Hořava Gravity and Foliation-Preserving Symmetry

Hořava gravity treats a foliation—not the full spacetime metric alone—as fundamental structure. Its reduced gauge symmetry permits arbitrarily high spatial derivatives while retaining only two time derivatives, changing both ultraviolet power counting and the constraint system. The phrase “Hořava gravity” therefore does not name one action: projectable, nonprojectable, and covariant khronometric formulations have different lapse variations and must be kept distinct.

Required background. Frames, Tetrads, and Spinors supplies the frame decomposition used to define the unit normal to the preferred leaves.

Helpful background. Einstein Equations provides the GR comparison; Gravitational Actions and Boundary Terms supplies the action variation; Formulations of General Relativity supplies ADM constraints; and Gravity Beyond General Relativity supplies the theory-family context.

Foliation symmetry and anisotropic scaling

Use the site signature $(-,+,+,+)$ and ADM line element

$$ds^2=-N^2dt^2+h_{ij}(dx^i+N^i dt)(dx^j+N^j dt).$$

With future unit normal $n_a=-N\nabla_a t$, define

$$K_{ij}=\frac{1}{2N}(\dot h_{ij}-D_iN_j-D_jN_i).$$

Equivalently, $K_{ab}=h_a{}^ch_b{}^d\nabla_cn_d$, with $h_{ab}=g_{ab}+n_an_b$. This sign belongs to the displayed ADM package. Reversing $K_{ij}$ consistently leaves the quadratic combination $K_{ij}K^{ij}-\lambda_HK^2$ unchanged, but changes expressions linear in $K$, including the corresponding boundary terms and momenta. Translate the full package when comparing source conventions.

The gauge transformations preserve the leaves:

$$t\mapsto t'(t),\qquad x^i\mapsto x'^i(t,\mathbf{x}).$$

They are foliation-preserving diffeomorphisms, not arbitrary maps $t'(t,\mathbf{x})$. The ultraviolet fixed-point scaling is

$$\mathbf{x}\mapsto b\mathbf{x},\qquad t\mapsto b^z t.$$

In three spatial dimensions, $z=3$ lets a kinetic term with two time derivatives scale like a potential with six spatial derivatives. Schematically, the propagator becomes $(\omega^2-k^6/M_*^4)^{-1}$ at high momentum. This is the power-counting idea developed in Hořava 2009, §§2.1–2.4, especially Eqs. (2.2), (2.5), and (2.10)–(2.11), PDF. Power-counting renormalizability is a motivation and structural result, not by itself a proof of unitarity or a viable infrared flow.

Projectable and nonprojectable actions

A useful nonprojectable action through the infrared operators is

$$\begin{aligned} S_H=\frac{1}{16\pi G_H}\int dt\,d^3x\,N\sqrt h\bigg[ &K_{ij}K^{ij}-\lambda_H K^2+\xi\,{}^{(3)}R+\eta\,a_i a^i\\ &+M_*^{-2}\mathcal V_4+M_*^{-4}\mathcal V_6\bigg], \end{aligned}$$

where

$$a_i=D_i\ln N.$$

$\mathcal V_4$ and $\mathcal V_6$ contain all allowed four- and six-spatial-derivative scalars consistent with the foliation symmetry. Boundary data and coefficients must be chosen so that integrations by parts used to define this basis are valid. The theory is an effective derivative expansion below its stated completion scale; the two-derivative infrared truncation cannot be extrapolated through momenta where $\mathcal V_4$ or $\mathcal V_6$ dominate.

In the projectable theory, $N=N(t)$, hence $a_i=0$ identically. Varying the lapse gives one integrated Hamiltonian constraint,

$$\int_{\Sigma_t}d^3x\,\sqrt h\,\mathcal H=0,$$

rather than $\mathcal H(\mathbf{x})=0$ at every point. In the nonprojectable theory, $N=N(t,\mathbf{x})$; $a_i$ and its spatial derivatives are allowed operators, and lapse variation is local. The shift still supplies local momentum constraints, but the lapse constraint and its preservation differ from GR. The reduced gauge symmetry generically leaves an additional scalar gravitational mode.

The $a_i a^i$ term is not cosmetic. Around flat space it repairs the scalar quadratic action in a parameter region where its kinetic and gradient coefficients are positive; the original restricted nonprojectable model omits an operator that the symmetry allows. See Blas, Pujolàs, and Sibiryakov 2010, pp. 1–3, Eqs. (1), (6)–(19), PDF. Their analysis is a flat-background, quadratic, power-counting argument. It does not establish every nonlinear branch or every coefficient choice.

