Vector–Tensor and Preferred-Foliation Gravity

A preferred timelike direction can be a dynamical vector, the normal to a foliation, or the background value of a massive vector field. Those choices look similar in a symmetric ansatz but lead to different gauge groups, constraints, mode cones, and black-hole boundaries. This chapter develops the three principal constructions and then tests them in a fixed order: degrees of freedom, kinetic and gradient signs, propagation relative to matter, strong-field branches, and cosmological or radiative predictions.

Helpful background. Einstein Equations supplies the metric dynamics; Gravitational Actions and Boundary Terms supplies variational reasoning; Formulations of General Relativity supplies ADM and constraint language; and Gravity Beyond General Relativity supplies the volume-wide comparison criteria.

Enter this chapter

The central distinction is between a preferred frame and a preferred foliation. A unit timelike æther $u^a$ defines a rest frame locally. It defines a foliation only when its twist vanishes and it can be written as the normalized gradient of a scalar time $T$. Einstein–æther theory varies an unrestricted unit vector; khronometric Hořava gravity restricts the action to a hypersurface-orthogonal vector before variation. They share every spherical æther configuration but separate when rotation activates vorticity.

Generalized Proca theory asks another question. Its vector is massive and need not have unit norm or define preferred time. Special derivative self-interactions preserve a second-class constraint pair so that the temporal component remains auxiliary and only three vector polarizations propagate. A cosmological temporal expectation value can still select a rest frame on a solution.

Throughout, the site convention is $(-,+,+,+)$, $c=\hbar=k_B=1$, and explicit Newton or Planck scales. Several foundational sources use $(+,-,-,-)$. Each page translates the whole sign package—Einstein–Hilbert term, unit-vector norm, vector kinetic term, extrinsic curvature, dual tensor, and curvature convention—rather than reversing an isolated formula.

Reading order and central questions

The pages are arranged so that later phenomenology never precedes the constraint and characteristic analysis that makes it meaningful.

Order	Page	Question carried forward
1	Einstein–Æther Gravity	How do the four two-derivative couplings determine five coupled metric–æther modes and the locally measured $G_N$?
2	Hořava Gravity and Foliation-Preserving Symmetry	How do projectability, nonprojectability, and the khronometric restriction change lapse variation and the scalar mode?
3	Generalized Proca and Vector–Tensor Interactions	Which Hessian degeneracy and secondary constraint remove the unwanted fourth vector polarization?
4	Mode Speeds, Cherenkov Bounds, and Stability	Which inequalities test kinetic energy, gradients, and emission kinematics before data are applied?
5	Universal Horizons and Preferred-Foliation Black Holes	Under which foliation, dispersion, branch, and regularity assumptions can one surface trap arbitrarily fast modes?
6	Cosmological and Radiative Predictions	How do common action parameters enter expansion, compact-body charges, extra polarizations, and comparison with data?

The order is diagnostic, not historical. A black-hole or cosmological solution is not yet a predictive theory if the corresponding constraint rank changes, a kinetic eigenvalue is negative, or the solution lies beyond its derivative cutoff.

Three actions, three constraint mechanisms

In site conventions, the Einstein–æther action begins schematically as

$$S_{\mathrm{ae}}=\frac1{16\pi G_{\mathrm{ae}}} \int d^4x\sqrt{-g}\left[R-K^{ab}{}_{mn}\nabla_a u^m\nabla_bu^n +\lambda(u^2+1)\right].$$

The multiplier fixes one vector component, diffeomorphism constraints remove gauge directions, and the coupled linear system carries two spin-2, two spin-1, and one spin-0 modes. Their characteristic cones depend on $c_{13}$, $c_{14}$, and $c_{123}$, while the static Newton constant depends specifically on $c_{14}$; the explicit speed and energy map is given in Jacobson 2008, §4, Eqs. (4.1)–(4.11), PDF.

Hořava gravity instead takes ADM fields $(N,N^i,h_{ij})$ and foliation-preserving diffeomorphisms as fundamental. Its infrared kinetic term is $K_{ij}K^{ij}-\lambda_HK^2$, and its potential may contain up to six spatial derivatives at the $z=3$ fixed point. If $N=N(t)$, lapse variation gives one spatially integrated Hamiltonian constraint. If $N=N(t,\mathbf{x})$, the acceleration $a_i=D_i\ln N$ is an allowed building block and lapse variation is local. That difference cannot be undone by evaluating both theories on a constant-lapse background.

Generalized Proca interactions are engineered so that

$$\det\left(\frac{\partial^2\mathcal L} {\partial\dot A_\mu\partial\dot A_\nu}\right)=0.$$

The associated primary relation and its preserved secondary constraint remove $A_0$ as a propagating variable. Curved-space terms such as $G_4(X)R$ and $G_5(X)G_{\mu\nu}\nabla^\mu A^\nu$ are counterterms required to retain the intended derivative and constraint structure. Degeneracy must nevertheless be checked on the actual metric–vector background.

A consistency sequence for any model

A reliable analysis begins by fixing five pieces of input: the action, the physical matter metric, the background and branch, the gauge or foliation formulation, and the derivative regime. Then apply the following sequence.

1. Count after reduction. Identify primary and secondary constraints and their class. Do not infer a mode count from field components or second-order equations alone.

2. Diagonalize the kinetic matrix. Positive residues or kinetic eigenvalues are independent of real characteristic speeds.

