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Eur. Phys. J. C (2015) 75:448 DOI 10.1140/epjc/s10052-015-3671-7

Regular Article - Theoretical

Space- singularities in Weyl

I. P. Lobo1,3,a, A. B. Barreto2,b, C. Romero2,c 1 CAPES Foundation, Ministry of Education of Brazil, Brasilia, Brazil 2 Departamento de Física, Universidade Federal da Paraíba, C. Postal 5008, João Pessoa, PB 58051-970, Brazil 3 Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy

Received: 13 August 2015 / Accepted: 8 September 2015 / Published online: 25 September 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We extend one of the Hawking–Penrose singu- less than the Planck time. In fact, there has been a fair larity theorems in to the case of some scalar- deal of work on classical and quantum new approaches to theories in which the scalar field has a geomet- gravity in which mechanisms that naturally prevent the for- rical character and -time has the mathematical structure mation of a cosmological singularity are present. In fact, from of a Weyl integrable space-time. We adopt an invariant for- the standpoint of quantum the mere existence of malism, so that the extended version of the theorem does not a cosmological singularity would be in contradiction with depend on a particular frame. Heisenberg’s . In connection with this point, many efforts have been made to understand what could be called the quantum structure of space-time. Among the 1 Introduction numerous attempts to make progress in this rather difficult issue, we would like to call attention for a recent proposal Until the mid-1960s it was argued by some cosmologists which argues that the quantum structure of space-time may that the presence of space-time singularities in general rel- even be related to the possibility of the space-time geome- ativistic cosmological models was not an essential prop- try possessing a non-Riemannian character [6]. On the other erty of the model, being, in fact, a consequence of the high hand, non-Riemannian geometries have appeared in physics degree of of the distribution of assumed mainly as a way to modify Einstein’s gravity while keep- in these models [1]. Accordingly, it was believed that in ing the idea that the gravitational field is a manifestation of more realistic situations these singularities would disappear. the space-time geometry [7,8]. As is well known, one of the However, this scenario was to change drastically after a earlier attempts in this direction was the Weyl unified field series of general mathematical results, namely, the so-called theory [9–16]. Hawking–Penrose singularity theorems, were proved [2–5]. Concerning space-time singularities, we know that modi- As is widely known, these theorems use methods of global fied theories of gravity call for a new mathematical treatment analysis to show that, under the assumption of the validity of of the problem, as the approach provided by Hawking and general relativity and a reasonable physical behavior of mat- Penrose is suitable only for general relativity. It is our pur- ter, space-time singularities are general phenomena which pose here to pursue this question further in the of some occur in gravitational collapse and cosmology (such as the modified gravity theories formulated in a particular kind of big bang) irrespective of the of the models. space-time geometry, namely, the so-called Weyl integrable The investigation of space-time singularities has touched space-time (WIST) [17–29]. on some deep philosophical issues and it is the view taken by The present article is organized as follows. In Sect. 2,we many scientists that these issues seem to call for a concep- outline the fundamental ideas of the geometry developed by tual revision of general relativity that at least take quantum Weyl, which underlies his unified field theory. We then pro- mechanics into account. As far as cosmological singularities ceed to briefly review the main features of Weyl’s attempt are concerned it seems that the current view held by most to unify gravity and electrodynamics in a single geometric cosmologists is that general relativity must break down at framework. In Sect. 3, we consider the extension of Ray- chaudhuri equation from a Riemannian setting to the case a e-mail: [email protected] of a Weyl integrable space-time. This extension is invariant b e-mail: adrianobraga@fisica.ufpb.br under Weyl transformations. In Sect. 4, we work out a gen- c e-mail: cromero@fisica.ufpb.br 123 448 Page 2 of 9 Eur. Phys. J. C (2015) 75 :448 eralization of one of the Hawking–Penrose singularity theo- vanishes, we reobtain (1). In this way, we have a generalized rems. Some applications of the formalism are given in Sect. 5. version of the Levi-Civita theorem given by the following These correspond to cosmological solutions of scalar-tensor proposition (see Ref. [31] for a proof): theories framed in a Weylian space-time. We conclude with some remarks in Sect. 6. Theorem 2.1 (Levi-Civita extended) Let M be a differen- tiable endowed with a metric g and a differentiable one-form field σ defined on M, then there exists one and only one affine connection ∇ such that: (i) ∇ is torsionless; (ii) 2 Weyl geometry ∇ is W-compatible with g.

