On the existence of the Schwarzschild solution and Birkhoff's Theorem in Scalar-Tensor gravity

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On the existence of the Schwarzschild solution and Birkhoff’s Theorem in

Scalar-Tensor gravity

Sante CarloniESA-Advanced Concepts team, European Space Research Technology Center (ESTEC)

Keplerlaan 1, Postbus 299, 2200 AG Noordwijk The Netherlands,Institute of Theoretical Physics, MFF, Charles University,V. Holesovickach 2 180 00 Praha 8, Czech Republic

Peter K S DunsbyDepartment of Mathematics and Applied Mathematics, University of Cape Town, South Africa,

South African Astronomical Observatory, Observatory Cape Town, South Africa.

We clarify a number of aspects of spherically symmetric solutions of non-minimally coupled scalartensor theories using the 1+1+2 covariant approach. Particular attention is dedicated to the exis-tence of a Schwarzschild solution and the applicability of Birkhoff’s theorem in this case. We alsopresent some new exact solutions.

PACS numbers: 04.50.Kd

I. INTRODUCTION

Scalar-Tensor (ST) theories of gravity are among themost studied extensions of General Relativity (GR). Ini-tially introduced as a completion of GR by Brans andDicke [1], ST theories have found a wide range of uses,which include providing models of the inflation mecha-nism [2] and modelling the late-time acceleration of theUniverse [3]. These theories arise naturally in the con-text of quantum field theory on curved spacetime [4] andhave also played a role in the debate about the no-hairconjecture of black holes and its various realisations [5].

The name scalar-tensor theory is often used to describetheories in which both minimally and non-minimally cou-pled scalar fields appear. The first class of theories arethe most studied, due in part to their simplicity and con-nection to the original models of inflation. The secondclass has instead gained attention with the discovery oftheir connection with higher dimensional theories such asKaluza-Klein and (Super-)String theories) [6]. For boththese classes, we have now a relatively clear picture ofmany aspects of the cosmologies of these models [7, 8](including some unexpected phenomena [9]) and with thewealth of cosmological data currently available, there aremany ways to compare their predictions with more stan-dard models of the Universe.

It therefore comes as a surprise that in comparisonwith cosmology, the investigation of spherically symmet-ric solutions of both types of theories has been more lim-ited. This is particularly true for theories which involvea non-trivial self-interacting potential. Indeed, many ofthe most important models of ST theories are charac-terised by a non-trivial self-interaction of the scalar fieldpotential and in situations where there is a non-minimalcoupling to gravity, very few results are known other thanthe (non-independent) four types of Brans solutions [10].Determining detailed information about spherically sym-metric solutions for these theories could open the way

to a new generation of tests based on astrophysical datarather than those from cosmology.This paper is an attempt to move in this direction.

We aim to obtain a deeper insight into the properties ofspherically symmetric solutions of ST theories, focusingmainly on the non-minimally coupled theories (many ofour results turn out to also be valid in the minimallycoupled case). The class of non-minimally coupled the-ories we consider are those characterised by a standardkinetic term for the scalar field. However, the actionswe consider can be recast in the general form given byBergmann, Nordtveldt and others [11] by a simple repa-rameterisation of the scalar field.Key to our analysis is the powerful 1+1+2 covariant

approach, developed by Clarkson and Barrett [12]. Thisformalism is a natural extension to the 1+3 approachoriginally developed by Ehlers and Ellis [14] and it isoptimised for problems which have spherical symmetry.The 1+1+2 formalism has been applied to the study oflinear perturbations of a Schwarzschild spacetime [12]and to the generation of electromagnetic radiation bygravitational waves interacting with a strong magneticfield around a vibrating Schwarzschild black hole [15].In cosmology it has been used to investigate perturba-tions of several Locally Rotationally Symmetric space-times (LRS) [13].Using this approach we clarify some aspects of spher-

ically symmetric solutions of non-minimally coupled STtheories, in particular the applicability of Birkhoff’s the-orem. We also present some new exact solutions.The paper is organised in the following way. In section

II we give the general equations of ST gravity. In SectionIII we briefly review the covariant approach and deducethe 1+1+2 framework from the 1+3 one. We then drivethe 1+1+2 equations in general and relate the 1+1+2quantities to the metric components. In section IV wegive the 1+1+2 equations for a general non-minimallycoupled ST theory of gravity and we present some impor-tant results which are needed if one is to use the 1+1+2

2

formalism to find exact solutions. In section V we discussthe existence and the meaning of Birkhoff’s theorem inthis framework and finally section VII is dedicated to ourconclusions.Unless otherwise specified, natural units (~ = c =

kB = 8πG = 1) will be used throughout this paper andLatin indices run from 0 to 3. The symbol ∇ representsthe usual covariant derivative and ∂ corresponds to par-tial differentiation. We use the −,+,+,+ signature andthe Riemann tensor is defined by

