Degenerate higher-order Maxwell-Einstein theories

Aimeric Coll\'eaux; Karim Noui

REVIEW 3 major objections 4 minor 1 cited by

Degenerate higher-order Maxwell-Einstein theories

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper classifies degenerate higher-order Maxwell-Einstein theories and identifies a unique family, generalizing quadratic Horndeski scalar-tensor theory to a U(1) gauge field, with second-order metric equations and third-order gauge…

desk verdict Careful, mostly honest classification work with a genuinely new Horndeski-like gauge-field theory, but the uniqueness and ghost-free claims are stronger than what is actually proven. read the letter →

arxiv 2502.03311 v3 pith:BT46HQSN submitted 2025-02-05 gr-qc hep-th

Aimeric Colléaux , Karim Noui This is my paper · ORCID

classification gr-qchep-th

keywords degeneratehigher-ordertheoriesMaxwell-EinsteintheorygaugefieldHorndeskiOstrogradskiinstabilitydisformaltransformationmimeticgravitykineticmatrixdegeneracy

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

Gravity coupled to an electromagnetic field is usually described by Einstein-Maxwell theory or by Horndeski's non-minimal coupling, both having second-order field equations. This paper shows that if one only insists that the metric field equations remain second-order, while allowing the gauge field to appear at third order, there is a unique family of theories, given by equation (3.10) and its conjugate, parametrized by one free function of an electromagnetic invariant. This family is the true analogue, for a $U(1)$ gauge field, of quadratic Horndeski scalar-tensor theory, and it is derived from the degeneracy of the kinetic matrix rather than from the usual second-order requirement; General Relativity and Horndeski's non-minimal coupling are recovered as special choices of the free function. The paper also classifies degenerate non-minimally coupled interactions, identifies conformally invariant cases, and uses $U(1)$-preserving disformal transformations to generate new ghost-free degenerate theories, including a gauge-field analogue of mimetic gravity.

What carries the argument

The argument is carried by a covariant projector formalism: the Faraday tensor $F_{\mu\nu}$ and the metric $g_{\mu\nu}$ define four projectors $\{p, o, \bar p, \bar o\}$ that split spacetime into two orthogonal two-dimensional surfaces, making dimensionally dependent identities and the $U(1)$ Bianchi identities manifest. With these projectors, every invariant in the action (linear in Riemann, quadratic in $\nabla F$) is re-expressed in a small basis, and a 3+1 decomposition yields a kinetic matrix $M$ whose block structure $M = \begin{pmatrix} E & D \\ D^\intercal & C \end{pmatrix}$ controls degeneracy. The degeneracy equations, in particular the conditions $D=0$ and $E=0$ that enforce second-order metric equations, are solved exactly, selecting the unique family (3.10). A 'complex rotation' $x \leftrightarrow \bar x$, $p \leftrightarrow \bar p$, $o \to i\bar o$ organizes the invariants into conjugate pairs and simplifies the classification.

What would settle it

Perform a Hamiltonian constraint analysis of the disformally transformed General Relativity action (4.25) for a generic invertible disformal transformation; if the number of propagating degrees of freedom differs from the two graviton and two photon polarizations of the seed theory, or if an Ostrogradski ghost appears, the paper's ghost-free claim is falsified. A simpler target is the singular conformal transformation (4.19) applied to General Relativity, whose degree-of-freedom count decides whether the gauge-field 'mimetic' theory is healthy.

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Extended reading notes

Core claim

The central claim is that the most general degenerate higher-order Maxwell-Einstein theory whose metric field equations contain at most second derivatives of the metric (and hence at most third derivatives of the gauge field) is the two-conjugate family $L_*[\upsilon] + \bar L_*[\bar{\upsilon}]$ given in equation (3.10), where $\upsilon$ is an arbitrary function of one electromagnetic invariant. This generalizes quadratic Horndeski scalar-tensor theory to a $U(1)$ gauge field: General Relativity and Horndeski's unique non-minimal coupling are obtained for constant and quadratic $\upsilon$, respectively, while linear $\upsilon$ yields a totally degenerate theory invariant under both conformal and $U(1)$-preserving disformal transformations. The paper further classifies all degenerate non-minimally coupled theories linear in the curvature, organizing them into four classes with a two-dimensional kernel structure, and isolates conformally invariant and fully degenerate subclasses. Finally, applying $U(1)$-preserving disformal transformations to General Relativity and to Horndeski's coupling produces new degenerate higher-order Maxwell-Einstein theories that the paper argues are ghost-free by construction, together with singular 'mimetic' transformations that give a gauge-field analogue of mimetic gravity.

