Emerging Newtonian potential in pure R 2 gravity on a de Sitter background

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In Fortsch. Phys. 64, 176 (2016), Alvarez-Gaume et al established that pure R 2 theory propagates massless spin-2 graviton on a de Sitter (dS) background but not on a locally flat background. We build on this insight to derive a Newtonian limit for the theory. Unlike most previous works that linearized the metric around a locally flat background, we explicitly employ the dS background to start with. We directly solve the field equation of the action (2κ) −1´d4 x √ −g R 2 coupled with the stress-energy tensor of normal matter in the form Tµν = M c 2 δ( r) δ 0 µ δ 0 ν . We obtain the following Schwarzschild-de Sitter metric ds 2 which features a potential V (r) = − κc 4 96πΛ M r with the correct Newtonian tail. The parameter Λ plays a dual role: (i) it sets the scalar curvature for the background dS metric, and (ii) it partakes in the Newtonian potential V (r). We reach two key findings. Firstly, the Newtonian limit only emerges owing to the de Sitter background. Most existing studies of the Newtonian limit in modified gravity chose to linearize the metric around a locally flat background. However, this is a false vacuum to start with for pure R 2 gravity. These studies unknowingly omitted the information about Λ of the de Sitter background, hence incapable of attaining a Newtonian behavior in pure R 2 gravity. Secondly, as Λ appears in V (r) in a singular manner, viz. V (r) ∝ Λ −1 , the Newtonian limit for pure R 2 gravity cannot be obtained by any perturbative approach treating Λ as a small parameter.

