FFR-PHENO-2010-031
Self-Compactifying Gravity
Beyhan Puli¸ce and S¸ ¨ukr¨u Hanif Tanyıldızı Institut f¨ur Physik, Albert-Ludwigs Universit¨at Freiburg Hermann-Herder-Str. 3, 79104Freiburg i.B., Germany Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,141980, Dubna, Moscow Region, Russia
Abstract
We study the self-compactification of extra dimensions via higher curvature gravity, f ( R ), where f ( R ) is the generic function of the Ricci scalar R . First, we reduce pure f ( R ) theory to a scalar-tensor theory by a conformal transformation [1, 3]. Then weshow that, by a second conformal transformation, this scalar-tensor theory turns out toa nonminimal scalar-tensor theory. We find non-vanishing scalar field configurationsthat satisfy the conditions on the partially vanishing energy-momentum tensor andthe equations of motion of the nonminimal scalar-tensor theory. It is interesting thatwe find that the source of gravity, φ , has discrete spectrum. The minimum of thepotential changes according to the value of the coupling constant of the scalar field tothe curvature scalar. When the minimum is at zero for a vanishing scalar field, theentire spacetime is flat. When the minimum is at a nonzero value for a non-vanishingscalar field, the extra space is compactified. We thus show that a given f ( R ) theorycan self-compactify the extra dimensions.Keywords: Self-Compactification, Scalar-Tensor Theory, Higher Curvature Gravity, Extra Di-mensions a r X i v : . [ h e p - t h ] F e b Introduction
The theories which contain only generic f ( R ) function in the lagrangian is mathematicallyequivalent to Einstein gravity plus a scalar field theory [1, 2, 3].The scalar-tensor theories, i.e. Einstein gravity plus a scalar field theories, may havescalar fields which couple to gravity minimally or nonminimally. Dicke discussed [4] theconformal transformation from Brans-Dicke theory [5] to the minimally coupled case. Con-formal transformations relate a nonminimal scalar-tensor theory to a minimal scalar-tensortheory [6].In order to reduce f ( R ) theory to a scalar-tensor theory with the coupling of the scalarfield with the curvature scalar (or Ricci scalar; we will use both of these terms), we useconformal transformations; in other words we rescale the metric tensor. The rescaled metrictensor is not a redefinition of the metric tensor in a different coordinate system; each ofthe metric tensors describes different gravitational fields and different physics.Spontaneous compactification has been discussed using higher order invariant termsof the curvature tensor [7], Einstein gravity with the coupling to matter [8], antisymetrictensor fields [9] and Yang-Mills fields [10, 11]. Spontaneous compactification mechanismshas also been analyzed with sigma models [12] and conformally-coupled scalars [13]. Ithas been shown that for some toy models, dynamical compactification is realized if allsupersymmetries are spontaneously broken [14]. Spontaneous compactification of Lovelocktheory in vacuum has been shown in [15].Here, we give a brief summary of a spontaneous compactification mechanism via a singlescalar field [16] to make a complete discussion in our work. In [16], they consider a scalar-tensor theory which has a nonminimal coupling term that shows the coupling of the scalarfield with the Ricci scalar. They assume some specific conditions for the source term ofthe Ricci tensor so the scalar field gravitates only in a subset of dimensions. They find thescalar field configurations and the corresponding self-interaction potential that satisfy theseconditions and the equations of motion of the theory (which was also found in [17]). Theyfind specific scalar field configurations that do not gravitate in four-dimensional spacetimebut only in extra space. When the vacuum expectation value of the scalar field is zerothe entire space is completely flat but when it has a nonzero value, the extra dimensionsare compactified while the four dimensions remain flat. The resulting topology becomesthe ordinary four-dimensional spacetime times the compact manifold of extra dimensions, M ⊗ E d .Our aim is to