Impact of Star Pressure on γ in Modified Gravity beyond Post-Newtonian Approach

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We offer a concrete example exhibiting marked departure from the Parametrized Post-Newtonian (PPN) approximation in a modified theory of gravity.Specifically, we derive the exact formula for the Robertson parameter γ in Brans-Dicke gravity for spherical compact stars, explicitly incorporating the pressure content of the stars.We achieve this by exploiting the integrability of the 00−component of the Brans-Dicke field equation.In place of the conventional PPN result γ PPN = ω+1 ω+2 , we obtain the analytical expression γ exact = ω+1+(ω+2)Θ ω+2+(ω+1)Θ where Θ is the ratio of the total pressure P * ∥ + 2P * ⊥ and total energy E * contained within the star.The dimensionless quantity Θ participates in γ due to the scalar degree of freedom of Brans-Dicke gravity.Our non-perturbative formula is valid for all field strengths and types of matter comprising the star.In addition, we establish two new mathematical identities linking the active gravitational mass, the ADM (Arnowitt-Deser-Misner) mass, and the Tolman mass, applicable for Brans-Dicke gravity.We draw four key conclusions: (1) The usual γ PPN formula is violated for high-pressure mass sources, such as neutron stars, viz.when Θ ̸ = 0, revealing a limitation of the PPN approximation in Brans-Dicke gravity.(2) The PPN result mainly stems from the assumption of pressureless matter.Even in the weak-field star case, non-zero pressure leads to a violation of the PPN formula for γ.Conversely, the PPN result is a good approximation for low-pressure matter, i.e. when Θ ≈ 0, for all field strengths.(3) Observational constraints on γ set joint bounds on ω and Θ, with the latter representing a global characteristic of a mass source.If the equation of state of matter comprising the mass source approaches the ultra-relativistic form, entailing Θ ≃ 1, γ exact converges to 1 irrespective of ω.More generally, regardless of ω, ultra-relativistic matter suppresses the scalar degree of freedom in the exterior vacuum of Brans-Dicke stars, reducing the vacuum to the Schwarzschild solution.(4) In a broader context, by exposing a limitation of the PPN approximation in Brans-Dicke gravity, our findings indicate the significance of considering the interior structure of stars in observational astronomy when testing candidate theories of gravitation that involve additional degrees of freedom besides the metric tensor.

I. MOTIVATION
The parametrized post-Newtonian (PPN) framework has been an invaluable tool in the study of gravitational theories [1][2][3].It is founded upon the assumptions of weak field and slow motion, representing the post-Newtonian limit of gravity.In this limit, the framework characterizes a given metric theory, at the first post-Newtonian level, by a set of ten real-valued parameters.
One of the most powerful applications of the PPN formalism is the calculation of the Robertson (or Eddington-Robertson-Schiff) parameter γ.This important parameter governs the amount of space-curvature produced by a body at rest and can be directly measured via the detection of light deflection and the Shapiro time delay.For one of the simplest extensions beyond General Relativity (GR)-the Brans-Dicke (BD) theory-the PPN approach is known to yield where ω is the BD parameter.This formula recovers the result γ GR = 1 known for GR in the limit of infinite ω, in which the BD scalar field approaches a constant field.Current bounds using Solar System observations set the magnitude of ω to exceed 40, 000.Generalizations of the PPN γ result to other modified theories of gravity are available [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17].
