Revisiting the Brans solutions of scalar-tensor gravity
aa r X i v : . [ g r- q c ] S e p Revisiting the Brans solutions of scalar-tensor gravity
Valerio Faraoni, ∗ Fay¸cal Hammad,
1, 2, † and Shawn D. Belknap-Keet ‡ Physics Department and STAR Research Cluster, Bishop’s University,2600 College St., Sherbrooke, QC, Canada J1M 1Z7 Physics Department, Champlain College-Lennoxville,2580 College Street, Sherbrooke, QC J1M 0C8, Canada Physics Department, Bishop’s University, Sherbrooke, QC, Canada J1M 1Z7
Motivated by statements in the literature which contradict two general theorems, the static andspherically symmetric Brans solutions of scalar-tensor gravity are analyzed explicitly in both theJordan and the Einstein conformal frames. Depending on the parameter range, these solutionsdescribe wormholes or naked singularities but not black holes.
PACS numbers: 04.50.Kd, 04.20.Jb, 04.70.Bw
I. INTRODUCTION
Brans-Dicke theory [1] is the prototypical theory ofgravity alternative to Einstein’s General Relativity (GR).Not long after its introduction, it was generalized toscalar-tensor theories [2] and, with the advent of stringtheories, new interest was generated by the fact that thesimple bosonic string theory reduces to a Brans-Dicketheory with coupling parameter ω = − φ (acting approximately as the inverse of the gravita-tional coupling strength φ = G − ) and a dimensionlessparameter ω which would naturally be of order unity,but is constrained by Solar System experiments to sat-isfy | ω | > ω be-comes a function of the Brans-Dicke scalar field, whichalso acquires a mass or a self-interaction potential. Incosmology, f ( R ) theories of gravity, which are ulti-mately classes of scalar-tensor theories with Brans-Dicke-like scalar degree of freedom φ = f ′ ( R ), have become ex-tremely popular to explain the current acceleration of theuniverse without invoking an ad hoc dark energy (see thereviews [5–7]). It is natural, in this context, to search foranalogues of the Schwarzschild solution of GR. Shortlyafter Brans-Dicke theory was introduced [1], Brans pre-sented four families of geometries which are static, spher-ically symmetric, vacuum solutions of the Brans-Dickefield equations [8]. Although there is legitimate suspicionthat these solutions may not be very significant from thephysical point of view (but the literature has contradic-tory statements about this point), it is often necessaryto pick some simple ( i.e. , static, spherical, and asymp-totically flat) solutions of an alternative theory of grav-ity as toy models for theoretical purposes or as physical ∗ [email protected] † [email protected] ‡ [email protected] Here R is the Ricci scalar associated with the connection of thespacetime metric g ab . solutions to test a theory experimentally. Currently alarge amount of work is devoted to testing deviationsfrom GR in black hole environments (see, e.g. , [9]). TheBrans solutions, being the first of their kind discoveredin scalar-tensor or dilaton gravity, are a natural choice.However, they are surrounded by some ambiguity. Ac-cording to a theorem by Agnese and La Camera [10],all static and spherically symmetric solutions of (Jordanframe) Brans-Dicke theory are either naked singularitiesif the post-Newtonian parameter γ = ω + 1 ω + 2 (1.1)satisfies γ <
1, or wormholes if γ >
1. The Bransclasses I-IV solutions fall into this category and, there-fore, they can only describe naked singularieties or worm-holes. This result seems to be missed by several authorssince there are claims in the literature that certain static,spherical classes of solutions of Brans-Dicke theory de-scribe black holes, which would contradict the Agnese-LaCamera theorem. For example, the Campanelli-Loustosolutions [11] have been believed to be black holes for along time until it was shown recently that they indeed de-scribe either wormholes or naked singularities [12]. Sim-ilarly, reading the existing literature, because of explicitor implicit statements one is left with the impression thatBrans solutions can describe black holes for some rangeof their parameters [13–16]. Similar statements about“cold black holes” similar to the Campanelli-Lousto so-lutions are found in the literature [14, 15, 17, 18]. IfBrans geometries were black hole ones, they would alsocontradict a theorem by Hawking [19] (recently extendedto general scalar-tensor gravity [20, 21]) stating that allBrans-Dicke black holes are the same as in GR.Naked singularities are of little interest from the phys-ical point of view because they correspond to the break-down of the Cauchy problem. Wormholes are completelyspeculative objects [22], but there is plenty of astrophysi-cal evidence for, and interest in, black holes. It is of someinterest, therefore, to clarify the confusion existing in theliterature about the Brans geometries, which we set outto do.The Brans-Dicke action in the absence of matter is S BD = Z d x √− g π (cid:18) φ R − ωφ ∇ a φ ∇ a φ (cid:19) , (1.2)where φ is the Brans-Dicke scalar field (approximatelyequivalent to the inverse of the gravitational coupling), R is the Ricci scalar, and g is the determinant of thespacetime metric g ab . We follow the notation of Ref. [23].The Brans-Dicke field equations in vacuo derived fromthe action (1.2) are [1] R ab − R g ab = ωφ (cid:18) ∇ a φ ∇ b φ − g ab ∇ c φ ∇ c φ (cid:19) + 1 φ ∇ a ∇ b φ , (1.3) (cid:3) φ = 0 . (1.4)By performing the conformal transformation of the met-ric g ab → ˜ g ab = φ g ab , (1.5)and the scalar field redefinition φ → ˜ φ = r | ω + 3 | πG ln (cid:18) φφ ∗ (cid:19) , (1.6)where φ ∗ is a constant (Einstein frame quantities are de-noted by a tilde), the Brans-Dicke action (1.2) assumesits Einstein frame form S BD = Z d x p − ˜ g " ˜ R π −
12 ˜ g ab ∇ a ˜ φ ∇ b ˜ φ . (1.7)This action formally looks like the Einstein-Hilbert ac-tion of GR in the presence of a matter scalar field en-dowed with canonical kinetic energy. The Einstein framevacuum field equations are˜ R ab −
12 ˜ g ab ˜ R = 8 πG (cid:18) ∇ a ˜ φ ∇ b ˜ φ −
12 ˜ g ab ˜ g cd ∇ c ˜ φ ∇ d ˜ φ (cid:19) , (1.8)˜ g ab ˜ ∇ a ˜ ∇ b ˜ φ = 0 . (1.9)We now proceed to analyze the four classes of Brans solu-tions in the Jordan and in the Einstein conformal frames. II. BRANS CLASS I SOLUTIONS
Class I Brans solutions have been discussed in severalpapers [13, 15, 24–31]. It is found that these metricscan describe wormholes, which is not surprising since aBrans-Dicke-like scalar field in scalar-tensor gravity hasa non-canonical kinetic energy and its effective stress-energy tensor on the right hand side of eq. (1.3) canviolate all of the energy conditions. More recent solu-tions proposed in the literature [32] have been identifiedas special limits of Brans I solutions [15, 33].