Rabcd = Γabd,c − Γabc,d + ΓebdΓace − ΓebcΓ

ade , (1)

where the Γabd are the Christoffel symbols (i.e. symmet-ric in the lower indices), defined by

Γabd =1

2gae (gbe,d + ged,b − gbd,e) . (2)

The Ricci tensor is obtained by contracting the first andthe third indices

Rab = gcdRacbd . (3)

Symmetrisation and the anti-symmetrisation over the in-dexes of a tensor are defined as

T(ab) =1

2(Tab + Tba) , T[ab] =

1

2(Tab − Tba) . (4)

Finally the Hilbert–Einstein action in the presence ofmatter is given by

A =1

2

∫

d4x√−g [R+ 2Lm] . (5)

II. GENERAL EQUATIONS FOR SCALARTENSOR GRAVITY

The most general action for ST theories of gravity isgiven by (conventions as in Wald [16]):

A =

∫

dx4√−g

[

1

2F (ψ)R− Lψ + Lm

]

, (6)

where

Lψ =1

2∇aψ∇aψ − V (ψ), (7)

V (ψ) is a generic potential expressing the self-interactionof the scalar field and Lm represents the matter contri-bution.Varying the action with respect to the metric gives the

gravitational field equations:

F (ψ)

(

Rab −1

2Rgab

)

= Tmab +∇aψ∇bψ

−gab(

1

2∇cψ∇cψ + V (ψ)

)

+

(∇b∇a − gab∇c∇c)F (ψ) , (8)

and the variation with respect to the field ψ gives thecurved spacetime version of the Klein–Gordon equation

∇a∇aψ +1

2F ′(ψ)R − V ′(ψ) = 0 , (9)

where the prime indicates a derivative with respect to ψ.Both these equations reduce to the standard equations forGR and a minimally-coupled scalar field when F (ψ) = 1.Equation (8) can be recast as

Gab =TmabF (ψ)

+ Tψab = T

(eff)ab , (10)

where Tψab has the form

Tψab =

1

F (ψ)

[

∇aψ∇bψ − gab

(

1

2∇cψ∇cψ + V (ψ)

)

+∇b∇aF (ψ)− gab∇c∇cF (ψ)] . (11)

Provided that ψ,a 6= 0, equation (9) also follows from theconservation equations

∇bTψab = 0 . (12)

The reformulation above will be very important for ourpurposes. In fact, the form of (10) allows us to treatscalar tensor gravity as standard Einstein gravity in thepresence of two effective fluids and permits a straight-forward generalisation of the 1+1+2 formalism to theseequations.

III. 1+1+2 COVARIANT APPROACH

In the following we give a brief review of the 1+1+2covariant approach [12]. We will proceed first with thestandard 1+3 decomposition and then perform a furthersplit of the spatial degrees of freedom relative to a pre-ferred spatial direction. This allows us to derive a setof variables better suited to systems in which a spatialdirection is important (i.e., the radial one in the case ofspherical symmetry).

A. Kinematics

In 1+3 approach we define a time-like congruence, withunit tangent vector ua (uaua < 01). In this way, anytensor field can be projected along ua (extracting thetemporal parts) or into the 3-space orthogonal to ua usingthe projection tensor hab = gab + uaub.

1 Note that we are not assuming that ua is normalised. The compo-nents of ua will play an important role in connecting the 1+1+2variables and the metric.

3

In the 1+1+2 approach, we further split this 3-spaceby introducing the spatial unit vector ea orthogonal toua, so that

eaua = 0 , eae

a = 1 . (13)

Then the tensor

Nab ≡ ha

b − eaeb = ga

b + uaub − eae

b , Naa = 2 (14)

projects vectors into the 2-surfaces orthogonal to ea and

ua. It is obvious that eaNab = 0 = uaNab. Using Nab,any 3-vector λa = habλ

b can be irreducibly split into acomponent along ea and a sheet component Λa, orthogo-nal to ea, i.e.,

λa = Λea + Λa , Λ ≡ λaea , Λa ≡ Nabλb , (15)

A similar decomposition can be done for 3-tensors λab =hcah

dbΦcd, which can be split into scalar (along ea), 2-

vector and 2-tensor parts as follows:

λab = λ〈ab〉 = Λ

(

eaeb −1

2Nab

)

+ 2Λ(aeb) + Λab , (16)

where

Λ ≡ eaebλab = −Nabλab ,

Λa ≡ Nabecλbc ,

Λab ≡ λab ≡(

N c(aNb)

d − 1

2NabN

cd

)

λcd . (17)

The Levi-Civita 2-tensor is defined as

εab ≡ εabcec = ηdabce

cud , (18)

where εabc is the 3-space permutation tensor, which is thevolume element of the 3-space and ηabcd is the spacetime4-volume element. εab plays the usual role of 2 volumeelement for the 2-surfaces.With these definitions it follows that any 1+3 quantity

can be locally split into three types of objects: scalars, 2-vectors and 2-tensors defined on the 2-surfaces orthogonalto ea.