Weyl geometry arises from the weakening of one of the postu- It follows that in a coordinate {xa} one can express lates of the , similarly as non-Euclidean the components of the affine connection completely in terms geometry was conveived after the fifth postulate of Euclidean of the components of g and σ: geometry was relaxed. The postulate we are referring to is the so-called the Riemannian compatibility condition, which a a 1 ad  ={ }− g [gdbσc + gdcσb − gbcσd ] (4) states the following. bc bc 2

{a }=1 ad[ + − ] Postulate 2.1 Let M be a endowed where bc 2 g gdb,c gdc,b gbc,d denotes the with an affine connection ∇ and a g. For any of second kind. The next proposition vector fields U, V, W ∈ T (M), it is required that gives a helpful insight on the geometrical meaning of the Weyl parallel transport. V [g(U, W)]=g(∇V U, W) + g(U, ∇V W). (1) Corollary 2.1 Let M be a differentiable manifold with an ∇ As is well known, this condition is equivalent to the affine connection , a metric g and a Weyl field of one-forms σ ∇ α = α(λ) requirement that the of the metric vanish, .If is W-compatible, then for any smooth curve α, thereby implying that the length of a vector remain unaltered and any pair of two parallel vector fields V and U along by parallel transport. we have   On the other hand, in Riemannian geometry, it will also be d d assumed that the connection ∇ be torsionless (or symmetric), g(V, U) = σ g(V, U) (5) dλ dλ i.e., that for any U, V ∈ T (M) the following condition holds: d α where dλ denotes the vector tangent to . ∇V U −∇U V =[V, U]. (2) By integrating the above equation along the curve α from From these two postulates we are led to the important Levi- a point p0 = α(λ0), we get Civita theorem, which states that the affine connection is  λ σ( d ) ρ λ dρ d entirely determined from the metric [30]. In a certain sense, g(V (λ), U(λ)) = g(V (λ0), U(λ0))e 0 . (6) this theorem characterizes a Riemannian manifold. In 1918 the mathematician generalized Let us U = V and denote by L(λ) the length of the vector the geometry of Riemann by introducing the possibility of V (λ) at an arbitrary point p = α(λ) of the curve. It is easy to change in the length of a vector through parallel transport. check that in a local {xa} Eq. (5) reduces To implement this idea Weyl conceived the following new to compatibility condition. dL σ dxa = a L. (7) Postulate 2.2 (Weyl) Let M be a differentiable manifold dλ 2 dλ endowed with an affine connection ∇, a metric tensor g and a α :[ , ]∈ → one-form field σ, called a Weyl field. It is said that ∇ is com- Consider now the set of all closed curves a b R α( ) = α( ). patible (W-compatible) with g if for any vector fields U, V, M, i.e., with a b The equation W ∈ T (M), we have  b σ( d )dλ g(V (b), U(b)) = g(V (a), U(a))e a dλ (8) V [g(U, W)]=g(∇ U, W)+g(U, ∇ W)+σ(V )g(U, W). V V defines a holonomy . If we want the elements of this (3) group to correspond to an isometry, then we must require that    Clearly, this represents a generalization of the Rieman- d σ dλ = 0(9) nian compatibility condition. Naturally, if the one-form σ dλ 123 Eur. Phys. J. C (2015) 75 :448 Page 3 of 9 448 for any loop. From Stokes’ theorem it follows that σ must be as a frame (M, g,φ)that is only defined “up to a Weyl trans- an exact form, that is, there exists a scalar function φ, such formation”. In this way, when dealing with a certain Weyl that σ = dφ. In this case we have a Weyl integrable manifold. manifold we choose a particular frame in the equivalence Weyl manifolds are completely characterized by the set class and consider that only geometric entities defined in (M, g,σ), which will be called a Weyl frame. Let us remark that frame which are invariant are of interest, since they can that the Weyl compatibility condition (5) remains unchanged be regarded as representative of the whole class. From this when we go to another Weyl frame (M, g, σ)by carrying out point of view, it is more natural to redefine some Riemannian the following simultaneous transformations in g and σ: concepts to meet the requirements of invariance. This proce- dure is analogous to the one adopted in conformal geometry, ¯ = f , g e