On the Consistency of the Hořava Theory

Abstract

With the goal of giving evidence for the theoretical consistency of the Hořava Theory, we perform a Hamiltonian analysis on a classical model suitable for analyzing its effective dynamics at large distances. The model is the lowest-order truncation of the Hořava Theory with the detailed-balance condition. We consider the pure gravitational theory without matter sources. The model has the same potential term of general relativity, but the kinetic term is modified by the inclusion of an arbitrary coupling constant λ. Since this constant breaks the general covariance under space-time diffeomorphisms, it is believed that arbitrary values of λ deviate the model from general relativity. We show that this model is not a deviation at all, instead it is completely equivalent to general relativity in a particular partial gauge fixing for it. In doing this, we clarify the role of a second-class constraint of the model.

Role of the extra coupling in the kinetic term in Hořava-Lifshitz gravity

“…Comparing its low-energy limit with that of general relativity (GR) is therefore relatively straightforward, if one keeps in mind our earlier remark that a different functional form of the action does not necessarily signal an inequivalent theory. There are two apparently contradictory results [4,5] on the equivalence or otherwise of the low-energy limits of HLG and GR, which the present paper aims to resolve. As we will see, they are due to some subtleties in the comparison, including the fact that there are several versions of Hořava-Lifshitz gravity, whose classical limits are not the same.…”

“…We will refer to any theory with action (7) as a "λ-R model," a name coined by Bellorín and Restuccia in [5]. An assertion often made in this context is that in order for Hořava-Lifshitz gravity to reproduce general relativity in a low-energy limit, the parameter λ must flow towards its "relativistic" value λ ¼ 1, because only then the familiar-looking form of the action is recovered.…”

“…Omitting these terms, it was later argued in [5] that the resulting nonprojectable theory for generic values of λ reproduces GR. This is at first sight surprising, since the functional form of the action is then not that of the Einstein-Hilbert action, and it also appears to be in contradiction with the results by Giulini and Kiefer just described.…”

“…Section II contains our analysis of the constrained structure of the λ-R model derived from nonprojectable HLG, for closed spatial slices. This involves a careful reexamination of all steps made in the asymptotically flat case, which in [5] was shown to be equivalent to GR in a maximal slicing. We show that the tertiary constraint appearing in the spatially closed case has the general solution π ¼ a ffiffi ffi g p , where π is the trace of the momentum conjugate to the three-metric g ij , g denotes the determinant of the metric and aðtÞ is a function of time.…”

Non-perturbative analysis of the gravitational energy in Hořava Theory

“…For λ = 1 3 , G ijkl is not invertible. The theory with α = 0 for any value of λ is equivalent to General Relativity [8,9]. The terms which depends on N and their derivatives in the potential were introduced in [10].…”