discuss a self-compactification mechanism via a more general scalarfield in a scalar-tensor theory which is induced by f ( R ) theory. We make two conformaltransformations; applying the first one we obtain Einstein gravity plus a scalar field theorythat does not contain any nonminimal terms (minimal scalar-tensor theory) from f ( R )theory, then by applying the second one we reach a scalar-tensor theory that includesvarious nonminimal terms (nonminimal scalar-tensor theory).Considering the resulting nonminimal scalar-tensor theory, we study self compactifica-1ion of extra dimensions via a single scalar field which lives in the entire spacetime howevergravitates only in a subset of dimensions. We find the scalar field configurations that fulfillsthe conditions on the energy momentum tensors.In Sec. 2, we begin our analysis by considering f ( R ) theory. We apply a conformaltransformation to transform this theory to a minimal scalar-tensor theory. Then by apply-ing another conformal transformation, we obtain a nonminimal scalar tensor theory. Thistheory has a nonminimal kinetic term and a nonminimal coupling of the scalar field withthe curvature scalar.In Sec. 3, we study the self-compactification mechanism. First, we find the energymomentum tensor of the scalar field, φ . In Sec. 3.1, we want that all components of thesource term of the Ricci tensor vanish in four dimensions in order to have a scalar fieldthat does not gravitate in four-dimensional spacetime. In this manner, we find the explicitform of the scalar field and the corresponding self-interaction potential. In Sec. 3.2, weanalyze the compactification mechanism. We want the pure extra-dimensional componentsof the source term of the Ricci tensor not vanish. Hence, we find a non-vanishing curvaturescalar of the extra dimensions since the curvature tensor is proportional to its source term.This non-vanishing curvature scalar means extra dimensions are curved while ordinary fourdimensions remain flat, so there is a compactification effect. This is self-compactificationsince we derive the nonminimal scalar-tensor theory from f ( R ) by conformal transforma-tions, i.e. f ( R ) itself gives rise to compactification of extra dimensions. f ( R ) Theory to Scalar-Tensor Theory
We know that, as a direct generalization of the Einstein-Hilbert theory, the modified gravitytheories by a generic function of the curvature scalar, f ( R ), are equivalent to Einsteingravity (with the same fundamental scale) plus a scalar field theory. We derive a minimalscalar-tensor theory from f ( R ) applying a conformal transformation. We will show that thetheory constructed, following another conformal transformation, turns out to be a modelwith nonminimal coupling that realizes the coupling between the curvature scalar and thescalar field. The kinetic term of this nonminimal scalar-tensor theory has some function of φ , F ( φ ), as a factor. First of all, we consider the action of f ( R ) theory S = (cid:90) d D x √− gM D − (cid:63) f ( R ) (1)where M (cid:63) is the fundamental scale of gravity and f ( R ) is a generic function of the curvaturescalar R . We remind that the chosen function of R should be conformal invariant. Wefind the Einstein tensor by taking variation of (1) with respect to the metric g AB G AB = R AB − g AB R (cid:2) f (cid:48) ( R ) (cid:3) − (cid:40) g AB f ( R ) − g AB f (cid:48) ( R ) R − g AB (cid:50) f (cid:48) ( R ) + ∇ A ∇ B f (cid:48) ( R ) (cid:41) (2)where f (cid:48) ( R ) ≡ ∂f ( R ) ∂ R . (3)Then, we make the conformal transformationˆ g AB ( z ) = e z ) g AB ( z ) . (4)Considering this conformal transformation, we find the connection coefficientsˆΓ CAB = 12 ˆ g CD (cid:16) ∂ A ˆ g BD + ∂ B ˆ g DA − ∂ D ˆ g AB (cid:17) . (5)Substituting these new connection coefficients into the following Riemann tensor relationˆ R CADB = ∂ ˆΓ CAB − ∂ ˆΓ CAD + ˆΓ