g (10) a branch of geometry, in which the geometric objects of inter- σ¯ = σ + df, (11) est are those invariant under conformal transformation, such as, say, the angle between two directions [32]. Accordingly, where f is an arbitrary scalar function defined on M.The one should naturally generalize the definition of all invari- conformal map ( 10) and the gauge transformation (11) define ant integrals when dealing with the integration of exterior classes of equivalences in the set of Weyl frames. It is impor- forms. For example, the√ Riemannian q-dimensional volume tant to mention that the discovery that the compatibility con- form defined as = −gdx1 ∧···∧dxq is not invariant dition (5) is invariant under this group of transformations was under√ Weyl transformations, hence it should be replaced by − q φ q essential to Weyl’s attempt at unifying gravity and electro- = −γ e 2 dx1 ∧···∧dx , and so on.1 Likewise, in a magnetism, extending the concept of space-time to that of Weyl integrable manifold it is more natural to define the con- a collection of manifolds equipped with a conformal struc- cept of “length of a curve” in an invariant way. It follows that ture, leading to the notion that space-time might be viewed our familiar notion of as the arc length of world- [ ] as a class g of conformally equivalent Lorentzian metrics lines in four-dimensional Lorentzian space-time should be [9–13,15,16]. modified. Because of this, we shall redefine the proper time τ measured by a clock moving along a parametrized time- like curve xμ = xμ(λ) between xμ(a) and xμ(b), in such 3 Weyl integrable space-time a way, that τ is the same in all frames. This suggests the following definition: We now consider the particular case of a WIST, where σ =   μ ν  1 dφ. As already mentioned, the set (M, g,φ)consisting of a b 2 τ = dx dx λ differentiable manifold M endowed with a metric g and a gμν d a dλ dλ φ Weyl frame    1 Weyl scalar field will be referred to as a .Inthis b μ ν − φ dx dx 2 case (11) becomes = e 2 gμν dλ. (13) a dλ dλ φ = φ + . f (12) It must be noted that the above expression can also be obtained from the special relativistic definition of proper time −φ If we set f =−φ in the above equation, we get φ = 0. We if we adopt the prescription ημν → e gμν. It is clear that refer to the set (M, g = e− f g, φ = 0) as the Riemann frame, the right-hand side of this equation is invariant under Weyl because in this frame the manifold becomes Riemannian. On transformations and that, in the Riemann frame, it reduces to the other hand, it can easily be verified that Eq. (4) follows the definition of proper time in general relativity. We, there- directly from ∇α gμν = 0. This result has interesting and use- fore, take τ, as given above, as the extension to an arbitrary ful consequences. In fact, the metric γ = e−φ g, defined for Weyl frame of general relativistic clock hypothesis, i.e., the any frame (M, g,φ), is invariant under the Weyl transforma- assumption that τ measures the proper time measured by tions (10) and (11) any geometric quantity built exclusively a clock attached to the particle. with γ is invariant. More generally, geometric objects such It is now easy to see that the extremization of the functional α as the components of the curvature tensor Rβμν, the compo- (13) leads to the equations φ nents of the Ricci tensor Rμν, the scalar e R are invariant   2 μ   d x μ 1 μν under the Weyl transformations (10) and (11). + − g (gανφ,β + gβνφ,α − gαβ φ ,ν) dλ2 αβ 2 It is important to note that because the Weyl transforma- dxα dxβ tions (10) and (11) define an equivalence relation between × = , λ λ 0 (14) frames (M, g,φ)it seems more appropriate to look into the d d equivalence class of such frames rather than on a particular √ frame. In other words, a Weyl manifold should be regarded 1 Note that g in the expression −g denotes the determinant of gμν . 