Load-bearing premise

The argument relies on the unproven assumption that transforming the metric in a way that preserves the $U(1)$ symmetry also preserves the number and stability of the theory's propagating fields; if that fails, the theories labeled ghost-free could in fact propagate unstable higher-derivative fields.

Editorial extensions

If this is right

The unique theory (3.10) provides a new candidate for a healthy theory of gravity coupled to electromagnetism, with second-order metric equations and third-order gauge equations, on par with quadratic Horndeski in the scalar case.
Constants and quadratic functions of the free function reproduce General Relativity and Horndeski's non-minimal coupling, so the new family continuously connects known healthy theories to unexplored ones.
The degenerate non-minimally coupled classification yields conformally invariant theories, such as (3.42) and (3.46), that are fully degenerate in the electromagnetic sector and can serve as building blocks for conformal models.
Disformal transformations of General Relativity and of Horndeski's coupling produce new ghost-free degenerate higher-order Maxwell-Einstein theories, equations (4.25) and (4.35), which encompass previously known U(1)-conformal and disformal theories.
The singular conformal transformations (4.19) define a gauge-field analogue of mimetic gravity, opening the door to cosmology and black-hole solutions in this framework.

Reading between the lines

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

If the unproven disformal-invariance assumption is confirmed by a Hamiltonian analysis, the disformal generation technique would apply to any healthy seed theory, not just General Relativity and Horndeski's coupling, potentially producing a much larger landscape of ghost-free degenerate gauge-gravity theories.
The two-dimensional wave equation (F4) satisfied by the coupling functions in the alternative basis hints at an integrable structure; a hidden symmetry could lead to a simpler, fully antisymmetrized formulation of the theory (3.10) and to higher-order extensions.
A natural extension is to test the gauge-field mimetic theories (4.19) as models of dark matter or dark energy, in direct analogy to scalar mimetic gravity; the electromagnetic invariants play the role of the mimetic scalar's kinetic term.
The classification's dependence on the algebraic condition (3.36), a vanishing Jacobian in the space of electromagnetic invariants, suggests that degeneracy classes may be invariant under a wider class of field-redefinition symmetries, which could be explored systematically.

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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

The genuinely new thing here is the projector-based classification and the theory L* (3.10). That two-function family really does generalize quadratic Horndeski to a U(1) gauge field, with GR and Horndeski's non-minimal coupling as special cases, and the conjugate pair of totally degenerate even theories is a genuine find. The projector formalism is a real improvement over working directly with the 21-dimensional basis; it makes the 3+1 decomposition and the disformal transformations tractable. The appendices are detailed enough to check by hand, the companion papers [97,48] are cited naturally rather than self-promotionally, and the paper is explicit that the Hamiltonian analysis and the full classification of Family 2 remain open.

Two soft spots matter. First, the 'unique' claim for L* is load-bearing but not fully proven. The main text reduces the problem to D=E=0 and asserts uniqueness; Appendix F only exhibits an alternative expression and shows that a left-over equation reduces to a 2D wave equation on a generic patch with nonvanishing denominators. That is not a completeness proof for the original system. Additional branches—singular C, non-smooth solutions, or denominator-zero cases—are not excluded. The abstract's 'all theories' and Section V's 'unique' therefore outrun what is shown. Second, the disformal ghost-free claim rests on an unproven assumption. The paper says it has no general proof that invertible U(1)-preserving disformal transformations preserve the number and nature of degrees of freedom, but later calls the generated theories 'by construction free of Ostrogradski ghosts'. In the scalar-tensor case this equivalence is established; here it is an inference. The honest statement is: ghost-free if the field-redefinition equivalence holds. While I am at it, the abstract's "obtaining all conformally invariant ones" is also stronger than the partial Family 2 treatment shows.