In the case of GR, by virtue of Birkhoff's theorem, spherically symmetric vacuum solutions are static (discarding the black hole region in the case of a fully vacuum spacetime) and asymptotically flat.The vacuum spacetime exterior of a mass source is described by the Schwarzschild metric which is dependent on only one parameter, the Schwarzschild radius.All information regarding the internal structure and composition of the source, namely, the types of matter comprising it as well as the distribution profile of matter in the source, is fully encapsulated in the Schwarzschild radius.Since GR, as a classical theory, lacks an inherent length scale (such as the Planck length), the dimensionful Schwarzschild radius cannot participate in the dimensionless γ parameter.Thus, this parameter is independent of the source in GR (a fact compatible with γ GR = 1).BD gravity, however, has a richer structure due to the BD scalar field Φ, an additional degree of freedom besides the metric components.For the special case of black holes, the no-hair theorem applies [18,19], meaning that, in the non-rotating case, the vacuum exterior to a static BD black hole is the Schwarzschild solution.Consequently, black holes in BD gravity, irrespective of the value of ω, have γ = 1 rather than the usual ω+1 ω+2 result.Besides black holes, BD gravity exhibits other structures-normal stars and exotic ones, such as wormholes and naked singularities [20][21][22].For these structures, the BD scalar field in the exterior vacuum is generally non-constant.The PPN formalism makes use of the non-constancy in Φ to derive the usual PPN result for stars, given in Eq. ( 1), which explicitly depends on the BD parameter ω [23].Yet, it is important to note that the post-Newtonian (PN) γ parameter, per Eq. ( 1), contains no information about the star (aside from the fact that the star is regular at its center), as the BD parameter ω is a parameter of the theory, not one of the resulting exterior vacuum.
In contrast to the second-derivative GR, where all information about the stellar source is condensed into one single parameter-the Schwarzschild radius-BD gravity, as a higher-order theory, permits the internal structure of the star to influence higher-derivative characteristics of the exterior solution, potentially leaving its footprints on the PN parameters.In their seminal work [25], Brans and Dicke recognized the approximate nature of estimating the two parameters characterizing the exterior vacuum solution, later known as the Brans Class I solution [26].This recognition was evidenced in Eq. (34) of Ref. [25].The PPN formula in Eq. ( 1) should be regarded as an approximation applicable in the limit of weak field everywhere (including inside the star) and slow motion in BD gravity, whereby the higher-derivative features of the exterior vacuum are suppressed.An important question arises: Can the internal structure of a star in BD gravity, and in modified theories in general, manifest through the PN parameters?And if so, under what conditions?
An obvious course of action would be to lift the weak field hypothesis on the source of the field, enabling an exact calculation of the external field, from which the PN parameters can be extracted.An attempt in this direction has been made recently in [24] where it is suggested that the higher-order terms can impact the γ parameter.Yet, there exists another condition, albeit less explicit, related to the pressure within the star.Note that the PPN requirement of slow motion applies not only for macroscopic objects but also for their microscopic constituents.Per the post-Newtonian bookkeeping scheme in Ref. [1], this translates to a requirement for low pressure relative to the energy content.Whereas a star is stationary, the microscopic motion of the matter contained within its domain can be relativistic, resulting in appreciably high pressure compared with its energy content.
Despite the challenges of relaxing the weak-field and low-pressure constraints, significant progress can be achieved in one particular situation-the BD theory.In our current study, we focus on a specific case-the BD exterior vacuum surrounding a matter spherical distribution which has a finite domain of support; we shall collectively call this type of structure a compact star.Here, we rigorously account for the influence of the compact mass source on the exterior vacuum and the γ parameter without resorting to any approximations.We achieve this by making use of the integrability of the 00-component of the BD field equations, enabling us to circumvent the limitations of the weak-field and low-pressure approximations.Advancements in detection methods allow for the study of neutron stars, making our investigation relevant both for practical applications and theoretical inquiries into the formalism and methodologies employed.Our study shall shed light on the role of stellar pressure and provide a benchmark for assessing its impacts in theories of modified gravity.
This paper provides a comprehensive account of the technical aspects of our findings, and is structured as follows.Section II revisits the form of the energymomentum tensor (EMT) of static spherisymmetric star sources in BD gravity.Sections III and IV handle the field equations in the standard coordinates and transform the results to the isotropic coordinates.The uniqueness of Brans solutions is addressed in Section V.Sections VI and VII conduct the interior-exterior matching and derive the γ parameter.Section VIII obtains mass relations, valid for BD gravity.Section IX illustrates the fact that any PN parameter can be calculated, by providing the explicit expression of the 2PN parameter δ.Sections X and XI offer discussions and outlooks.A further exposition on the uniqueness of Brans solutions is given in Appendix A.

II. THE ENERGY-MOMENTUM TENSOR
Consider the BD action in the Jordan frame [25,26] with the metric signature convention (− + ++).Hereafter, we choose the units G = c = 1 with G being the far-field Newtonian constant.