I. MOTIVATION
The interest in pure R 2 gravity has recently experienced a resurgence. As a member of the f (R) family, it is known to be ghost-free. This is because when shifting from the Jordan frame to the Einstein frame, its auxiliary scalar degree of freedom only involves derivatives of second order and no higher. As a result, the theory escapes the Ostrogradsky instability that often plagues higher-derivative theories [1,2]. Unlike other siblings in the f (R) family, pure R 2 gravity enjoys yet an additional advantage-it is free of any inherent scale. Its action consists of a single term, (2κ) −1´d4 x √ −g R 2 , where the gravitational coupling κ is dimensionless. Pure R 2 gravity has a restricted scale symmetry [3], viz. it exhibits invariance under the Weyl transformation, g µν = Ω 2 (x)g µν , with the local conformal factor Ω(x) obeying the harmonic condition, Ω = 0. Theoretical aspects of pure R 2 gravity and its implications for black holes and cosmology have been the subject of active investigation .
It is worth noting that the pure R 2 action departs from the conventional Einstein-Hilbert paradigm by excluding the Einstein-Hilbert term c 4 16πG R. In an important work on graviton propagators [27], Alvarez-Gaume et al discovered that the pure R 2 action propagates a massless spin-2 tensor mode provided that the background metric is de Sitter rather than locally flat. This massless mode has a capacity to yield a long-range interaction instead of a short-range Yukawa exchange (the latter would be a * hoang.nguyen@ubbcluj.ro typical hallmark of a massive mode, otherwise). Consequently, it would be natural to expect that the massless spin-2 tensor mode in pure R 2 gravity should produce a potential with the correct Newtonian tail ∼ 1/r, akin to the behavior exhibited by a standard massless spin-2 tensor mode in General Relativity (GR). The significance of this mode lies in its potential to confer a proper Newtonian behavior to the pure R 2 theory, despite the absence of the Einstein-Hilbert term in the action.
For a theory to be a viable description of gravitational phenomena in the Solar System, it must exhibit a Newtonian limit with the tail of 1/r, with r being the distance from a mass source. This limit was well established in the case of the Einstein-Hilbert action, soon after the development of GR, through solving the Einstein field equations in the presence of normal matter. A typical example involves considering a static point mass M located at the origin, where the stress-energy tensor takes the form which has a non-vanishing trace, viz. T := g µν T µν = M c 2 δ( r). For this stress-energy tensor, the Einstein field equation, is known to admit the Schwarzschild metric as an exterior solution, given by where the Schwarzschild radius r s is equal to 2GM/c 2 . The term V (r) := − c 2 2 rs r represents a gravitational potential that exhibits the correct Newtonian falloff ∼ 1/r, with r s determined by the mass source M and the (Newton) gravitational constant G in the Einstein-Hilbert action, c 4 16πG´d 4 x √ −g R.
Establishing the Newtonian limit becomes intricate when considering alternative theories of gravity, and one illustrative example is conformal gravity. The theory is defined by the action (2κ where C µνρσ represents the Weyl tensor and κ CG denotes a dimensionless coupling parameter [28]. The field equation of conformal gravity, known as the Bach equation, is given by Interestingly, this equation also allows for a Schwarzschild metric as a valid solution in vacuum regions where T µν = 0. However, despite this property, establishing the Newtonian limit for conformal gravity remains an unresolved challenge [29,30]. One of the key obstacles arises from the traceless nature of the Bach tensor B µν : per Eq. (4), the (conformal) gravitational field can only interact with a stress-energy tensor that is also traceless, whereas the stress-energy tensor of ordinary matter, expressed in Eq. (1), has a non-zero trace. Consequently, connecting the Schwarzschild radius r s and the mass M of the source becomes problematic in conformal gravity.
In the case of the pure R 2 action, the corresponding field equation is given by It is known that this equation admits a Schwarzschild metric as a solution in the exterior region. However, as demonstrated in the case of conformal gravity, this still falls short of establishing the desired Newtonian limit. To achieve this goal, it is required to establish a connection between the Schwarzschild radius r s , the mass M , and the parameter κ of the action (2κ) −1´d4 x √ −g R 2 . The challenge lies in the fact that the gravitational coupling κ is dimensionless and does not resemble the dimensionful Newton constant G that would partake in the regular Newtonian potential, V ( r) = − GM r . In other words, since the pure R 2 theory lacks the (dimensionful) G in its action, one must "construct" G from the (dimensionless) κ. This crucial issue has not been addressed in the existing literature. The objective of our paper is to solve Eq. (5) for the stress-energy tensor expressed in Eq. (1) and establish the desired connection between r s and κ. The presence of the Dirac delta function in Eq. (1) makes the task delicate. To tackle it, we will introduce a new method utilizing the Gauss-Ostrogradsky theorem.
It is important to highlight that previous research on the existence of a Newtonian limit in pure R 2 gravity has predominantly yielded negative results, as documented in comprehensive studies such as Refs. [31][32][33]. These works shared a common characteristic: they employed a weak-field approximation around a locally flat background. However, as we will demonstrate in this paper, this practice is not applicable for pure R 2 gravity, which inherently possesses a de Sitter background. By utilizing a locally flat background, these previous studies inadvertently neglected the crucial information regarding the scalar curvature of the de Sitter background. As highlighted by Alvarez-Gaume et al in [27], it is precisely the de Sitter background that accommodates the massless spin-2 graviton, offering a potential avenue for a longrange Newtonian potential to emerge.
Our paper is organized as follows: In Section II, we introduce a new method which utilizes the Gauss-Ostrogradsky theorem and, as a proof of concept, apply it to (re-)derive the Newtonian potential for the Einstein-Hilbert action. Section III encompasses the central outcome of our research, as we employ the method to solve the field equation of pure R 2 gravity, viz. Eq. (5), for the stress-energy tensor given in Eq. (1). We present a comprehensive, step-by-step calculation in full detail, with additional assistance provided in Appendix A. Section IV delves into the reasons why previous works have been unable to establish the Newtonian limit for pure R 2 gravity. We shed light on the limitations and shortcomings that hindered their progress in achieving this goal. Finally, Section V summarizes the key steps of our work and offers an outlook.

