatives, as appropriate for the low energy limit, is given by [18,20] …”

“…Hořava gravity can be usefully embedded into standard GR via a Stückelberg-like mechanism [34,18,20]. This formalism makes the extra degree of freedom explicit by coupling Einstein gravity to an additional scalar field φ.…”

Stability of spin-0 graviton and strong coupling in Horava-Lifshitz theory of gravity

“…Despite all of these attractive features, the theory plagues with two serious problems: the instability of the Minkowski background [4,11,[20][21][22], and the strong coupling [23][24][25][26][27]. To solve these problems, various modifications were proposed [28,29].…”

Weighted power counting and perturbative unitarity

“…Unfortunately, Lifshitz type UV-completions of these theories are not necessarily without problems. The proposed UV completion of 5d QED exhibits a fine tuning problem [12], and in the case of Hořava-Lifshitz gravity there has been concern over the consistency of various versions of the theory [13][14][15][16][17][18]. Some of these versions become strongly coupled at a certain scale and there is a breakdown of the perturbative expansion (for recent reviews related to this problem see [19,20]).…”

“…Some of these versions become strongly coupled at a certain scale and there is a breakdown of the perturbative expansion (for recent reviews related to this problem see [19,20]). One way to see this breakdown is to check the bound for perturbative unitarity [13,21]. Since making a theory power-counting renormalizable does not guarantee the absence of strong coupling, it is interesting to ask what happens to perturbative unitarity for an arbitrary theory at a Lifshitz point.…”