It is well documented [27] that upon the Weyl mapping gµν := Φ g µν , Φ := ln Φ , the gravitational sector of the BD action can be brought to the Einstein frame as R−(ω + 3/2) ∇µ Φ ∇µ Φ .The Einstein-frame BD scalar field Φ has a kinetic term with a signum determined by (ω + 3/2).Unless stated otherwise, we shall restrict our consideration to the normal ("non-phantom") case of ω > −3/2, where the kinetic energy for Φ is positive.
In this paper, we work exclusively in the Jordan frame.For convenience, let us denote a rank-two tensor The BD field equation for the metric components is and the equation for the BD scalar field is where the energy-momentum tensor (EMT) of the matter sector is For a coordinate system that is static and spherically symmetric, the metric can be written as (with g 01 = g 10 = 0) With this metric, the only non-vanishing components of the tensor X µν are the diagonal ones, namely, µ = ν.
With respect to the off-diagonal components, µ ̸ = ν, since g µν = 0, the field equation requires meaning the EMT has to be diagonal.The trace is merely a sum T = T 0 0 + T 1 1 + T 2 2 + T 3 3 .Furthermore since g 33 = g 22 sin 2 θ (9) one has Therefore, the EMT must adopt the following form The only assumptions we made are the stationarity and spherical symmetry for the metric components and the BD scalar field.The energy density ϵ, the radial pressure p ∥ and the tangential pressure p ⊥ are functions of r.The trace is simplified to We shall not impose any further constraints on the EMT, which can be anisotropic.

III. INTEGRABILITY OF THE 00−FIELD EQUATION
Let us start with the standard areal coordinate system In this form, a star center exists at r = 0, at which the metric components, A(r) and B(r) and the BD field Φ(r) are regular (including their first derivatives with respect to r).The scalar field equation ( 5) yields Multiply both sides with dV := dr dθ dφ, then integrate with the integral domain V being a ball of radius r, centered at the origin.
Since A, B, Φ, A ′ , Φ ′ are finite at the star center, r = 0, (viz.regularity conditions), upon integration of the left hand side, the above equation becomes Next, the 00−component of X µν can be written as It is imperative to note that the X 00 term is expressible in a neat "integrable" form, Eq. (20).Consequently, the 00−component of the field equation ( 4) yields Integrating from the star center (assuming regularity) The rest of this section deals with the terms in the right hand side of Eqs. ( 17) and (22).Consider a compact star of finite radius r * and denote V * the domain of the star, namely, V * being a ball centered at the origin with radius r * .The following integrals are defined for the domain V * : Note that outside the star, ϵ, p ∥ , and p ⊥ vanish.Therefore for a ball V that is centered at the origin and encloses V * (namely, its radius r exceeds r * ), the following identities hold: For a ball V enclosing V * , the integrals in the right hand side of Eqs. ( 17) and ( 22) are = and Remark 1.The integrability property that we exploited in deriving Eqs. ( 17) and ( 22) is not limited to the standard coordinates.If we used the metric in Eq. ( 7), the left hand side of Eqs. ( 17) and ( 22) would read and CΦ √ AB dA ar respectively, and the √ −g term in the right hand side of these equation would be calculated using the latter metric.Nevertheless, the standard-coordinate metric in (14) explicitly reveals the presence of the point r = 0 at which the surface area of a 2-sphere vanishes.This point naturally corresponds to the star's center and serves as the lower bound in the integrals defined in (23).
Remark 2. In the integrals defined in Eq. ( 23), the element dV The term √ A, equal to √ −g 00 , is a "redshift factor".The combination r 2 √ Bdr sin θdθ dφ, equal to r 2 √ g 11 dr sin θdθ dφ, is the spatial volume element in the spatial part of the metric given in Eq. ( 14).Note that in the region occupied with matter, e.g.where ϵ(r) ̸ = 0, the space in general is not Euclidean; consequently, for the standard-coordinate metric, B(r) deviates from 1.
It should also be noted that the combination r 2 √ g 11 dr sin θdθ dφ is equal to g (3) dV , where g (3) is the determinant of the spatial section of the metric in Eq. ( 14).The combination is thus invariant with respect to a twice-differentiable transformation of the radial coordinate.The quantities E * , P * ∥ , and P * ⊥ can be interpreted as the total energy, radial pressure, and tangential pressure contained within the compact star.

