Dilatonic effects on a falling test mass in scalar-tensor theory
aa r X i v : . [ g r- q c ] M a y Dilatonic effects on a falling test mass in scalar-tensor theory
J.R. Morris
Physics Dept., Indiana University Northwest,3400 Broadway, Gary, Indiana 46408, USA ∗ Abstract
Effects of a 4d dilaton field on a falling test mass are examined from the Einstein frame perspectiveof scalar-tensor theory. Results are obtained for the centripetal acceleration of particles in circularorbits, and the radial acceleration for particles with pure radial motion. These results are appliedto the specific case of nonrelativistic motion in the weak field approximation of Brans-Dicke theory,employing the exact Xanthopoulos-Zannias solutions. For a given parameter range, the resultsobtained from Brans-Dicke theory are qualitatively dramatically different from those of generalrelativity. Comments are made concerning a comparison with the general relativistic results in thelimit of an infinite Brans-Dicke parameter.
PACS numbers: 04.50.Kd, 04.20.Jb, 04.50.-hKeywords: Brans-Dicke theory, scalar-tensor theory, dilaton gravity, exact solutions ∗ Electronic address: [email protected] . INTRODUCTION Scalar-tensor theories form a class of candidates for a modified description of gravity, andsome type of modified gravity at large distances could give rise to observable deviationsfrom general relativity. Brans-Dicke theory[1] is a prototypical scalar-tensor theory where,in a Jordan frame representation, a massless scalar field couples nonminimally to the Riccicurvature scalar. However, more general scalar-tensor theories allow different couplingsof the scalar “dilaton” field to the curvature, as well as accommodating nonzero scalarfield potentials. Four dimensional scalar-tensor theories arise from a variety of theoreticalapproaches aimed at achieving unification and/or explaining certain types of observations.Such approaches include Kaluza-Klein type models, string theory, and brane-world modelsinvolving extra space dimensions, and result in effective four dimensional models of gravitywith a nonminimally coupled scalar field[2]. Therefore, a study of the effects presented bya general form of scalar-tensor theory will include the effects that emerge from a variety ofhigher dimensional models, as well as four dimensional scalar-tensor theories that may notrequire extra dimensions.A fairly general form of a scalar-tensor theory is considered here, and we concentrate onthe Einstein frame representation of the theory where the dilatonic effects and the metrictensor field effects can be distinguished more easily. We then proceed to find expressionsfor the motion of a test particle moving in a static, spherically symmetric background. Ex-pressions are obtained for (1) the angular speed of a test mass in circular motion, and (2)the radial acceleration of a particle undergoing pure radial motion. Simplification resultswhen we consider nonrelativistic motion. As an example, we apply these expressions to theexact analytical vacuum solutions of Brans-Dicke theory[1], i.e., the Xanthopoulos-Zanniassolutions[3], which solve the Einstein frame field equations. The differences between theBrans-Dicke results and the general relativity (GR) results are seen, and for a given param-eter range, are dramatically different in a qualitative sense. Comments are also offered toillustrate in a concrete way, that, as pointed out by Faraoni[4],[5], when the matter stress-energy vanishes, GR is not generically recovered from the Brans-Dicke theory in the limit ofan infinite Brans-Dicke parameter.
