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Space-Time Singularities in Weyl Manifolds

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Space-Time Singularities in Weyl Manifolds Eur. Phys. J. C (2015) 75:448 DOI 10.1140/epjc/s10052-015-3671-7 Regular Article - Theoretical Physics Space-time singularities in Weyl manifolds 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- times less than the Planck time. In fact, there has been a fair larity theorems in general relativity to the case of some scalar- deal of work on classical and quantum new approaches to tensor gravity theories in which the scalar field has a geomet- gravity in which mechanisms that naturally prevent the for- rical character and space-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 mechanics 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 uncertainty principle. 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 symmetries of the distribution of matter 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 light 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 symmetry 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 manifold 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 basis {xa} one can express lates of the Riemannian geometry, 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 differentiable manifold endowed where bc 2 g gdb,c gdc,b gbc,d denotes the with an affine connection ∇ and a metric tensor g. For any Christoffel symbols 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 covariant derivative 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 Hermann Weyl generalized Let us set 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 coordinate system {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 group. 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.
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