B. Derivatives and the kinematical variables

Using ua and hab we can obtain two derivative opera-tors: one defined along the time-like congruence:

Xa..bc..d = ue∇eX

a..bc..d (19)

and the projected derivative D:

DeXa..b

c..d = hafhpc...h

bghqdhre∇rT

f..gp..q . (20)

Applying the covariant derivative to ua we can obtain thekey 1+3 quantities:

∇aub = −uaub +1

2Θhab + σab + ωab , (21)

where ua is the acceleration, Θ is the expansion parame-ter, σab the shear and ωab is the vorticity.In the same way as before we can now split the D

operator using ea and Nab:

Xa..bc..d ≡ efDfXa..b

c..d , (22)

δfXa..bc..d ≡ Na

e..NbgNi

c..NkdN j

fDjXe..gi..k .(23)

The covariant derivative of ea can be split in the directionorthogonal to ua into it’s irreducible parts to give:

Daeb = eaab +1

2φNab + ξεab + λab . (24)

For an observer that chooses ea as a special direction inspacetime, φ = δae

a represents the expansion of the sheet,λab = δaeb is the shear of ea (i.e., the distortion of the

sheet), ξ = 12εabδaeb is a representation of the “twisting”

or rotation of the sheet and aa = ea its acceleration.Using equations (15) and (16) one can split the 1+3

kinematical variables and Weyl tensors as follows:

ua = Aea +Aa , (25)

ωa =1

2εabcωbc = Ωea +Ωa , (26)

σab = Σ

(

eaeb −1

2Nab

)

+ 2Σ(aeb) +Σab , (27)

Eab = E(

eaeb −1

2Nab

)

+ 2E(aeb) + Eab , (28)

Hab = H(

eaeb −1

2Nab

)

+ 2H(aeb) +Hab , (29)

where Eab and Hab are the electric and magnetic partof the Weyl tensor respectively. It follows that the keyvariables of the 1+1+2 formalism are:

Θ,A,Ω,Σ, E ,H,Aa,Ωa,Σa, Ea,Ha,Σab, Eab,Hab .(30)

Similarly, we may split the general energy momentumtensor in (10) as:

Tab = µuaub + phab + 2q(aub) + πab , (31)

where µ is the energy density and p is the pressure.The anisotropic fluid variables qa and πab can be fur-

ther split as:

qa = Qea +Qa , (32)

πab = Π

[

eaeb −1

2Nab

]

+ 2Π(aeb) +Πab . (33)

C. 1+1+2 equations for LRS-II Spacetimes

Because of its structure, the 1+1+2 formalism is ide-ally suited for a covariant description of all the LRS space-times. These spacetimes possess a continuous isotropygroup at each point [17] and exhibit locally a unique,preferred, covariantly defined spatial direction.

4

Since LRS space-times are constructed to beisotropic, there are no preferred directions in thesheet and consequently all 1+1+2 vectors and ten-sors vanish. It follows that the only non-zero 1+1+2variables are the covariantly defined scalars [12]A,Θ, φ, ξ,Σ,Ω, E ,H, µ, p,Π, Q .A subclass of the LRS spacetimes, called LRS-II, con-

tain all the LRS spacetimes that are rotation free. Asa consequence, the variables Ω, ξ and H are identicallyzero in LRS-II spacetimes and

A,Θ, φ,Σ, E , µ, p,Π, Q

fully characterise the kinematics. The propagation andconstraint equations for these variables are obtained bythe Ricci and (twice contracted) Bianchi identities andcan be found in [12].Let us now turn to the case of spherically symmetric

static spacetimes which belong naturally to LRS class II.The condition of staticity implies that the dot derivativesof all the quantities vanish. Therefore the expansion van-ishes, (Θ = 0), and this implies Σ = 0. The same holdsfor the heat flux Q. Hence the set of 1+1+2 equationswhich describe the spacetime become:

φ = −1

2φ2 − 2

3µ− 1

2Π− E , (34)

E − 1

3µ+

1

2Π = −3

2φ

(

E +1

2Π

)

, (35)

0 = −Aφ+1

3(µ+ 3p)− E +

1

2Π , (36)

p+ Π = −(

3

2φ+A

)

Π− (µ+ p)A , (37)

A = − (A+ φ)A+1

2(µ+ 3p) , (38)

K = −φK, (39)

K =1

3µ− E − 1

2Π +

1

4φ2 . (40)

Eliminating E and using the constraints (35) and (40) thesystem above can be reduced to

φ = Aφ− µ− p−Π− φ2

2, (41)

A = −A (A+ φ) +1

2(µ+ 3p) , (42)

Π + p = −A(µ+ p+Π)− 3Πφ

2, (43)