A. Jordan frame
In the Jordan frame representation, the Brans class Iline element and scalar field are, respectively, ds I ) = − (cid:18) − B/r
B/r (cid:19) /λ dt + (cid:18) Br (cid:19) (cid:18) − B/r
B/r (cid:19) λ − C − λ (cid:16) dr + r d Ω (cid:17) , (2.1) φ ( I ) = φ (cid:18) − B/r
B/r (cid:19)
C/λ , (2.2)in polar coordinates ( t, r, θ, ϕ ), where r is an isotropicradius and d Ω = dθ + sin θ dϕ is the line element onthe unit 2-sphere. It must be r ≥ λ = ( C + 1) − C (cid:18) − ωC (cid:19) > . (2.3)Here ω is a parameter of the theory and B, C , and λ areparameters of this family of solutions. B plays the role ofa mass parameter and, in analogy with the Schwarzschildgeometry of GR, it makes sense to consider only non-negative values of this parameter. There are actuallytwo other reasons why we restrict our study here to non-negative values of B . The first reason is that one can stillinclude the case B = 0, but then class I solutions simplyreduce to the trivial Minkowski space. The second reasonis that one can also include the case B <
0, but then, aswe will shortly see, one just recovers the case of positive B by taking the mass parameter of the theory to be − B instead of B . Therefore, we shall hereafter assume B > ω of the the-ory is fixed, one has two independent parameters ( B, C )or (
B, λ ), since eq. (2.3) relates λ and C . It is useful to re-strict the parameter space and, to this end, we note that,according to our assumption that B >
0, only positivevalues of λ will be relevant here. In fact, consider as anexample the simple case C = 0, in which the Brans-Dickescalar φ reduces to a constant and eq. (2.3) yields λ = ± φ is constant, Brans-Dicke theory reduces to GRand the Schwarzschild solution, which is the unique vac-uum, static, and spherically symmetric solution of theEinstein equations must be recovered. By setting λ = 1the line element (2.1) reduces to ds = − (cid:18) − B/r
B/r (cid:19) dt + (cid:18) Br (cid:19) (cid:16) dr + r d Ω (cid:17) (2.4)which is the Schwarzschild metric in isotropic coordinates[13]. If instead λ = −
1, one obtains ds − = − (cid:18) B/r − B/r (cid:19) dt (2.5)+ (cid:18) − Br (cid:19) (cid:16) dr + r d Ω (cid:17) . (2.6)This is again just the Schwarzschild solution providedthat, either r → − r which we don’t consider here [13], orthat B < − B as the mass parameter instead of B ;a case we have already chosen to exclude. Therefore, wealso assume λ > ω → ∞ , it is λ ≃ ω C / → ∞ and, if C = 0(the case C = 0 having already been discussed), the lineelement (2.1) reduces to ds ∞ ) = − dt + (cid:18) − B r (cid:19) (cid:16) dr + r d Ω (cid:17) , (2.7)while the Brans-Dicke scalar becomes constant. If C = 0,the corresponding solution of GR is not recovered fromBrans I solutions in the ω → ∞ limit. Instances in whichsolutions of scalar-tensor theories do not reduce to thecorresponding GR limit have been discussed in [34] andpossible reasons for this behaviour have been identifiedin the anomalous asymptotic dependence of φ on ω as ω → ∞ [34–36].The condition (2.3) amounts to imposing that C issuch that points on the parabola of equation λ ( C ) = (cid:0) ω + 1 (cid:1) C + C + 1 lie in the λ > λ ( C ) = 0 are C ± = − ± p − (2 ω + 3) ω + 2 . (2.8)and they are real only if ω ≤ − /
2. By looking at thesign of the coefficient (1 + ω/
2) of this parabola, it is easyto establish that: • If ω < −
2, the parabola has concavity facing down-ward and intersects the C -axis at C ± >
0. It mustbe C − < C < C + . • If ω = −
2, then the parabola degenerates into thestraight line λ = C + 1 and it must be C > − • If − < ω < − /
2, the parabola has concavityfacing upward and it must be
C < C − or C > C + ,where C − < C + < • If ω = − /
2, then λ = [( C + 2) / and theparabola has concavity facing upward and touchesthe C -axis only at C = −
2, therefore the only re-striction is C = − • If ω > − /
2, the concavity still faces upward butthere are no intersections between the C -axis andthe parabola, which always lies above it. There isno restriction on the values of C .Let us consider now the Ricci scalar R : by contractingthe Brans-Dicke field equations (1.3) and using eq. (2.2), one obtains R = ωφ ∇ c φ ∇ c φ = 4 ωB C λ r (cid:18) Br (cid:19) − − ( C +1) λ (cid:18) − Br (cid:19) − ( C +1) λ . (2.9)If ω = 0 and C = 0, then the Ricci scalar is singularat r = B when ( C + 1) /λ <
2. Whether this value ofthe isotropic radius is physically significant is discussedcase-by-case below.The areal radius is read off the line element (2.1) andis R ( r ) = (cid:18) Br (cid:19) ( C +1) λ (cid:18) − Br (cid:19) − ( C +1) λ r, (2.10)and its derivative is dRdr = (cid:18) B/r − B/r (cid:19) C +1 λ (cid:20) r − B ( C + 1) λ r + B (cid:21) r . (2.11)In the following it is useful to know the roots of the equa-tion dR/dr = 0, which are r ( ± ) = B ( C + 1) λ ± s − (cid:18) λC + 1 (cid:19) . (2.12)In order to make our discussion of the various regionsof the parameter space more compact, we focus on thepossible values of the parameter combination ( C + 1) /λ ,which is relevant for both the roots of the equation dR/dr = 0 and in the search for horizons. The hori-zons (which, when existing, are both apparent and eventhorizons), are located by the roots of the equation [37, 38] ∇ c R ∇ c R = 0 , (2.13)which is equivalent to (cid:20) r − B ( C + 1) λ r + B (cid:21) = 0 . (2.14)Its roots coincide with those of the equation dR/dr = 0and, when they exist in the real domain, they are alwaysdouble roots. Let us consider separately the various rel-evant cases.
