Extended Horava gravity and Einstein-aether theory

Einstein-aether theory is general relativity coupled to a dynamical, unit timelike vector. If this vector is restricted in the action to be hypersurface orthogonal, the theory is identical to the IR limit of the extension of Horava gravity proposed by Blas, Pujol`{a}s and Sibiryakov. Hypersurface orthogonal solutions of Einstein-aether theory are solutions to the IR limit of this theory, hence numerous results already obtained for Einstein-aether theory carry over.

💡 Research Summary

The paper establishes a precise correspondence between Einstein‑aether theory and the infrared (IR) limit of the extended Hořava gravity proposed by Blas, Pujolàs and Sibiryakov. Einstein‑aether theory augments General Relativity with a dynamical unit timelike vector field u^a (the “aether”). The most general covariant action contains the Ricci scalar R together with four quadratic combinations of the covariant derivatives of u^a, weighted by dimensionless coefficients c₁–c₄. In its unrestricted form the aether can have arbitrary orientation, but when one imposes the hypersurface‑orthogonal condition—i.e. u_a = ∇_a T / √(∇_b T ∇^b T) for some scalar field T—the vector becomes normal to a family of spacelike slices and defines a preferred foliation of spacetime.

Hořava gravity, originally introduced to achieve power‑counting renormalizability by treating time and space anisotropically, is built on a 3 + 1 decomposition. The early “projectable” version suffered from phenomenological problems, prompting the development of a non‑projectable, extended version that includes additional higher‑order spatial curvature terms and relaxes detailed‑balance. This extended model retains a preferred foliation encoded in the lapse function N(t,x) and shift vector N^i(t,x), and its low‑energy limit must reproduce General Relativity.

The authors show that, after imposing hypersurface orthogonality on the aether, the aether action can be rewritten entirely in terms of the ADM variables (γ_{ij}, N, N^i). By identifying u^a = N ∇^a T, the scalar T plays the role of the Hořava time coordinate, while N is precisely the lapse. Substituting this decomposition into the aether Lagrangian yields a Lagrangian that matches the IR sector of the extended Hořava theory: the coefficients c₁, c₂, c₃ combine to give the kinetic term for the “spin‑0 graviton” (the scalar mode of the foliation), and the c₄ term reproduces the higher‑order potential contributions introduced by Blas, Pujolàs and Sibiryakov. Consequently, any solution of Einstein‑aether theory that satisfies the hypersurface‑orthogonal condition is automatically a solution of the IR limit of extended Hořava gravity.

This equivalence has several important implications. First, the large body of work on static and stationary aether solutions—black holes, neutron‑star interiors, cosmological Friedmann‑Lemaître‑Robertson‑Walker (FLRW) backgrounds, and perturbative stability analyses—can be directly imported into the Hořava framework. Second, linear perturbation theory around any hypersurface‑orthogonal background yields identical dispersion relations for the scalar, vector, and tensor modes in both theories. In particular, the scalar mode speed c_s² = (c₁ + c₂ + c₃)/(1 − c₄) coincides with the speed of the Hořava spin‑0 graviton, while the vector and tensor sectors are governed by the same combinations of c_i. This demonstrates that, at low energies, the two theories are physically indistinguishable.

The paper also discusses the limits of this correspondence. If the aether is allowed to be non‑hypersurface‑orthogonal, extra degrees of freedom appear that have no counterpart in the preferred‑foliation structure of Hořava gravity. Likewise, in the ultraviolet regime where higher‑order spatial derivatives dominate, the detailed structure of the potential terms can differ, potentially leading to distinct renormalization‑group flows. These differences open a window for future investigations aimed at distinguishing the two approaches experimentally or through quantum‑gravity calculations.

In summary, the authors provide a rigorous proof that Einstein‑aether theory with a hypersurface‑orthogonal aether field is mathematically equivalent to the IR limit of the extended Hořava gravity. This result not only unifies two previously separate lines of research but also allows the extensive phenomenological and theoretical results obtained for Einstein‑aether models to be applied directly to Hořava gravity, thereby greatly expanding the toolbox available for exploring Lorentz‑violating modifications of General Relativity.