Mimetic Horava Gravity

Ali H. Chamseddine; Tobias B. Russ; Viatcheslav Mukhanov

REVIEW 3 major objections 4 minor 1 cited by

Mimetic Horava Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper constructs a fully diffeomorphism-invariant action whose synchronous-gauge restriction is Horava gravity, using only the constrained mimetic scalar and a projection operator, without adding new propagating degrees of freedom.

desk verdict A genuinely neat covariantization of Horava terms in mimetic gravity, with the load-bearing no-DOF claim unproven—send it to a referee, but expect the authors to have to work for it. read the letter →

arxiv 1908.01717 v2 pith:VJ7SORQV submitted 2019-08-05 hep-th gr-qc

Ali H. Chamseddine , Viatcheslav Mukhanov , Tobias B. Russ This is my paper · ORCID

classification hep-thgr-qc

keywords mimeticgravityHoravadiffeomorphisminvariancesynchronousgaugeprojectionoperatorextrinsiccurvatureCottontensorrenormalizability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that Horava gravity—a theory that improves ultraviolet behaviour by adding higher spatial derivatives and explicitly breaking Lorentz symmetry—can be written as a fully diffeomorphism-invariant theory, using only the scalar field already present in mimetic gravity. The mimetic constraint $g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi=1$ makes $\partial_\mu\varphi$ a timelike unit normal, so it selects a preferred time coordinate. The paper constructs four-dimensional tensors with a projection operator that reduce, in the synchronous gauge $\varphi=t$, $N=1$, $N^i=0$, to exactly the spatial curvature, extrinsic curvature, Cotton tensor, and Chern–Simons term of Horava gravity. If correct, the explicit breaking of Lorentz invariance in Horava gravity is a gauge choice, and no new ghost-like degrees of freedom are introduced. This matters because previous covariant versions of Horava gravity required extra fields that spoiled renormalizability.

What carries the argument

The workhorse is the constrained mimetic scalar $\varphi$, obeying $g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi=1$, whose gradient is a timelike unit vector $n_\mu=\partial_\mu\varphi$. The associated projection operator $P^\nu_\mu=\delta^\nu_\mu-\partial_\mu\varphi\,\partial^\nu\varphi$ sends spacetime vectors into the three-dimensional slices $\varphi=\text{const.}$. Using the identities $\nabla_i\nabla_j\varphi=-\kappa_{ij}$ and $\Box\varphi=\kappa$, together with the projected Riemann tensor (15), every spatial object in Horava gravity is rebuilt as a covariant combination. This machinery is what makes the synchronous gauge a gauge choice rather than a symmetry-breaking input.

What would settle it

A Hamiltonian or Dirac constraint analysis of action (31), or a linearized fluctuation count around flat space, would settle it: if the theory propagates more than the two tensor polarizations of Horava gravity, the no-new-degrees-of-freedom claim is false. Conversely, a two-loop or unitarity computation in the synchronous gauge that shows the $(q\cdot n)^{-\alpha}$ singularities do not decouple would falsify the renormalizability expectation.

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Extended reading notes

Core claim

The central discovery is that every ingredient of Horava gravity can be expressed as a covariant four-dimensional tensor built from the mimetic scalar and the projection operator $P^\nu_\mu=\delta^\nu_\mu-\partial_\mu\varphi\,\partial^\nu\varphi$. In the synchronous slicing, identities such as $\nabla_i\nabla_j\varphi=-\kappa_{ij}$ and $\Box\varphi=\kappa$ turn projected Riemann and Ricci tensors into the spatial curvature $\tilde{R}^l{}_{kij}={}^3R^l{}_{kij}$ and $\tilde{R}_{ij}={}^3R_{ij}$; the projected covariant derivative gives $D_k{}^3R_{ij}$, and the expressions (22) and (30) reproduce the Cotton tensor and the three-dimensional Chern–Simons form. An exemplary action with couplings $c_1,\dots,c_7$ and a Lagrange multiplier enforcing the mimetic constraint therefore reduces exactly to a Horava-gravity action in the synchronous gauge. The paper asserts that no new propagating degrees of freedom appear beyond those of Horava gravity, because the mimetic field is subject to the constraint and plays the role of the time coordinate.

Load-bearing premise

The argument assumes the synchronous gauge $\varphi=t$, $N=1$, $N^i=0$ is always reachable and that the mimetic constraint eliminates exactly one field without creating a new physical degree of freedom; if that counting fails, the claimed equivalence and ghost-freedom are not established.