123 448 Page 4 of 9 Eur. Phys. J. C (2015) 75 :448   μ τ where αβ designates the Christoffel symbols calculated invariant proper time defined in (13). Hence, the tangents to with gμν. Recall that in the derivation of the above equations the congruence generate a tangent vector field V normalized the parameter λ must be chosen such that to unit length. In order to keep the formalism invariant under Weyl transformations we shall choose the affine parameter of α β −φ dx dx the congruence as the Weyl invariant arc length, i.e., we nor- e gαβ = K = const (15) γ = −φ dλ dλ malize V with respect to the invariant metric μν e gμν, along the curve, which, up to an affine transformation, per- γ(V (τ), V (τ)) = 1. (17) mits us to identify λ with the proper time τ. Surely, these equations are exactly those that yield the affine in Therefore, in a local coordinate system, a curve μ(τ) μ∇ α = α = dxα a Weyl integrable space-time, since they can be rewritten as described by x satisfies V μV 0, where V dτ and we have the affine geodesics in a Weyl integrable space- 2 μ α β d x μ dx dx time shown in (16). Furthermore, once we have normalized + αβ = 0, (16) dτ 2 dτ dτ the tangent vector field with the invariant metric γμν,we   obtain μ = μ − 1 μν( φ + φ − φ ) where αβ αβ 2 g gαν ,β gβν ,α gαβ ,ν , μ ν according to (4), are identified with the components of the Vμ∇α V = 0 = V ∇α Vν, (18) Weyl connection. Thus the extension of the geodesic postu- late by requiring that the functional (13)beanextremumis since ∇αγμν = 0. equivalent to assuming that the particle must follow Let us now consider, at some point p of M, the hypersur- μ affine geodesics defined by the Weyl connection αβ . Let us face  orthogonal to the vector field V . As in the standard note that, as a consequence of the Weyl compatibility condi- procedure, we define the operator of projection  onto the tion (3) between the connection and the metric, Eq. (15) holds hypersurface  as automatically along any affine geodesic determined by (16). μ Since both the connection components αβ and the proper μν = γμν − VμVν. (19) time τ are invariant when we switch from one Weyl frame to the other, Eq. (16) are invariant under Weyl transformations. As is well known, μν represents to the first fundamental As is well known, the geodesic postulate is not only con- form of the hypersurface induced by the metric γ and its role cerned with the motion of particles, but it also determines is to project any vector of Tp M at p onto Tp, the tangent the propagation of light rays in space-time. On the other space to the submanifold .2 hand, since the paths of light rays are null curves, one cannot We proceed with the derivation of the Raychaudhuri equa- use the proper time as a parameter to describe these curves. tion in this new setting. We first need to consider a smooth Thus light rays are supposed to follow null affine geodesics, one-parameter subfamily αs(τ) of geodesics in the congru- which cannot be defined in terms of the functional (13), but, ence of V and then define a deviation vector η that represents instead, they must be characterized by their behavior with an infinitesimal spatial displacement from a given geodesic respect to parallel transport. We naturally extend this pos- αo(τ) to a neighboring geodesic in this subfamily. Once we tulate by simply assuming that light rays follow Weyl null have been given η in , we define the vector field η along this affine geodesics. subfamily by Lie dragging it along V , that is, by requiring that

4 The ŁV (η) = 0. (20) The Raychaudhuri equation played a fundamental role in the From the definition of and the fact that the derivation of the Hawking–Penrose singularity theorems. In connection ∇ isassumedtobetorsionlessweareledtothe the derivation, however, it is assumed right from the begin- following equation: ning that the geometry underlying the space-time is Rieman- nian. In this section, we investigate the extension of this equa- μ α μ α V ∇μη = η ∇μV , (21) tion to the case of a Weyl integrable space-time. Let us first remark that the extension of the geodesic pos- tulate to WIST assumes that the particle motion must follow 2 Note that we are using the invariant metric γ in order to ensure a invariant projection tensor. Accordingly, the operation of raising and Weyl time-like geodesics. In the following we shall consider γ  lowering tensorial indices must be always carried out with . This guar- a smooth congruence of time-like geodesics corresponding antees that the duality between covariant and contravariant vectors is to the worldlines of a class of observers, parametrized by the not modified by Weyl transformations. 123 Eur. Phys. J. C (2015) 75 :448 Page 5 of 9 448