IV. TRANSFORMING TO ISOTROPIC COORDINATES
With the total energy and pressures defined via Eq.( 24), the set of equations ( 17), ( 22), (26), and (28) essentially provide the "conservation" rules for the metric components and the BD scalar in the exterior vacuum, i.e., for r > r * , per and ) Note that the right hand sides of Eqs. ( 29) and (30) are integration constants, induced by the matter distribution in the interior of the compact mass source.Our next step is to relate the parameters of the exterior vacuum to these constants.The vacuum solution for BD gravity is best known in the isotropic coordinate, in the form of the Brans Class I solution.(The issue with the generality and uniqueness of the Brans Class I solution shall be addressed in Section V.) It is thus necessary to bring the left hand sides of Eqs. ( 29) and (30) to the isotropic coordinate ρ, namely, using the following metric and the BD scalar ϕ(ρ).Transforming ( 14) into (31) requires the following identifications in addition to a mapping for the BD scalar In the far-field region, it is expected that the relation between r and ρ is monotonic, viz.dρ dr > 0. The quantities of interest thence become and We thus arrive at and Remark 3. As we stated at the end of the preceding section, the energy density and pressure quantities are invariant upon a coordinate transformation in the radial coordinate.This can be verified for the case at hand.For example, the total energy in the standard coordinates is whereas in the isotropic coordinates with the identification ϵ ′ (ρ) = ϵ (r(ρ)).It is straightforward to see that Hence, the energy integral E * is the same for both systems-the standard and the isotropic coordinates.Likewise, the same conclusion applies for the pressure integrals P * ∥ and P * ⊥ .

V. BRANS CLASS I AS THE UNIQUE VACUUM SOLUTION FOR ω > −3/2
Equations ( 36) and ( 37) stand handy for us to relate the parameters of the exterior vacuum solution to the energy-pressure integrals.It is well documented that the Brans Class I solution is a vacuum solution in BD gravity.
In this section, we shall further demonstrate that, for the case of non-phantom kinetic energy for the BD field, viz.ω > −3/2, the Brans Class I solution is the most general and unique vacuum solution.That is to say, the Brans Class I solution is the vacuum solution in BD gravity, when ω > −3/2.
Historically, Brans discovered the solutions during his PhD thesis and reported them in [26] without offering a derivation, although the solutions can be verified via direct inspection.The earliest public account for an analytical derivation of these solutions can be traced back to Ref. [27] in which Bronnikov discovered the more general Brans-Dicke-Maxwell electrovacuum solution.
The Brans solutions are comprised of 4 different classes (or types).In the exposition of Bronnikov [27], the Brans-Dicke theory is first mapped from the Jordan frame to the Einstein frame via a Weyl mapping.The transformed BD scalar field becomes uncoupled (thus free) scalar field which sources the Einstein-frame spacetime metric.This transformed theory is known to admit the Fisher-Janis-Newman-Winicour (FJNW) solution.
However, the existence of multiple Brans classes has sidetracked a recognition of their "uniqueness".There is a degree of redundancy in these classes, however.In [34] Bhadra and Sarkar pointed out that Class III and Class IV are equivalent via a coordinate transformation, ρ ↔ 1/ρ, reducing the count of classes to 3.These authors also uncovered a "symmetry" in terms of parameters of Brans Class I and Brans Class II, upon making certain replacement in the parameters (and a coordinate transform) 1 .Consequently, it was deemed that the two classes, I and II, were equivalent, leaving only Class I and Class III to be truly independent.Nomenclature aside, it should be noted that diffeomorphism alone cannot transform Class I into Class II and vice versa.In [33] Faraoni and colleagues revisited this issue and correctly branded the "symmetry" alluded above a "duality" rather than an "equivalence".As we shall show momentarily, the 3 Brans classes (I, II, and IV) remain separate solutions (while fully covering) in the parameter space.Yet, the 3 classes can be "unified" upon an appropriate parametrization.That is to say, all 3 Brans classes of solution can be brought into a single form, as we shall do below.Consider the metric and scalar field with M 1 ∈ R + (a parameter of length dimension) and dimensionless parameters Λ ∈ R and κ ∈ R. It is straightforward to verify by direct inspection that the metric and scalar field satisfy the BD field equations, with the parameter κ ∈ R obeying (ω ∈ R and Λ ∈ R) Three cases to consider, depending on the signum of κ: Case 1: For κ > 0, it is evidently Brans Class I solution as originally reported in [26].