These are not fatal to the paper's value. The classification work is careful and the new theories are worth taking seriously. But the central uniqueness claim and the ghost-free label need either proof or qualification.

Who should read it: anyone working on vector-tensor or DHOST-type modified gravity, and people interested in mimetic constructions for gauge fields. It deserves a serious referee; I would send it out, with the clear request that the uniqueness proof be completed or weakened and that the ghost-free language be aligned with what is proven.

Referee Report

3 major / 4 minor

Summary. The paper develops a projector-based classification of four-dimensional higher-order Maxwell-Einstein (HOME) theories that are linear in the Riemann tensor and quadratic in derivatives of the field strength, with a focus on degenerate kinetic matrices. The central claim is that the condition D=E=0 (no quadratic kinetic terms for the gauge-field accelerations and no cross terms with the extrinsic curvature) admits a unique pair of conjugate theories, given in Eq. (3.10), whose field equations involve second derivatives of the metric and at most third derivatives of the gauge field. This is presented as the U(1)-gauge analogue of quadratic Horndeski theory, reproducing GR and Horndeski's non-minimal coupling for special choices of the coupling functions. The paper also classifies degenerate non-minimally coupled interactions, identifies conformally invariant subclasses, and studies U(1)-preserving disformal transformations, including singular "mimetic" transformations and the disformal images of GR and Horndeski's non-minimal coupling, which are claimed to yield new ghost-free DHOME theories.

Significance. If the uniqueness claim for Eq. (3.10) is established, this is a substantial and useful result: it provides the first systematic generalization of quadratic DHOST ideas to U(1) gauge fields, and the explicit projector formalism (Appendices A, B, E) is a valuable tool for subsequent work. The paper is unusually candid about its limitations, explicitly deferring the Hamiltonian analysis and acknowledging that Family 2 of degenerate non-minimal couplings is not fully classified. It also contains several concrete and falsifiable sub-results, such as the conformally invariant degenerate theory (3.46) and the explicit disformal images (4.25) and (4.35). However, two load-bearing gaps prevent me from recommending acceptance in the present form: the completeness proof for the uniqueness statement is not fully supplied, and the ghost-freeness of the disformally generated theories is asserted more strongly than the paper's own caveats justify.

major comments (3)

[§ III A 2 and Appendix F] The central uniqueness claim for Eq. (3.10) is not fully proven. The text reduces the problem to imposing D=E=0, and states that the unique solutions are the two conjugate families (3.10), but the main text does not present the solution of these conditions. Appendix F gives an alternative expression and derives the left-over equation ∂²_uv[(u+v)α]=0, whose general solution is α=(f(u)+g(v))/(u+v), with β then fixed by (F5). This does not demonstrate that all components of D and E vanish if and only if this single scalar equation holds, and it assumes nonvanishing denominators such as S1 and u+v throughout. Additional branches—for example solutions with different dependence of the coupling functions, or solutions near S1=0 or u+v=0—could invalidate the uniqueness claim. Because the abstract and Section V state "all" and "unique", this gap is load-bearing. Please either provide a complete branch analysis of D=E=0 or qualify the uniqueness claim explicitly.
[§ IV and § IV C] The ghost-freeness claim for the disformally generated theories is in tension with the paper's own caveat. In Section IV, the authors write that for invertible disformal transformations "although we do not have a general proof, the nature of their degrees of freedom should also be preserved", yet Section IV C and the abstract describe the resulting theories as "by construction free of Ostrogradski ghosts". The latter is a much stronger statement than the admitted hypothesis. If the degree-of-freedom preservation fails, the "new ghost-free DHOME theories" of Section IV C are not established as ghost-free. Please either prove the preservation statement (or cite a complete proof) or weaken the ghost-free wording to "candidate ghost-free" and adjust the abstract accordingly.
[§ III B 5 and Abstract] The abstract claims that the paper classifies degenerate non-minimally coupled interactions "obtaining all conformally invariant ones", but Section III B 5 explicitly states that the second family (m0≠0, m1=0) "is more complicated to fully classify" and restricts to an example, the conformally invariant theory (3.49). Thus "all conformally invariant ones" is not supported by the presented analysis. This completeness overstatement appears in the abstract and should be corrected, e.g. by limiting the claim to the fully classified Family 1, or by stating that Family 2 is partially classified.