Editorial extensions

If this is right

Horava gravity can be interpreted as a gauge-fixed version of a diffeomorphism-invariant mimetic theory, so existing Horava calculations, including renormalization analyses, carry over unchanged.
The same mimetic scalar and projection operator can covariantize any term invariant under spatial diffeomorphisms, not only the terms displayed in action (31).
Couplings such as the Cotton tensor and the three-dimensional Chern–Simons form are included covariantly without adding extra fields.
In the synchronous gauge, noncovariant-gauge singularities of the form $(q\cdot n)^{-\alpha}$ remain, and the paper expects them to decouple from the physical $S$-matrix.
If renormalizability of projectable Horava gravity holds, the proposed model should be power-counting renormalizable while preserving full diffeomorphism invariance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test is a full Hamiltonian analysis of action (31) to verify that the mimetic constraint removes exactly one degree of freedom in the presence of higher spatial derivatives; the paper asserts this without showing the count.
The construction suggests a general recipe: any preferred-frame or spatial-diffeomorphism-invariant theory might be covariantized by choosing a timelike unit covector derived from a constrained scalar, which could apply to other Lorentz-violating models.
If the equivalence is exact, the physical content of Horava gravity is not a fundamental breaking of Lorentz invariance but a choice of time slicing; observable signatures would then be tied to the dynamics of the mimetic scalar rather than to a fixed preferred frame.
The renormalizability claim rests on an analogy with projectable Horava models; a covariant background-field calculation that integrates out the mimetic constraint would be the concrete way to test it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the covariant tensors in Section 2 are real: Eqs. (9), (15), (19), (22), (26), and (30) do what the authors claim, and the reduction to 3R, 3Rij, Cotton, and Chern-Simons in the synchronous gauge is shown carefully. This is a genuine technical contribution. Second, the paper's advertised physical implication—that this formulation introduces no new propagating degrees of freedom—is not established. The DOF question is not addressed; the renormalizability remark is explicitly an expectation.

What is new: within mimetic gravity, the projection operator built from the constrained scalar is used to write Horava-style terms covariantly. This specific construction is not in the prior mimetic literature, and it is a neat observation. The inverse-mapping nature is fine: the paper is showing existence of a covariant formulation, not fitting data. The identities in Eqs. (9), (15), and (19) are explicit and appear correct. The paper is honest that the renormalizability is expected, not proven.

Where the soft spots are: The central claim in the abstract and Section 3 is that no new DOFs are needed. That is asserted, not demonstrated. In standard mimetic gravity the Lagrange multiplier constraint generically adds a scalar mode; fixing φ=t is a coordinate choice, not evidence that δφ has no conjugate momentum. A Hamiltonian or phase-space count is needed to show the theory's propagating content matches Horava gravity (which itself has a scalar graviton). The omission matters because the whole interest of the construction is a covariant Horava theory without extra DOFs. The authors cite their own mimetic papers appropriately, but they do not cite khronometric theory, where a unit timelike vector is used to build covariant Horava actions; the relation to those models should be discussed, even if the mimetic constraint differs. The statement that 'any expression invariant under spatial diffeomorphisms' can be covariantized is illustrated but not proven; for a letter that is acceptable, but it is a sweeping claim.

Bottom line: This is a useful constructive paper for someone working on mimetic gravity or covariant Horava formulations. It deserves serious refereeing: the construction should be available, but the DOF claim needs either a proof, a sketch, or a clear demotion to 'conjecture.' I would send it to a referee, expecting major revision or a reframed conclusion. I would not cite the no-DOF claim as established.

Referee Report

3 major / 4 minor

Summary. This paper proposes a fully covariant formulation of Horava gravity built on the mimetic scalar field φ. The authors define a projection operator using n_μ = ∂_μφ and construct four-dimensional tensors (Eqs. 9, 15, 19, 22, 30) whose nonvanishing components in the synchronous gauge coincide with the spatial Ricci scalar, Ricci tensor, Cotton tensor, and Chern-Simons form, respectively. They then write an action (31) containing these tensors and a Lagrange multiplier enforcing g^{μν}∂_μφ∂_νφ=1, and claim that this action reduces to Horava gravity in the synchronous gauge without introducing new propagating or ghost-like degrees of freedom. The paper also briefly discusses the possibility of renormalizability, explicitly labeling it as an expectation rather than a proof.