II. CONFORMAL FRAMES
Consider a Jordan frame representation of a scalar-tensor theory of the form S = Z d x p ˜ g ( F ( ˜ φ )2 κ ˜ R [˜ g µν ] + 12 ˜ g µν ∂ µ ˜ φ∂ ν ˜ φ − V ( ˜ φ ) ) + S m [˜ g µν ] (2.1)2here κ = 16 πG , ˜ g = | det ˜ g µν | , the scalar field ˜ φ is identified as a 4d dilaton with a potential V ( ˜ φ ), and a metric signature (+ , − , − , − ) is used. The Jordan frame metric and line elementare given by d ˜ s = ˜ g µν dx µ dx ν . The matter action S m [˜ g µν ] is constructed from the metric ˜ g µν and matter terms. For instance, a classical particle action can be written as S m,cl = − X A Z m ,A d ˜ s A = − X A Z m ,A [˜ g µν ( x A ) dx µA dx νA ] / (2.2)where m ,A is the mass of particle A in the Jordan frame, assumed to be a constant. A fieldtheoretic matter action is S m = Z d x p ˜ g ˜ L m (˜ g µν , ψ ) (2.3)where ψ labels matter fields. A classical matter Lagrangian density can be defined by[6],[7] p ˜ g ˜ L cl = − X A Z m ,A [˜ g µν ( x A ) dx µA dx νA ] / δ (4) ( x − x A ) (2.4)The associated stress-energy tensors for field theoretic or classical actions˜ T µν = 2 √ ˜ g ∂ ( √ ˜ g ˜ L m ) ∂ ˜ g µν , ˜ T µνcl = − √ ˜ g ∂ ( √ ˜ g ˜ L cl ) ∂ ˜ g µν (2.5)then give ˜ T > g µν → g µν = Ω ˜ g µν , Ω = q F ( ˜ φ ) , ˜ φ → φ ( ˜ φ ) , dφd ˜ φ = 1 F (cid:26) F + 316 πG h F ′ ( ˜ φ ) i (cid:27) / (2.6)where F ′ ( ˜ φ ) = dF/d ˜ φ , giving an Einstein frame representation S = Z d x √ g (cid:26) κ R [ g µν ] + 12 g µν ∂ µ φ∂ ν φ − U ( φ ) (cid:27) + S m (Ω − g µν ) (2.7)The potential U ( φ ) depends upon the functions F ( ˜ φ ) and V ( ˜ φ ( φ )), U ( φ ) = V Ω = V [ ˜ φ ( φ )] F [ ˜ φ ( φ )](See, for example, [9].) The Einstein frame line element is ds = g µν dx µ dx ν = Ω d ˜ s = F ( ˜ φ ) d ˜ s . In the Einstein frame a particle has a mass m , which is generally position dependent3ue to its dependence on the scalar field ˜ φ . Consider, for example, a classical matter actionof the form in (2.2), − S m = Z m d ˜ s = Z m (Ω − ds ) = Z m F − / ds = Z mds (2.8)so that the Einstein frame mass m is related to the Jordan frame mass m by[8],[9] m = Ω − m = F − ( ˜ φ ) m (2.9)Therefore, a particle having a constant mass m in the Jordan frame will have a mass m = F − / m in the Einstein frame. Since the fields ˜ φ and φ generally depend on spacetimeposition, then the Einstein frame mass m = m ( x µ ) in general. The matter Lagrangiandensity in the Einstein frame, L m , is related to that in the Jordan frame, ˜ L m , by[8],[9] L m = Ω − ˜ L m (˜ g µν ) = F − ˜ L m (˜ g µν ) (2.10)A particular example is that of Brans-Dicke (BD) theory[1], with a Jordan frame action( G = 1) S = 116 π Z d x p ˜ g (cid:26) ˜ φ ˜ R + ω BD ˜ φ ˜ g µν ∂ µ ˜ φ∂ ν ˜ φ (cid:27) + S m (˜ g µν ) (2.11)A conformal transformation to the Einstein frame is given by[11] g µν = ˜ φ ˜ g µν , g µν = ˜ φ − ˜ g µν , √ g = ˜ φ p ˜ g, φ = √ a ln ˜ φ, a = ω BD + 32 (2.12)and the action in the Einstein frame then takes the form S = 116 π Z d x √ g (cid:26) R + 12 g µν ∂ µ φ∂ ν φ (cid:27) + S m ( ˜ φ − g µν ) (2.13)where R is built from g µν and Einstein gravity is coupled to a massless Einstein frame scalardilaton field φ . Using g µν = Ω ˜ g µν as in (2.6), we identify Ω = ˜ φ / and from (2.9) we have m = Ω − m = ˜ φ − / m (2.14)(The kinetic term in (2.11) is in noncanonical form, but a rescaling of the scalar field[9]˜ φ → ¯ φ / (8 ω BD ) would put the kinetic term into a canonical form as in (2.1), with F ( ¯ φ ) ∝ ¯ φ / (8 ω BD ).) Terms in the matter Lagrangian L m = Ω − ˜ L m (˜ g µν ) = ˜ φ − ˜ L m (˜ g µν ) pick up ananomalous coupling to the dilaton ˜ φ . 4 classical test particle of mass m moving in a gravitational field described by ds = g µν dx µ dx ν has an action like that in (2.8), S = − Z m [ g µν u µ u ν ] / ds (2.15)where u α = dx α /ds is subject to the “on shell” constraint u α u α = 1. The “geodesic” equationof the motion (in an otherwise matter-free region) obtained from (2.15) can be written inthe form[8] dds ( mg µν u ν ) − m ( ∂ µ g αβ ) u α u β − ∂ µ m = 0 (2.16)or in the form du ν ds = − Γ ναβ u α u β + ∂ µ (ln m )( g µν − u µ u ν ) (2.17)The first term on the right hand side of (2.17) is recognized as the gravitational accelerationdue to the metric field g µν , while the second term on the right hand side represents thedilatonic acceleration due to the scalar field, and therefore a deviation from pure, unforced,geodesic motion. Since m ( x µ ) ∝ Ω − ( x µ ) = F − / ( x µ ), the motion of a particle in theEinstein frame of a scalar-tensor theory where ∂ µ m = 0, will differ from that described bygeneral relativity (GR) where m = const. This reflects the fact that the Jordan frame metric˜ g µν for a scalar-tensor theory will generally be different from the metric of GR. Since theacceleration of a test mass in the Einstein frame depends upon the tensor field g µν as well asthe dilatonic acceleration due to the scalar field φ , it is not enough to consider the asymptoticform of the metric alone, e.g., g −
1, for the case of an asymptotically flat spacetime.