minor comments (4)

[§ II A] There is a typo: "respectly" should be "respectively" in the sentence "We will investigate both cases where F is respectly an arbitrary two-form field or the curvature associated with a U(1) potential".
[Appendix B] The "Pontying vector" should be "Poynting vector" (in the definition after Eq. (B8) and in the related discussion).
[Throughout] The spelling of "Ostrogradski" is inconsistent; the paper uses both "Ostrogradski" and "Ostrograski" (e.g. in the Introduction and Section II D 4). Please standardize to one spelling.
[§ III A 2] The phrase "The unique two conjugated classes of theories" is grammatically awkward; consider rephrasing to "The unique pair of conjugate theories".

Circularity Check

0 steps flagged · score 1.0 of 10

Central DHOME classification and uniqueness derivation are self-contained; no circular reduction found, only an explicitly conjectural step in the disformal ghost-free claim.

full rationale

The central classification and uniqueness claims are derived rather than assumed. The paper constructs electromagnetic projectors from g and F, classifies invariants with explicit DDI, Bianchi, and boundary identities, defines the kinetic building blocks C, D, E from a 3+1 decomposition, and solves D=E=0 in Appendix F. The stated left-over equation ∂²_{uv}[(u+v)α]=0 has the general solution α=(f(u)+g(v))/(u+v), giving exactly two free functions of one invariant, which matches the two conjugate families (3.10). This is a genuine differential calculation, not a fit or a definitional identity. GR and Horndeski's non-minimal coupling are recovered as special choices of those functions, not imposed as inputs. Prior work [97] is used as a starting basis, but the paper re-derives the 21-dimensional basis independently in Appendix A, so the self-citation is not load-bearing for the main result. The only notable caveat is in Section IV, where the claim that invertibly disformed theories are 'by construction free of Ostrogradski ghosts' rests on the explicitly unproved assumption that U(1)-preserving disformal transformations preserve the nature and number of degrees of freedom. The paper flags this as conjectural ('although we do not have a general proof'), so it is a missing-proof/correctness risk rather than a circular reduction. Appendix F's brevity about the completeness of the D=E=0 solution is a presentation gap, not circularity. No fitted parameter is relabelled as a prediction, and no known result is merely renamed.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim relies on the algebraic structure of four-dimensional two-forms, the restriction to non-null field configurations, and the standard degeneracy condition from DHOST. No new physical entities are introduced. The free functions parameterize the classified theories rather than being fitted to data.

free parameters (5)

υ( x̄) in L* (3.10) = arbitrary function
Free function defining the new Horndeski-like theory. Not fitted; it parameterizes the classified theory.
ῡ(x) in conjugate = arbitrary function
Conjugate partner to υ.
α0(x, x̄) in Class 0/1 non-minimally coupled theories = arbitrary function
Free functions in the degenerate non-minimal couplings (3.27, 3.35).
α1(x, x̄) = arbitrary function
Free function in the Weyl-coupling theory (3.42).
γ, δ in (3.47) = constants
Coupling constants in the two-parameter family C1III±.

