Probing the minimal geometric deformation with trace and Weyl anomalies

The minimal geometric deformation, MGD, and the MGD-decoupling methods consist of well-succeeded procedures that can construct analytical solutions of the brane Einstein’s effective field equations, in AdS/CFT and its membrane paradigm [1–5]. Our universe, the codimension-1 brane with intrinsic tension, is assumed to be embedded in a bulk [6]. The minimal anisotropic procedure onto the brane, has been thrivingly employed to engender exact inner solutions to Einstein’s field equations for static and nonuniform stellar configurations, containing local and nonlocal bulk terms. The MGD is a formal approach that generates holographic and realistic varieties of not only any solution in General Relativistic (GR) but extended ones, on fluid branes [7]. Weyl functions into stellar distributions can create effective physical signatures of the bulk. Gravitational field equations, in GR, take into account the regime of a rigid brane, where the tension is infinite. However, to match recent observational data, the brane tension, that emulates the vacuum energy itself, must be finite. The current range of the brane tension, σ, is given by σ ' 2.813 × 10 GeV [8]. The brane tension controls the MGD of the Schwarzschild spacetime, that is a solution of the brane Einstein equations, and also drives other deformed solutions that include tidal charge [9–13]. The brane tension is proportional to the universe temperature, shown by WMAP regarding the CMB anisotropy [14, 15]. The MGD was employed to construct compact stellar distributions on the brane [16–18]. Refs. [19, 20] introduced the bridge between braneworld models and the holographic AdS/CFT setup [19–21]. The MGD and extended MGD solutions were studied in Refs. [22–24] under different phenomenological setups. Analog MGD models of gravity, using acoustics in a moving fluid, were proposed [25]. Refs. [26–35] have paved a robust way to construct solutions of the brane Einstein effective field equations, using MGD methods. Moreover, MGD-decouping procedure has been employed to engender anisotropic solutions that describe anisotropic stellar

The MGD-decoupling procedure iteratively constructs, upon a given isotropic source of gravitational field, anisotropic compact sources of gravity, that are weakly coupled. One starts with a perfect fluid, then coupling it to more elaborated stress-energy-momentum tensors that underlie realistic compact configurations [47][48][49][50][51][52]. Ref. [53] demonstrated that for positive anisotropy, when the radial pressure is smaller than the tangential pressure, the stellar distribution exerts a force that is repulsive and compensates for the pressure. In this way, anisotropic compact stars are more plausible stellar configurations to happen, as proposed and studied in Refs. [54][55][56][57][58][59][60][61][62][63][64][65]. Current observations of anisotropy in compact stellar distributions, via gravitational waves, have contributed for MGD-decoupling to occupy a successful role, as a framework that describe high density, anisotropic, astrophysical entities, that comprise X-ray sources, X-ray pulsars and X-ray bursters as well [66,67]. The MGD-decoupling has been also employed to study strange stellar configurations, as the astrophysical object SAX J1808.4-3658 [53], as the brightest X-ray burst ever observed, as spotted by the Neutron Star inner Composition Explorer (NICER). In addition, anisotropic neutron compact stellar configurations were used to describe the compact astrophysical objects 4U 1820. 30 and 1728.34, RX J185635.3754, PSR J0348+0432 and 0943+10, for instance, [68,69]. Strange quark stellar configurations were also explored in Ref. [70]. The extension of isotropic compact solutions to anisotropic ones, through the MGD-decoupling, was also proposed in Refs. [71,72].
The so called Weyl anomaly represents the fact that conformal invariance under Weyl rescaling displayed by gravity-interacting classical fields, is no longer present after quantization. This anomalous behaviour was first pointed out by Ref. [73], back in 1973, and since then has been applied to many areas of theoretical physics (for a review see [74]). Of particular interest for this work is its application to AdS/CFT correspondence, where this anomaly is related to black holes on braneworlds, and, in this context, it is called holographic Weyl anomaly [75]. Other relevant aspects of trace anomalies were comprehensively developed in the seminal Refs. [76][77][78][79].
Comparison of the holographic Weyl anomaly to the trace anomaly of the energy-momentum tensor from 4D field theory leads to a coefficient [80] that measures the back-reaction of the brane on the bulk geometry. Therefore it is a good way to understand how accurate the AdS/CFT description of the on-brane boundary theory is, given a particular solution of the codimension-1 bulk.