CDE ˆΓ EAB − ˆΓ CBE ˆΓ EAD (6)we obtain the transformed Riemann tensor in terms of the untransformed Riemann tensorˆ R CADB = R CADB + (cid:34) ∇ D ∇ A Φ − ( ∇ A Φ)( ∇ D Φ) + ( ∇ E Φ)( ∇ G Φ) g EG g AD (cid:35) δ CB − (cid:34) ∇ B ∇ A Φ − ( ∇ A Φ)( ∇ B Φ) δ CD + ( ∇ E Φ)( ∇ G Φ) g EG g AB (cid:35) δ CD + (cid:34) ∇ B ∇ F Φ − ( ∇ B Φ)( ∇ F Φ) (cid:35) g CD g AD − (cid:34) ∇ D ∇ F Φ − ( ∇ F Φ)( ∇ D Φ) (cid:35) g CF g AB . (7)Then, contracting the Riemann tensor above with the transformed metric (4), we get thetransformed Ricci tensorˆ R AB = R AB − ( D − (cid:40) ∇ A ∇ B Φ − ∇ A Φ ∇ B Φ + g AB g CD ∇ C Φ ∇ D Φ (cid:41) − g AB g CD ∇ C ∇ D Φ (8)and contracting this Ricci tensor with the transformed metric (4), we obtain the Ricciscalar ˆ R = e − (cid:40) R − D − g AB ∇ A ∇ B Φ − ( D − D − g AB ∇ A Φ ∇ B Φ (cid:41) . (9)3ubstituting relations (8) and (9) intoˆ G AB = ˆ R AB −
12 ˆ g AB ˆ R (10)we obtain the conformally transformed Einstein tensor in terms of the untransformed Ein-stein tensor (2) ˆ G AB = G AB − ( D − (cid:34) ∇ A ∇ B Φ − g AB (cid:50) Φ (cid:35) + ( D − (cid:34) ∇ A Φ ∇ B Φ + D − (cid:16) ∇ Φ (cid:17) g AB (cid:35) (11)We choose Φ to be Φ ≡ D − f (cid:48) ( R ) (12)in order to have all terms in (11) in terms of f ( R ). In other words, a Weyl transformation isperformed such that terms containing derivatives of the scalar cancel. In the transformedmodel (22), in principle, the algebraic equation of motion can be solved for the scalar.Plugging back the solution into the action results in f ( R ) gravity. This gives the scalar asa function of the scalar curvature (12). This connection was known, before [2, 3].We substitute the derivatives ∇ A ∇ B Φ = 1 D − (cid:40) f (cid:48) ( R ) ∇ A ∇ B f (cid:48) ( R ) − ∇ A f (cid:48) ( R ) ∇ B f (cid:48) ( R )[ f (cid:48) ( R )] (cid:41) (13)and ∇ A Φ ∇ B Φ = 1( D −
2) [ f (cid:48) ( R )] ∇ A f (cid:48) ( R ) ∇ B f (cid:48) ( R ) (14)into (11) and obtainˆ G AB = G AB − ∇ A ∇ B f (cid:48) ( R ) f (cid:48) ( R ) − D − D − g AB g CD ∇ C f (cid:48) ( R ) ∇ D f (cid:48) ( R )[ f (cid:48) ( R )] + D − D − ∇ A f (cid:48) ( R ) ∇ B f (cid:48) ( R )[ f (cid:48) ( R )] + g AB g CD ∇ C ∇ D f (cid:48) ( R ) f (cid:48) ( R ) . (15)Here, we make a new scalar field definition to have the Einstein tensor (15) only in termsof the new scalar field ϕ ≡ . (16)4o, (15) becomesˆ G AB = 14 ( D − D − ∇ A ϕ ∇ B ϕ −
18 ( D − D − g AB ∇ C ϕ ∇ C ϕ − ˆ g AB V ( ϕ ) (17)where the following expression for the potential, V ( ϕ ), have been introduced V ( ϕ ) ≡ e − D ϕ (cid:32) R e D − ϕ − f ( R ) (cid:33) . (18)The kinetic term in (17) becomes canonical with a definition of a new scalar field φ [3] φ ≡ M D − (cid:63) (cid:112) ( D − D − ϕ = ⇒ φ = M D − (cid:63) (cid:114) D − D − f (cid:48) ( R ) . (19)Hence, the substitution of the differentiation of ϕ ∇ A φ ∇ B φ = M D − (cid:63)
14 ( D − D − ∇ A ϕ ∇ B ϕ (20)into (17) gives the transformed Einstein tensor in terms of the scalar field φ ˆ G AB = ( ∇ A φ )( ∇ B φ ) −
12 ˆ g AB ( ˆ ∇ φ ) − ˆ g AB ˆ V ( φ ) . (21)We know that the Einstein tensor (21) corresponds to Eintein gravity plus a scalar fieldtheory [1] S = (cid:90) d D x (cid:112) − ˆ g (cid:26) M D − (cid:63) ˆ R − (cid:16) ˆ ∇ φ (cid:17) − ˆ V ( φ ) (cid:27) . (22)This is a minimally-coupled scalar-tensor theory. φ couples to gravity via only its kineticterm. We now want to pass a nonminimally coupled scalar-tensor theory to be able torealize vanishing energy-momentum tensor as in [16, 17]. Considering the Einstein gravityplus a scalar field theory (22) which is derived from the higher curvature gravity, f ( R ), wemake the conformal transformation˜ g AB ( z ) = e ω ( z ) ˆ g AB ( z ) (23)where e ω ( z ) ≡ (cid:16) − ζM − D(cid:63) φ ( z ) (cid:17) / (2 − D ) . (24)The Ricci scalar after the conformal transformation (23) is˜ R = (cid:16) − ζM − D(cid:63) φ (cid:17) − / (2 − D ) ˆ R − D − D − ζ M − D ) (cid:63) φ (cid:16) ˆ ∇ φ (cid:17) (cid:16) − ζM − D(cid:63) φ (cid:17) . (25)5esides, we need to substitute the following expressions (cid:112) − ˜ g = (cid:16) − ζM − D(cid:63) φ (cid:17) D/ (2 − D ) (cid:112) − ˆ g (26)and ( ˜ ∇ φ ) = (1 − ζM − D(cid:63) φ ) − / (2 − D ) ( ˆ ∇ φ ) . (27)into (22). Finally, after the conformal tranformation (23), the theory we have is Einsteingravity plus a scalar-tensor theory with a nonminimal coupling of the scalar field φ withthe Ricci scalar R S = (cid:90) d