assumptions (5)

domain assumption Four-dimensional spacetime and Lorentzian signature
The projector construction and the classification of invariants rely on dimension four via the Cayley-Hamilton identity (A1).
domain assumption Algebraically general electromagnetic field configurations: x ≠ 0, x̄ ≠ 0
Assumed in Section II A footnote 9 and II B; null configurations are excluded from the classification.
domain assumption U(1) gauge field satisfying the Bianchi identity ∇[μFνσ]=0
This is the definition of the field strength for a gauge potential, used throughout the main classification (Section II A).
domain assumption Existence of a surface-forming time-like unit normal n for the 3+1 decomposition
The kinetic matrix and degeneracy conditions are computed using such a foliation (Section II D 1).
ad hoc to paper The degeneracy condition E = D^T C^{-1} D is the relevant criterion for maximal degeneracy
Taken from DHOST literature (ref [5]); the paper acknowledges it is a sufficient but not necessary condition for ghost-freeness (Section II D 4).

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Cite this review

Pith. "Pith review of Degenerate higher-order Maxwell-Einstein theories." pith.science (2026). https://pith.science/paper/BT46HQSN

@misc{pith2026250203311,
  author       = {Pith},
  title        = {Pith review of: Degenerate higher-order Maxwell-Einstein theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BT46HQSN}},
  note         = {Machine review of arXiv:2502.03311}
}

read the original abstract

We classify higher-order Maxwell-Einstein theories linear in the curvature tensor and quadratic in the derivatives of the electromagnetic field strength whose kinetic matrices are degenerate. This provides a generalisation of quadratic degenerate higher-order scalar-tensor theories for a U(1) gauge field. After establishing a classification of the independent Lagrangians, we obtain all the theories with at most third order field equations involving only second order derivatives of the metric, thus generalising Horndeski's quadratic theory for a gauge field. Some of these are shown to be conformally invariant. We then classify degenerate non-minimally coupled interactions, obtaining all conformally invariant ones. Finally, we investigate the effect of U(1)-preserving disformal transformations on these degenerate Lagrangians. The ``mimetic" singular transformations are obtained and new ghost-free degenerate theories are generated.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Works this paper leans on

137 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

There are several possibly interesting routes to follow from this gravitational action

A general discussion on the action Let us consider the following action Idisf = Igrav + Z d4x√−g Lm [Ψ, gµν] + Z d4x√−g λµνσ ∇[µFνσ ] , (4.23) 29 where Lm contains matter fields Ψ minimally coupled to gµν ∈ {gµν, ˜gµν}, in particular it may contain a non-linear electrodynamics term V (F , G), while the gravitational action is given by Igrav = Z d4x LGR [˜...
[2]

Notice also that det ( C ) = 0 for this class, so that the gravitational sector is degenerate

Gravitationally degenerate theories The degenerate Lagrangian associated with the Class 3(= ¯2) is given by RC3 = α0 (x, ¯x) R0 + R2 + x ¯x R1 + α3 (x, ¯x) R3 (3.27) and remark that R0 + R2 = −Gµν ¯pµν. Notice also that det ( C ) = 0 for this class, so that the gravitational sector is degenerate. Although not ideal, it does not automatically imply the pre...
[3]

coordinate

Partially degenerate embeddings of GR and Horndeski’s non-minimal coupling We continue with the Class 0, excluding its intersection with the Classes 2 and 3 that we just studied. The Lagrangian is given by RC0 = 1 2 α0 (x, ¯x) R − 1 4x¯x α1 (x, ¯x) ⋆F µν⋆F ρσRµνρσ (3.35) and in this case det ( C ) ̸= 0, so that the gravitational sector is generically not ...
[4]

Fully degenerate & conformally invariant theories The last class, summarised in Table III and given by the Lagrangian RC1 = α0 (x, ¯x) (R0 + R2 + R3) + α1 (x, ¯x) R1 (3.38) is the only one which yields fully degenerate theories in the electromagnetic sector, meaning that rank (M) = rank (C ) = 6, other than General Relativity + Horndeski’s NMC, as we are ...
[5]

24 relations between the coupling functions

Family 2 : Degenerate conformal invariant theories The second family of degenerate theories, which satisfies m0 ̸= 0 and m1 = 0, is more compli- cated to fully classify because, contrary to Family 1, it does not seem to imply necessary algebraic 25 Of course, the argument of the function α1 can be any conformally invariant quantity such as F√ F 2+G2 . 24 ...
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disformality