Significance. If the central claim were established, the paper would offer an elegant way to present Horava gravity as a diffeomorphism-invariant theory, using the mimetic scalar as a dynamical preferred time coordinate. The tensor identities and the systematic construction of covariant analogs of purely spatial objects (extrinsic curvature, spatial curvature, Cotton tensor, and CS form) are clearly derived and are potentially useful for future work on covariantizations of Lorentz-violating gravity. However, the advertised result—absence of additional propagating degrees of freedom—is not demonstrated. Because standard mimetic gravity is known to add a scalar mode, the paper's main physical conclusion rests on an unverified assumption, which substantially limits the significance of the present version.

major comments (3)

[Abstract and §1] The central claim that the construction works 'without introducing ghost-like degrees of freedom' is asserted but never demonstrated. The constraint g^{μν}∂_μφ∂_νφ=1, written explicitly as Eq. (4), is a constraint on the 11 variables (g_{μν}, φ). Solving it in the synchronous gauge and setting φ=t+A, as done in Eq. (5), is a choice of coordinates; it does not remove the scalar perturbation δφ and its conjugate momentum from the physical phase space. In standard mimetic gravity (ref. [1]) the same constraint produces an additional scalar (dust-like) propagating degree of freedom. The paper offers no Hamiltonian analysis, no count of physical degrees of freedom, and no computation of the kinetic signs of the modes. Since the advertised result is precisely the absence of new degrees of freedom, this omission is load-bearing.
[After Eq. (3)] The statement that out of the 11 variables there are 'only 10 independent fields' is a configuration-space counting, not a determination of the number of propagating degrees of freedom. The physical DOF count must be obtained from the constraint structure of the full action (31). This is especially important because the higher-curvature terms ~R^2 and ~R_{μν}~R^{μν} and the topological terms c6 and c7 can alter the constraint algebra and the number of propagating modes. The paper instead asserts that existing Horava calculations can be reused without repeating them; this transfers conclusions from a gauge-fixed theory to a covariant one only if the extra χ-field dynamics are shown to be trivial or to match exactly. The required canonical analysis is absent.
[Eq. (31) and final discussion] The sentence 'There is no need to repeat calculations done for the Horava models, as those could be thought of as a gauge fixed version' is too quick. Even if the action (31) reduces algebraically to Horava gravity in the synchronous gauge, the mimetic constraint is not a pure gauge-fixing condition on the metric alone; it is a dynamical constraint on a separately varied scalar field. Whether the constrained scalar carries an independent degree of freedom is precisely the point that needs checking, and it cannot be inherited from Horava calculations without a dedicated analysis.

minor comments (4)

[Notation] The symbol ~R is used both for the scalar in Eq. (9) and for the tensor in Eq. (19); please use different notation (e.g., ~R^{(3)} or a different letter) to avoid confusion.
[Conventions] The paper does not state its metric signature or the sign conventions for the curvature tensors and extrinsic curvature. Specifying these would make the tensor identities easier to verify.
[Eq. (26)-(30)] The relationship between the three-dimensional Chern-Simons form ^3ω_P defined in Eq. (27) and the covariant object ~ω_P in Eq. (30) should be explained more explicitly, in particular why the correction term ∇_λ dφ ∧ R^τ_λ ∇_τφ removes the unwanted κ-dependent terms in Eq. (26).
[Final paragraph] The renormalizability statement is explicitly presented as an expectation ('Even though an actual proof could be quite demanding, we expect...'). To avoid an unsupported claim being attributed to the letter, it would be helpful to state at the outset that a proof of renormalizability is outside the scope of this paper.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the covariant-to-Horava dictionary is explicitly constructed term by term, while the unproven DOF count is a correctness gap rather than a circular reduction.

full rationale

The paper's central construction is a deliberate dictionary: covariant tensors built from the mimetic scalar and the projection operator are engineered so that, in the synchronous gauge, they reduce to the Horava terms. Every such reduction is stated as an identity under the synchronous-gauge conditions, e.g. Eq. (9) 'coincides with the spatial curvature scalar 3R of synchronous slices', and Eq. (19) is defined so that its 'non-zero components coincide with 3Rij in the synchronous gauge'. These are explicit mathematical equivalences under a specified gauge choice, not empirical predictions or fits. The paper is transparent that the final action (31) is an 'exemplary Horava action in mimetic gravity', i.e. a reformulation of known Horava terms in covariant language. No fitted parameter is renamed as a prediction, and no centrality is imported solely from a self-citation: the mimetic relations (1)-(2) are re-derived in the text, and the cited previous mimetic works appear as background and motivation rather than as the proof of the present construction. The main physical assertion that no new propagating degrees of freedom appear is asserted rather than demonstrated through a Hamiltonian or constraint count; the paper notes 10 independent fields among g_mu nu and phi but does not count propagating modes or kinetic signs. That is an unverified assumption or correctness risk, not a circular step under the rules of this review. The renormalizability claim is explicitly hedged as an expectation ('we expect the mimetic Horava model presented here to be renormalizable'), so it is not presented as a derived result. Hence the derivation chain is self-contained for what it actually proves; the unsupported DOF claim should be judged as a rigor gap, not as circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No fitted parameters; only free couplings in the effective action. The novel construction uses the existing mimetic scalar field and defines projection operators, not new physical entities. The main extra assumptions are the global time slicing by φ and the unproven absence of extra degrees of freedom and ghosts.