1. Parameter range ( C + 1) /λ < In this case dRdr = (cid:18) B/r − B/r (cid:19) C +1 λ (cid:20) r − B ( C + 1) λ r + B (cid:21) r > (cid:18) Br (cid:19) C +1 λ (cid:18) − Br (cid:19) − C +1 λ ( r − B ) > , (2.15)for all values of r > B . Moreover, R ( r ) = (cid:18) Br (cid:19) C +1 r (cid:18) − Br (cid:19) | − C +1 λ | r , (2.16)shows that r = B corresponds to areal radius R = 0,hence the range 0 < r < B is unphysical. The Ricciscalar (2.9) is singular at R = 0. In this parameter rangethe spacetime always hosts a naked central singularity if ω = 0. The details of the geometry near this singularityvary with the value of ( C + 1) /λ as described below. • If 0 < ( C + 1) /λ < dR/dr → + ∞ as thespacetime singularity is approached ( R → + (or r → B + ). • If ( C + 1) /λ = 0, then R ( r ) = (cid:18) − B r (cid:19) r → , (2.17) dRdr = 1 + B r → , (2.18)as the singularity at R = 0 is approached. • If ( C + 1) /λ <
0, then dRdr = (1 − B/r ) | C +1 λ | (1 + B/r ) | C +1 λ | (cid:20) r − B C + 1 λ r + B (cid:21) (2.19)tends to zero at the singularity R = 0 or r = B .
2. Parameter range ( C + 1) /λ = 1 We have R ( r ) = r (1 + B/r ) and R ( B ) = 4 B > < r < B of the isotropic radius isnow physically meaningful. Note that R ( r ) → + ∞ as r → + and that dRdr = (cid:18) Br (cid:19) (cid:18) − Br (cid:19) , (2.20)therefore the function R ( r ) decreases if 0 < r < B , hasthe absolute minimum R ( B ) = 4 B >
0, and increases for r > B . The equation ∇ c R ∇ c R = 0 locating the horizonsis equivalent to (1 − B/r ) = 0, with r = B a doubleroot. If ω = 0, there is a would-be wormhole throat at r = B (or, at R = 4 B ) where, however, the Ricci scalaris singular. This finite radius singularity separates twodisconnected spacetimes.If ω = 0, λ >
0, and 0 < C <
1, also the Brans-Dickescalar diverges at r = B , which means that the effec-tive gravitational constant vanishes. If C < C ≥ φ vanishes and the gravitational coupling strengthdiverges. The case C = 0 has already been discussedfor all values of ω . In the context of black holes, the divergence or the vanishing of the Brans-Dicke scalar de-notes “maverick” black holes which are contrived, unsta-ble, or pathological and are usually discarded as unphys-ical ( e.g. , [20]) and the same criterion should be adoptedfor wormholes (naked singularities are already unphysi-cal).
3. Parameter range ( C + 1) /λ > In this case the equation dRdr = (cid:18) B/r − B/r (cid:19) | C +1 λ | (cid:20) r − B ( C + 1) λ r + B (cid:21) · r = 0 (2.21)has the two roots (2.12), which are both positive. It isstraightforward to see also that0 < r ( − ) < B < r (+) . (2.22)The areal radius R ( r ) = (1 + B/r ) | C +1 λ | r (1 − B/r ) | C +1 λ − | → + ∞ (2.23)as r → B + , hence the range r < B of the isotropicradius is unphysical and we ignore the root r ( − ) < B .The apparent horizons are located at the roots of theequation (cid:18) − Br (cid:19) − (cid:20) r − B ( C + 1) λ r + B (cid:21) = 0 . (2.24)Ignoring the root r = B , which corresponds to R = + ∞ , r = r (+) > B is a double root and we have a wormholethroat at r (+) . As seen earlier, the Ricci scalar (2.9) issingular at r = B if ( C + 1) /λ < ω = 0, but thissingularity is actually pushed to infinity since r → B + corresponds to infinite physical radius R hence this isan acceptable solution. The Ricci scalar is regular for( C + 1) /λ ≥ B. Einstein frame class I solutions
The Einstein frame metric and free scalar field are d ˜ s I ) = φ ( I ) ds I ) = − (cid:18) − B/r
B/r (cid:19) C +2 λ dt + (cid:18) Br (cid:19) C +2 λ (cid:18) − Br (cid:19) − ( C +2) λ (cid:16) dr + r d Ω (cid:17) (2.25)˜ φ ( I ) = r | ω + 3 | πG Cλ ln (cid:18) − B/r
B/r (cid:19) + const. (2.26)The areal radius and its derivative are˜ R ( r ) = φ R = (cid:18) Br (cid:19) C +22 λ (cid:18) − Br (cid:19) − C +22 λ r , (2.27) d ˜ Rdr = (cid:18) B/r − B/r (cid:19) C +22 λ (cid:20) − (cid:18) C + 2 λ (cid:19) Br + B r (cid:21) , (2.28)while the Einstein frame Ricci scalar (obtained by con-tracting the field equations (1.8)) is˜ R = 8 πG ˜ g rr d ˜ φdr ! = 2 B C | ω + 3 | λ r (cid:18) Br (cid:19) − − C +2 λ (cid:18) − Br (cid:19) C +22 λ − . (2.29)If ( C + 2) /λ <
4, the Ricci scalar is singular at r = B (and it is always singular at r = 0 unless C = 0, in whichcase it is ˜ R = 0).The equation ∇ c ˜ R ∇ c ˜ R = 0 locating the horizons is (cid:18) − B r (cid:19) − (cid:20) − (cid:18) C + 2 λ (cid:19) Br + B r (cid:21) = 0 (2.30)and has the same roots r ( ± ) = B ( C + 2)2 λ ± s − λ ( C + 2) ! (2.31)as the equation d ˜ R/dr = 0. When these roots exist andare real and positive, they are always double roots and,therefore, the solutions always contain either wormholethroats or naked singularities. Assuming that
B > λ > C + 2) /λ ] < C +2) /λ = ± r = B . If instead [( C + 2) /λ ] > r ( ± ) . Further, if C > − r ( ± ) are both positive; if C = − C < −
2, there areno horizons. Let us examine the situation in more detail.
