Conformal and kinetic couplings as two Jordan frames of the same theory

Non-minimal scalar-tensor (ST) theories may admit an Einstein frame representation, where gravity is described by the Einstein-Hilbert action plus the scalar sector. Some STs become just {\em minimal} Einstein-scalar (MES) theories, notable examples are Brans-Dicke and $R\phi^2$. ST theories with {\em derivative} coupling can also be reduced to an Einstein frame by disformal transformations, but, as a rule, their scalar sector will contain higher derivative terms. Here we draw attention to a new Palatini kinetically coupled theory which can be reduced to pure MES by an invertible disformal transformation. This theory can then be converted into the Jordan frame of another ST, $R\phi^2$, which also admits an invertible transformation to MES. Two theories, each of which is dual to MES, will be {\em sequentially dual} to each other and can be considered as two different Jordan frames of the same theory. Both chosen theories violate null energy condition. Transforming the same singular MES solutions, we find the desingularization signs in both Jordan frames, but these are more pronouned in the framework of kinetic theory, leading, in the cosmological case, to Genesis-type behavior.

Proliferation of derivatively coupled theories led to attempts to describe general properties of the ST landscape [4,49]. An important tool for this is provided by disformal dualities.
Here we want to draw attention to group property of invertible transformations both conformal and disformal: two successive transformations generate another invertivle transformation (up to subtleties with their domains). Consequently, two different ST theories, each of which can be transformed to an Einstein frame, in which the metric sector is described by the Einstein-Hilbert action, and scalar sector is the same, will be sequentially dual to each other. If the scalar sector in the Einstein frame is described by the equations of the second order, both dual STs will be free from Ostrogradsky instabilities. Of particular interest is the class of STs which are invertibly reduced in their Einstein frames just to minimal Einstein-scalar theory (MES). Then you can use frame transformations as solution generating technique to explore new theories in situations which are considered problematic in the General theory of relativity, especially near singularities.
Recently, a new type of behavior has been discovered in cosmological solutions of STs with higher derivatives, such as Galileon theories [64] and DHOST [65]. The univers starts (or passes through the previous evolution) from Minkowskyspace  later discussed by many people [72][73][74][75][76]. Conversion to an Einstein frame (but not to MES) was found for non-minimal models including arbitrary functions F (φ)R and F (R, φ) [77], including the cosmological constant [78] or potentials [79] in the MES frame, in higher dimensions [80].
Later, a Palatini version of this theory was discussed, for relationship to the metric approach and references, see [21].
At the same time, physical nonequivalence of the Jordan frame and the Einstein frame was subject of long discussion, for a review of papers prior to 1994 see [79,81,82], for more recent aspects and references see [83][84][85][86][87]. As a rule, two dual forms of scalar-tensor theory differ significantly when matter terms are added to them. This actually determines which frame has to be considered as physical one. Another aspect of (non)equivalence is related to issues of stability and the quantum-level properties, tihis also remain the subject of discussion [88].
Here we will explore the difference of two Jordan frames of sequentially dual theories near the MES-frame singularities revealing that derivative coupling ensures stronger violation of NEC, than conformal coupling. Namely, the static scalar MES singularity becomes a horizon in the Rφ 2 Jordan frame, but demonstrates regular behavior without a horizon in the new kinetic theory frame. The cosmological MES singularity becomes just the start of the universe from but in the kinetic theory it exhibits sharp violation of NEC at the very beginning of expansion generating the Genesis-type behavior.
The plan of the paper is as follows. In Section II we revisit the non-derivative ξRφ 2 discussing transformations to the Einstein frame, NEC violation and other aspects. In Section III we consider the two-coupling derivative theory, which for some particular ratio of the couplings reduces to Horndeski class in the metric approach. We then adopt Palatini formulation, showing that the theory is ghost-free for arbitrary couplings while for another ratio of two couplings the theory it is disformally dual to MES and, therefeore, sequentially dual to the theory ξRφ 2 .
In Section IV we use dualities as a generating technique to construct Jordan frame duals for the static FJNW solution and the stiff-matter FRW cosmology in the Jordan frames of both theories, comparing their desingulariziation features. The results are summarized in Section V.