free parameters (1)

c1..c7 = free couplings, not fitted to data
Coefficients in the exemplary action (31). The reduction to Horava gravity in the synchronous gauge does not depend on specific values; they are hand-chosen couplings of an effective action. To match GR at low curvature, the authors set c1=c2=1 and others to 0 or functions of □φ.

assumptions (3)

standard math Standard differential-geometry identities (Gauss-Codazzi, Ricci decomposition) are used to relate 4D curvature to 3D curvature and extrinsic curvature.
Used throughout, e.g., in Eqs. (11)-(19), to derive the covariant expressions for 3R, 3R_ij, and the Cotton tensor.
domain assumption The mimetic scalar φ can be used as a global synchronous time coordinate with N=1, N_i=0 and spacelike hypersurfaces φ=const.
The authors state after Eq. (5): 'φ can always be used as one particular synchronous time coordinate, fixing a unique 3+1 slicing.' This requires global conditions that are assumed but not proven.
ad hoc to paper The gauge-fixed theory has the same number of propagating degrees of freedom as Horava gravity and no ghost-like modes.
Asserted in the introduction ('without introducing ghost-like degrees of freedom') and later as an expectation for renormalizability, but no DOF count or Hamiltonian analysis is provided. This is a key unproven assumption.

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Cite this review

Pith. "Pith review of Mimetic Horava Gravity." pith.science (2026). https://pith.science/paper/VJ7SORQV

@misc{pith2026190801717,
  author       = {Pith},
  title        = {Pith review of: Mimetic Horava Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJ7SORQV}},
  note         = {Machine review of arXiv:1908.01717}
}

read the original abstract

We show that the scalar field of mimetic gravity could be used to construct diffeomorphism invariant models that reduce to Horava gravity in the synchronous gauge. The gradient of the mimetic field provides a timelike unit vector field that allows to define a projection operator of four-dimensional tensors to three-dimensional spatial tensors. Conversely, it also enables us to write quantities invariant under space diffeomorphisms in fully covariant form without the need to introduce new propagating degrees of freedom.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Two-field mimetic gravity revisited and Hamiltonian analysis
gr-qc 2019-09 conditional novelty 6.0 of 10

A two-field mimetic gravity model has two scalar degrees of freedom, not one, and the extra entropy mode is a ghost when the fields have opposite-sign kinetic terms.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A. H. Chamseddine and V. Mukhanov, Mimetic Dark Matter, JHEP 1311 (2013) 135. 6

work page 2013

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A. Golovnev, On the Recently Proposed Mimetic Dark Matter, Phys. Lett. B728 (2014) 39

work page 2014

[3] [3]

A. H. Chamseddine, V. Mukhanov and A. Vikman, Cosmology with Mimetic Matter, JCAP 1406 (2014) 017

work page 2014

[4] [4]

A. H. Chamseddine and V. Mukhanov, Resolving Cosmological Singu- larities, JCAP 1703 (2017) 009

work page 2017

[5] [5]

A. H. Chamseddine and V. Mukhanov, Nonsingular Black Hole, Eur. Phys. J. C77 (2017) 83

work page 2017

[6] [6]

Barvinsky, D

A. Barvinsky, D. Blas, M. Herrero-Valea, S. Sibiryakov and C. Steinwachs, Renormalization of Hoˇ rava Gravity,Phys. Rev. D93 (2016) 064022

work page 2016

[7] [7]

Barvinsky, M

A. Barvinsky, M. Herrero-Valea and S. Sibiryakov, Towards the renor- malization group ﬂow of Hoˇ rava gravity in (3 + 1) dimensions, Phys. Rev. D100 (2019) 026012

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Germani, A

C. Germani, A. Kehagias and K. Sfetsos, Relativistic Quantum Gravity at a Lifshitz point, JHEP 009 (2009) 060

work page 2009