1. Parameter range ( C + 2) /λ < − In this case it is˜ R ( r ) = (cid:18) Br (cid:19) C +22 λ (cid:18) − Br (cid:19) | C +22 λ | r (2.32)and ˜ R ( r ) → + as r → B , hence the range 0 < r < B is unphysical, while ˜ R ( r ) → + as r → + ∞ . The roots r ( ± ) are negative and the Ricci scalar diverges at ˜ R = 0,where there is a naked singularity.
2. Parameter range ( C + 2) /λ = − In this case ˜ R ( r ) = ( r − B ) r (2.33)vanishes as r → B and diverges in both limits r → + and r → + ∞ . It could seem that there is a wormholethroat at r = B but the Ricci scalar diverges there. Alsothis geometry hosts a naked singularity at ˜ R = 0.
3. Parameter range − < ( C + 2) /λ < Then˜ R ( r ) = (cid:18) Br (cid:19) C +22 λ (cid:18) − Br (cid:19) | − C +22 λ | r (2.34)and ˜ R ( r ) → + as r → B while ˜ R ( r ) → + ∞ as r → + ∞ .There are no real roots r ( ± ) and no horizons. The Ricciscalar diverges at ˜ R = 0, where we have again a nakedsingularity.
4. Parameter range ( C + 2) /λ = 2 r = B is is a quadruple root and˜ R ( r ) = (cid:18) Br (cid:19) r (2.35)has the limits ˜ R → + ∞ as r → + and ˜ R → + ∞ as r → + ∞ . There is a wormhole throat at ˜ R = 4 B , theminimum value of ˜ R .
5. Parameter range ( C + 2) /λ > Both double roots r ( ± ) are positive and˜ R ( r ) = (cid:18) Br (cid:19) C +22 λ (cid:18) − Br (cid:19) − | − C +22 λ | r (2.36)diverges in both limits r → B + and r → + ∞ , hence therange 0 < r < B is unphysical. In this parameter rangeit is 0 < r ( − ) < B < r (+) and there is a wormhole throatat r (+) . III. BRANS CLASS II SOLUTIONSA. Jordan frame class II solutions
There is a duality relating class II and class I solutions[14, 15], so these two classes are not independent. Weshall come back to this duality in Sect. VI below. TheJordan frame Brans class II line element and scalar fieldare ds II ) = − e arctan( r/B ) dt + e − C +1)Λ arctan( r/B ) (cid:18) B r (cid:19) (cid:16) dr + r d Ω (cid:17) , (3.1) φ ( II ) = φ e C Λ arctan( r/B ) , (3.2)where Λ = C (cid:18) − ω C (cid:19) − ( C + 1) > . (3.3)This implies that C = 0, hence this value of the parame-ter C will not be considered in the following even thoughit is clear that it would play a role if the inequality (3.3)is forgotten. Indeed, note that if Λ and B are allowed totake simultaneously imaginary values, then setting C = 0will just turn the metric (3.1) into the Schwarzschild met-ric (2.4) written in isotropic coordinates. We shall comeback to this remark in Sect. VI.Let us examine the possible range of the parameters B, C , and Λ. The points of the parabola Λ ( C ) = − (cid:0) ω + 1 (cid:1) C − C − half-plane. This parabola has concavity facing downwardsand it intersects the C -axis at C ± = − ∓ p − (2 ω + 3) ω + 2 . (3.4)There are no such intersections if ω > − / ω = − /
2; we conclude that itmust be ω < − / • If − < ω < − / − (1 + ω/ < C must lie in the range C − 2, the parabola degenerates into thestraight line Λ ( C ) = − ( C + 1) and it must be C < − • If ω < − 2, it must be C < C − or C > C + .The Ricci scalar is R = ωφ ∇ c φ ∇ c φ = 4 ω B C r e C +1)Λ arctan( r/B ) Λ ( r + B ) = 4 ω B C Λ e C +1)Λ arctan( r/B ) R . (3.5) The only possible singularity of the Ricci scalar R canoccur as R → 0. The areal radius is R ( r ) = (cid:18) B r (cid:19) e − C +1)Λ arctan( r/B ) r (3.6)and its derivative is dRdr = e − C +1)Λ arctan( r/B ) (cid:20) r − B ( C + 1)Λ r + B (cid:21) . (3.7)Note that R > r and that R → + ∞ as r → + ∞ and also as r → + . Since the Ricci scalar (3.5)can only diverge as R → + , there are no singularities ofthe Ricci scalar in Brans class II spacetimes.The roots of the equation dR/dr = 0 are r ( ± ) = B Λ (cid:16) C + 1 ± p ( C + 1) + Λ (cid:17) . (3.8)Let us examine their sign, keeping in mind that p ( C + 1) + Λ + C + 1 > , (3.9) C + 1 − p ( C + 1) + Λ < . (3.10) • If Λ B > 0, the parabola ψ ( r ) ≡ r − B ( C +1)Λ r − B has concavity facing upwards and crosses the r -axisat r ( − ) and r (+) , with r ( − ) < < r (+) . Therefore dR/dr < R ( r ) decreases if 0 B < 0, the parabola ψ ( r ) still has concavityfacing upward but now r (+) < < r ( − ) and thediscussion is the same as in the previous case pro-vided that the switch r (+) ↔ r ( − ) is made.The equation ∇ c R ∇ c R = 0 locating the horizons be-comes (cid:20) − B r − B ( C + 1)Λ r + B (cid:21) = 0 . (3.11)The roots are the same as for the equation dR/dr = 0and, when they are real and positive, they are alwaysdouble roots. This fact implies that there are no blackhole horizons and that class II solutions do not describeblack holes but only wormhole throats or naked singular-ities. Further, the roots r ( ± ) can be written as r ( ± ) = B Λ C + 1 ± s C (cid:18) − ω C (cid:19) ! (3.12)and the inequality (3.3) implies that C (cid:0) − ω C (cid:1) > ( C + 1) ≥ 0, hence there are always two real roots r ( ± ) of the equation ∇ c R ∇ c R = 0 locating the hori-zons in the allowed range of parameters. Are these rootspositive? In order to answer this question, note that p ( C + 1) + Λ + C + 1 > | C + 1 | + C + 1 ≥ 0, hencesign (cid:0) r (+) (cid:1) = sign (Λ B ) , (3.13)while C + 1 − p ( C + 1) + Λ < C + 1 − | C + 1 | ≤ (cid:0) r ( − ) (cid:1) = − sign (Λ B ) . (3.14)We can now analyze all the possibilities for the two pa-rameters B and Λ. 1. Parameter range B > , Λ > When B > > 0, it is r ( − ) < < r (+) and thereis a double root r (+) marking the location of a wormholethroat. The same situation occurs when B < < 2. Parameter range B > , Λ < When B > < 0, it is r (+) < < r ( − ) andthere is a wormhole throat at r ( − ) . The same situationoccurs when B < > 3. Limit to GR Finally, let us consider the limit to GR of Brans IIsolutions. Since ω < − / 2, the limit should be ω → −∞ ,which implies that Λ ≈ − ω C / → + ∞ (rememberthat C = 0). In this limit the Brans-Dicke scalar (3.2)becomes constant but the line element reduces to ds ∞ ) = − dt + (cid:18) B r (cid:19) (cid:16) dr + r d Ω (cid:17) . (3.15)The areal radius is R ( r ) ≈ r + B r ; (3.16)by inverting this relation one obtains r − Rr + B = 0and there are the two values of the isotropic radius r , = 12 (cid:16) R ± p R − B (cid:17) (3.17)for each value of the physical areal radius R , which im-plies that it must be R ≥ | B | . The equation locatingthe apparent horizons ∇ c R ∇ c R = g rr (cid:18) dRdr (cid:19) = (cid:18) − B /r B /r (cid:19) = 0 (3.18)has the double root r = | B | (corresponding to R = 2 | B | ).There is always a wormhole throat in this spacetime,which is not the spherically symmetric, static, asymp-totically flat, vacuum solution of GR ( i.e. , Schwarzschildspace). Therefore, the limit ω → −∞ fails to reproducethe GR limit even though the Brans-Dicke scalar becomesconstant. B. Einstein frame class II solutions The Einstein frame class II line element and scalar fieldare d ˜ s II ) = φ ( II ) ds II ) = − e C Λ arctan( r/B ) dt + e − C +2)Λ arctan( r/B ) (cid:18) B r (cid:19) (cid:16) dr + r d Ω (cid:17) , (3.19)˜ φ ( II ) = r | ω + 3 | πG C Λ arctan (cid:16) rB (cid:17) + const. (3.20)The areal radius and its derivative are˜ R ( r ) = (cid:18) B r (cid:19) e − ( C +2)Λ arctan( r/B ) r , (3.21) d ˜ Rdr = e − ( C +2)Λ arctan( r/B ) (cid:20) − ( C + 2)Λ Br − B r (cid:21) , (3.22)while the Ricci scalar is˜ R = 2 B C | ω + 3 | Λ r (1 + B /r ) e C +2)Λ arctan( r/B ) (3.23)and is never singular. The equation ∇ c ˜ R ∇ c ˜ R = 0 locat-ing the horizons is (cid:20) r − B ( C + 2)Λ r − B (cid:21) = 0 (3.24)and, when its roots r ( ± ) = B ( C + 2)Λ ± s (cid:18) C + 2 (cid:19) (3.25)are real and positive they are always double roots, hencethere can only be either wormhole throats or naked sin-gularities. 1. Parameter range B ( C + 2) / Λ < It is r (+) < < r ( − ) and ˜ R ( r ) → + ∞ as r → + , while˜ R ( r ) → + ∞ as r → + ∞ . There is a wormhole throat at r ( − ) . 2. Case C = − It is r ( ± ) = ± B and ˜ R ( r ) = (cid:0) B /r (cid:1) r has the lim-its ˜ R ( r ) → + ∞ as r → + and ˜ R ( r ) → + ∞ as r → + ∞ .There is a wormhole throat at r = | B | (corresponding tophysical radius ˜ R = 2 | B | ). 3. Parameter range B ( C + 2) / Λ > In this case we have r ( − ) < < r (+) and ˜ R ( r ) → + ∞ in both limits r → + and r → + ∞ . There is a wormholethroat at r (+) . IV. BRANS CLASS III SOLUTIONS Although it is claimed that the class III family doesnot admit wormholes [24], this is not the case, as shownbelow. A. Jordan frame class III solutions The line element and Brans-Dicke scalar of Jordanframe class III Brans solutions are, respectively, ds III ) = − e − rB dt + B r e C +1) B r (cid:16) dr + r d Ω (cid:17) , (4.1) φ ( III ) = φ e − Cr/B , (4.2)where C = − ± p − (2 ω + 3) ω + 2 (4.3)and, clearly B = 0 , ω ≤ − / , ω = − 2. The areal radiusand its derivative are R ( r ) = B r e ( C +1) B r , (4.4) dRdr = B ( C + 1) r e ( C +1) B r (cid:18) r − BC + 1 (cid:19) . (4.5)We can rewrite the line element (4.1) using the arealradius R instead of the isotropic radius r by means ofthe substitution dr = r B ( C + 1) (cid:16) r − BC +1 (cid:17) e − ( C +1) B r dR , (4.6)which yields ds III ) = − e − rB dt + B ( C + 1) (cid:16) r − BC +1 (cid:17) dR + R d Ω . (4.7)The horizons, when they exist, are located by the equa-tion ∇ c R ∇ c R = 0, which becomes simply g RR = 0, or (cid:18) C + 1 B (cid:19) (cid:18) r − BC + 1 (cid:19) = 0 . (4.8)There is a double root r H = BC + 1 (4.9) when this quantity is positive, with corresponding arealradius R H = e B ( C + 1) . (4.10)Therefore, there are either zero or two coincident realroots and there can not be black holes: Brans class IIIsolutions always describe naked singularities or worm-holes.The Ricci scalar is R = ωφ ∇ c φ ∇ c φ = ω C B e − C +1) B r r . (4.11)Let us examine the various possibilities for the range ofparameters B and C . 1. Parameter range C < − , B > In this case we have R ( r ) = B r e − | C +1 B | r , (4.12) dRdr = − B r e − | C +1 B | r (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) BC + 1 (cid:12)(cid:12)(cid:12)(cid:12) r (cid:19) , (4.13)and the function R ( r ) is monotonically decreasing with R ( r ) → r → + ∞ and R ( r ) → + ∞ as r → + . Since R H = e B ( C + 1) < φ = φ e | CB | r → R → r → + ∞ ) and the Ricciscalar R = ω C B e | C +1 B | r r → + ∞ (4.14)as R → + . Therefore, there is a naked singularity at R = 0. 