II. NON-DERIVATIVE THEORY ξRφ 2
For the reader's convenience, we briefly review the main features of this theory, which is one of the oldest ST with non-minimal non-derivative coupling [69,[72][73][74][75][76]: where we set 8πG N = 1. Variation of this action with respect to the metric and the scalar field gives the Euler-Lagrange equations: where the stress energy tensor is The Weyl transformation φ → Ω −1 φ, g µν → Ω 2 g µν , leaves the Eqs. (2) invariant if ξ = 1/6.
Then the trace of T φ µν vanishes on shell [10][11][12]: and R = 0 as expected for a conformal field, and so φ = 0 on shell.
Attributing the Einstein tensor term in (3) to the left hand side of the Einstein equation, we obtain the effective stress tensor: To pass to Einstein frame we recale the metric [69,73] g µν = Ω 2 g µν , arriving at the following action whereR is the Ricci scalar of the new metric, are the new potential and the kinetic prefactor.
To put the kinetic term into the standard form one has to pass to a new scalar fieldφ, This redifinition results in the Einstein frame action where the potential has to be expressed through the new scalar field. The Eq. (9) can be integrated explicitly as follows [73]: where ν = 1 − 6ξ.
For ξ = 1/6, V = 0 these transformations reduces to the original form of conformal transformation found by Bekenstein [69] and suggested as generating technique to construct solutions Rφ / 6 theory from the solutions of MES: from any solutionĝ µν ,φ of the theory, a solution g µν , φ to the theory Rφ / 6 theory is obtained via the transformation This transformation is invertible, provided the value φ 2 = 6 is not reached, an inverse map beingĝ µν = cosh 2 (φ/ Maeda [77]) had shown that a more general theory with the non-minimal functional coupling F (R, φ can be reduced to the Einstein-Hilbert term plus scalar fields (but not MES).

B. Generating Mexican hat potential
Now let's start with the MES theory with the cosmological constant: and apply the inverse duality transformations (14). The cosmological term then generats in the Jordan frame action a potential term [78]: which has a Mexican hat shape where in dimensionful units and G N is the Newton constant. Note that the vacuum expectation value v of Higgs is not a free parameter, but up to a factor is equal to the Planck's mass. In particular, one can not set V = 0, so the resulting theory is not conformal. The case of more general potentials in MES-frame was considered in [79].

D. Palatini
In the Palatini (or metric-affine) version [21], connection is treated as independent field which has to be fixed by varying the action S P (Γ, g): Generically, independent variation of the connection generates non-metricity and torsion. In this case the Ricci tensor is not symmetric. However, the action (21) includes only symmetric part of it. As a result, it is invariant under projective transformation of the connection (for a recent discussion see [89] in which case torsion can be consistently set to zero [90,91]. Then the Ricci tensorR µν (Γ) should be varied as where the covariant derivative with respect to the Palatini connection is understood. Variation of (21) with respect toΓ, after integration by parts, leads to the following equation With the field redefinition (6), one can rewrite this aŝ showing that the Palatini connection is nothing but the Levi-Civita connection of the Einstein frame metric.
Variation of (21) with respect to metric g µν gives the Einstein equation which can be written in terms of the Einstein frame metric as followŝ

III. DERIVATIVE COUPLING
A. The metric theory Consider the action with non-minimal coupling of the scalar filed to Ricci tensor and Ricci scalar defined by the Levi-Civita connection where φ µ = φ n ug µν and two coupling constants have dimension of inversed mass square. The Ricci scalar is defined though the Levi-Civita connection of the metric g µν , its variation is given by Applying this to (27) and commuting some covariant derivatives one obtains the equation where the first terms is the minimal energy-momentum tensor, , while the other terms correspond to separate contributions of two nonmininmal couplings where φ αβ ∇ α φ β and G µν is the Einstein tensor. Variation over φ gives the scalar equation Obviously, for generic values of the coupling constants κ 1 κ 2 both the Einstein and the scalar equations contain higher derivatives of φ. Collecting the third derivative terms, we find: These terms vanish in the case corresponding to the Einstein tensor in the Lagrangian (27). The Ricci-terms in the scalar equation combine in the Einstein tensor too, so, in view of the Bianchi identity ∇ µ G µν = 0, which holds in the metric theory, the Eq. (32) becomes the second order eqiation