K = −p−Π+1

4φ(A + φ) . (44)

E =1

3(µ+ 3p) +

1

2Π−Aφ . (45)

One could, of course, decide to eliminate other variables.In particular one might try to retain (39), due to its sim-plicity. However, as we will see in section IV, this choice

has to be taken with great care, especially when attempt-ing to find exact solutions. Once these equations havebeen solved it is useful to connect the 1+1+2 quantitiesto the metric coefficients and hence reconstruct the met-ric.Consider now the general spherically symmetric static

metric:

ds2 = −A(ρ)dτ2+B(ρ)dρ2+C(ρ)(dθ2+sin2 θdφ2) . (46)

Using the definition of covariant derivative one obtains:

A = eaua =1

2A√B

dA

dρ, (47)

and

φ = δaea =

1

C√B

dC

dρ. (48)

Note that we have two equations for three metric compo-nents, so at first sight it might seem that given a solutionof the 1+1+2 potentialsA and φ, there is no way to deter-mine the metric. One needs to remember, however, thatthe form of the coefficient B depends on the choice of thecoordinated ρ, so that the factor

√B can be reabsorbed

into the definition of ρ and effectively the metric (46) hasonly two unknown functions. Therefore a p coordinatedirectly associated with the “hat” derivative, would haveX = X,p which implies B(p) = 1.The formulae above reveal an interesting connection

between the 1+1+2 formalism and the Takeno variables[18]. In fact one can see that many of the theorems provedby Takeno have a correspondence in the 1+1+2 formal-ism.

IV. SPHERICALLY SYMMETRIC STATICSPACE-TIMES IN SCALAR TENSOR GRAVITY

The simplest way to write the 1+1+2 equations forthe case of ST gravity is to use the recasting of the field

equations that we gave in Section II. In particular Tψabcan be decomposed as in (31) with

µψ =1

F (ψ)

[

1

2ψ2 + V + F ′

(

ψ + φ ˆψ)

+ F ′′ ψ2

]

,(49)

pψ =1

F (ψ)

[

− ψ2

6− V − 2

3F ′′ψ2 − 2

3F ′(

ˆψ + φψ)

−F ′A ψ]

, (50)

Πψ =1

F (ψ)

[

2

3ψ2 +

2

3F ′′ψ2 +

2

3F ′(

ˆψ − φψ)

]

. (51)

In this way it is possible to write (34-38) as

5

F[

2φ+ (φ− 2A)φ]

+ 2ψ2 = 2AF ′ψ − 2F ′′ψ2 − 2F ′ ˆψ , (52)

2F[

A +A(A+ φ)]

+ 2V = F ′′ψ2 + (3A+ φ)F ′ψ − F ′ ˆψ , (53)

ψ[

3 (F ′)2+ 2F

]

ˆψ + F ′ (3F ′′ + 1) ψ2 + (A+ φ)[

3 (F ′)2+ 2F

]

ψ + 4V F ′ − 2FV ′

= 0 , (54)

K =ψ2

2F+V

F+F ′

Fψ(A+ φ) +

1

4φ(A+ φ) , (55)

E =ψ2

3F− 2V

3F− F ′

Fψ(A +

1

2φ)−Aφ , (56)

where we have assumed F 6= 0. The above equationscharacterise the static and spherically symmetric solu-tions of a general ST theory of gravity. Note that inspite of the fact that (55) contains second derivatives ofthe scalar field ψ, it does not correspond exactly to theKlein-Gordon equation. In fact, equation (54) can beshown to correspond to a combination of the Klein Gor-don equation and the trace of the general field equations.Let us now show how (52-56) can be used to obtain

exact solutions. The first thing to do is to choose a suit-able radial coordinate. A clever choice is to proceed in away that equation (39) has a trivial solution like in the

case of the coordinate r for which X = − 12rφ∂rX in [12].

The Gauss curvature is therefore just K = r−2. However,one must be careful in this respect to check that (55) isfulfilled, because the choice above decouples K from φ.This can be clearly seen by considering the theory

F =2F1e

φ0ψ2

φ0+ F2 +

2ψ

φ0, (57)

V = −1

4ψ20

[

3φ0

(

4F1eφ0ψ2 + F2φ0 + 2ψ

)

+ 10]

,(58)

with the solution

A =1

2φ0 , φ = φ0 and ψ = ψ0 + 2 ln(φ0r) (59)

of (52-54), which corresponds to the following solutionfor the metric

A = A0r2φ0|φ0| , B = ψ2

0r2 , C = r−2 . (60)

Although the above solution satisfies (52-54) and (39), itdoes not satisfy (55) and is not a solution of the field equa-tions (8). This happens because the coordinate changeremoves the connection between K and φ so that (39)does not guarantee that (55) is satisfied.On the other hand, the theory