2. Parameter range C > − , B > In this case the areal radius R ( r ) → + ∞ as r → + ∞ and R ( r ) → + ∞ as r → + . Its derivative dR/dr isnegative, and R ( r ) decreases, for 0 < r < r H . R ( r ) hasthe absolute minimum R H = e B ( C + 1) > r H , andincreases for r > r H . The double root r H of the equation ∇ c R ∇ c R = 0 is positive and there is a wormhole throatat r H , where the Brans-Dicke field (4.2) assumes the finitevalue φ H = φ e − CC +1 and it becomes constant if C = 0(the C = 0 solution is treated below in the discussion ofthe limit to GR).Special subcases are: • C > B > 0, in which the Brans-Dicke scalar is afinite and decreasing function of r for all values ofthis coordinate. Its derivative with respect to theareal radius is dφdR = dφdr drdR = dφdr (cid:18) dRdr (cid:19) − = − φ Cr B ( C + 1) ( r − r H ) e − (2 C +1) B r → ∞ (4.15)as r → r H . Therefore, for r < r H (or R > R H in the “left branch” of R ), it is dφ/dR > 0, with dφ/dR → + ∞ as r → r − H . For r > r H (or R > R H in its “right branch”), instead, it is dφ/dR < dφ/dR → −∞ as r → r +H . The Brans-Dicke scalarhas a cusp, but remains finite, at the horizon R H where it attains its maximum value, which meansthat the effective gravitational coupling G eff ∼ φ − is maximum there). • − < C < B > 0: in this case the Brans-Dickescalar φ = φ e | CB | r (4.16)is an increasing function of the isotropic radius r and its derivative with respect to the areal radiusis dφdR = | C | r B ( C + 1) ( r − r H ) e − C +1) B r . (4.17)We need to further distinguish the situation − 2, in which dφdR = | C | r B | C + 1 | ( r − r H ) e | C +1) B | r → ∞ (4.18)as the wormhole throat is approached when r → r H .In this case dφ/dR is negative for 0 < r < r H , van-ishes at r H , and is positive for r > r H . The Brans-Dicke scalar is minimum and finite, but has a cusp(and G eff is maximum) at the wormhole throat. 3. Parameter range C > − , B < In this case it is R H = e B ( C + 1) < dR/dr is alwaysnegative the areal radius is the decreasing function of theisotropic radius R ( r ) = B r e − | C +1 B | r . (4.19)The limit r → + corresponds to R → + ∞ , while r → + ∞ corresponds to R → + . The Ricci scalar is R = ω C B e | C +1 B | r → + ∞ (4.20)as R → + . Therefore, there is central naked singularityfor these parameter values. 4. Parameter range C = − , B = 0 In this case we have R = B /r and the line elementbecomes ds III ) = − e − r/B dt + B r (cid:16) dr + r d Ω (cid:17) = − e − r/B dt + dR + R d Ω . (4.21)The spatial sections are flat and there are no horizons.The limits r → + and r → + ∞ corresponds to R → + ∞ and R → + , respectively. Both the Ricci scalar and theBrans-Dicke scalar field R = ω B R , (4.22) φ = φ e B/R , (4.23)diverge as R → + : there is a naked singularity at R = 0. 5. Parameter range C < − , B < This situation is identical to the case C > − B > 6. Limit to GR Finally, let us discuss the limit to GR ω → −∞ , whichyields C → 0. In this limit the Brans-Dicke scalar (4.2)becomes constant and the line element reduces to ds ∞ ) = − e − r/B dt + (cid:18) Br − B (cid:19) dR + R d Ω . (4.24)There is a wormhole throat at the horizon R = R H = e B .Also for Brans III solutions, the limit in which φ becomesconstant does not reproduce the corresponding solutionof GR. B. Einstein frame class III solutions In the Einstein frame the line element and scalar ofclass III solutions are, respectively, d ˜ s III ) = φ ( III ) ds III ) = − e − ( C +2) rB dt + B r e ( C +2) rB (cid:16) dr + r d Ω (cid:17) , (4.25)˜ φ ( III ) = − r | ω + 3 | πG CrB + const. (4.26)0The areal radius and its derivative are˜ R ( r ) = B r e ( C +2) r B , (4.27) d ˜ Rdr = B r e ( C +2) r B (cid:18) C + 22 B (cid:19) (cid:18) r − BC + 2 (cid:19) , (4.28)while the Einstein frame Ricci scalar is˜ R = | ω + 3 | C r B e − ( C +2) rB . (4.29)The equation ∇ c ˜ R ∇ c ˜ R = 0 becomes( C + 2) (cid:18) r − BC + 2 (cid:19) = 0 (4.30)and has the double root r ∗ = 2 BC + 2 . (4.31)(if C = − 1. Parameter range B/ ( C + 2) > In this range of parameters the double root r ∗ is pos-itive and the areal radius ˜ R ( r ) diverges in both limits r → + and r → + ∞ . There is a wormhole throat at r ∗ ,corresponding to ˜ R ∗ = e B ( C + 2) / 2. Parameter range B/ ( C + 2) < In this case there are no horizons, the areal radius ˜ R ( r )tends to zero value as r → + ∞ , where the Ricci scalardiverges, and to infinity as r → + . There is a nakedcentral singularity. 3. Case C = − In this case there are no horizons and the areal ra-dius ˜ R ( r ) = B /r behaves as in the previous case. TheRicci scalar diverges again at ˜ R = 0 and there is a nakedcentral singularity. V. BRANS CLASS IV SOLUTIONS There is another duality relating class III and class IVsolutions [15]. We shall come back again to this dualityin Sect. VI. Brans IV solutions were examined, for a re-stricted range of parameters, in the recent Ref. [39] in Specifically, for the situations B > B > , C > − both the Jordan and Einstein frames. There it is shownthat, for a certain range of parameters, the formal so-lution is a wormhole in the Jordan frame and a nakedsingularity in the Einstein frame, and the detailed reasonwhy this happens was pointed out [39]. For completeness,we briefly revisit also those cases. A. Jordan frame class IV solutions The Jordan frame line element and Brans-Dicke scalarfield for Brans class IV solutions are, respectively , ds IV ) = − e − Br dt + e B ( C +1) r (cid:16) dr + r d Ω (cid:17) (5.1) φ ( IV ) = φ e − BCr , (5.2)where C = − ± p − (2 ω + 3) ω + 2 . (5.3)Clearly, the Brans-Dicke parameter is limited to therange ω = − , ω < − / 2. The parameter B has thedimensions of a length and r > 0. The areal radius andits derivative are R ( r ) = e B ( C +1) r r , (5.4) dRdr = e r H r (cid:16) − r H r (cid:17) , (5.5)where r H = B ( C + 1) (5.6)is the root of the equation dR/dr = 0 and R H = e r H is the corresponding value of the areal radius. Theisotropic radius (5.6) is also the double root of the equa-tion locating the horizons ∇ c R ∇ c R = 0, which becomes(1 − r H /r ) = 0. When r H is real and positive the solu-tion describes a wormhole, otherwise there are no hori-zons and no black holes. The Ricci scalar is R = ωφ ∇ c φ ∇ c φ = ω B C r e − B ( C +1) r . (5.7) Note that in the original Brans’ class IV metric and the corre-sponding scalar field, introduced in Ref. [8], the parameter B appears in the denominator of the exponents and, hence, hasthere the dimensions of inverse length. We chose here to put B in the numerators in order for it to have the same dimensions oflength as it does within the other three classes. Furthermore, itis only under these forms of the metric and the scalar field thatthe dualities we are going to discuss in Sect. VI appear to bemore than just mathematical transformations of the label r . 