The khronometric covariant form

A scalar $T$ whose level sets are the preferred leaves defines

$$u_a=-\frac{\nabla_aT}{\sqrt{-g^{bc}\nabla_bT\nabla_cT}}, \qquad T\mapsto f(T),\quad f'\gt0.$$

Thus $u^a u_a=-1$ and the æther is hypersurface orthogonal. In unitary gauge $T=t$, its acceleration is $a_i=D_i\ln N$. Substituting this restricted æther into the Einstein–æther action gives the infrared khronometric action. Translating Jacobson’s $(+,-,-,-)$ formulas as a whole, the parameter map to the action above is

$$\xi=\frac{G_H}{G_{\mathrm{ae}}}=\frac{1}{1-c_{13}},\qquad \frac{\eta}{\xi}=c_{14},\qquad \frac{\lambda_H}{\xi}=1+c_2.$$

The decomposition $\nabla_a u_b=K_{ab}-u_a a_b$ in our sign package shows why only three combinations of the four æther couplings survive when the twist vanishes. Jacobson 2010, pp. 1–4, Eqs. (3), (8)–(10), and (19), PDF establishes the action relation and the fact that the khronon equation remains second order in preferred time even though it contains an additional spatial derivative.

Every hypersurface-orthogonal Einstein–æther solution satisfies the khronometric infrared equations. The converse is not automatic because varying an unrestricted vector and restricting afterward is stronger than restricting the action before variation. Spherical and homogeneous-isotropic æther configurations are hypersurface orthogonal, which explains the substantial overlap of those branches without making the full theories equivalent.

Checks and limiting cases

For $\lambda_H=\xi=1$, $\eta=0$, and vanishing higher-spatial-derivative coefficients, the ADM integrand is $K_{ij}K^{ij}-K^2+{}^{(3)}R$: the Einstein–Hilbert bulk action in the declared convention. The GR constraint symmetry is then restored. This is a genuine infrared GR limit only if matter couples to the same reconstructed metric and the extra scalar is not rendered strongly coupled before the limit is reached.

Projectability provides a discriminating check. Starting from the nonprojectable field equations and evaluating a solution with $a_i=0$ does not recover the projectable theory: the former was varied with arbitrary $\delta N(t,\mathbf{x})$, whereas the latter permits only $\delta N(t)$. The local versus integrated constraint survives even when the background lapse happens to be spatially constant.

Common pitfalls

Using “Hořava gravity” without a lapse domain. State whether $N$ is projectable, nonprojectable, or derived from a khronon before counting constraints or quoting a scalar dispersion relation.

Equating low-energy khronometric theory with the ultraviolet theory. The two-derivative covariant action omits the higher-spatial-derivative operators responsible for the $z=3$ scaling. It describes the infrared branch only.

Inferring equivalence from a symmetric solution. Shared spherical solutions do not prove equality of the unrestricted variational problems. Rotation can activate æther twist and separate the branches.

Exercises

Show directly why projectable lapse variation gives only a global Hamiltonian constraint, and explain why setting $N=N(t)$ after varying a nonprojectable action is not the same operation.

Solution

Write the lapse-dependent variation as

$$\delta S=\int dt\,d^3x\,\sqrt h\,\mathcal H(t,\mathbf{x})\,\delta N.$$

For a nonprojectable lapse, $\delta N(t,\mathbf{x})$ is arbitrary, so the fundamental lemma gives $\mathcal H(t,\mathbf{x})=0$ pointwise. For a projectable lapse, $\delta N$ depends only on $t$ and factors out of the spatial integral:

$$\delta S=\int dt\,\delta N(t) \left[\int d^3x\,\sqrt h\,\mathcal H(t,\mathbf{x})\right].$$

Stationarity therefore imposes only the bracketed integral equal to zero. Restricting a nonprojectable solution to spatially constant $N$ retains the already-derived local equation; restricting the variation first never generated it. The order “restrict, then vary” versus “vary, then restrict” changes the theory.

References

• Blas, D., O. Pujolàs, and S. Sibiryakov. “A Healthy Extension of Hořava Gravity.” Physical Review Letters 104 (2010) 181302. DOI. Open PDF.
• Hořava, P. “Quantum Gravity at a Lifshitz Point.” Physical Review D 79 (2009) 084008. DOI. Open PDF.
• Jacobson, T. “Extended Hořava Gravity and Einstein-Aether Theory.” Physical Review D 81 (2010) 101502. DOI. Open PDF.