III. MOTION IN A STATIC, SPHERICALLY SYMMETRIC BACKGROUND
We now focus upon the motion of a classical test particle of mass m ( r ) moving under theinfluence of a metric field g µν in the Einstein frame of a scalar-tensor theory that can be writ-ten in the form of eq.(2.1). We assume that g µν and m are static and spherically symmetricfunctions, independent of t and azimuth angle ϕ , with g µν being diagonal, and consider mo-tion in the equatorial plane, θ = π/
2. The special cases of circular motion and pure radialmotion will be considered by using (2.16) or (2.17), along with the constraint u α u α = 1.Different coordinate systems can be used (Schwarzschild-like or isotropic), but we take themetric to have a general form ds = e f ( r ) dt − e − h ( r ) dr − ρ ( r ) r d Ω (3.1)where ρ ( r ) = e − h ( r ) for isotropic coordinates, and d Ω = dθ + sin θdϕ .5irst, we point out that if the motion is initially within the equatorial plane θ = π/
2, then itremains in this plane, so that u θ = dθ/ds = 0. This is seen from the θ component of (2.16),which reduces to d ( mu θ ) ds − m ρ ( r ) r sin θ cos θ ( u ϕ ) = 0, so that if θ = π/ u θ = 0 initially,then d ( mu θ ) /ds = 0 initially (no θ component of acceleration), so that motion remains inthe θ = π/ t component equation, ∂ g αβ = 0 and ∂ m = 0, so dds ( m u ) = 0, which gives p = mu = E ; u = Emg (3.2)where the energy E is a constant parametrizing the particular orbit. For example, a givencircular orbit has a fixed value of E , but this value will generally depend upon the orbitalradius r , so that E = E ( r ) is a constant on the orbit. For pure radial motion, E mightcharacterize the asymptotic energy of the test mass, and different values of E characterizedifferent radial orbits (e.g., different turning points).Similarly, for the ϕ equation, dds ( mu ϕ ) = 0, with p ϕ = mu ϕ = mg ϕϕ u ϕ = − L ; u ϕ = − Lm , u ϕ = − Lmg ϕϕ (3.3)where the angular momentum L is a constant of motion characterizing the orbit, and g ϕϕ isevaluated at θ = π/ L = 0, but for a circularorbit L depends upon the orbital radius, as with Newtonian gravity.The radial equation reduces to dds ( mu r ) − m (cid:2) ( ∂ r g )( u ) + ( ∂ r g rr )( u r ) + ( ∂ r g ϕϕ )( u ϕ ) (cid:3) − ∂ r m = 0 (3.4)and the constraint equation is u α u α = g ( u ) + g ϕϕ ( u ϕ ) + g rr ( u r ) = 1. Using (3.2) and(3.3), this constraint gives ( u r ) = 1 | g rr | (cid:20) E g m − L | g ϕϕ | m − (cid:21) (3.5)The kinematically allowed regions where the test mass can propagate are defined by ( u r ) ≥
0, with radial turning points given by ( u r ) = 0.We could also define an effective potential V for the radial motion by (see, forexample,[12],[13]) ( u r ) + V = E /m , where V = 1 | g rr | (cid:18) L | g ϕϕ | m (cid:19) + E m (cid:18) − g | g rr | (cid:19) (3.6)with radial turning points determined by E/m = V .6 . Circular motion For circular motion u r = 0 and L and E are constants of the particular orbit. We write u ϕ = − Lmg ϕϕ = dϕds = u dϕdt = u ω = Emg ω (3.7)where the angular speed ω = dϕ/dt , and we have used (3.2) and (3.3). Therefore the angularspeed ω can be written as ω = LE g | g ϕϕ | (3.8)with E and L related by (3.5) with u r set to zero. For example, consider the case ofnonrelativistic circular motion of a test particle with constant mass m due to Newtoniangravity in a Minkowski spacetime, where g → | g ϕϕ | → r in the equatorial plane,and E → m . We then have from (3.8) ω = L/mr , the ordinary Newtonian relation.Now the relation between E and L , given by (3.5) with u r set to zero, is equivalent to theconstraint equation p µ p µ = m with p r = 0, and leads to Em = √ g (cid:20) L | g ϕϕ | m (cid:21) / (3.9)The second term within brackets on the right is recognized as an orbital kinetic energy (perunit mass) term. Eq. (3.9) allows E to be eliminated from the expression in (3.8), leaving ω to be determined by L , along with the metric g µν and mass function m ( r ). The radialequation (3.4), ( ∂ r g ) (cid:18) Emg (cid:19) + ( ∂ r g ϕϕ ) (cid:18) Lmg ϕϕ (cid:19) = − ∂ r mm (3.10)then allows a determination of L in terms of the g µν and m . Thus ω , evaluated on the orbitwith radius r , will ultimately depend not only upon the metric g µν , but also upon the massfunction m ( r ) and its rate of change ∂ r m , evaluated on the orbit with radius r . Nonrelativistic limit: | ~p | /m ≪ : The above procedure and eq.(3.8) simplifies in the nonrelativistic limt of low velocities, v ≪ | p i p i | ≪ m . Then p µ p µ = g E + p i p i = m ≈ g E , or E ≈ m √ g (3.11)7This is also obtained from (3.9) when we drop the orbital kinetic energy term.)The radial equation (3.10), with the use of (3.11) and some rearrangement, yields Lm ≈ | g ϕϕ | (cid:26) ∂ r | g ϕϕ | ∂ r (cid:2) ln( m g ) (cid:3)(cid:27) / (3.12)in the nonrelativistic limit. Using (3.11) and (3.12), the angular speed in (3.8) is given inthe nonrelativistic limit by ω ≈ ∂ r ( m g ) m ∂ r | g ϕϕ | (3.13)To test this, we use the Schwarzschild solution in Schwarzschild coordinates, with m = const, g = 1 − GM/r , and | g ϕϕ | = r in the equatorial plane. Eq.(3.13) then gives the Newtonianlimit, ω = GM/r and a centripetal acceleration | a c | = ω r = GM/r , i.e., the ordinaryNewtonian gravitational field produced by a static, spherically symmetric body of mass M .However, it is possible that a scalar-tensor theory with nonconstant mass and a differentmetric field could yield a dramatically different result. An example is provided later forBrans-Dicke theory. B. Radial Motion
We take θ = π/ ϕ = const , so that u θ = u ϕ = 0 for pure radial motion, and the constraintequation becomes g ( u ) − | g rr | ( u r ) = 1, with u = E/m . We also have the radialcomponent of equation (2.17), which becomes du r ds = − (cid:2) Γ r ( u ) + Γ rrr ( u r ) (cid:3) + ∂ r mm (cid:2) g rr − ( u r ) (cid:3) (3.14)with Γ r = − g rr ∂ r g and Γ rrr = g rr ∂ r g rr . The constraint equation( u r ) = 1 | g rr | (cid:20) E g m − (cid:21) (3.15)and u = g E/m can be used in (3.14), allowing the proper radial acceleration to be givenby g rr du r ds = 12 ( ∂ r g ) (cid:20) E g m (cid:21) − g rr ( ∂ r g rr ) (cid:20) − E g m (cid:21) + ∂ r mm (cid:20) E g m (cid:21) (3.16)8 . Nonrelativistic limit: | ~p | /m ≪ : In the nonrelativistic limit | p r p r | /m ≪
1, or g rr ( u r ) ≪
1, then (3.15) implies that E g m ≈ ds ≈ √ g dt . The geodesic equationthen takes the simplified form d rdt ≈ (cid:18) g g rr (cid:19) (cid:20)
12 ( ∂ r g ) + ∂ r mm (cid:21) (3.17)in describing the radial acceleration of a test mass m in the nonrelativistic limit, where for ascalar-tensor theory m = m ( r ) in the Einstein frame. As an example, we again apply this tothe Schwarzschild case, where m is constant and g = − g − rr = (1 − GM/r ), to get a radialacceleration a r = a c = − GM/r , the usual Newtonian limit. However, we will also consideran example from Brans-Dicke theory. IV. APPLICATION TO BRANS-DICKE THEORY