1. Parameter range B > , C > − In this case r H = | B ( C + 1) | > R ( r ) = e | B ( C +1) r | r diverges as r → + and as r → + ∞ ; itdecreases for 0 < r < r H , assumes its minimum value at r H and increases for r > r H . We have a wormhole throatat R H = e B ( C + 1), at which φ and R are finite (see [39]for further discussion). 2. Parameter range B < , C > − The root r H = − | B ( C + 1) | is negative and there areno horizons. The areal radius R ( r ) increases monoton-ically from zero value as r → + reaching infinity as r → + ∞ . The Ricci scalar R = ω B C r e | B ( C +1) r | (5.8)diverges as r → 0+ (or R → + ), signaling a centralnaked singularity. 3. Parameter range B < , C < − The double root r H = | B ( C + 1) | is positive, R ( r ) → + ∞ as r → + and as r → + ∞ . There is a wormholethroat at r H , with φ and R finite. 4. Parameter range B > , C < − It is r H = − | B ( C + 1) | < R ( r ) increases monotonically from zerovalue at r = 0 to infinity as r → + ∞ . The Ricci scalar R = ω B C r e | B ( C +1) r | (5.9)diverges as r → + ( i.e. , as R → + ), signaling again acentral naked singularity. 5. Parameter range C = − , B = 0 This situation corresponds to ω = − 2, which is ex-cluded by eq. (5.3). However, one could think of consid-ering the formal line element (5.1) without reference toits derivation in [8], that is, ds IV ) = − e − Br dt + dr + r d Ω , (5.10)for which areal and isotropic radius coincide, and whichhas flat spatial sections. The Ricci scalar R = ω B C R (5.11)diverges as R → 6. The GR limit The GR limit should correspond to ω → −∞ , whichimplies that C → r H → B . The scalar field be-comes constant in this limit, the line element reduces to ds ∞ ) = − e − Br dt + e Br (cid:16) dr + r d Ω (cid:17) , (5.12)and the areal radius is R ( r ) = r e Br . The Ricci scalar is R = ω B C r e − B ( C +1) r ≈ − B r e − Br . (5.13)If B > 0, it is r H > R ( r ) → + ∞ as r → + and as r → + ∞ . The Ricci scalar is finite at R = 0 and thesolution describes a wormhole.If B < 0, it is r H < R ( r ) → r → + and R ( r ) → + ∞ as r → + ∞ . The Ricci scalar diverges at R = 0: there is acentral naked singularity.In either case the limit in which φ becomes constantfails to reproduce the corresponding GR solution. B. Einstein frame class IV solutions The Einstein frame line element and scalar field forclass IV solutions are d ˜ s IV ) = φ ( IV ) ds IV ) = − e − B ( C +2) r dt + e B ( C +2) r (cid:16) dr + r d Ω (cid:17) , (5.14)˜ φ ( IV ) = r | ω + 3 | πG BCr + const. (5.15)The areal radius is simply˜ R = e B ( C +2)2 r (5.16)and the Ricci scalar is˜ R = | ω + 3 | B C r e − B ( C +2) r . (5.17)The equation ∇ c ˜ R ∇ c ˜ R = 0 becomes ( r − r ∗ ) = 0,where r ∗ = B ( C + 2)2 (5.18)is the only root and a double root. 1. Parameter range B ( C + 2) < In this case r ∗ < R ( r ) → + as r → + , wherethe Ricci scalar diverges, and ˜ R ( r ) → + ∞ as r → + ∞ .There is a naked central singularity.2 2. Parameter range B ( C + 2) > In this case r ∗ > 0, and ˜ R ( r ) → + ∞ in both limits r → + and r → + ∞ . There is a wormhole throat atphysical radius ˜ R ∗ = e B ( C + 2) / 3. Case C = − In this case areal radius and isotropic radius coincideand the Ricci scalar diverges as ˜ R → + . There is anaked central singularity. VI. THE DUALITIES As mentioned above, Brans solutions are not actuallyall independent as there are dualities relating pairs of thesolution classes [14, 15]. There is a duality relating classesI and II and there is another duality relating classes IIIand VI. It is therefore not surprising to find the samepattern concerning the existence of wormholes and/ornaked singularities within a pair of solutions related bysuch a duality. It is also not a coincidence that all theBrans classes fail to recover the GR limit as ω → ∞ .Furthermore, as we shall see shortly, these dualities areakin to the duality one finds for the Schwarzschild blackhole solution when the latter is written in isotropic coor-dinates. Indeed, it is well known that the Schwarzschildmetric in isotropic coordinates is self-dual under the in-version r ↔ B /r , as it can easily be verified using themetric (2.4). This fact might actually have been expectedas the Schwarzschild solution in isotropic coordinates isrecovered either from class I or class II when the param-eter C vanishes as we saw in eq. (2.4) and in the remarkbelow eq. (3.3), respectively, for then the Brans-Dickefield φ becomes a constant. The same observation alsoapplies to the case of classes III and IV, as the latter re-duces to the Minkowski spacetime for B = 0 while theformer reduces, for B → ∞ ( C cannot vanish for theseclasses), to a Minkowski spacetime whose radial coordi-nate r has been inverted to 1 /r . All these observationsremain valid when going to the Einstein frame.The duality