Case 2: For κ < 0, utilizing the identity tan The change of variable ρ resulting in Brans Class II solution in the new radial coordinate ρ ′ [26]: Case 3: For κ = 0, the limit produces Brans Class IV solution [26]: Let us emphasize that the recasting exercise thus far represents nothing essentially new.The important point which our recasting offers is that the signum of κ elects the "class" that the "unified" Brans solution, Eq. ( 43), belongs.Moreover, a pair of {ω, Λ} uniquely determines the signum for κ, and hence the form of the solution.Figure 1 shows the categorization.In the {ω, Λ} plane, the two branches of the solid line correspond to the loci of κ = 0, viz.(1 + Λ)2 − Λ 1 − ωΛ/2 = 0. Above and below the solid line are the domains for Brans Class I and Brans Class II, respectively.The 3 classes do not overlap while fully covering the {ω, Λ} plane.The ambiguity in selecting the solution is removed.The uniqueness of the "unified" Brans solution, Eq. ( 43), is thereby established.Remark 4. For ω > −3/2, the function κ is positivedefinite for all values of Λ in R, as can be seen by rewriting κ as Consequently, only Brans Class I is admissible for the non-phantom action, ω > −3/2, a fact evident in Fig. 1.
Remark 5.It is worth noting that Brans Class I recovers the Schwarzschild metric when Λ = 0 forcing κ = 1.On the other hand, Brans Class II and Class III do not have this property and are often relegated as "pathological" solutions.Conveniently, they only exist for the phantom action, i.e., ω ⩽ −3/2.
Another way to recognize the unified nature of the 3 Brans solutions is via the harmonic coordinate instead of the isotropic coordinate [27].For completeness, we shall revisit this representation in Appendix A.

VI. MATCHING THE EXTERIOR SOLUTION WITH ENERGY-PRESSURE INTEGRALS
As established in the preceding section, for the nonphantom kinetic action (i.e., ω > −3/2), the exterior vacuum solution is exclusively Brans Class I. We are now equipped to perform the matching of its parameters.

VII. THE ROBERTSON PARAMETERS
The Robertson expansion in isotropic coordinates is [23]: in which β and γ are the Robertson (or Eddington-Robertson-Schiff) parameters.The metric in Eq. ( 43) can be re-expressed in the expansion form Comparing Eqs.(65) against Eq.(66), we obtain where we have added in the subscript "exact" as emphasis.Note that Λ measures the deviation of the γ parameters from GR (γ GR = 1).From Eq. ( 62), Λ depends on both ω and Θ.Finally, we arrive at which can also be conveniently recast as by recalling that γ PPN = ω+1 ω+2 .With the aid of Eqs. ( 60) and (61), the active gravitational mass is where the contribution of pressure to the active gravitational mass is evident [36,37].

VIII. MASS RELATIONS IN BRANS-DICKE GRAVITY
In GR, the Tolman mass was defined as [38][39][40] In our EMT form (12), this renders For a metric that has the following asymptotic form (i.e., as ρ → ∞) [40]: the quantity M grav is the active gravitation mass of the source, whereas M ADM is the ADM mass.In GR, it is known that For BD gravity, besides Expression (72) for M grav , viz.
we can also calculate the ADM mass, with the aid of Eqs. ( 65) and ( 69) The difference for ω > −3/2 and normal matter, viz.p ∥ ⩽ 1 3 ϵ and p * ⊥ ⩽ 1  3 ϵ.It is interesting to note that although M grav , M ADM , and M T have different values in BD gravity, the mean value of M grav and M ADM precisely equals M T .Remark 6.In the limit of infinite ω, we recover the usual relation in GR Furthermore, in GR, the ADM mass has been established within the Tolman-Oppenheimer-Volkoff framework (using the standard coordinates) to be Using Eq. ( 23) in the standard coordinates ( 14), we rewrite the right hand side of Eq. ( 83) as We thus have re-derived the Tolman relation in GR, as a by-product of our study [38,40].