II. RE-DERIVING THE NEWTONIAN LIMIT FOR EINSTEIN-HILBERT ACTION BY WAY OF THE DIVERGENCE THEOREM
To solve the complete field equation (5), we design a specialized method to target the stress-energy tensor T µν given in (1) which involves the Dirac delta function. This section serves as a proof-of-concept demonstration, where we introduce our method and utilize it to re-derive the well-established Newtonian limit of the standard Einstein-Hilbert theory. In Section III, we will apply our method to the pure R 2 action.
------∞------Consider the following static spherically symmetric line element (with the speed of light c being explicit): For the Einstein-Hilbert action the Einstein field equation is with the stress-energy tensor for a static point mass M : The 00-component of the Einstein field equation is thus The relevant terms of the Ricci tensor: give Expressing leads to in which only the linear term is V is retained.
To proceed, our strategy is to express G 00 as a divergence of a vector field A then make use of the Gauss-Ostrogradsky theorem. Below is how to do so.
Let us define a vector field A( r) which has only radial component, i.e.
with A(r) judiciously chosen as The divergence of A( r) in the spherical coordinate: Comparing (17) and (21), we have and, using (11), a divergence equation Multiply both sides of Eq. (23) with the 3-volume element d 3 V = r 2 dr dΩ and integrate over the ball V of radius r and center at the origin: Applying the Gauss-Ostrogradsky theorem turns the lefthand-side of Eq. (24) into a surface integral on the sphere: which, by virtue of spherical symmetry, yields Combining (26) with (19), we readily obtain which is precisely the Newton law of universal gravitation. This concludes our proof-of-concept exercise.

III. DERIVING THE NEWTONIAN POTENTIAL FOR PURE R 2 ACTION
Let us apply the Gauss-Ostrogradsky procedure designed in the preceding section to the case at hand in which κ is a dimensionless coupling. For brevity, let us define the following tensor: 1 The pure R 2 field equation reads with the stress-energy tensor given in Eq. (10). The equation to be solved is the 00-component of Eq. (30): 1 Recall that, for 1 2κ f (R) + Lm action, the field equation is: Tµν and the stress-energy tensor: Our strategy is analogous to the preceding Section: we shall express X 00 as the divergence of a (to-bedetermined) vector field B( r), then apply the Gauss-Ostrogradsky theorem to Eq. (31) in order to link B with the mass source M .
To proceed, we first need to compute two additional terms in X 00 , namely, g µν R and ∇ µ ∇ ν R. Since R only depends on r, the following expressions hold: 2 and (noting that ∂ 0 R = 0) Substituting (12), (13), (33), and (35) into Eq. (29): see Eq. (A13) in Appendix A. It is straightforward to verify that a function f chosen to be f = 1 − Λ 3 r 2 would result in R ≡ 4Λ per Eq. (13), and force X 00 in Eq. (37) to vanish everywhere, thus obeying the vacuo field equation.
Let us then substitute with V (r) to be determined, into Eq. (37). We get (see Eq. (A19) in Appendix A for step-by-step calculations) in which only the terms linear in V are retained (i.e., the weak-field approximation). Note, however, that we retained all orders of Λ in Eq. (39). 2 Recall that for a scalar field φ: φ = 1 √ −g ∂µ( √ −gg µν ∂ν φ) and To find the vector field B( r), we would need to find the analytical formula for the anti-derivative´dr r 2 X 00 (r). From Eq. (39), the anti-derivative can be expressed as c 2ˆd r r 2 X 00 (r) := I(r) − Λ 3 J(r) (40) in which The anti-derivative J(r) can be split further into in which The latter three expressions, together with Eq.
Remarkably, the anti-derivative I(r) − Λ 3 J(r) exists in closed-form. The terms´dr (r 2 V ) in Eqs. (49) and (50) have managed to cancel themselves out.

Conversely, Eq. (40) is equivalent to
Next, let us define a vector field B( r) which only has radial component, i.e.

B( r) := B(r)r
with B(r) judiciously set equal to The divergence of the radial vector field B( r) in the spherical coordinate is then Comparing Eqs. (53) and (57), the X 00 component is indeed the divergence of the vector field B: thus, by virtue of Eq. (31), giving Multiply both sides of Eq. (59) with the 3-volume element d 3 V = r 2 dr dΩ then integrate over the ball V of radius r and center at the origin: The the left-hand-side of which is described by Eq. (52). Since V (r) = A/r automatically makes X 00 = 0, by substituting V (r) = A/r into Eq. (52), we get which, together with Eq. (63), produces With V (r) = A/r, we thereby obtain the Newtonian law for pure R 2 gravity: This concludes our derivation of the SdS metric for pure R 2 gravity.