D x (cid:112) − ˜ g (cid:26) M D − (cid:63) ˜ R − F ( φ ) 12 (cid:16) ˜ ∇ φ (cid:17) − ζ ˜ R φ − ˜ V ( φ ) (cid:27) (28)where ˜ V ( φ ) = (cid:16) − ζM − D(cid:63) φ (cid:17) − D/ (2 − D ) ˆ V ( φ ) (29)and F ( φ ) = 11 − ζM − D(cid:63) φ (cid:26) − D − D − ζ M − D(cid:63) φ + (cid:16) − ζM − D(cid:63) φ (cid:17) (cid:27) (30)is the explicit form of the function in the nonminimal kinetic term. The nonminimal scalar-tensor theory (28) is conformal invariant since it is reduced from a conformal invarianttheory, f ( R ), by conformal transformations. One should notice that there is a function F ( φ ) (30) which makes the kinetic term nonlinear. We will keep this function and study thecompactification mechanism with this form of the action. Here, ζ is the coupling betweenthe scalar field and the curvature scalar. We consider a real scalar field φ living in D -dimensional spacetime with coordinates z A =( x µ , y i ) where A = 0 , , , , · · · , N, µ = 0 , , , , i = 1 , , , ¯ i = 4 , · · · , N . Themetric is ˜ g AB = η µν + ˜ g ij , the metric signature η AB = diag( − , +1 , +1 , +1 , · · · ), number ofdimensions= N + 1 and D ≡ d .In this section, we will analyze the compactification mechanism with the action S = (cid:90) d D x (cid:112) − ˜ g (cid:26) M D − (cid:63) ˜ R − F ( φ ) 12 (cid:16) ˜ ∇ φ (cid:17) − ζ ˜ R φ − ˜ V ( φ ) (cid:27) (31)6here M (cid:63) is the fundamental scale of gravity and ˜ R is the curvature scalar. The fieldconfigurations that extremize the action (31) satisfy the following equations of motion˜ R AB = ˜ T AB M D − (cid:63) − ζφ (32) F ( φ )˜ g AB ˜ ∇ A ˜ ∇ B φ = ζ ˜ R φ + ˜ V (cid:48) ( φ ) + 12 F (cid:48) ( φ )( ˜ ∇ φ ) (33)where the source term of the Ricci tensor is˜ T AB ≡ ˜ T AB + 12 − D ˜ g AB ˜ T = ˜ ∇ A φ ˜ ∇ B φF ( φ ) − ζ ˜ ∇ A ˜ ∇ B φ − − D (cid:16) V ( φ ) − ζ ˜ (cid:50) φ (cid:17) ˜ g AB (34)and the energy-momentum tensor of the scalar field φ is˜ T AB = ˜ ∇ A φ ˜ ∇ B F ( φ ) − ˜ g AB (cid:16)
12 ˜ g CD ˜ ∇ C φ ˜ ∇ D φF ( φ ) + ˜ V ( φ ) (cid:17) + ζ (cid:16) ˜ g AB ˜ (cid:50) − ˜ ∇ A ˜ ∇ B (cid:17) φ . (35)It is obvious that the coupling of the scalar field to the curvature scalar induces theterm with the zeta coupling in (35). In this and the following subsection, we deal with (34), the source term of the Ricci tensor,in four dimensions and in extra dimensions to find the form of the scalar field φ , itsself-interaction potential and the curvature scalar in extra dimensions. We analyze thepartially gravitating scalar fields which gravitate only in extra dimensions. We studiedthe non-gravitating scalar fields that have been already analyzed in [17] and the partiallygravitating scalar fields in [16] to help us for this work.The first thing we do here is to put conditions on ˜ T AB , so it vanishes partially, i.e. ˜ T µν = 0 , ˜ T µ ¯ j = ˜ T ¯ iν = 0 and ˜ T ¯ i ¯ j (cid:54) = 0 and the corresponding metric tensor structure shouldbe g µν = η µν , g µ ¯ j = g ¯ iν = 0 and g ¯ i ¯ j = g ¯ i ¯ j ( (cid:126)y ) as discussed in [16].We write the source term of the Ricci tensor in four dimensions˜ T µν ( φ ) = ˜ ∂ µ φ ˜ ∂ ν φF ( φ ) − ζ ˜ ∂ µ ˜ ∂ ν φ − − D (cid:16) V ( φ ) − ζ ˜ g αβ ˜ ∂ α ˜ ∂ β φ (cid:17) η µν (36)where the following expression for the self-interaction potential of the scalar field, φ , havebeen introduced ˜ V ( φ ) ≡ ˜ V ( φ ) − ζ ˜ g ij ˜ ∇ i ˜ ∇ j φ (37)7o make all the terms look in four dimensions. From now on, we remind that all the termswhich include the potential, ˜ V ( φ ), depend on all of the coordinates.We require ˜ T µν = 0 for all µ, ν = 0 , , , i.e. the flow of the energy-momentum tensor in thedirection of the four dimensions should vanish. The conditions on ˜ T µν are˜ T µ (cid:54) = νµν = 0 = ˜ ∂ µ φ ˜ ∂ ν φF ( φ ) − ζ ˜ ∂ µ ˜ ∂ ν φ , (38)˜ T = 0 = ( ˜ ∂ φ ) F ( φ ) − ζ ˜ ∂ φ − − D (cid:16) V ( φ ) − ζη µν ˜ ∂ µ ˜ ∂ ν φ (cid:17) η , η = − T ii = 0 = ( ˜ ∂ i φ ) F ( φ ) − ζ ˜ ∂ i φ − − D (cid:16) V ( φ ) − ζη µν ˜ ∂ µ ˜ ∂ ν φ (cid:17) η ii , η ii = +1 . (40)We want to find the nontrivial configurations of the scalar field that nullify all componentsof ˜ T µν and the specific form of the self-interaction