transformation that relates class III toclass IV is the following inversion, r ←→ B r . (6.19)Notice that the form of the duality transformationdisplayed here is slightly different from that given inRef. [15]. The dimensions here are correct as the pa-rameter B has the same dimensions of length as r . Infact, by a straightforward substitution, one easily recov-ers in the Jordan frame the metric (4.1) from the metric(5.1), and vice-versa , thanks to this transformation ofthe coordinate r . In the Einstein frame one also recov-ers the metric (5.14) from the metric (4.25) using such an inversion. The effect of this inversion is easily seenby comparing the location of the would-be wormholes’throats (4.9) and (5.6) in the Jordan frame and (5.18)and (4.31) in the Einstein frame; one just being the in-verse of the other up to the factor B as dictated by theinversion (6.19).The duality transformation that relates class I to classII is, r ←→ B r , λ ←→ − i Λ , B ←→ iB. (6.20)Here also, our notation differs from that of Refs.[14, 15]in that we used B for the r -inversion in order to get thedimensions right. In fact, by a straightforward substi-tution, one easily recovers in the Jordan frame the met-ric (3.1) from the metric (2.1), and vice-versa , thanksto these three transformations. The same applies to themetrics (2.25) and (3.19) in the Einstein frame.In contrast to the case of classes III and IV, however,the effect of the duality transformation (6.20) is not tomake the radii (2.12) and (3.8) of the would-be wormholethroats in the Jordan frame inverse of each other. Thesame applies to the case of the radii (2.31) and (3.25) ofthe would-be wormholes in the Einstein frame. In bothcases, one is recovered from the other just by makingthe substitutions (6.20) on Λ and B but the resultingradii are not inverse of each other. This could easily beunderstood by the fact that, in contrast to classes IIIand IV which admit as a limit the Minkowski spacetime,classes I and II admit as a limit the Schwarzschild metricwhich is already self-dual under the inversion r ↔ B /r in isotropic coordinates.Now, since the parameter B has the dimensions oflength and the parameters λ and Λ both appear in ex-ponents inside the metrics (3.1) and (2.1), it might seemunphysical to change these parameters into imaginaryentities. Note, however, that the meaning we give tothe parameter B cannot be taken independently fromthe parameter λ in class I or independently from the pa-rameter Λ in class II. In fact, as we saw in eq. (2.4),for C = 0 the parameter B becomes the mass parame-ter of the Schwarzschild solution if λ = 1, whereas for λ = − − B that should be inter-preted as the Schwarzschild mass parameter in eq. (2.6).The same applies for class II whose metric (3.1) repro-duces the Schwarzschild metric in isotropic coordinatesonly when both Λ and B become imaginary.Finally, one might wonder at this point if there still ex-ists another duality transformation that might relate onepair of classes to another pair. The answer is no and thereason is the following. The fact that one pair of solutions(classes I and II) reduces to the Schwarzschild metric fora specific value of the parameter C and the other pair(classes III and IV) reduces to the Minkowski spacetimefor a specific value of the parameter B is what forbidsthe existence of any duality between the two pairs. Inother words, as there is no duality between the curvedSchwarzschild spacetime and the flat Minkowski space-3time, no duality could exist between the first pair andthe second pair either. VII. CONCLUSIONS We have verified explicitly that the Brans classes I-IVof solutions [8] of Brans-Dicke theory [1] always describeeither wormholes or else horizonless geometries contain-ing naked singularities, and they never describe blackholes, in agreement with the Agnese-La Camera theorem[10] and with Hawking’s theorem on Brans-Dicke blackholes [19–21]. Ambiguity and confusion lingering in theliterature about the nature of these solutions and appar-ent contradictions with the theorems mentioned aboveare thus eliminated once and for all. Spacetimes harbor-ing naked singularities are unphysical since one cannotprescribe regular initial data in the presence of a nakedsingularity and the initial value problem fails, leavingthe theory void of predictability and wormholes as theonly remaining Brans solutions. Wormholes are exoticobjects which require the energy conditions to be vio-lated. The Brans solutions are vacuum solutions andthe Brans-Dicke scalar acts as the only form of effective“matter” once the field equations are written as effectiveEinstein equations, as in eq. (1.3). It is well known that a non-minimally coupled scalar field and the Brans-Dickescalar can violate all of the energy conditions, thereforeit is no surprise that one can obtain wormholes as solu-tions of vacuum Brans-Dicke theory, as has already beenremarked in the literature.No Jordan frame solution of any Brans class has thecorrect limit to GR, which lends a word of caution onusing these solutions as toy models, since they are ratherpathological. The reason for their failure to reproducethe corresponding GR solution ( i.e. , the Schwarzschildgeometry) is by now well understood, as recalled above.The Einstein frame versions of the Brans solutions,naturally, describe only spacetime geometries containingeither wormholes or naked singularities. 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