We now apply the results of (3.13) and (3.17) to the case of a static, spherically symmetricbackground of Brans-Dicke (BD) theory. The Jordan frame representation of the BD theoryis given by (2.11). Exact static, spherically symmetric vacuum solutions in the Jordan framewere provided by Brans[14]. The conformal transformations described by (2.12) allows thetheory to be rewritten in the Einstein frame representation, given by (2.13). The BD vacuumsolutions in the Einstein frame, as well as the higher dimensional generalizations, have beenprovided by Xanthopoulos and Zannias[3]. Cai and Myung[11] have also studied these solu-tions, explicitly relating the Jordan frame solutions and the Einstein frame solutions throughthe transformations of (2.12). We apply these solutions to describe the region exterior tosome neutral, nonrotating astrophysical object of BD theory, and look at the asymptoticlimit r ≫ r . (There is a naked singularity at r = r , except in the case of the Schwarzschildlimit, where the solution coincides with the Schwarzschild solution[3],[11].) However, thesolution inside the astrophysical object will not be a vacuum solution, so that we do notgenerally expect a physical singularity to exist. For an astrophysical object like a star orplanet, we expect that r/r ≫ ds = e f dt − e − h ( dr + r d Ω ) (4.1) e f = g = ξ γ ; ξ = (cid:18) r − r r + r (cid:19) (4.2)9 − h = | g rr | = (cid:18) − r r (cid:19) ξ − γ = e − f (cid:18) − r r (cid:19) (4.3) φ = ± ˜ γ ln ξ = √ a ln ˜ φ ; ˜ γ = [4(1 − γ )] / (4.4a)˜ φ = ξ Γ ; Γ = ± ˜ γ √ a = ± (cid:20) a (1 − γ ) (cid:21) / = ±| Γ | (4.4b)where r and γ are integration constants ( r > ξ = (cid:18) r − r r + r (cid:19) ≤ , ˜ γ = [4(1 − γ )] / , Γ = ± ˜ γ √ a = ± (cid:20) a (1 − γ ) (cid:21) / (4.5)These are the Einstein frame fields and solutions, with 0 ≤ γ ≤ r = r where R = g µν R µν → ∞ unless γ = 1 and φ = 0 (the Schwarzschild solution). Note : In the set of solutions presented in ref.[3], only the solution with the + sign in (4.4a),i.e., φ = +˜ γ ln ξ , is presented. However, the second solution φ = − ˜ γ ln ξ is seen to exist dueto the invariance of the action and equations of motion (EoM) under the transformations g µν → g µν , φ → − φ . Thus if φ is a solution to the EoM, then so is − φ (see, for example,refs.[11] and[15]). Therefore φ can be positive or negative, and the Brans-Dicke scalar ˜ φ = ξ Γ = ξ ±| Γ | can be either a decreasing or an increasing function of r and ξ . The Einsteinframe mass m of a test particle is given by (2.14) and (4.4b), m = m ˜ φ − / = m ξ − Γ / (4.6)where m is the constant Jordan frame mass.We now want to consider the asymptotic forms of these solutions for which r /r ≪
1. Inthis case we have the following approximations to O ( r /r ). ξ ≈ − r r , g ≈ − γ r r , | g rr | ≈ /g , | g ϕϕ | θ = π/ ≈ r (cid:0) γ r r (cid:1) , ∂ r | g ϕϕ | θ = π/ ≈ r (cid:0) γ r r (cid:1) , m m ≈ (cid:0) r r (cid:1) , m m g ≈ − γ − Γ) r r , (4.7)For the case of nonrelativistic particle motion, applying (4.7) to (3.13) for the case of circularmotion yields the result ω ≈ (2 γ − Γ) r r , a c = − ω r ≈ − (2 γ − Γ) r r (4.8)10he Schwarzschild case is obtained for γ = 1, Γ = 0, and the identification r = GM/ M is the mass of the gravitating object[3],[12]. The Schwarzschild limit thereforegives ω → GM/r and a c → − GM/r , i.e., the Newtonian limit of the gravitational fieldfar from the Schwarzschild radius. (The Schwarzschild radial coordinate R is related to theisotropic coordinate r by[3],[12] R = r (1 + r /r ) , with R → r asymptotically.) Similarly,applying (4.7) to (3.17) for the case of radial motion, we have d rdt ≈ − (2 γ − Γ) r r (4.9)We therefore have the same gravitational acceleration, a c = a r , for circular or radial motion,with the expected Newtonian limit for the Schwarzschild case.A qualitative distinction between GR and BD in the weak field limit is seen for the case whenΓ > γ , in which case the radial acceleration a r ≈ (Γ − γ ) r /r given by (4.9), becomes positive rather than negative , indicating a repulsion rather than an attraction . Similarly,(4.8) implies that ω < repulsion ,rather than attraction . This is seen as an example where there is a dilatonic repulsion thatdominates the metric field attraction, since, from (4.7), the metric field produces a g − < g ( ∂ r g ) >