This paper is organized as follows: Sect. II is dedicated to review the MGD derivation as a complete method to deform the Schwarzschild solution and to describe realistic stellar distributions on finite tension branes. In Sect. III, the trace anomalies are computed for MGD solutions, from the point of view of 4D QFT and compared to that predicted by the AdS-CFT correspondence. Sect. IV is devoted to conclusions and important perspectives.

II. MGD: FRAMEWORK AND METRIC
The MGD method provides bulk corrections to well known solutions of GR, including high energy and nonlocal corrections [3,12]. Underlying the MGD procedure, fluid branes are endowed with an intrinsic tension, mimicking the vacuum energy [14,81].
The brane effective Einstein field equations are given by where g µν is the brane metric, G stands for the brane Newton coupling constant and G µν is the well known Einstein tensor; Λ brane is the cosmological constant on the brane. The stress-energy-momentum tensor, appearing in Eq. (1), can be decomposed as [82] T αβ = T αβ − E αβ + σ −1 κ 4 5 Π αβ + K αβ + M αβ , (2) where κ 2 5 = 48πG. The T αβ term represents the brane stress-energy-momentum tensor, that encodes the brane energy (including dark energy) and brane matter (including dark matter) content. Given the bulk Weyl tensor C αβρσ , its projection onto the brane, E αβ ≡ C αβρσ n ρ n σ , where n σ is a unitary vector field out of the brane, brings nonlocal ingredients to the brane effective Einstein field equations (1) and, in general, is σ −1 -dependent. Clearly, when the brane is infinitely rigid, corresponding to the GR σ → ∞ case, the tensor E αβ is equal to zero. It is worth to mention that the brane Weyl tensor contains the so called Weyl functions, namely, the Weyl scalar, U, and the anisotropy, P, encoded into any stellar configuration that is solution of (1), being both proportional to the stellar configuration compactness. The Weyl functions arise from the MGD undertaken by the g rr component of the metric, due to AdS bulk effects. In addition, generalized models, modifying the pressure by bulk effects, also encompass nonlocal terms encoding the bulk Weyl curvature [9]. The Π αβ component of the stress-energymomentum tensor encrypts quadratic terms involving the stress-energy tensor, arising from the extrinsic curvature terms in the Einstein tensor projection onto the brane. In fact, the brane matching conditions, applied to the extrinsic curvature tensor, K µν , makes it to be expressed [82]. Also denoting T = T µ µ , one can explicitly write The tensor K αβ in Eq. (2) describes an eventual asymmetric embedding to the brane into the AdS bulk and the M αβ tensor include bulk gravitons and moduli fields [81,83]. Compact stellar configurations represent analytical solutions of the gravitational field equations (1), with metric where dΩ 2 denotes the solid angle element. One denotes A(r) = e ν(r) and B(r) = e ξ(r) , for the sake of conciseness. Let one defines the integral denoting by a prime the derivative with respective to the radial coordinate. The MGD asserts that the B(r) −1 radial component of the metric (4) can be deformed as [9] e −ξ(r) = µ(r) + κ(r, σ −1 ) , where [7] κ(r, for In Eq. (8), p(r) denotes the pressure and ρ(r) the density of the compact star. The GR infinitely rigid brane, σ → ∞ case, yields κ(r) → 0. Moreover, the function appearing in Eq. (6) reads where R denotes the star surface radius and M Schw is the Schwarzschild star GR mass. The function L(r) = L(ν(r), p(r), ρ(r)) encodes bulk-induced anisotropy. The function b = b(r, σ) in Eq. (7) will be derived soon. The MGD κ(r), in vacuum, where p(r) = ρ(r) = 0, will be hereon represented byh(r) [2]: The MGD can be split into terms that are factors of σ −1 , encompassing high energy terms, and terms that encode nonlocal effects of the Weyl fluid. Junction conditions match the inner MGD metric, meaning that r < R, for κ(r, σ) given in Eq. (6), making L(r) = 0, in the so called outer region, r > R. The Weyl fluid, that moistens the brane, can be represented by Weyl functions. The on-brane Weyl tensor, being inversely proportional to the brane tension, can be split off as where the vector field u µ is the velocity that describes the flow of the Weyl fluid and the tensor h µν = g µν − u µ u ν projects quantities into the flow direction. In addition, the σ-dependent Weyl scalar, explicitly given by U = − 1 2 σE αβ u α u β represents the energy density, whereas denotes the (nonlocal) anisotropic energy-stress-momentum tensor. Besides, the brane nonlocal flux of energy is given by The trace of the anisotropic energy-stress-momentum tensor and the Weyl scalar in the outer region are, respectively, given by In the vacuum, p(r) = ρ(r) = 0 in the r > R outer sector. Therefore, the outer metric can be written as [2,18] ds 2 = −e ν + (r) dt 2 + dr 2 1− 2GM c 2 r +h(r) Junction conditions that match the outer to the inner stellar sector, at the stellar configuration surface r = R, imply that [2] ν ± (R) = ln 1 − 2GM Regarding Eq. (16) where a α denotes a radial vector field and one denotes the matching function bẏ averaging any quantity on the star surface by its values in both the inner and the outer surface neighbourhoods. Thus, Eq. (17) implies⌊T ⊺ µν a ν⌋ = 0, yielding [10] 2 [σ + ρ(r)] p(r) + ρ 2 (r) + 4G 2 U(r) + 2G 2 P(r) ˙ = 0 .