μν μν which shows how the deviation vector changes along the where we denote σ 2 = σμνσ , ω2 = ωμνω , and the term μ ν congruence. An important role played in the investigation of Rμν V V is usually referred to as the Raychaudhuri scalar. ˙ μ the behavior of neighboring geodesics as we go along the Setting  = V ∇μ, we can write (30) in the following congruence  is played by the so-called deformation tensor form, known as the Raychaudhuri equation: Qμν, defined as ˙ 1 2 2 2 μ ν  +  + 2(σ − ω ) = Rμν V V . (31) Qμν =∇ν Vμ. (22) 3 It is worth noting that (31) has the same form as in the case of In terms of Qμν we can rewrite (21)as a Riemannian space-time, although it must be recalled that the Ricci tensor is built with the Weylian connection. In fact, μ∇ ηα ≡ α ημ, V μ Q μ (23) this is not surprising as we have redefined the proper time in −φ an invariant way, using the invariant metric γμν = e gμν. which clearly means that the deformation tensor measures the In the next section, we shall analyze the conditions that lead μ failure of η to be parallel transported [33]. Furthermore, it is to singularities in the space-time. μ easy to see that this tensor is purely spatial, since Qμν V = ν 0 = Qμν V . Thus, Qμν is a tensor defined in the subspace of the tangent space perpendicular to V . Finally, in order to 5 Extending the singularity theorem have a physical interpretation of some kinematical aspects of the congruence , we can decompose Qμν in its irreducible Because of the form of the Raychaudhuri equation takes in parts: a Weyl integrable space-time, the description of conjugate points is the same as in Riemannian geometry. Thus, we 1 θ = θ(τ )< Qμν ≡ μν + σμν + ωμν, (24) have the following statement: if 0 0 0forsome 3 μ ν τ = τ0, Rμν V V ≤ 0, then in a finite invariant proper time τ ≤ 3/|θ |, the congruence will develop a conjugate point where the parameters , σμν, and ωμν are known, respec- 0 θ →−∞ tively, as the expansion, shear and vorticity of the congruence . As a matter of fact, the extension of any the- (for instance, Ref. [33]). From the above equation we have orem from Riemannian geometry to a Weyl integrable can be trivially carried out by simpling considering the invariant μν −φ  =  Qμν, (25) metric γ = e g. For instance, as we have already men- 1 tioned the extremization of the functional (13) leads directly σμν = Q(μν) − μν, (26) 3 to the Weylian geodesics. On the other hand, results coming ωμν = Q[μν]. (27) from differential topology and concerning the causal struc- ture of space-time that are valid in a Riemannian space-time It is easy to see that in the case where the congruence is are also valid in a Weyl integrable space-time since conformal locally orthogonal to the hypersurface ,wehaveωμν = 0. transformations do not affect the light-cone structure nor the We are now particularly interested in the behavior of , manifold orientability. Indeed, the proof of one of the most which measures the expansion of the congruence and can tell important results we are going to enunciate now, namely, the us about the existence of conjugated points. Thus, in order generalized Jacobi theorem, proceeds along the same lines to obtain the Raychaudhuri equation, we need to know the of reasoning employed in the Riemannian case, where we −φ rate of change of  along the congruence. Therefore, since, merely replace g by the invariant metric γ = e g [33].  =∇ α μ∇  = by definition, α V , we must compute V μ Theorem 5.1 (Jacobi) Let γ :[0, 1]→M a differential μ∇ [∇ α]. V μ α V On the other hand, from the definition of the time-like curve connecting points p, q ∈ M. Then a nec- curvature tensor, we have essary and sufficient condition for γ to locally maximize the invariant arc length between p and q is that γ be a Weyl ∇ ∇ μ −∇ ∇ μ = μ λ β ν V ν β V R λνβ V (28) geodesic without conjugate points between p and q. Note the relation between the Raychaudhuri scalar  where and the extrinsic curvature of the submanifold orthogonal μ μ μ ρ μ ρ μ to geodesic congruence, which is represented by the mixed R λνβ = ∂β λν − ∂νλβ + λνρβ − λβ ρν, (29) μ μ β α tensor Q ν =  α ν∇β V . For the case of a congruence of geodesics orthogonal to hypersurface  and parameterized one can readily obtain μ μ by the invariant arc length we have Q ν =∇ν V . Thus, the μ α trace of Q ν is equal to , i.e., Q = Q α = .Nowweare μ 1 2 2 2 μ ν V ∇μ =−  − 2(σ − ω ) + Rμν V V , (30) 3 ready to prove the following proposition. 123 448 Page 6 of 9 Eur. Phys. J. C (2015) 75 :448