potential which corresponds to theseconfigurations of the scalar field.We consider the ansatz for the scalar field φ ( z ) ≡ σ ( z ) α ( z ) (41)where σ ( z ) and α ( z ) also depend on all coordinates z . We make this ansatz to find themost general form of the scalar field that satisfies the conditions (38, 39 and 40), so wehave introduced the scalar field in terms of two different functions which depend on thecoordinates z .Firstly, we find the form of σ by summing (39) and (40)0 = ˜ T + ˜ T ii = F ( φ ) σ α (cid:32) α u (cid:48) + 2 αα (cid:48) u (cid:48) + u α (cid:48) (cid:33)(cid:32) ( ∂ σ ) + ( ∂ i σ ) (cid:33) − ζσ α (cid:32) u α (cid:48) + 8 uu (cid:48) αα (cid:48) + 4 α u (cid:48) + 2 uα (cid:48)(cid:48) + 4 u (cid:48) α (cid:48) − αu (cid:48) (cid:33)(cid:32) ( ∂ σ ) + ( ∂ i σ ) (cid:33) − ζσ α (cid:32) αu (cid:48) − uα (cid:48) (cid:33)(cid:32) ∂ σ + ∂ i σ (cid:33) (42)where we have introduced u ≡ ln σ , u (cid:48) ≡ ∂ ln σ∂σ = σ − . (43)8rom the last line of (42) it is obvious that σ ( z ) is a second order polinomial function σ ( z ) = 12 aη µν x µ x ν + η µν x µ p ν + b ( (cid:126)y ) (44)where a , b and p µ are integration constants. Here, b ( (cid:126)y ) is, in general, a function of extracoordinates.Secondly, in order to find the explicit form of α ( z ) we analyze the first condition on˜ T µν T µ (cid:54) = νµν = ˜ ∂ µ σ α ˜ ∂ ν σ α F ( σ α ) − ζ ˜ ∂ µ ˜ ∂ ν σ α . (45)We substitute the derivations˜ ∂ µ σ α ˜ ∂ ν σ α = σ α (cid:32) α u (cid:48) + 2 αα (cid:48) uu (cid:48) + u α (cid:48) (cid:33) ˜ ∂ µ σ ˜ ∂ ν σ (46)and ˜ ∂ µ ˜ ∂ ν σ α = 2 σ α (cid:32) u α (cid:48) + 4 uu (cid:48) αα (cid:48) + 2 u (cid:48) α (cid:48) + 2 u (cid:48) α + uα (cid:48)(cid:48) − u (cid:48) α (cid:33) ∂ µ σ∂ ν σ + 2 σ α (cid:32) u (cid:48) α + α (cid:48) u (cid:33) ˜ ∂ µ ˜ ∂ ν σ (47)into (45). Then, ˜ T µ (cid:54) = νµν takes the form0 = ˜ T µ (cid:54) = νµν = σ α (cid:110)(cid:16) F ( σ α ) − ζ (cid:17) ( uα (cid:48) + u (cid:48) α ) − ζ (2 u (cid:48) α (cid:48) − u (cid:48) α + uα (cid:48)(cid:48) + u (cid:48)(cid:48) α − u (cid:48)(cid:48) α ) (cid:111) ˜ ∂ µ σ ˜ ∂ ν σ = σ α (cid:110)(cid:16) F ( σ α ) − ζ (cid:17) t − ζt (cid:48) + 2 ζ ( u (cid:48) α + u (cid:48)(cid:48) α ) (cid:111) ˜ ∂ µ σ ˜ ∂ ν σ = σ α (cid:110)(cid:16) F ( σ α ) − ζ (cid:17) t − ζt (cid:48) (cid:111) ˜ ∂ µ σ ˜ ∂ ν σ (48)where t ≡ uα (cid:48) + u (cid:48) α and t (cid:48) ≡ ˜ ∂t/ ˜ ∂σ = uα (cid:48)(cid:48) +2 u (cid:48) α (cid:48) + u (cid:48)(cid:48) α . All primes refer to differentiationswith respect to σ and we have used ( u (cid:48) α + u (cid:48)(cid:48) α ) = 0 in the last line.It is obvious from (48) that (cid:16) F ( σ α ) − ζ (cid:17) t − ζt (cid:48) = 0. We integrate this expressionside by side two times. Firstly, we integrate for σ at one side and for t on the other side dσ = 2 ζ ( F ( φ ) − ζ ) t dt , σ = − ζ ( α (cid:48) ln σ + σ − α ) ( F ( φ ) − ζ ) (49)9hen, we integrate α terms at one side and σ terms at the other side both for σ and obtain α α = 1ln σ (cid:90) σ (cid:18) − ζF ( (cid:101) σ α ) − ζ (cid:19) d (cid:101) σ (cid:101) σ (50)which is a function of σ ( z ).In the presence of F ( φ ) (cid:54) = 1, the solution for the field profile reads as in (41) with (44)where a , b and p µ are constants which are, in general, functions of the extra coordinates.So, according to our ansatz (41), the form of the scalar form is φ ( z ) = (cid:32) aη µν x µ x ν + η µν x µ p ν + b (cid:33) σ (cid:82) σ (cid:16) − ζF ( (cid:101) σα ) − ζ (cid:17) d (cid:101) σ (cid:101) σ . (51)The effects of F ( φ ) (cid:54) = 1 are collected in α ( σ ), which reads as (50) whose right-handside is an indefinite integral over (cid:101) σ . One notices that for F ( φ ) = 1, α = − ζ − ζ followsautomatically [17]. One notices that this integral relation generalizes that of [17, 16].We integrate the right hand side of (50) by substituting the function F ( φ ), (30), to seethe relations between the parameters better α ( σ ) = 1ln σ (cid:90) σ − ζ − ζM − D(cid:63) (cid:101) σ α (cid:26) − D − D − ζ M − D(cid:63) (cid:101) σ α + (cid:16) − ζM − D(cid:63) (cid:101) σ α (cid:17) (cid:27) − ζ d (cid:101) σ (cid:101) σ (52)Then, this yields − D − D − ζ M − D(cid:63) e z + (cid:16) − ζM − D(cid:63) e z (cid:17) − ζ (cid:0) − ζM − D(cid:63) e z (cid:1) − ζ (cid:16) − ζM − D(cid:63) e z (cid:17) dz = 1 σ dσ (53)where z ≡ α ( σ ) ln σ . (54)Hence, the integration side by side gives the following relation α ( σ ) = 2 ζ ζ − (cid:34) − Bσ α ( σ ) ζ ln σ − A ln( − Bσ α ( σ ) )4 Bζ ln σ (cid:35) (55)where, we have introduced A ≡ − D − D − ζ M − D(cid:63) B ≡ ζM − D(cid:63) . φ , (41), by analyzingthe second condition on ˜ T µν T = ( ˜ ∂ σ α ) F ( σ α ) − ζ ˜ ∂ σ α − − D (cid:16) V ( σ α ) − ζη µν ∂ µ ∂ ν σ α (cid:17) η , η = −
