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On the Extra Mode and Inconsistency of Horava Gravity

D. Blas, O. Pujolas, S. Sibiryakov

TL;DR

Horava gravity introduces an extra scalar mode by breaking full diffeomorphism invariance, and this mode is governed by a first-order-in-time equation on generic inhomogeneous, time-dependent backgrounds, with a background-dependent dispersion. Using both the original ADM formulation and the Stueckelberg covariantization, the paper shows this mode experiences fast short-distance instabilities and very low strong-coupling scales, undermining IR consistency; the dispersion and strong-coupling depend on background curvature, vanishing in flat or cosmological settings. The non-projectable theory thus faces severe pathologies, while the projectable version maps to ghost condensate dynamics with caustics and similarly low cutoffs. The work highlights fundamental obstacles in Horava gravity as a quantum gravity candidate and outlines possible routes—either reducing symmetry to ghost-condensate-like theories or promoting the extra mode to a full scalar with nonlinear lapse terms—each with its own conceptual and phenomenological challenges (e.g., Lorentz violation, black hole thermodynamics).

Abstract

We address the consistency of Horava'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 Stueckelberg 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 Horava'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 Stueckelberg formalism to Horava's model due to its non-relativistic nature.

On the Extra Mode and Inconsistency of Horava Gravity

TL;DR

Horava gravity introduces an extra scalar mode by breaking full diffeomorphism invariance, and this mode is governed by a first-order-in-time equation on generic inhomogeneous, time-dependent backgrounds, with a background-dependent dispersion. Using both the original ADM formulation and the Stueckelberg covariantization, the paper shows this mode experiences fast short-distance instabilities and very low strong-coupling scales, undermining IR consistency; the dispersion and strong-coupling depend on background curvature, vanishing in flat or cosmological settings. The non-projectable theory thus faces severe pathologies, while the projectable version maps to ghost condensate dynamics with caustics and similarly low cutoffs. The work highlights fundamental obstacles in Horava gravity as a quantum gravity candidate and outlines possible routes—either reducing symmetry to ghost-condensate-like theories or promoting the extra mode to a full scalar with nonlinear lapse terms—each with its own conceptual and phenomenological challenges (e.g., Lorentz violation, black hole thermodynamics).

Abstract

We address the consistency of Horava'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 Stueckelberg 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 Horava'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 Stueckelberg formalism to Horava's model due to its non-relativistic nature.

Paper Structure

This paper contains 8 sections, 99 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Cauchy problem for Hořava gravity
  3. The elusive mode
  4. Stückelberg formalism
  5. Instability and strong coupling
  6. Hořava gravity with projectable lapse as a ghost condensate
  7. Conclusions and discussion
  8. WKB expansion for the extra mode

Figures (1)

  • Figure 1: The energy scales appearing in the Hořava model: the Planck mass $M_P$ and the strong coupling scale $\Lambda_{sc}$. The theory can be trusted only in the narrow window of energies $0<E<\Lambda_{sc}$ (dashed region). The scale $\Lambda_{sc}$ is much smaller than $M_P$, it depends on the background curvature and goes to zero for flat / cosmological backgrounds, see Eq. (\ref{['omegacut']}) below.