0, but m ( r ) is a decreasing function with ∂ r m <
0. Specifically, from (3.17)and (4.7), d rdt ≈ − (cid:18) g | g rr | (cid:19) (cid:20)
12 ( ∂ r g ) + ∂ r mm (cid:21) ≈ ( g ) (cid:20) − (cid:16) γ r r (cid:17) + | ∂ r m | m (cid:21) , (Γ > γ ) (4.10)showing that the metric field (due to the first term on the right) produces a negative accel-eration, but the dilatonic acceleration due to the mass (due to the second term on the right)produces a positive radial acceleration, which overwhelms the negative metric contribution.(An interesting occurrence of repulsive gravity in GR has also been reported[13].)For the case Γ = 2 γ , then ω → a r →
0, so that a test mass at rest remains at rest.However, for the parameter range Γ < γ , which is satisfied for Γ > | Γ | < γ , and forall Γ <
0, the radial acceleration is negative, with an overall attraction.The solar system constraint on the massless Brans-Dicke theory requires ω BD > , m BD . The conclusion reachedis that for a mass m BD & × − GeV, all values of ω BD ≥ − / ω BD → ∞ , in which case a → ∞ , Γ →
0, and thescalar field is removed from the theory ( φ → φ → a r ≈ − γr /r , which is a factor of γ times the Schwarzschildvalue obtained from GR. The reason for this can be seen from the Xanthopoulos-Zanniassolutions (4.1) - (4.4), noting that the metric g µν in this case does not collapse to theSchwarzschild metric for γ = 1 [5]. This illustrates in a concrete way the point made byFaraoni[4],[5] that BD theory does not always reduce to GR in the ω BD → ∞ limit whenthe matter stress-energy vanishes, with T µν = 0. V. SUMMARY
A fairly general form of scalar-tensor theory has been considered, with a focus on the Einsteinframe representation of the theory, where scalar field dilatonic effects and metric tensor fieldeffects become distinguishable. Expressions for the motion of a test particle moving in astatic, spherically symmetric background are found (1) for the case of circular motion, and(2) for the case of pure radial motion. Simplified expressions are obtained for nonrelativisticparticle motion. As an example, these expressions have been applied to the exact analyticalvacuum solutions to Brans-Dicke theory, by using the Xanthopoulos-Zannias solutions for thefield equations in the Einstein frame. The differences between the Brans-Dicke results andthe general relativity results are seen. For a given parameter range, namely, for Γ > γ , theseare dramatically different qualitatively, as the dilatonic repulsion of a test mass is greaterthan the gravitational attraction due to the tensor field. Furthermore, it is illustrated in aconcrete way that, as pointed out previously by Faraoni[4],[5], when the matter stress-energyvanishes, T µν = 0, GR is not automatically recovered from the Brans-Dicke theory in thelimit of an infinite Brans-Dicke parameter, ω BD → ∞ . The Xanthopoulos-Zannias solutionsin this limit do not coincide with the Schwarzschild solution, unless the Xanthopoulos-Zannias parameter is unity, γ = 1. [1] C. Brans and R.H. Dicke, Phys. Rev. 124, 925 (1961)[2] Y. Fujii and K. Maeda, “ The Scalar-Tensor Theory of Gravitation ” (Cambridge UniversityPress, 2003). See, for example, Chapter 1 and references therein.[3] B.C. Xanthopoulos and T. Zannias, Phys. Rev. D40, 2564 (1989)[4] V. Faraoni, Phys. Lett. A245, 26-30 (1998) [e-Print: gr-qc/9805057]
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