1= 22 − D ˜ V ( σ α ) + 2 ζσ α (cid:16) uα (cid:48) + u (cid:48) α (cid:17)(cid:16) aD − a (cid:17) + 2 ζD − σ α (cid:16) u α (cid:48) + 4 uu (cid:48) αα (cid:48) + 2 u (cid:48) α + uα (cid:48)(cid:48) + 2 u (cid:48) α (cid:48) − u (cid:48) α (cid:17)(cid:16) aσ + p − ab (cid:17) . (56)Eq. (56) yields the self-interaction potential as˜ V ( σ α ) = 2 σ α − ζa u (cid:48) × (cid:40)(cid:16) u (cid:48) uα (cid:48) + u (cid:48) α (cid:17) D + 22 + 2 u α (cid:48) + 4 uu (cid:48) αα (cid:48) + 2 u (cid:48) α + uα (cid:48)(cid:48) + 2 u (cid:48) α (cid:48) − u (cid:48) α (cid:41) + σ α ζ (cid:16) p − ab (cid:17)(cid:40) u α (cid:48) + 4 uu (cid:48) αα (cid:48) + 2 u (cid:48) α + uα (cid:48)(cid:48) + 2 u (cid:48) α (cid:48) − u (cid:48) α (cid:41) (57)We make an abbreviation in (50) α ( σ ) ≡ σ X ( σ ) (58)and rewrite the potential as the following˜ V ( σ α ) = 2 σ α − aζ (cid:104) σX (cid:48) D + 22 + 2 X (cid:48) σ + X (cid:48)(cid:48) σ (cid:105) + σ α − ζ (cid:16) p − ab (cid:17)(cid:16) X (cid:48) σ + X (cid:48)(cid:48) σ (cid:17) . (59)Consequently, the potential takes the form˜ V ( φ ) = 8 φ α − α aζ (cid:16) D + 2 (cid:17)(cid:16) F ( φ ) − ζ (cid:17) (cid:16) ζ − ζ crit( φ ) (cid:17) + 2 φ α − α ζ (cid:16) p − ab (cid:17)(cid:16) F ( φ ) − ζ (cid:17) (cid:16) F ( φ ) + φ α F (cid:48) ( φ ) (cid:17) (60)11here ζ crit( φ ) = 14 F ( φ ) D − φ α F (cid:48) ( φ ) D + 2 , F (cid:48) ( φ ) ≡ ∂F ( φ ) ∂σ (61)is the critical value of ζ for which the theory becomes conformal. The potential (60), whichis felt by the scalar field φ that has the special form as we have shown in (51), must havethis form in order to have a compactification effect in the theory. ζ crit( φ ) takes values inthe range of the limit values of (61) for d → d → ∞ . One realizes that for F ( φ ) = 1these values of ζ crit( φ ) reduces to the ζ crit( φ ) values in [16, 17] which are 1 / d → / d → ∞ .The potential has two different minima, according to the sign of ( ζ − ζ crit( φ )), whichcorrespond to uncompactified and compactified spacetime structures. This fact will beexplained in Sec. 3.2. It is useful here to remember (19). It shows the relation between the scalar field φ and thefunction f ( R ). We will see all the equations in the following sections still in terms of φ but it should be kept in mind that they are also in terms of f ( R ) or f (cid:48) ( R ) because of (19).We discuss the compactification mechanism via the scalar-tensor theory (28). Hereby westate that the f ( R ) theory causes the compactification because the scalar-tensor theory isderived from it.The equations of motion in extra dimensions are˜ R ij = ˜ T ij ( φ ) M D − (cid:63) − ζφ (62) F ( φ )˜ g ij ˜ ∇ i ˜ ∇ j φ = 12 F (cid:48) ( φ )˜ g ij ˜ ∂ i φ ˜ ∂ j φ + 12 F (cid:48) ( φ ) η µν ˜ ∂ µ φ ˜ ∂ ν φ + ζ ˜ Rφ + ˜ V (cid:48) ( φ ) − F ( φ ) η µν ˜ ∂ µ ˜ ∂ ν φ . (63)Here the source term of the Ricci tensor is0 (cid:54) = ˜ T ij = F ( φ ) ˜ ∂ i φ ˜ ∂ j φ − ζ ˜ ∇ i ˜ ∇ j φ − − D (cid:16) V ( φ ) − ζη µν ˜ ∂ µ ˜ ∂ ν φ (cid:17) ˜ g ij . (64)We will use (64) and the equations of motion (62, 63) to find the Ricci scalar. We firstwant to write the explicit form of the term in the paranthesis in (64). So, we find thefollowing double-derivative of φ by using the ansatz (41) as φ = σ α η µν ˜ ∂ µ ˜ ∂ ν σ α = (cid:32) α (cid:48) σ α ln σ + 8 σ α − αα (cid:48) ln σ + 4 α (cid:48) σ α −