IV. WHY HAVE MOST OTHER EXISTING STUDIES FAILED TO PRODUCE THE NEWTONIAN LIMIT IN PURE R 2 GRAVITY?
The question of whether pure R 2 gravity, a theory that excludes the Einstein-Hilbert term from its action, exhibits a Newtonian limit has been a contentious topic, although the prevailing consensus leans towards the negative. It is generally believed that pure R 2 gravity lacks a proper Newtonian limit, as supported by the comprehensive studies [31][32][33], for example. Consequently, it is commonly accepted that the inclusion of the Einstein-Hilbert term as a leading term in the action is necessary to ensure a Newtonian limit, while higher-order terms such as R 2 serve as small supplements to the action [34]. However, our result expressed in Eq. (66) strongly challenges these longstanding beliefs.
To understand why previous studies have fallen short, let us first review their approaches and identify the shortcomings. Typically, when addressing this question in a given theory of gravitation, the common practice was to linearize the metric around the flat background, η µν := (− + ++), viz.
with h µν treated as a small perturbation. The essence of the Newtonian limit boils down to determining whether or not h 00 satisfies a second-order Poisson equation in the presence of normal matter. The rationale for employing η µν stemmed from the requirement of asymptotic flatness, which states that the metric g µν should approach a Minkowski metric when far away from all mass sources. Since the expansion around η µν , as expressed in Eq. (67), successfully yields the Newtonian limit in General Relativity (GR), most studies in modified gravity have adopted this conventional practice without questioning the domain or limitation of its applicability. However, in the context of pure R 2 gravity, the metric should morph into a de Sitter cosmic background at large distances, rendering the use of η µν inadequate.
A prevalent approach in this area has been to establish the Newtonian limit -or the lack thereof -for the full quadratic action: and subsequently set α and γ to zero in the final analytical result. Among these endeavors, the comprehensive investigation conducted in Refs. [31][32][33] can be quoted as a representative example. Their rationale was that a second-order Poisson equation that is required for the Newtonian tail, naturally arises from the second-order GR theory but is incompatible with the fourth-order nature of the R 2 theory. Instead, this latter theory would be expected to support a fourth-order Poisson equation: The solution to this equation yields a linear potential, U ( r) = − 1 8π r, which lacks a desired Newtonian tail 1/r. However, a critical issue emerges. In Ref. [27], Alvarez-Gaume et al highlighted an subtle problem with the full quadratic action. In the absence of matter, the background of the full action (68) (when γ = 0) is Ricciscalar-flat, i.e., R = 0. In contrast, the background of the pure R 2 action (i.e., action (68) with γ = 0) is characterized by a constant Ricci scalar, R = 4Λ, where Λ ∈ R. The transition from γ = 0 to γ = 0 poses a problem: the former background contains no information about Λ inherent in the latter background. Consequently, all analytical results obtained for γ = 0 cannot be smoothly extrapolated to the case where γ = 0, as they -by design -lack knowledge about Λ. In other words, when dealing with the pure R 2 theory, the locally flat metric is a false background to start with. Instead, one must directly begin with a de Sitter background. Rather than using the conventional expansion (67), one must perturb the metric g µν around a de Sitter background, denoted byḡ µν , per In their work [27], Alvarez-Gaume et al employed both expansions (67) and (70). They discovered that the pure R 2 theory propagates a massless spin-2 tensor mode on a de Sitter background but not a flat background. This is a significant result: a de Sitter background is essential to support the propagation of massless spin-2 gravitons, leading to the emergence of the long-range Newtonian potential. This is precisely where previous studies have gone astray. While using a standard locally flat metric η µν in the weak-field expansion is appropriate for GR as well as the quadratic gravity action (68) when γ = 0, it becomes inadequate when considering the quadratic gravity action (68) with γ = 0. The vacua of these theories differ, with GR and quadratic gravity with γ = 0 featuring Ricciscalar flat vacua (R = 0), whereas the quadratic gravity action with γ = 0 acquires vacua with non-vanishing scalar curvature.
Aligned with Alvarez-Gaume et al's work, previous studies that took a similar approach of starting with a de Sitter background have also observed indications of Newtonian behavior in certain modified theories of gravity [35][36][37]. One particularly interesting study is by Nojiri and Odintsov [35], where they intentionally departed from the conventional method and reported a Newtonian limit for a modified gravity theory that incorporates R 2 .