F = F0ψ2 V = V0ψ

β (61)

does satisfy the system (52-55) for this convenient choiceof radial coordinate and it is easy to find the exact solu-

tion

A =A0

r, φ =

2

r(62)

K =F0 [2A0(β − 4) + β − 10]− 2

r2(β − 2)F0+

V0σβ0

r2F0σ20

, (63)

ψ = ψ0r2

β−2 , (64)

which, in terms of the metric coefficients is given by:

A = r2A0 , B = 1 , C = K−1 . (65)

The above solution satisfies all the Einstein equationsupon direct substitution. Since this solution does not re-duce to Minkowski spacetime in any limit of the parame-ters, it is clearly not asymptotically flat. The associatedNewtonian potential can be calculated in the usual wayand contains a constant term of the same order of thegravitational constant2.

V. THE EXISTENCE OF A SCHWARZSCHILDSOLUTION AND BIRKHOFF’S THEOREM

An important question one can address using the sys-tem (52-56) is whether or not the theory (6) admits ingeneral a Schwarzschild solution and if the Birkhoff the-orem is at all satisfied. In what follows we discuss thisproblem in detail.The Schwarzschild solution is obtained when φ and A

satisfy

φ+φ2

2−Aφ = 0 , A +A2 +Aφ = 0 . (67)

2 On using a conformal transformation gab = Ω2gab with Ω2 =F (ψ), the theory (61) is mapped into General Relativity with aminimally coupled scalar field with the potential

V (ψ) = e

√

F0ψ√

6F0+1 . (66)

An exact solution for a similar theory has been found by Chant al. in [19] and this means that the two solutions are related.Incidentally, this solution is also related to the ones found in[20, 21] and that have been found in other contexts.

6

In addition, since the Ricci scalar is identically zero, wefind that the standard Klein-Gordon equation holds

ˆψ + 2(A− φ)ψ − V ′ = 0 . (68)

Substituting the above equations in (52-54) and assumingF 6= 0, we obtain:

ψ2 (F ′′(ψ) + 1)−AψF ′(ψ) + ˆψF ′(ψ) = 0 , (69)

ˆψF ′(ψ) + ψ[

ψF ′′(ψ) + (3A+ φ)F ′(ψ)]

+ 2V (ψ) = 0 , (70)

ψF ′(ψ)[

ψ(

2(A+ φ)F ′(ψ)− ψ)

+ 2V (ψ)]

= 0,(71)

It is easy to see that this system has a (double) solution

F ′′(ψ) = −1 , F ′(ψ) = 0 , V (ψ) =1

2ψ2 , (72)

which is clearly inconsistent. This means that the classof scalar tensor theories of gravity discussed in section(6) have no Schwarzschild solution if Ψ is not constant.

This result clearly has consequences for Birkhoff’s the-orem. There are a number of formulations of this the-orem in the context of General Relativity[22–29]. Wewill adopt the one of Schutz [30] which can be stated asfollows:

The Schwarzschild solution is the only static,

spherically symmetric, asymptotically flat so-

lution of General Relativity in vacuum.

There are other more mathematically precise definitionsof the Birkhoff’s theorem, but for our purposes theone given above will be sufficient. Since there is noSchwarzschild solution associated to the theory (6), it isclear that the classical formulation of Birkhoff’s theoremgiven above does not apply. However, one could define ageneralised version, for example:

Scalar tensor gravity must possess a unique

static, spherically symmetric and asymptoti-

cally flat solution in vacuum.

Let us adopt this as an Extension of Birkhoff’s Theorem(EBT) and let us see what impact it has on the system(52-54).

• Staticity and Spherical symmetry

Staticity and spherical symmetry is guaranteed bythe construction of the 1+1+2 formalism so thatall the solutions of the (52-54) have this propertyby definition.

• Vacuum condition

In ST gravity there is not a clear definition of theconcept of a vacuum: one could argue that the

scalar field ψ is a form of matter or a scalar part ofthe gravitational field [31]. In the first case one can-not have a proper vacuum solution, however in theclassical treatment there is no reason to consider φto be a matter field. In [22], Birkhoff’s theorem wasgeneralised to the case of a static matter source. Ifwe use this result, as far as the scalar field is static,we can always consider this hypothesis to be satis-fied.

• Uniqueness of the solution

Proving the uniqueness of the solution of the sys-tem (52-54) is however more tricky: one has toprove that in the explicit form the L.H.S. of thissystem is Lifschitz continuous so that the Picard-Lindelof theorem is satisfied. This implies that thefunctions F and V need to be Lifschitz continuousand A and φ need to be continuos in the variableassociated with the hat derivative. In addition, thecondition

F (ψ)[2F (ψ) + 3F ′(ψ)] 6= 0 (73)

needs to be satisfied. This last condition is evidentonly if the (52-54) is expressed in standard form.Note that this condition also implies that F shouldnot have any zeros i.e. that the gravitational inter-action cannot change sign.