IX. EXTENSION TO OTHER PPN LEVELS
In principle, once the Brans Class I solution for the exterior vacuum is fully determined by the energy and pressure integrals, all PN parameters (of any PPN level) can be readily calculated.These PN parameters would be useful for estimating the values of ω and Θ, since a sole measurement of γ is insufficient to fix two unknowns.In this section, we illustrate the utility of the complete Brans Class I for a second-order PN parameter δ.It is worth noting that ongoing and future space missions, such as the Gaia Mission, require the second-order for light propagation in some specific situations [41], thereby underscoring the importance of deriving δ in this context.The 2PN extension in the isotropic coordinate system is Note that for Schwarzschild metric, the expansion above yields γ = δ = 1.The expansion of the Brans Class I gives ) with κ given in Eq. (44).Taken together and in conjunction with Eq. ( 62), these expansions render When ω → ∞, this formula recovers the GR value It should also be noted that, for ultra-relativistic matter, viz.Θ → 1 − , as Λ → 0, the formula gives X. DISCUSSIONS Formula (69) is one of the most important outcomes of our paper.This section aims to clarify a number of logical and technical steps taken in the derivation, and discusses the implications of Formula (69).
Our derivation proceeded in the following steps: 1. Assuming the metric and the mass source to be stationary and spherically symmetric, we deduced from the BD field equation that the most general EMT of the source can be put in the form T ν µ = diag −ϵ(r), p ∥ (r), p ⊥ (r), p ⊥ (r) .Whereas the EMT can be anisotropic, we imposed no further conditions on the EMT, such as being a perfect fluid.See Section II.
Using the scalar field equation for Φ and the 00−component of the field equation, and imposing regularity conditions at the star's center, we derived two ODE's-Eqs.( 17) and ( 22)-which express A(r), B(r) and Φ(r) in the exterior vacuum in terms of the total energy E * and total pressure P * ∥ + 2P * ⊥ contained within the star.The standard coordinate system allows for the existence of the point r = 0 at which the surface area of the 2sphere vanishes, which naturally serves as the star's center.This point acts as the lower bound for the domain of integration for E * and P * ∥ + 2P * ⊥ .See Section III.
3. We next transformed the two aforementioned ODE's into the isotropic coordinate system, The resulting ODE's in terms of F (ρ), G(ρ) and ϕ(ρ) are Eqs.(36) and (37).The advantage of the isotropic coordinates is that the exterior solution is best known in this system (i.e., the Brans solutions).Furthermore, the energy and pressure integrals, E * and P * ∥ + 2P * ⊥ , are unchanged when moving to this system.See Section IV.
4. We next presented the "unified" Brans solution, Eq. (43), which cover all Brans Classes I, II and IV (with Class III being equivalent to Class IV).For a non-phantom action, i.e. ω > −3/2, Brans Class I is the most general and unique static spherisymmetric vacuum solution.For a phantom action, i.e. ω < −3/2, all 3 Brans classes are admissible; however, only one single class is selected for a given set of parameters of the "unified" Brans solution.The uniqueness of the "unified" Brans solution is thus established.See Section V.
5. We employed the Brans Class I solution to perform the matching of the functions F (ρ), G(ρ) and ϕ(ρ) with the energy and pressure integrals, E * and P * ∥ + 2P * ⊥ .See Section VI.
6. Expressing the Brans Class I metric in the Robertson expansion, we obtain the γ and δ PN parameters in terms of ω and a (dimensionless) Θ, defined as the ratio of P * ∥ + 2P * ⊥ and E * .See Sections VII and IX.
7. As a by-product, we obtained two mathematical relations linking the active gravitational mass and the ADM mass with the energy and pressure integrals for BD gravity.In addition, we (re)-derived the Tolman relation in GR.See Section VIII.