V. SUMMARY AND OUTLOOK
We have successfully established the existence of a proper Newtonian limit in pure R 2 gravity, a theory that excludes the Einstein-Hilbert term at the outset: There were two principal challenges for us to overcome. Firstly, unlike General Relativity (GR), which is a second-order theory, pure R 2 theory is inherently fourthorder, thus naturally resulting in a fourth-order Poisson equation instead of a second-order Poisson equation that would be needed for a Newtonian tail. Secondly, the action (71) lacks a Newton constant G to start with; its gravitational coupling κ is dimensionless due to scale invariance.
To derive our result, we solved the pure R 2 field equation in the presence of a static point mass M , representing normal matter with the stress-energy tensor T µν = M c 2 δ( r) δ 0 µ δ 0 ν . Guided by Alvarez-Gaume et al's discovery regarding the role of the de Sitter background, we cast the metric in the form with V (r) being a function to be determined. In the absence of matter, V (r) ≡ 0 which results a de Sitter background.
The treatment of the Dirac delta function in the stressenergy tensor required special handling. To do so, we introduced a new method via the Gauss-Ostrogradsky theorem, where we cast the geometric side of the 00component of the field equation as the divergence of an auxiliary vector field B( r) which is expressible in terms of the potential V (r). When applying the Gauss-Ostrogradsky theorem on the divergence term, the field equation, including the stress-energy tensor of the point mass M , is transformed into an inhomogenous ODE of V (r). From the ODE, the potential V (r) can be summarily obtained. We illustrated the use of this divergencebased method for the Einstein-Hilbert action to recover the classic result V (r) = −GM/r, and then went on to apply the method to the pure R 2 action. We carried out step-by-step calculations, as detailed in Section III and the Appendix.
For the vacuo outside of the mass source, the function V (r) was found to have an exact analytical form The resulting metric is Schwarzschild-de Sitter, with the Schwarzschild radius parameter r s being fully determined by the mass source M and the (dimensionless) gravitational coupling κ. The term V (r) therefore represents a potential with the correct Newtonian falloff.
The parameter Λ plays a vital role. It appears in two places: (i) in the background term −Λr 2 /3 and (ii) in the Newtonian potential V (r). The de Sitter background allows the emergence of the Newtonian potential naturally. Its scalar curvature 4Λ helps transform the dimensionless gravitational coupling κ into a dimensionful Newton constant per Note that the Newtonian constant G is not a parameter of the pure R 2 action. Rather it is generated from κ and Λ.
It is evident why our result is beyond the reach for the conventional approach that linearizes the metric around a locally flat background, as explained in Section IV. Whereas a linearization around η µν is legitimate for the Ricci-scalar-flat vacua, such as those in GR, the locally flat metric η µν is a false background to be used in pure R 2 gravity. Previous works that relied on the locally flat background inadvertently omitted crucial information, namely, the curvature 4Λ of the background metric.
Moreover, the participation of Λ in the Newtonian potential is singular, as it appears in the denominator of expression (74). Consequently, perturbative techniques that treat Λ order-by-order cannot achieve this analytical form of V (r). Our derivation fully incorporates the effects of Λ non-perturbatively, hence able to unveil the singular relationship between the potential V (r) and Λ. In contrast, the conventional method of linearization around η µν fails to capture this singular relation.
In summary, building upon Alvarez-Gaume et al's discovery of the crucial role played by the de Sitter background in enabling the existence of massless spin-2 gravitons [27], our study strengthens their findings by establishing Newtonian behavior for pure R 2 gravity.
Outlook.-In a broader context, the capability of the R 2 term alone to produce a Newtonian potential opens up the possibility of exploring theories that dispense the Einstein-Hilbert term from the outset. This advancement paves the way for investigating classically scale invariant theories of gravity that incorporate the Glashow-Weinberg-Salam model of particle physics, showcasing significant potential for further exploration [38,39]. Acknowledgments I thank Timothy Clifton for his constructive correspondences regarding his joint work with the late John Barrow [36]. I further thank Richard Shurtleff and Tiberiu Harko for their helps and comments. Maciej Dunajski helpfully pointed out an affinity between our work and his recent proposal in [40], opening up potential avenues of exploration and enriching the scope of this study. ------∞------

Appendix A: DETAILED CALCULATIONS
In support of Section III, below is the detail calculation being carried out step-by-step.
In this Appendix we shall suppress the speed of light c for the sake of clarity. The metric is The terms that are relevant for our calculations in this Appendix are (with prime denoting derivative with respect to r):