• Asymptotic flatness

The most difficult hypothesis to prove is that ofasymptotic flatness. The general proof of this prop-erties for a given metric requires very refined theo-retical tools like the Penrose’s conformal compact-ification [16]. The covariant approaches offers aninteresting alternative, although a somewhat lessgeneral, approach to this problem. In fact decom-posing the Riemann tensor in terms the 1+3 vari-ables one obtains [14]

Rabcd = RabP cd +RabI cd +RabE cd +RabH cd , (74)

RabP cd =2

3(µ+ 3p)u[a u[c h

b]d] +

2

3µha[c h

bd] , (75)

RabI cd = − 2 u[a hb][c qd] − 2 u[c h[ad] q

b]

−2 u[a u[c πb]d] + 2 h[a[c π

b]d] , (76)

RabE cd = 4 u[a u[cEb]d] + 4 h[a[cE

b]d] , (77)

RabH cd = 2 ηabe u[cHd]e + 2 ηcde u[aHb]e , (78)

which in terms of the 1+1+2 variables and thestatic and spherically symmetric case reduces to

7

RabP cd =2

3(µ+ 3p)u[a u[c h

b]d] +

2

3µha[c h

bd] , (79)

RabI cd = −2 u[a u[c

(

eb]ed] −1

2N b]

d]

)

Π+ 2 h[a[c

(

eb]ed] −1

2N b]

d]

)

Π , (80)

RabE cd = 4 u[a u[c

(

eb]ed] −1

2N b]

d]

)

E + 4 h[a[c

(

eb]ed] −1

2N b]

d]

)

E ,

RabH cd = 0 . (81)

If a metric is asymptotically flat, there will be alimit in which ua, ea and Nab are constant tensorsand the Riemann tensor is identically zero. Thisimplies, by definition, that in this limit A has tobe zero and φ has to be zero and that the aboverelations become equations for µ, p, Π and E . Using(36) it is easy to see that

E =1

3(µ+ 3p) +

1

2Π−Aφ , (82)

which means

E → 1

3(µ+ 3p) +

1

2Π . (83)

Therefore, the behaviour of E is determined by µ,p and Π: if these last quantities tend individuallyto zero then Riemann tensor will also tend to zero.Using (49-51), one obtains that in the case of (6)this is realised if

ψ → const ,V (ψ)

F (ψ)→ 0, (84)

which is compatible with what is found in [32]. Itis interesting to note that in this limit (41) and(42) reduce to the equations that give rise to theSchwarzschild solution.

A. The relation with f(R)-gravity and conformaltransformations.

The form of the Birkhoff theorem given above (EBT) isclearly unsatisfactory. It implies that no general conclu-sion about this issue can be made for ST gravity and con-sequently forces us to check whether this theorem holdson a case by case basis. To alleviate this situation, onecould think of using what has have learned in the case off(R)-gravity (see [33]). In this paper it was found thatthe validity the original Birkhoff’s theorem is guaranteedif

f(R)∣

∣

R=0= 0, f ′(R)

∣

∣

R=06= 0 . (85)

Since we know that f(R)-gravity can be mapped into aBrans-Dicke-like theory with a non-trivial potential

ψ = f ′(R), R(ψ) = V ′(ψ) (86)

V (ψ) = R(ψ)ψ − f(R(ψ)), (87)

we can ask if the results of [33] lead to any insight onthe validity of the EBT for ST gravity. Unfortunatelythe answer is negative. In fact, since the Schwarzschildsolution is characterised by R = 0, (86-87) implies imme-diately that the scalar field is constant. In other words,the conditions found in [33] effectively correspond to GRvia (86-87).

B. Conformal transformations.

Another interesting way to attack this problem is tolook at it from the point of view of conformal transforma-tions. It is well known that under a conformal transfor-mation ST theories of gravity of the type (6) are mappedinto GR minimally coupled to a scalar field [34]. Canwe then use conformal transformations to discover EBTcomplying theories?Let us consider the conformal transformation

gab = Ω2gab , (88)

with Ω2 = F (ψ). It is well known that under this trans-formation, equation (6) in vacuum can be recast as

A =

∫

dx4√−g

[

1

2R− 1

2∇aΨ∇aΨ−W (Ψ)

]

, (89)

where

Ψ =

∫(

3F ′(ψ)2 − 2F (ψ)

2F (ψ)2

)1/2

dψ (90)

and

W (Ψ) =V (Ψ)

F (Ψ)

∣

∣

∣

∣

Ψ=Ψ(ψ)

, (91)

As mentioned earlier, it has been recently shown that ifthe scalar field is static, the (Jebsen-)Birkhoff theoremholds for these theories [22]. It would be interesting todetermine that if we have a solution of (89) satisfying theEBT, it is possible to obtain a solution of (6) satisfyingthe EBT. In other words, we need to check that the EBTholds under a conformal transformations.In terms of the key 1+1+2 quantities, the conformal

transformation (88) correspond to the transformations

ua = Ωua, ea = Ωea Nab = Ω2Nab . (92)

8

Let us see now examine how the conditions of the EBTtransform under (88).