To derive the function b(σ), firstly Eq. (19) must be rewritten as [2], It means that the functionh(R) -evaluated at the star surface -attains negative values. Therefore, the MGD coordinate singularity, r MGD = 2GM /c 2 , is spotted near the center of the MGD stellar configuration, when compared to the Schwarzschild coordinate singularity r Schw = 2GM Schw /c 2 . In fact, recall that M = M Schw + O(σ −1 ). Therefore, Weyl fluid effects on the brane induce a weaker gravitational force, when compared with standard Schwarzschild solutions [2,27]. Eqs. (20,21) imply that [2] b where d 0 is given by the awkward expression in Eq. (31) in Ref. [9]. Ref. [23] derived the corresponding experimental and observational signatures of a bulk Weyl fluid, obtained from the Solar system classical tests, encompassing the perihelion precession of Mercury, the deflection of light by the Sun and the radar echo delay. The bound d0 σ 2.8 × 10 −11 has been obtained in Ref. [23].
The MGD metric can be then written as [2] A(r) = e ν Schw (r) = 1 − 2GM where In the GR, rigid brane limit σ → ∞, the MGD metric is clearly lead to the Schwarzschild metric, as M = M Schw + O(σ −1 ). Hence, using the classical tests of GR and replacing the bound d0 This universal bound holds for MGD compact stellar configurations of any mass. In fact, varying the mass in Eq. (24) also makes the star surface radius, R, and hence the functions in Eq. (22) to me modified, accordingly.

III. MGD ANOMALIES
As stated briefly in the introduction, our goal is to compare the trace anomalies from the field theory side against the one found in the CFT. This was introduced in Ref. [80] and consists in defining the coefficient are the trace and holographic Weyl anomalies, respectively, and n b is the number of gauge bosons. The quantity T 4D is obtained using only field theoretic methods in curved spacetimes [84], and does not vanish in general for curved backgrounds (even for Ricci flat ones), as it depends on the Kretschmann scalar,K. The holographic Weyl anomaly appears when one evaluates the effective action of a CFT, via the AdS/CFT procedure. When computing the effective action of the boundary theory on the brane one is forced to choose one amongst the equivalence class of metrics forming the conformal structure of the boundary, therefore explicitly breaking the conformal symmetry in order to obtain a finite value. This anomaly is perceived as an UV effect, since it is present in the boundary theory, but arises from a divergence whose origin is on the IR scale. Such a divergence is present in the bulk [75]. The explicit form of this anomaly depends on the dimension of the spacetime where the CFT boundary is placed. For odd dimensions, the anomaly always vanishes, whereas for even dimensionality the expressions get more intricate as the number of dimensions increase [75]. We only present the 4D case, which is the case we are interested in, the anomaly reads 1 Here Rµν is the Ricci tensor and R the scalar curvature.
considering a stack of N branes, where E [4] is the Euler density and I [4] is the conformal invariant. In 4D, there is only one conformal invariant, which is the Weyl tensor contracted with itself. The explicit expressions for the invariants are It is clear from Eq.
Eq. (28) is then obtained by using the AdS/CFT dictionary on braneworlds, that relates N degrees of freedom to the Planck length and brane tension [20,80].