3. Inspect every characteristic sector. State the background and frequency window. In a preferred frame, superluminal relative to the metric is not synonymous with acausal, while subluminal relative to matter can open vacuum Cherenkov emission.

4. Check exceptional parameter surfaces separately. Vanishing $c_{14}$, $c_{123}$, a zero Proca mass, or the GR value $\lambda_H=1$ can change the rank of the principal or constraint matrix. Substitution into a formula derived off that surface may be invalid.

5. Construct the strong-field or cosmological branch. Enforce regular characteristic horizons, boundary conditions, and a cutoff. Symmetric backgrounds can suppress twist or field-strength operators that return in perturbations.

6. Map theory quantities to observables. Only after the preceding checks should mode fluxes, sensitivities, background couplings, or horizon quantities be passed to empirical inference.

This sequence explains why there is no single “healthy-coupling region” independent of formulation and background. It also prevents a GR-like limit of one observable—such as $G_{\mathrm{cosmo}}=G_N$ or a quadrupole coefficient equal to one—from being mistaken for equality of theories.

Characteristic cones and matter

For an æther mode of rest-frame speed $s_I$, the effective inverse metric can be written

$$g_{(I)}^{ab}=g^{ab}+(1-s_I^{-2})u^a u^b.$$

This compact expression organizes the chapter. On flat space it yields $\omega^2=s_I^2\mathbf{k}^2$. On a stationary black hole it defines a finite-speed characteristic horizon. In the formal $s_I\to\infty$ limit, the stationary horizon condition approaches $u\cdot\chi=0$, the candidate universal-horizon equation. Relative to minimally coupled matter, $s_I\lt1$ also supplies the kinematic possibility of vacuum Cherenkov emission.

These three uses share a cone but not an evidential status. The dispersion relation and horizon are theory calculations. A Cherenkov exclusion additionally needs particle energy, composition, path length, coupling strength, and EFT validity. Numerical observational bounds therefore remain in Tests of Gravity.

The two research frontiers

Universal horizons are established on specified static spherical infrared branches, and classical Smarr and first-law relations exist for special families. Their generic status is unresolved: khronometric instantaneous modes motivate a nonspherical singularity argument, slow Hořava rotation preserves the khronon at first order, while slow Einstein–æther rotation produces vorticity and no analogous hypersurface. Higher-spatial-derivative propagation and quantum thermodynamics add further conditional layers.

Cosmological and radiative predictions are likewise formulation dependent. In æther theory,

$$\frac{G_{\mathrm{cosmo}}}{G_N} =\frac{1-c_{14}/2}{1+(c_{13}+3c_2)/2},$$

while compact-body sensitivities can activate dipole radiation even when this ratio is one. In generalized Proca cosmology, a temporal background has $F_{\mu\nu}=0$, hiding intrinsic vector interactions from the background equations; those interactions return in vector perturbations. The frontier question is not whether such signatures can be written, but whether one controlled branch passes background, perturbative, compact-object, cutoff, and empirical checks simultaneously.

Review the chapter

Consider a proposal with a spacetime-dependent lapse, an $a_i a^i$ operator, $z=3$ ultraviolet scaling, and a hypersurface-orthogonal field $u_a=-N_T\nabla_aT$. It has $c_{13}=0.06$, $c_2=-0.02$, and $c_{14}=0$. The proposal claims that equality of cosmological and local Newton constants proves equivalence to GR and guarantees a regular rotating universal horizon. Diagnose both claims and list the next calculations required.

Solution

The spacetime-dependent lapse and $a_i a^i$ term identify the nonprojectable formulation; $z=3$ refers to the higher-spatial-derivative ultraviolet completion, while the normalized gradient identifies its khronometric infrared description.

Here $c_{13}+3c_2=0.06-0.06=0=-c_{14}$, so indeed

$$G_{\mathrm{cosmo}}=G_N=G_{\mathrm{ae}}.$$

That equality tests only two background/static coefficients. It does not remove the scalar graviton, set all mode speeds and kinetic residues to their GR values, determine compact sensitivities, or control the ultraviolet operators. The claimed equivalence is therefore false.

Hypersurface orthogonality makes $u\cdot\chi=0$ meaningful, but regular rotation is not implied by the equality of Newton constants. One must construct the rotating khronometric branch, verify that $T$ remains timelike, locate the outermost root, impose regularity at every finite-speed characteristic horizon, and test nonspherical perturbations. Because $c_{14}=0$ is an exceptional surface where an æther speed can diverge, its constraint and principal symbols require a direct limiting analysis. Finally, one must check whether the $z=3$ operators modify the background or high-frequency rays before calling the boundary universal.

References

• Blas, D., O. Pujolàs, and S. Sibiryakov. “A Healthy Extension of Hořava Gravity.” Physical Review Letters 104 (2010) 181302. DOI. Open PDF.
• Heisenberg, L. “A Systematic Approach to Generalisations of General Relativity and Their Cosmological Implications.” Physics Reports 796 (2019) 1–113. DOI. Open PDF.
• Hořava, P. “Quantum Gravity at a Lifshitz Point.” Physical Review D 79 (2009) 084008. DOI. Open PDF.
• Jacobson, T. “Einstein-æther gravity: a status report.” Proceedings of Science QG-PH (2007) 020. DOI. Open PDF.