12 4 α σ α − − ασ α − + 2 α (cid:48)(cid:48) σ α ln σ (cid:33) η µν ˜ ∂ µ σ ˜ ∂ ν σ + (cid:32) α (cid:48) σ α ln σ + 2 ασ α − (cid:33) η µν ˜ ∂ µ ˜ ∂ ν σ (65)using the definition (58) η µν ˜ ∂ µ ˜ ∂ ν σ α = 2 σ α − (cid:0) σ X (cid:48) + σ X (cid:48)(cid:48) (cid:1) η µν ˜ ∂ µ σ ˜ ∂ ν σ + 2 σ α X (cid:48) η µν ˜ ∂ µ ˜ ∂ ν σ = 2 σ α − (cid:0) σ X (cid:48) + σ X (cid:48)(cid:48) (cid:1) (cid:0) aσ + p − ab (cid:1) + 2 σ α X (cid:48) a (66)where η µν ˜ ∂ µ σ ˜ ∂ ν σ = − ( ax + p ) + (cid:80) i ( ax i + p i ) = 2 aσ + p − ab and η µν ˜ ∂ µ ˜ ∂ ν σ = 4 a .We find the explicit form of the term in the paranthesis in (64) by using (59) and (66)2 ˜ V ( φ ) − ζη µν ˜ ∂ µ ˜ ∂ ν φ = − aζ ( D − F ( φ ) − ζ φ α − α (67)So, we use the relation (37) for the second term in (64) and (67) for the term in theparathesis to find the trace of this source term of Ricci tensor in the following form˜ T = F ( φ ) ∂ i φ∂ i φ + 2 (cid:16) ˜ V ( φ ) − ˜ V ( φ ) (cid:17) − (cid:16) D − (cid:17) aζ φ α − α F ( φ ) − ζ . (68)We want to write the first term of (68) explicitly. For this purpose, we use the followingdifferential rule g ij ∇ i ∇ j φ = 2 g ij ∇ i φ ∇ j φ + 2 g ij φ ∇ i ∇ j φ (69)and write the left-hand side of it from (37). For the second term on the right hand side of(69), we use the second equation of motion (63) by replacing the last term of it with η µν ˜ ∂ µ ˜ ∂ ν φF ( φ ) = F ( φ ) (cid:34) ˜ V (cid:48) ( φ ) ζ + 2 aζ ( D − α − α φ α − α ( F ( φ ) − ζ ) − φ α − α F (cid:48) ( φ )( F ( φ ) − ζ ) (cid:35) . (70)So, the differential rule (69) turns out to − ζ (cid:16) ˜ V ( φ ) − ˜ V ( φ ) (cid:17) = (cid:32) F (cid:48) ( φ ) φF ( φ ) (cid:33) g ij ∂ i φ∂ j φ + F (cid:48) ( φ ) φF ( φ ) η µν ∂ µ φ∂ ν φ + 2 ζ R φ F ( φ ) + 2 φ ˜ V (cid:48) ( φ ) F ( φ ) − V (cid:48) ( φ ) ζ − aζ ( D − α − α φ α − α ( F ( φ ) − ζ ) − φ α − α F (cid:48) ( φ )( F ( φ ) − ζ ) . (71)13e see that the first term on the right hand side of (71) contains what we need for thefirst term in (68). Hence, plugging the term we need from (71) in (68), we obtain the traceof the source of the Ricci tensor in extra dimensions˜ T = F ( φ )2 F ( φ ) + φF (cid:48) ( φ ) (cid:40) − F ( φ ) (cid:16) ˜ V ( φ ) − ˜ V ( φ ) (cid:17) ζ − F (cid:48) ( φ ) φη µν ∂ µ φ∂ ν φ − ζ ˜ R φ − φ ˜ V (cid:48) ( φ ) + F ( φ ) 2 ˜ V (cid:48) ( φ ) ζ + 4 aζ (cid:16) D − (cid:17) F ( φ ) (cid:16) F ( φ ) − ζ (cid:17) (cid:34) α − α φ α − α (cid:16) F ( φ ) − ζ (cid:17) − F (cid:48) ( φ ) φ α − α (cid:35)(cid:41) + 2 (cid:16) ˜ V ( φ ) − ˜ V ( φ ) (cid:17) − (cid:16) D − (cid:17) aζ φ α − α F ( φ ) − ζ . (72)Finally, taking the trace of the first equation of motion (62)˜ T = ˜ R M D − (cid:63) − ζφ (73)and equaling the right hand sides of (72) and (73), we find the Ricci scalar in the extradimensions˜ R = 1 M D − (cid:63) − ζφ + ζφ φ F (cid:48) ( φ ) F ( φ ) (cid:40) F ( φ )2 F ( φ ) + φF (cid:48) ( φ ) (cid:34) − F ( φ ) ζ ( ˜ V ( φ ) − ˜ V ( φ )) − F (cid:48) ( φ ) η µν φ ˜ ∂ µ φ ˜ ∂ ν φ − φ ˜ V (cid:48) ( φ ) + 2 F ( φ ) ζ ˜ V (cid:48) ( φ )+ 4 aζ ( D − F ( φ ) α − α φ α − α ( F ( φ ) − ζ ) − φ α − α F (cid:48) ( φ )( F ( φ ) − ζ ) (cid:35) + 2( ˜ V ( φ ) − ˜ V ( φ )) − D − aζ φ α − α F ( φ ) − ζ (cid:41) (74)which is the expression that gives information about how extra space is curved. In otherwords, it characterizes the curvature of the extra space.Here, we look into the properties of the potential (60) in detail. It is not the true self-interaction potential of the scalar field but the potential that is felt by the generic scalar fieldin four dimensions. The true self interaction potential of φ is ˜ V ( φ ), i.e. (4 + d )-dimensionalpotential. The special form of the potential, ˜ V ( φ ), which satisfies the equations of motion,(32) and (33), takes role in compactification process and keeps the four-dimensional space-time flat. The warped compactified spacetime is energetically chosen structure of spacetime14nstead of M d for specific values of φ that makes the potential the potential ˜ V ( φ ) mini-mum.Figure 1: Left: The minimum of the potential for ζ > ζ crit which corresponds to M d .Right: The minimum of the potential for ζ < ζ crit which corresponds to the warpedcompactified spacetime.The potential (60) have two minima at φ = 0 with ζ > ζ crit and φ (cid:54) = 0 with ζ < ζ crit.For both