• Staticity and Spherical Symmetry

It is obvious that a time independent transforma-tion will map a static metric into a static metric. Inaddition, it it is easy to see that under the confor-mal transformation above, the 1+1+2 vector quan-tities are mapped to zero. For example:

Aa =Aa

Ω+δaΩ

Ω. (93)

If the quantities on the RHS are subject to spher-ical symmetry, it is clear that Aa = 0. The samereasoning applies to all the other quantities.

• Vacuum condition

This aspect of the conformal transformation can beconfusing. With respect to the definition of a vac-uum given above, one effectively passes from vac-uum theory to a theory in which matter is present.However, as we have seen, this additional form ofmatter is static and we can use the results of [22]to conclude that the change of nature of the scalarfield does not affect the transformation propertiesof the EBT.

• Uniqueness of the solution.

In order to prove that uniqueness is preserved, onehas to ensure that the conditions of the Picard–Lindelof theorem are still satisfied in the trans-formed system. The Lifschitz continuity of the func-tions F and V is guaranteed by the fact that theyare not modified by the conformal transformation.However, in the field re-definition part of the trans-formation, the scalar field ψ is changed by (90), soone has to prove that the continuity is preservedunder re-definition of the field. However, this oper-ation involves an integral of a continuous function(F is assumed to be always different from zero) andtherefore it is continuous.

For A and φ, under conformal transformation onehas

A =AΩ

+Ω

Ω, φ =

φ

Ω+

Ω

Ω, (94)

so that if Ω is continuous they preserve their con-tinuity. Remarkably, when we substitute (94) into(52-54), (73) is preserved and no other conditionsneed to be added. Consequently, for a regular con-formal transformation, the uniqueness of the solu-tion is not modified.

• Asymptotic flatness

It is easy to see that the relations (75-78) are in-variant under conformal transformations. There-fore, the conditions for asymptotic flatness found

from these equations remain the same. However,the thermodynamic quantities are rescaled via theconformal factor:

µ = Ω2µ, p = Ω2p, Π = Ω2Π , (95)

therefore we require that if a tilded thermodynamicquantity goes to zero this behaviour is guaranteedalso for the un-tilded quantities. It is clear that thiscan happen only if the conformal factor asymptoti-cally approaches a constant. In terms of the scalarfield, these conditions amount to Ψ → Ψ0 = const.and W (Ψ) = F (Ψ(ψ))−1V (Ψ(ψ)) → 0. The firstcondition is satisfied only if (90) converges to a con-stant when F and its first derivative do so and thisgives us a constraint on the ST theories that satisfythe EBT.

We can summarise what we have found above as fol-lows:

Given a conformal transformation in which

the conformal factor is static, continuous and

asymptotically approaches a constant, one

can use a solution of GR with a minimally

coupled scalar field to obtain a ST theory with

an accompanying solution satisfying the EBT.

Let us verify this result explicitly3. Consider the min-imally coupled theory4

A =

∫

dx4√−g

[

1

2R− 1

2∇aσ∇aσ

]

. (96)

The spherically symmetric solutions for these theory arewell known [35–38]. A solution which is also asymptoti-cally flat is given by

ds2 = −A(r)dτ2+B(r)dρ2+C(r)(dθ2+sin2 θdφ2)], (97)

where

A(r) =

(

1− b

r

)γ

, (98)

B(r) =

(

1− b

r

)−γ, (99)

C(r) =

(

1− b

r

)1−γr2, (100)

3 Some consideration regarding solutions in the Jordan framewhich involve a constant scalar field are in order. As we haveseen in this case any scalar tensor theory of gravity becomeseffectively GR plus a cosmological constant (if a non trivial po-tential is there). In this respect therefore, the above theorem andthe other considerations made below are to be considered onlyin cases in which the scale field is not constant i.e. in which thetheory is different form standard GR.

4 It is clear that since this theory conformably related to he Brans-Dicke theory all the solutions given below are related conformallyto the Brans solution. The real difference would arise if a non-trivial potential was considered. Exact solutions for this case are,however, very rare and the purpose of these example was morethe illustration of the point made earlier in the section.