Generality of our derivation-(i) Non-perturbative approach:
Our derivation is non-perturbative in nature.It makes use of the integrability of the 00−component of the BD field equation, along with the scalar field equation involving □ Φ. (ii) Minimal assumptions: The physical assumptions employed are the regularity at the star's center and the existence of the star's surface separating the interior and the exterior.Our derivation relies solely on the scalar field equation and the 00-component of the field equation, without the need for the full set of equations, specifically the 11− and 22− components of the field equation.Consequently, the conservation equation (by way of the Bianchi identity) is not required.(iii) Universality of result: The final formula, Eq. ( 69), holds for all field strengths and all types of matter (whether convective or non-convective, for example).We do not assume the matter comprising the stars to be a perfect fluid or isentropic.The only constraints on matter come from the stationary and spherical symmetry through Eq. ( 12).
Higher-derivative characteristics-In contrast to the one-parameter Schwarzschild metric, the Brans Class I solution depends on two parameters, i.e. the solution is not only defined by its gravitational mass, but also by a scalar mass besides the gravitational one [27].This is because the BD description of gravity involves more fields than the only metric involved in GR, a scalar field being also part of the gravitational sector.(The same is also to be expected in the higher order theories framework, like f (R) theories, since f (R) theories can be recast as ω = 0 BD theories endowed with a scalar dependent potential.)The exterior BD vacuum should reflect the internal structure and composition of the star.This expectation is confirmed in the final result, Eq. (69), which underscores the participation of the parameter Θ.
Role of pressure-Figure 2 shows contour plots of γ exact and δ exact as functions of γ PPN (i.e, ω+1 ω+2 ) and Θ.The parameter ω lies in the range (−3/2, +∞) (to avoid the "phantom" kinetic term for the BD scalar field when moving to the Einstein frame, an issue mentioned at the beginning of Section II) which translates to ω+1 ω+2 ∈ (−1, 1).The parameter Θ lies in the range [0, 1) in which 0 corresponds to "Newtonian" stars, and 1 corresponds ultra-relativistic matter where p ∥ = p ⊥ = 1 3 ϵ.It is important to note that the strict upper bound Θ = 1 is excluded since a linear equation of state for matter cannot produce a well-defined star surface, at which location p = 0, when solving the Tolman-Oppenheimer-Volkoff (TOV) equation for GR and BD stars.
• For Newtonian stars, i.e. low pressure (Θ ≈ 0), the PPN γ result is a good approximation regardless of the field strength.
• To evaluate ω and Θ for a given star system, measurements of both γ and δ are necessary.Since the contour plots of γ and δ in Fig. 2  √ ω) anomaly can exist even for non vanishing trace of the EMT [52] (see also [53,54]).For this anomaly to occur, the "remnant" BD scalar field needs to exhibit a singularity or time-dependence.However, these conditions are not satisfied for static stars, where regularity requirements are imposed.Therefore, the violation of γ reported in this current article is not related to the O (1/ √ ω) anomaly.
On the loss of Birkhoff's theorem-It can be argued that, in BD gravity, the loss of Birkhoff's theorem and the dependence of γ on a star's internal structure may be interconnected.Indeed, let us consider, in BD gravity, a static spherisymmetric Newtonian star (initial state).Its exterior vacuum is described by a Brans Class I solution given by Eq. ( 43), characterized by 2 parameters M 1 and Λ (or equivalently κ per Eq. ( 44)).Note that the parameter Λ depends on the pressure content of the star, as is evident in Eq. (62).For the initial Newtonian star, since p ≪ ϵ inside the whole star, Θ ini is approximately zero, rendering Λ ini ≈ − 1 ω+2 .Let us now consider that this star starts collapsing, and that the collapse ends at some (final) compact state.The pressure is no longer negligible with respect to ϵ in this final state, in such a way that the final value Θ fin is significant, making Λ fin significantly differing from Λ ini .On the other hand, Birkhoff's theorem in GR mandates that any spherisymmetric vacuum must be independent of the coordinate t regardless of the (time-) evolution of the source.If Birkhoff's theorem were valid for BD gravity, the vacuum exterior to the collapsing star would have been left unchanged during the process (i.e. a time independent solution), which is incompatible with the observation that the final Λ fin differs from the initial one Λ ini .Thence, the fact that Λ explicitly depends on Θ, as described by Eq. (62), implies that Birkhoff's theorem cannot hold for BD gravity.Reciprocally, the strong necessity to revisit BD gravity's PPN γ expression could have been anticipated from the mere fact that Birkhoff's theorem is not valid in BD gravity.