Theorem 5.2 Let (M, g,φ) be a globally hyperbolic where R denotes the Weylian scalar curvature, φ represents μ ν Weylian space-time with Rμν V V ≤ 0 for all time-like the Weyl scalar field, ξ is a dimensionless parameter, while vectors V . Suppose there exists a space-like Cauchy sur- Lm stands for the Lagrangian of the matter fields. The form of face  such that the trace of its extrinsic curvature (for the Lm is determined from the corresponding Lagrangian in spe- orthonormal congruence of past directed geodesics) satisfies cial relativity by replacing the ordinary derivatives by covari- B ≤ C < 0 over the whole surface, B and C being constants. ant derivatives with respect to the Weyl connection. The field Then no past directed time-like curve coming from  can equations are given by /| | have an invariant arc length greater than 3 C . In particular, 1 all past directed time-like geodesics are incomplete. Rμν − gμν =−∇μφ,ν 2 ,α −φ Proof Let us prove this theorem by contradiction. Suppose +(2ξ − 1)φ,μφ,ν − ξφ,αφ gμν − e Tμν, (35) λ = λ(τ)  there exists a time-like curve coming from ,α ,α 1 −φ τ ∇αφ + 2φ,αφ =− e T, (36) whose value of the invariant arc length at some point p 2λ ∈ λ is greater than 3/|C|. It is well known that as the set where we have set λ = 3 − 2ξ. of curves joining two points in a globally hyperbolic mani- 2 fold is compact, the invariant arc length function must have a maximum value for a given curve. Then this curve must be a From the above equations it is not difficult to verify that   γ 2 2 Weyl geodesic. Therefore, there must be a geodesic (with μ ν d φ dφ /| |  Rμν V V =− + (2ξ − 1) invariant arc length greater than 3 C ) joining p to .This ds¯2 ds¯ means there are no conjugated points between p and .But,   −φ μ ν 1 2 1 2 from Raychaudhuri’s inequality, we know that γ must have −e Tμν V V − |V | T + |V | T . 2 4λ conjugated points between p and , which is a contradiction. We then conclude that the original curve λ cannot exist. (37) Clearly, in this theory (32) does not imply the violation of the In the case of general relativity where the space-time math- strong energy condition (33), but rather we have the following ematical structure is that of Riemannian geometry, which is situation. Consider the conditions below: a special case of Weyl geometry when φ is a constant of d2φ motion, the geometric condition ≡ φ¨ ≥ 0, (38) ds¯2 μ ν ξ − ≤ , Rμν V V ≤ 0(32)2 1 0 (39) μ ν 1 2 1 2 Tμν V V − |V | T + |V | T ≥ 0. (40) is equivalent to the so-called strong energy condition 2 4λ If any one of these conditions is violated, then the solution μν Tμν V − T/2 ≥ 0, (33) may correspond to a non-singular space-time. Consider, as an example, the vacuum solution of the field equations (35) and requiring that Q < 0 is equivalent to assuming that in and (36) obtained in Ref. [34]. By assuming homogeneity the course of the cosmic evolution the Universe underwent and isotropy we can write an expansion period, which seems to be a rather reasonable   dr 2 assumption. In view of the above, this leads to the conclusion ds2 = dt2 − a2(t) + r 2d 2 , (41) that the Universe, as modeled by GR, must necessarily have 1 − κr 2 0 . ˙ 0 had a beginning starting from a singular state. Let us now φ,α = φ,0δα = φδα. (42) consider some different scenarios offered by two alternative This leads to the equations gravitational theories, namely, the Novello–Oliveira–Salim– Elbaz’s theory (N) [17,34] and a recent geometrical approach (4ξ − 3) a˙2 + κ + (φ˙a)2 = 0, (43) to scalar-tensor theory (GST), both inspired by the idea that 6 space-time can be described by Weyl integrable geometry 4ξ − 3 2aa¨ +˙a2 + κ − (φ˙a)2 = 0, (44) [35]. 4 3 ˙ (a φ),0 = 0. (45) 5.1 Novello’s theory By integrating the last of these equations we get φ˙ = ζa−3, ζ ξ − > Novello’s theory starts with the action where is a constant. Now, if 4 3 0, it follows that  √ 4 4 ,μ −2φ 2 a0 S = d x −g{R − 2ξφ φ,μ + e L }, (34) a˙ (t) = 1 − , (46) m a(t) 123 Eur. Phys. J. C (2015) 75 :448 Page 7 of 9 448