Beyond non-minimal couplings : a ten-functions family of DHOME theories As explained previously, the main reason to restrict to non-minimally coupled Lagrangians is because of the very large number of degenerate theories existing in the generic case where B ̸= 0, as we will illustrate here. Interestingly, it can be shown that the factorisation (3.21), div...
[7]

gauge-mimetic

U(1) frame of General Relativity & “gauge-mimetic” gravity Under a generic U(1) preserving disformal transformation, General Relativity is mapped into the following higher-order Maxwell-Einstein theory, LGR = p −˜g ˜R = s¯s√−g φR − ∆Rµνpµν + L + ¯L + ∇[µ h ˜g−1 ρσ δΓµ ρ]σ i (4.25) where we recall that ∆ = φ − ¯φ and L = − ∆2 2φ Q¯0 2 − Q0 3 − ∆ ¯φ ¯φxQ+ 1...
[8]

U(1) frame of Horndeski’s non-minimal coupling We now turn to the unique U(1) non-minimal coupling yielding second order field equations in four dimensions, written in terms of the disformal metric ˜g, and consider the following Lagrangian LNMC = − 1 16 δµ [αδν βδρ γδσ δ] p −˜g ˜g−1 κα ˜Rµνκ βFρσ ˜F γδ + BNMC , (4.32) where we simply added the following b...

Show all 137 references

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F0 = g , F µν I = F µαF αν I−1

Independent electromagnetic matrices One can easily obtain an upper bound on the number of independent matrices built from the curvature F and metric field g using the Cayley-Hamilton identity in four dimensions 32, F µν 4 = (x¯x)2gµν + x2 − ¯x2 F µν 2 , (A1) where we introduc...
[10]

However, these derivatives are not independent

Derivatives of projectors A priori, any derivatives of F can be trivially decomposed as a sum of the derivatives {∇x, ∇¯x, ∇qI } where the matrices qI ∈ {o, ¯o, p,¯p} have been defined in (2.18). However, these derivatives are not independent. Indeed, from (2.14), it is clear ...
[11]

Dimensionally dependent identities As we said above, the parametrisation of the tensors F in terms of o and ¯o, and the decom- position of these antisymmetric tensors in terms of a tetrad (as shown in appendix B) drastically simplifies the treatment of DDIs. Indeed, it becomes...
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Bianchi identity So far, the analysis of the previous subsections apply to any two-form fieldF in four dimensions. Considering the classification of higher order electromagnetic invariants, imposing that F is the curvature of a U(1) connection, reduces further the dimensionali...
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Boundary terms Let us now classify the possible boundary terms relating the previous invariants together. In order to remain within the class of theories linear in the Riemann tensor or quadratic in the derivative of the field strength, the boundary terms must be of the form ∇...
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This should not be viewed as a restriction on 39 their specific dependence on ( x, ¯x)

Kinetic matrix of non-minimally coupled theories Choosing a basis for EM velocities given by {J+, J−, J}, the non-vanishing building blocks of the kinetic matrix of the non-minimally coupled theories (3.20) can be written as follows, starting from the cross terms, D =   ζ1 ...
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F ull degeneracy in the electromagnetic sector We now focus on theories which are fully degenerate in the electromagnetic sector, meaning rank(M) = rank(C ) = 6. a. RC1II In order to prove that the theory (3.42), given by RC1II = α1 (x, ¯x) R1 = α1 (x, ¯x) oµν ¯oρσRµρνσ (C7) a...
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(Non-)invertibility conditions of disformal transformations Let us present an alternative way to compute the (non-)invertibility conditions of disformal transformations, which gives a nice interpretation of the relations (4.17) and (4.18). Let us recall that these transformati...
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“Non-conformal component” of U(1) disformal transformations Let us start with the transformation law of the triad {eµ, bµ, rµ} introduced in section II D. As- suming the positive branch in the transformation of the normal (4.7) and that one of the conditions (4.8) is satisfied...
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Christoffel symbols and Riemann curvature In this section, we give the transformation laws of the Christoffel symbols and the Riemann curvature under disformal transformations. Christoffel symbols transform according to δΓσ µν = ˜Γσ µν − Γσ µν = ωσ µν + ¯ωσ µν , (D13) where we...
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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