9

with the scalar field

ψ =

√

1− γ2

2log

(

1− b

r

)

. (101)

and 0 < γ < 1. Using the results above we can generatea set of theories with accompanying solutions satisfyingthe EBT.For example, choosing

Ω =1

2P 2

(

1− b

r

)

P√

1−γ22√

3

+1

2

(

1− b

r

)−P√

1−γ22√

3

,

(102)or

Ω =1

2

(

1− b

r

)

P√

1−γ22√

3

+1

2P 2

(

1− b

r

)−P√

1−γ22√

3

,

(103)with α > 0. We obtain:

F (ψ) =1

2P 2+

1

4e−

√23P +

e√

23Pψ

4P 4+

1

4e−

√23Pψ, (104)

and

F (ψ) =1

2P 2+e−

√23Pψ

4P 4+

1

4e√

23Pψ , (105)

respectively. In this way, the coefficients of the accompa-nying solution satisfying the EBT are

A =A

Ω2=

4P 4(

1− br

)γ+

√1−γ2P√

3

[

(

1− br

)

√1−γ2P√

3 + P 2

]2 , (106)

B =B

Ω2=

4P 4(

1− br

)

√1−γ2P√

3−γ

[

(

1− br

)

√1−γ2P√

3 + P 2

]2 , (107)

C =C

Ω2=

4P 4r2(

1− br

)−γ+√

1−γ2P√3

+1

[

(

1− br

)

√1−γ2P√

3 + P 2

]2 , (108)

(109)

with the scalar field solution

ψ =

√

1− γ2

2P 2log

(

1− b

r

)

. (110)

A second possibility is

Ω =

√α− 1

(

1− br

)

√(α−1)(1−γ2)

2√

3α

√6

, (111)

with α > 1. We obtain:

F (ψ) = 1− ψ2

6. (112)

In this way, the coefficients of the accompanying solutionsatisfying the EBT are

A =A

Ω2=

4P 4(

1− br

)

√1−γ2P√

3+γ

(α− 1) (γ2 − 1) log2(

1− br

) (113)

B =B

Ω2=

4P 4(

1− br

)

√1−γ2P√

3−γ

(α− 1) (γ2 − 1) log2(

1− br

) (114)

C =C

Ω2=

4P 4r2(

1− br

)

√1−γ2P√

3+1−γ

(α− 1) (γ2 − 1) log2(

1− br

) (115)

with the scalar field solution

ψ =

(

1− b

r

)

√α−1

√1−γ2

2√

3√α

. (116)

Finally a third possibility would be

Ω = cosh2

(√

1− γ2 log(

1− br

)

4√3

)

, (117)

with α > 1. We obtain:

F (ψ) =

(

1− ψ2

24

)2

. (118)

In this way, the coefficients of the accompanying solutionsatisfying the EBT are

A =A

Ω2=

4P 4(

1− br

)

√1−γ2P√

3+γ

[

P 2 sinh2(√

1−γ2 log(1− br )

4√3

)

+ 1

]2 ,(119)

B =B

Ω2=

4P 4(

1− br

)

√1−γ2P√

3−γ

(

[

P 2 sinh2

(√1−γ2 log(1− b

r )4√3

)

+ 1

]2 ,(120)

C =C

Ω2=

4P 4r2(

1− br

)

√1−γ2P√

3+1−γ

[

P 2 sinh2(√

1−γ2 log(1− br )

4√3

)

+ 1

]2 ,(121)

with the scalar field solution

ψ = 2√6 sinh

(√

1− γ2 log(

1− br

)

4√3

)

. (122)

It is clear from the above examples that this process canbe repeated with any exact solution in the Einstein frame.

10

VI. CONCLUSIONS.

In this paper we have used the 1+1+2 formalism toanalyse the spherically symmetric metrics in the contextof non-minimally coupled Scalar-Tensor (ST) gravity. Asin the case of the 1+3 covariant approach, our methodcan be easily applied if one treats the non-Einsteinianparts of the gravitational interaction as an effective fluid.The key 1+1+2 equations form a closed system of three

differential equation plus two constraints, which can besimplified considerably by carefully choosing the radialcoordinate. Note however, that this choice can result ina decoupling of the key equations, leading to solutions ofthe 1+1+2 equations which are not solutions of the fullEinstein equations.The main result of this paper relates to the existence

of the Schwarzschild solution in ST gravity and on howthis impacts on the original formulation of Birkhoff’s the-orem. Using the 1+1+2 equations it is easy to show thatno ST theory admits a Schwarzschild solution unless thescalar field is trivial. It follows that one cannot define a

Birkhoff theorem in the usual way. Instead we proposedan extension to this theorem (EBT) in which the role ofthe Schwarzschild solution is taken by the general staticand spherically symmetric solution for these theories. Us-ing the conformal relation between GR and ST gravity,we demonstrated that the EBT is preserved under a con-formal transformation. In this way, the knowledge of aunique static and spherically symmetric and asymptoti-cally flat solution in GR minimally coupled to a scalarfield leads to the derivation of a number of theories forwhich the EBT is satisfied.

The investigation of the detailed properties of thesesolutions and how they relate to Astrophysics will be thesubject of future work.

Acknowledgments

We wish to thank Valerio Faraoni and Timothy Cliftonfor useful comments. PKSD thanks the NRF (SouthAfrica) for financial support.

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