The developments in the beginning of Sect. II can be now equivalently implemented in the context of AdS/CFT. Firstly, the brane Einstein equations can be expressed as [75,[85][86][87][88]: for a = 4K −1 , where K = K µ µ is the extrinsic curvature trace. In addition, the quantity Γ CFT carries the CFT action on the boundary. Its trace anomaly reads [75,87]: The term S • has R 2 counterterms that yield a finite action, whereas the trace of the term δS • /δg µν equals zero, for Now, taking the trace of all terms in Eq. (31) yields R = − 8πG c 4 T + a 2 4 1 3 R 2 − R µν R µν . Therefore, up to linear order, the CFT stress-energy-momentum tensor emulates the brane Weyl tensor part, that can be read off as [85,87,88] Hence, it appropriately locates the on-brane Weyl tensor (11), and its consequences to the Einstein effective brane equations (1), in the AdS/CFT setup. The coefficient (26) measures the reliability of results obtained through the AdS/CFT correspondence, when a given spacetime metric is under investigation in the following sense: by measuring the trace anomalies, one can check how the back-reaction of bulk perturbations affect results on the brane or vice-versa, that is, if the presence of the brane has any effect on the bulk geometry. The value of Υ CFT ranges from 0 to infinity, and the predictions from AdS/CFT become less reliable the higher the value of the coefficient is [80].
If one considers the Schwarzschild spacetime [80], setting ℓ = 0 in Eq. (23b), the immediate results T 4D ∝ K and T CFT = 0 are found. The second identity leads to the conclusion that Υ CFT → ∞, and therefore results from the AdS/CFT correspondence are, at best, questionable.
We proceed to extract numerical values for Υ CFT for two known values of the ℓ parameter, that appears in the MGD metric radial component (23b), respectively for two regimes of the ADM mass M [23,43]. In the following calculations, we will use SI units for all quantities 2 . Ref. [9] has shown that ℓσ = −0.0042572. This is particularly interesting because we can then eliminate the brane tension in Eq. (39) and write it only in terms of known quantities: The first set of data to be considered is the already discussed value (25) [23], the other relevant values are 2 ℓp = 1.616 × 10 −35 m, G = 6.674 × 10 −11 m 3 kg −1 s −2 .
given above that equation. Besides, in Ref. [43] the MGD solution was applied to modelling gravitational lensing effects, where the Sagittarius A * black hole of mass M = 4.02 × 10 6 M ⊙ was considered, therefore R = 2M is the event horizon radius. For this case, the observational value of ℓ = 0.06373m was obtained [43]. For both sets, |ℓ| ≪ 1 leading Eq. (40) to imply that When ℓ → 0, all the results regarding the Schwarzschild solution are recovered, however the current range of the brane tension, σ 2.813 × 10 −6 GeV 4 [8], together with the equality ℓσ = −0.0042572, makes ℓ not to attain a null value. Therefore, it places the MGD, in the AdS/CFT formulation on the brane, into a trustworthy position to be a realistic model to describe, in this context, stellar distributions that are compatible to AdS/CFT.

IV. CONCLUSIONS
The MGD, usually employed to obtain static, strongly gravitating, spherically symmetric and compact stellar distributions was here explored with the tools of trace and Weyl anomalies. Contrary to the Schwarzschild solution, for which the implausible result Υ CFT → ∞ in Eq. (26) yields the AdS/CFT correspondence to be difficult to be implemented in this context, the MGD has been shown to be a reliable attempt to describe realistic models, in the AdS/CFT setup. In fact, the parameter in Eq. (26) quantifies how safe AdS/CFT is when bulk/brane back-reaction effects are taken into account. Since the value of the Υ CFT coefficient, for the MGD case, was shown to be near unity, it means that the MGD solutions may occupy a privileged place and can play a prominent role on emulating AdS/CFT on braneworld scenarios.
Similarly to AdS/QCD models, where the extra dimension is interpreted as an energy scale in QCD, in the setup here established, phenomena regarding CFT coupled to gravity can be exclusively interpreted from the point of view of the brane. Bulk gravitons that propagate in the bulk correspond to 4D gauge bosons on the boundary. The difference of these two countenances, in the AdS/CFT setup, cannot be identifiable, from any phenomenological point of view.