B. Palatini
In the Palatini version the action will read Similarly to the conformally coupled theory, this action includes only symmetric part of the Ricci tensor, and it is projective invariant under (22). We therefore set torsion to zero and make variation with respect to connection according to (23). This gives the following equation for an unknown connection: where we have denoted To solve the Eq. (37) with respect toΓ we would like to cast it into the form∇ λĝµν = 0 for some second metric, or to some equivalent equation. Indeed, since Z µν √ −g is the tensor density we will try to introduce such a metric via an identification so that the determinant would be of the same metric. To proceed, we first construct the matrix W µν , an inverse of the matrix Z µν : It can be obtained as linear combination of g µν and φ µ φ ν as follows where µ = 1 − (κ 1 + κ 2 )X. To find the ratio of the determinants, we rewrite this in the form where the matrix M has the property M 2 ∼ M . For such matrices the determinant is given by Then from (41) we obtain Since the determinant of Z µν is inverse to det W , we finally find from (39) : and, using this, we obtain the second metric explicitely aŝ Now the Eq. (37) becomes∇ so the Palatini connection will be the Levi-Civita connection of the new metric: Now we turn to other equations of motion. Variation of the action (59) with respect to the metric leads to the Einstein-Palatini equation where the Lagrangian can be concisely presented as Finally, a variation over φ gives rise to a scalar equation which, in principle, could contain higher-derivative terms.

C. Einstein frame
So far we have obtained the second metricĝ µν as an auxiliary one, needed to generate the Palatini connection. Note that it is related to the physical metric g µν by a disformal transformation (45). The inverse ofĝ µν can be read off from the Eq.(39) with account for the ratio of determiants (44):ĝ The functions λ and Λ depend on the initial metric through the norm of the gradient of the scalar field X = φ µ φ ν g µν , so to invert the transformation one has to express X through the norm with respect to the second metricX =ĝ µν φ µ φ ν . Contracting the Eq. (51) with φ µ φ ν we obtain the equationX Clearly, we have to restrict physical domain by the conditions µ > 0, λ > 0. One must also avoid the critical point of the functionX(X) where the derivative is zero. This occurs at where the inverse derivative will diverge. But in the regions of monotonicity ofX(X) the Eq.
(52) is a cubuc equation obtained by squaring (52) whose roots can be found explicitly (for more details see [43]), so with such precautions, we can say that the transformation between two metrics is reversible.
Noticing the relation and the representation (49) of the Lagrangian, it is now an easy task to express it entirely in terms of the second metric: We have obtained the Einstein-Hilbert term plus a modified scalar kinetic term without higher derivatives. In view of invertibility of the transformation to the Einstein frame, this means that the initial Palatini theory (59) is free of Ostrogradsky ghosts for general generic coupling constants κ 1 , κ 2 . Recall that in the metric formalism it belongs to Horndeski class only for

D. New Palatini kinetic coupling
Now we see that in the Palatini formalism another particular relation, namely, defines an exceptionally simple derivetively couples ST theory, in which case µ = 1 so it is disformally dual to MES is in the Einstein frame [43]: In this dual theory the Einstein equation reads and the scalar obeys the covariant d'Alembert equation Note, that for the Einstein-Hilbert lagrangian both the metric and the Palatini variations lead to the same equations, therefore, one can replace the Palatini Ricci scalar built with the Levi-Civita connection of the Einstein frame metric, by the usual metric scalar curvaturê One can verify that Eqs. (48) and (50)  in Eq. (50), we reduce the latter to (62). For this one-parametric family of Lagrangians (note that both signs of κ are relevant, depending on whether the φ m u is timelike or spacelike in the Einstein frame [43]).
We will be interested in an inverse disformal transformation from Einstein metricĝ µν to Jordan metric g µν . For this, one has to express the factor λ through the Einstein-metric norm X = φ µ φ νĝ µν . From the Eq. (55) with account for (58) one obtains the following cubic equation which has a real solution where A = 2z √ z 2 − 1 + 2z 2 − 1. Then the Jordan metric will read:

A. FJNW in the Einstein frame
Minimal scalar gravity (12) has a satic spherically symmetric solution, which was first found Fisher [92] and later rediscovered by many people including Janis, Newman and Winicour [93], nowadays mostly abbreviated as FJNW solution where q is the scalar charge and It is asymptotically flat and has a singularity at r = b.

B. Conformal theory
Consider the case γ = 1/2 when all irrational powers are square roots. Then q = b 3/8 and Bekenstein's transformation reads Now perform the coordinate transformation In terms of the new coordinates the solution takes the BBMB form The metric conincides with the extremal Reissner-Nordstrom solution, while the scalar field diverges on the horizion. As was shown by Bekenstein, the singulatity is unseen by a particle interacting with this scalar, so the solution as a whole can be regarded as a regular black hole.
Thus a naked sungularity of MES solution was converted to a horizon in the frame of Rφ2 theory. But the singularity insode the horizon remained.