There are several possibly interesting routes to follow from this gravitational action

A general discussion on the action Let us consider the following action Idisf = Igrav + Z d4x√−g Lm [Ψ, gµν] + Z d4x√−g λµνσ ∇[µFνσ ] , (4.23) 29 where Lm contains matter fields Ψ minimally coupled to gµν ∈ {gµν, ˜gµν}, in particular it may contain a non-linear electrodynamics term V (F , G), while the gravitational action is given by Igrav = Z d4x LGR [˜...

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Notice also that det ( C ) = 0 for this class, so that the gravitational sector is degenerate

Gravitationally degenerate theories The degenerate Lagrangian associated with the Class 3(= ¯2) is given by RC3 = α0 (x, ¯x) R0 + R2 + x ¯x R1 + α3 (x, ¯x) R3 (3.27) and remark that R0 + R2 = −Gµν ¯pµν. Notice also that det ( C ) = 0 for this class, so that the gravitational sector is degenerate. Although not ideal, it does not automatically imply the pre...

[3] [3]

coordinate

Partially degenerate embeddings of GR and Horndeski’s non-minimal coupling We continue with the Class 0, excluding its intersection with the Classes 2 and 3 that we just studied. The Lagrangian is given by RC0 = 1 2 α0 (x, ¯x) R − 1 4x¯x α1 (x, ¯x) ⋆F µν⋆F ρσRµνρσ (3.35) and in this case det ( C ) ̸= 0, so that the gravitational sector is generically not ...

[4] [4]

Fully degenerate & conformally invariant theories The last class, summarised in Table III and given by the Lagrangian RC1 = α0 (x, ¯x) (R0 + R2 + R3) + α1 (x, ¯x) R1 (3.38) is the only one which yields fully degenerate theories in the electromagnetic sector, meaning that rank (M) = rank (C ) = 6, other than General Relativity + Horndeski’s NMC, as we are ...

[5] [5]

24 relations between the coupling functions

Family 2 : Degenerate conformal invariant theories The second family of degenerate theories, which satisfies m0 ̸= 0 and m1 = 0, is more compli- cated to fully classify because, contrary to Family 1, it does not seem to imply necessary algebraic 25 Of course, the argument of the function α1 can be any conformally invariant quantity such as F√ F 2+G2 . 24 ...

[6] [6]

disformality

Beyond non-minimal couplings : a ten-functions family of DHOME theories As explained previously, the main reason to restrict to non-minimally coupled Lagrangians is because of the very large number of degenerate theories existing in the generic case where B ̸= 0, as we will illustrate here. Interestingly, it can be shown that the factorisation (3.21), div...

[7] [7]

gauge-mimetic

U(1) frame of General Relativity & “gauge-mimetic” gravity Under a generic U(1) preserving disformal transformation, General Relativity is mapped into the following higher-order Maxwell-Einstein theory, LGR = p −˜g ˜R = s¯s√−g φR − ∆Rµνpµν + L + ¯L + ∇[µ h ˜g−1 ρσ δΓµ ρ]σ i (4.25) where we recall that ∆ = φ − ¯φ and L = − ∆2 2φ Q¯0 2 − Q0 3 − ∆ ¯φ ¯φxQ+ 1...

[8] [8]

U(1) frame of Horndeski’s non-minimal coupling We now turn to the unique U(1) non-minimal coupling yielding second order field equations in four dimensions, written in terms of the disformal metric ˜g, and consider the following Lagrangian LNMC = − 1 16 δµ [αδν βδρ γδσ δ] p −˜g ˜g−1 κα ˜Rµνκ βFρσ ˜F γδ + BNMC , (4.32) where we simply added the following b...