Brans-Dicke stars with ultra-relativistic matter-Generally speaking, the limit Θ → 1 − renders Λ → 0 and κ → 1 (per Eq. ( 44)) regardless of ω in the range (−3/2, +∞).From Eq. ( 43), the Brans Class I solution degenerates to the Schwarzschild solution in isotropic coordinates The degeneracy infers that ultra-relativistic Brans-Dicke stars and GR stars are indistinguishable, as far as their exterior vacua are concerned.Furthermore, as mentioned in the Introduction section, per Hawking's no-hair theorem [18,19], non-rotating uncharged BD black holes also correspond to the Schwarzschild solution.Therefore, considering a static spherisymmetric exterior vacuum, one cannot distinguish among a GR star, an ultrarelativistic BD star, and a BD black hole.
The degeneracy of the Brans Class I solution to the Schwarzschild solution can be explained by the following observation: For ultra-relativistic matter, the trace of the EMT vanishes, per Eq. ( 13).The scalar equation ( 5) then simplifies to □ Φ = 0 everywhere.Coupled with the regularity condition at the star center, this ensures a constant Φ throughout the spacetime which is now described by the Schwarzschild solution.Consequently, the scalar degree of freedom in BD gravity is suppressed in the ultra-relativistic limit.This prompts an intriguing possibility whether Birkhoff's theorem is fully restored in this limit.
Interestingly, the degeneracy appears to manifest in a special type of Bergmann-Wagoner theory [55,56].In this theory, ω is promoted to be a function of the scalar field, ω(Φ), and a potential term of the scalar field Λ(Φ) is introduced.The scalar equation in this theory is where the prime denotes derivative with respect to Φ.For the choice Λ(Φ) = λ 0 Φ with λ 0 ∈ R, the scalar field equation then simplifies to ∇ µ 2ω(Φ) + 3 ∇ µ Φ = 0 for ultra-relativistic matter, T = 0.This equation obviously admits a constant field Φ ≡ Φ 0 as a solution, thereby reducing the metric equation to the standard Einstein-Hilbert equation augmented with a cosmological term λ 0 g µν .The case of λ 0 = 0 corresponds to the massless Bergmann-Wagoner theory.

XI. CLOSING AND OUTLOOKS
In closing, we have derived the exact analytical formulae, Eqs.(69) and 90, for the PN parameters γ and δ for spherical compact mass sources in BD gravity.The derivation's success hinges on the integrability of the 00−component of the field equation, rendering it nonperturbative and applicable for any field strength and type of matter constituting the source.The derivation is parsimonious, requiring only a subset of the Brans-Dicke field equations, whereas the full set would be needed to determine the interior of the stars.A complete account of the technical insights was provided in the preceding section.
Outlook #1 -The conventional PPN result for BD gravity γ PPN = ω+1 ω+2 lacks dependence on the physical features of the mass source, a trait shared by other alternative theories beyond GR, such as the Bergman-Wagoner theory.In the light of our exact result, the γ PPN should be regarded as an approximation for stars in modified gravity under low-pressure conditions.Our findings underscore the limitations of the PPN formalism, particularly in scenarios characterized by high star pressure.It is plausible to expect that the role of pressure may extend to other modified theories of gravitation, such as in the massive Brans-Dicke gravity considered in [57,58].
Outlook #2 -The energy and pressure integrals, E * and P * ∥ +2P * ⊥ defined in Eq. ( 23), can be evaluated within the TOV framework for BD gravity.This approach entails developing the TOV equations and integrating them from the star's center outward.The resulting exterior solution is fully determined by factors such as the equation of state of the star's constituent matter and the central pressure of the star.Accordingly, the matching between the interior spacetime solution and the exterior vacuum is automatically handled when the integration crosses the surface at which the pressure vanishes.However, it should be noted that the integration procedure is numerical in nature, as there is currently no exact analytical solution available for the interior of stars in BD gravity.Inspired by our exact γ derivation presented in this paper, details regarding the TOV equation for BD gravity, presented in a new optimal gauge choice, are currently underway [59].