( )4 = ( 4ξ−3 )ζ 2 ( ) ≥ with a0 12 , thus implying that a t a0.The By taking into account (52) one easily obtains model describes a non-singular bouncing universe, a scenario   μ ν ∗ μ ν 1 2 in which after undergoing a period of contraction, dominated Rμν V V ≡−κ Tμν V V − |V | T by the scalar field, the universe reaches a minimum value, 2 φ and then starts to expand, at an inflationary rate, until the − ω(φ)φ˙2 +|V |2e V(φ). (54) radiation dominates over the scalar field and the scale factor begins to follow the standard cosmic evolution [34]. Clearly, As we can see, here again the (32) does not require the viola- the non-singular behavior of the model is a consequence of tion of the strong energy condition (33). Thus, if we assume the fact that the condition ξ ≤ 1/2 is violated. In order to that (33) holds, then any space-time described by these equa- μ ν display the bouncing character of this model, it is convenient tions will satisfy Rμν V V ≤ 0, as long as to solve (46) in terms of the conformal time η, defined as φ dη = dt/a(t). It can be shown that the scale factor and the ω(φ)φ˙2 − e |V |2V(φ) ≥ 0. (55) scalar field are given, respectively, by a(η) = a0 cosh 2(η − η0), (47) We conclude that in this case the singularity theorem applies. ζ On the other hand, if (55) is violated, then the solution may φ(η) = (η − η ) −1 . arccos [cosh 2 0 ] (48) correspond to a non-singular space-time. 2a2 0 In the following let us consider some solutions to (52, 53), for some choices of the potential V(φ) in the case ω(φ) = 5.2 Geometrical scalar-tensor theory constant and Tμν = 0. These solutions correspond to homo- geneous and isotropic models in GST theory and are obtained The geometrical approach to scalar-tensor theory starts with in the Riemann frame (M, g = e− f g, φ = 0), in which case the Brans–Dicke action |V |2 = eφ. If we take the line element written as in (42), then  √ the field equations (52) reduce to 4 −φ ,α ∗ −φ −φ S = d x −ge [R + ωφ φ,α + k e Lm(e g)], ˙2  ω 2φ a + = φ˙2 + e V(φ), (49) 3 2 3 2 (56) a  a 2 2 a¨ a˙ 2  ω e2φ where φ is assumed apriorito be a geometrical field, i.e., 2 + + =− φ˙2 + V(φ), (57) a a a2 2 2 an intrinsic part of the space-time geometry, ω is a dimen- k∗ = 8π sionless parameter and c4 . By applying the Palatini while (53)gives variational method, one obtains the Weyl integral compati-   bility condition [35], a˙ e2φ 1 dV φ¨ + 3 φ˙ =− V(φ) + , (58) a ω 2 dφ ∇α gμν = gμνφ,α. (50) where  = 0, ±1, according to the curvature of the spatial Naturally, the action (49) can easily be extended to accom- section.3 modate a scalar potential V(φ) and to allow for a functional Solutions to Eqs. (56)–(58) with flat spatial section  = 0 dependence of ω on φ, thus leading to [36]  were obtained for the following choices of the scalar poten- √ −2φ −(2+λ)φV −2φ( φ2 + ) λ −2φ(φ2 + 4 −φ μν tial: e , e o, e m , and 2 e S = d x −g{e [R + ω(φ)g φ,μφ,ν] 2 2ω/3) , where , Vo, λ, and m are constants. These, in the ∗ −2φ Riemann frame, correspond to the cosmological constant, −V(φ) + κ e Lm}. (51) the dilaton field, the massive scalar field, and a field with The field equations obtained from (49)aregivenby a quartic interaction. These three cases are displayed in the   tables below, where the presence of a singular behavior is 1 1 ,α Rμν − gμν R = ω(φ) gμνφ,αφ − φ,μφ,ν determined according to whether or not (55) holds (Table1). 2 2 1 φ ∗ − e gμνV(φ) − κ Tμν, (52) 3 2 It is interesting to note here that from the above we can obtain the   φ   following equation: 1 dω ,α e 1 dV φ =− 1 + φ,αφ − + V , (53) 2ω dφ ω 2 dφ ω  H˙ =− φ˙2 + . (59) 2 a2 where  denotes the d’Alembertian operator calculated with the Weyl connection. This equation might be useful to set the possible values of ω. 123 448 Page 8 of 9 Eur. Phys. J. C (2015) 75 :448