cases, p − ab > a > ζ values determinesthe structure of the entire spacetime, i.e. the structure spontaneously changes from theuncompactified spacetime structure M d to the warped compactified spacetime structure. We have shown a self-compactification mechanism via higher curvature gravity, f ( R ). First,by a conformal transformation, we have mapped f ( R ) theory into Einstein gravity plus ascalar field theory with a minimal coupling and then by another conformal transformationwe have mapped the resulting theory to Einstein gravity plus a scalar field theory with anonminimal coupling. We have shown that there are non-vanishing scalar field configura-tions that satisfy the conditions on the partially vanishing source term, ˜ T AB , of the Riccitensor.Compactification mechanisms were studied in the nonminimal scalar tensor theories. Inthese works, the Lagrangian of the theory is written by hand, and there is no any function,which depends on the scalar field which is the source of gravity, in front of the kinetic term.In our work, it is interesting and different that we didn’t put our nonminimal scalar tensoraction by hand, such that we derived it from a pure higher curvature gravity f ( R ) viaconformal transformations. We showed that, besides the nonlinear term which shows the15oupling between the curvature scalar and the scalar field, our nonminimal scalar tensortheory includes also a nonminimal kinetic term. Hereby, we say that, if the thing whichcauses compactification is a higher curvature gravity, then the nonminimal scalar tensortheory which is obtained from the higher curvature gravity must have a non-minimal kineticterm with F ( φ ).The scalar field floats in the bulk without coupling to any field, however when thescalar field has a specific configuration, it couples to gravity and this coupling term in theaction plays a role for the compactification. So, the scalar field gravitates only in a subsetof dimensions, i.e. in extra dimensions. This means that extra space has a non-vanishingcurvature scalar which we have already shown its form explicitly in (74). Curved space ofextra dimensions may possess compact form [8, 11, 18, 19] or not [20].The special form of the potential, ˜ V ( φ ), which satisfies the equations of motion, (32)and (33), determines the structure of the entire spacetime, i.e. it plays role for the compact-ification of extra dimensions and keeps the four-dimensional spacetime flat. The warpedcompactified spacetime is the energetically chosen structure of the spacetime instead of M d for specific configurations of the scalar field φ and the corresponding potential ˜ V ( φ ).The solutions of the equations of motion, (62, 63), give information about the topologyand the shape of the extra space, but it is not easy to have an analytic solution sincethe equations of motion depend on functions of extra dimensions b ( (cid:126)y ) and this functiondepends on ˜ g ij as already mentioned in [16].In our compactification mechanism all the results are in terms of the scalar field φ ,however it is important to keep in mind that according to the relation (12), all the resultscan be rewritten in terms of f ( R ). So the whole mechanism is described only in terms of f ( R ). Additionally, differently from other works, we realized that the scalar field whichis the source of gravity is forced to have discrete spectrum via the equation (55), and oneobtains different spectrums for each different values of the parameters D , ζ and M (cid:63) .Ultimately, under all these illuminations, we have shown that f ( R ) theory in (4 + d ) dimensions can self-compactify extra dimensions while the four-dimensional spacetimeremains flat. Manifestly, the result of the paper is that the theory with non-canonicalkinetic term (28) also accommodates self-compactification. The relation to f ( R ) gravityis new, however a product compactification corresponds to a warped compactification. We thank to Prof. Dr. D.A. Demir and Prof. Dr. V.V. Nesterenko for illuminating discus-sions. This work was partially supported by DFG GRK1102.
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