C. New kinetic theory
Now transform FJNW to the Jordan frame of the new kinetically coupled theory (59). In the static case, interesting solutions arise for κ 1 = −κ 2 > 0, so here we denote κ = κ 1 (or invert the sign of κ in (59) taking κ positive again). The disformal transformation (66) generates now the new metric aacording to the rules where the factor λ is obtained from (66): 1 3 arccos(x/w)], x < w, For large r the variable x ∼ 1/6 4 , so λ = 1 + O(r −4 ) and the solution remains asymptotically flat: Near the MES singularity r = b one can expand in terms of ξ = (r − b)/b, denoting κq 2 /b 4 = ν 3 : In the case γ = 1/2, making the coordinate change z = µb ln ξ with ∞ < z < ∞, one obtains This metric represent the product of a two-dimensional Minkowsky space and a sphere. Note that the scalar field is not transformed and remains singular. But the disformal transformation appropriately subtracts divergence from the metric.
The component (χχ) does not contain the scalar field and admits the first integral a 4ȧ2 using which we findN We still have freedom to fix the gauge, the convenient one beingâ = 2a 0 t. Then For more recent MES solutions which can be used as seed to probe non-minimal STs, see [94][95][96].

B. Minkowsky start in Rφ 2
Performing Bekenstein's transformations in the case Λ = 0, k = 0 one obtains the following exact cosmological solution of the theory (16): In terms of the synchronous time τ = a 0 t 2 (t 2 + 2), or we obtain Thus the univers starts from Minkowsky space. The Hubble parameter and its derivative are The universe is always decelerating.
When k = ±1, Λ = 0 the very beginning of the expansion is the same. Now transform the MES cosmological solution to the Jordan frame of the Palatini kinetically coupled theory (59). In this case the relevant sign of the coupling constant κ is positive. We will be interested by behavior of the scale factor near the singularity of the MES solution. Since in this case both the cosmological constant and curvature terms are negligible, we start with k = 0, Λ = 0 in synchronous gauge whereâ as was found by Zel'dovich in 1972 for the "stiff-matter" [97,98]. Obviously, this metric is singular at t = 0 and describes a decelerating expansion.
Now we transorm to the Jordan frame of the kinetic theory. From (66) we obtain an algebraic equation for N : Its real solution is smooth, although in terms of real functions it looks piecewise: 2 cos 1 3 arccos(x) , z < 1, where A = z + √ z 2 − 1 1/3 . For large z (small t) one has: for small z (large t), In terms of time this gives g tt = (αt) −2 1 + (αt) 4/3 , α = 3 2κ For the scale factor we obtain: Now need to go to the synchronous time t → τ (t) solving the equation N dt = dτ. For small t, keeping the leading term in (97), one finds: dt/dτ = α t → t = e ατ , so that t → 0 corresponds τ → −∞.
Now compute the Hubble parameter differentiating with respect to synchronous time in the vicinity of t = 0: Its derivative readsḢ and satifies condition of strong NEC violation: the ratiȯ H H 2 = 9 2 (αt) −4/3 = 9 2α 4/3 e −4ατ /3 , diverges exponentially in terms of the synchronous time as τ → −∞. Such behavior is typical for Genesis scenario [64,65]. So, NEC violation is even more pronounced in the kinetic theory.

VI. CONCLUSIONS
Our goal was to discuss sequential duaities in non-minimal scalar-tensor theories which arise when two or more theories coincide in their respective Einstein frames into which they can be transformed using invertible mappings. Such dualities are especially useful if the Einstein frame theory is simply the minimally coupled Einstein-scalar theory. By the group property of reversible mappings, two theories, each of which is dual to the MES, are dual to each other; therefore, in a sense, they can be considered as two Jordan systems of the same theory. It is not difficult to find such theories bewtween the subset of non-derivatively coupled STs. As an example of derivative coupling, we have chosen the recently proposed new Palatini kinetc theory. Transforming static and cosmological solutiopn of MES into Jordan frames of these two theories we have found that the second one drastically change behavior of solutions near the singularity and in the cosmologal case leads to Genesis-like behavior.
Class of sequential dualities can be extended taking MES with potentials, which also allow for exact solutions. These will generate non-minimal STs which will be ghost-free as well, though presumably they will not have such a simple form in their Jordan frame as our example here.
We realize, of course, that adding matter generically will destroy suquential dualities in STs, but still such property of their pure gravitational sectors (including scalar degree of freedom) seems useful in understanding the landscape of STs.