Table 1 The other parameters Potential Solution to φ(t) Where {Vo,λ,m,,andφo} are constants  √  −2φ 2 6 e φ(t) = φo ± |ω| arctan sinh (t − t0) ω<0and>0  3  2 −(λ+ )φ H λ2 λ φ ωV 2 V φ( ) = 2 o + 2 o =± o e o t λ ln ω t e Ho ω−λ2 2 6 −2φ( φ2 + ) φ( ) = φ − 2α α =± − ω e m t o ω t 4   −2φ λ(φ2 − / ω)2 φ( ) = φ − 4A =± λ e 2 2 3 t o exp ω t A 3

Table 2 The parameters are the Potential Solution to a(t) Singularity in finite time same as in Table 1 and ao is also  a constant √ 1/3 −2φ ( ) = 6 ( − ) e a t a0 cosh 2 t t0 None  2 2 λ 2ω/λ −(λ+2)φ λ Ho − φo 2 e Vo a(t) = ao ω e 2 t + 1 Singular if ω>λ /6 2    α e−2φ(mφ2 + ) a(t) = a exp αt φ − t Non-singular if ω<0 o o ω   2 ωφ2   −2φ λ φ2 − 2 ( ) = − o − 8A − + 2 ω> e 2 3ω a t ao exp 8 exp ω t 1 3ω At Singular if 0

6 Final remarks Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm The Hawking–Penrose singularity theorems are a direct con- ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit sequence of Einstein’s theory of gravity. Given the important to the original author(s) and the source, provide a link to the Creative role they have played in our understanding of the Universe, as Commons license, and indicate if changes were made. modeled by general relativity, it is also of interest to find the Funded by SCOAP3. analogous of these theorems in alternative gravity theories. Singularity theorems and energy conditions have been stud- ied in connection with Brans–Dicke theory, perhaps the most References popular scalar-tensor gravity theories. However, their mean- ing remains still controversial due to the question of whether 1. E.M. Lifshitz, V.V. Sudakov, I.M. Khalatinikov, Sov. Phys. JETP 13 or not the Einstein frame and the Jordan frame are physically , 1298 (1961) 2. R. Penrose, Phys. Rev. Lett. 14, 57–59 (1965) equivalent [37,38]. In the present geometrical approach, this 3. S.W. Hawking, R. Penrose, Proc. R. Soc. Lond. A 314, 529 (1970) controversy does not arise, as the physical entities defined 4. S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space- in the theory are naturally invariant under frame transforma- Time (Cambridge University Press, Cambridge, 1973) 32 tions [39–42] (Table 2). 5. J.M.M. Senovilla, D. Garfinkle, Class. , 124008 (2015) Finally, it has been shown that Weyl integral space-time 6. M. Novello, J.M. Salim, F.T. Falciano, Int. J. Geom. Methods Mod. seems to be the natural geometrical scenario that arises in Phys. 8, 87–98 (2011) the context of scalar-tensor theories, at least, whenever one 7. H.F.M. Gönner, Living Rev. Relativ. 7, 2 (2004) 17 has a theory that assumes a non-minimal coupling of the 8. H.F.M. Gönner, Living Rev. Relativ. , 5 (2014) 9. H. Weyl, Sitzungesber Deutsch. Akad. Wiss. Berl. 465 (1918) scalar field to the [35]. The basic principle 10. H. Weyl, Space, Time, Matter (Dover, New York, 1952) which underlies the determination of the space-time geom- 11. A nice account of Weyl’s ideas as well as the refutation of his etry would be, in this case, the adoption of the Palatini gravitational theory may be found in W. Pauli, variational principle, which has played an important role (Dover, New York, 1981) 12. L. O’Raiefeartaigh, N. Straumann, Rev. Mod. Phys. 72, 1 (2000) in recently proposed modified theories of gravity [43]. The 13. N. Rosen, Found. Phys.12 , 213 (1982) investigation of the behavior of space-time singularities in 14. For a more formal mathematical treatment, see G.B. Folland, J. this new framework was one of our main motivations to write Differ. Geom. 4, 145 (1970) the present paper. 15. For a comprehensive review on Weyl geometry see E. Scholz. arXiv:1111.3220 [math.HO] 16. E. Scholz. arXiv:1206.1559 Acknowledgments The authors thanks CNPq/CAPES for financial 17. M. Novello, H. Heintzmann, Phys. Lett. A 98, 10 (1983) support. I. P.Lobo is supported by the CAPES-ICRANet Program (BEX 18. For gravitational theories in general formulated in WIST and 14632/13-6). related topics, see K.A. Bronnikov, Yu.M. Konstantinov, V.N.Mel- nikov, Graviy. Cosmol. 1, 60 (1995) 123 Eur. Phys. J. C (2015) 75 :448 Page 9 of 9 448

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