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Abstract. We define the notion of a finite-time singularity of a vector field and then discuss a technique suitable for the asymptotic analysis of vector fields and their integral curves in the neighborhood of such a singularity. Having in mind the application of this method to cosmology, we also provide an analysis of the time singularities of an isotropic universe filled with a perfect fluid in general relativity. IntroductionThere are two approaches to characterizing spacetime singularities in a cosmological context. The first approach may be called geometric and consists of finding sufficient and/or necessary conditions for singularity formation, or absence, independently of any specific solution of the field equations under general conditions on the matter fields. Methods of this sort include those based on an analysis of geodesic congruences in spacetime and lead to the well known singularity theorems, cf.[1], as well as those which are depend on an analysis of the geodesic equations themselves and lead to completeness theorems such as those expounded in [2], and the classification of singularities in [3].The second approach to the singularity problem can be termed dynamical and refers to characterizing cosmological singularities in a geometric theory of gravity by analysing the dynamical field equations of the theory It uses methods from the theory of dynamical systems and can be global, referring to the asymptotic behaviour of the system of field equations for large times, or local, giving the behaviour of the field components in a small neighborhood of the finite-time singularity.In this latter spirit, we present here a local method for the characterization of the asymptotic properties of solutions to the field equations of a given theory of gravity in the neighborhood of the spacetime singularity 1 . We are interested in providing an asymptotic form for the solution near singularities of the gravitational field and understanding all possible dominant features of the field as we approach the singularity. We call this approach the method of asymptotic splittings.

We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric.We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature K are integrable. This last condition is required only for the tracefree part of K if the universe is expanding.

The singularity structure and the corresponding asymptotic behavior of a 3-brane coupled to a scalar field or to a perfect fluid in a five-dimensional bulk is analyzed in full generality using the method of asymptotic splittings. In the case of the scalar field, it is shown that the collapse singularity at a finite distance from the brane can be avoided only at the expense of making the brane world-volume positively or negatively curved. In the case where the bulk field content is parametrized by an analogue of perfect fluid with an arbitrary equation of state P = γρ between the 'pressure' P and the 'density' ρ, our results depend crucially on the constant fluid parameter γ: (i) For γ > −1/2, the flat brane solution suffers from a collapse singularity at finite distance, that disappears in the curved case. (ii) For γ < −1, the singularity cannot be avoided and it becomes of the big rip type for a flat brane.(iii) For −1 < γ ≤ −1/2, the surprising result is found that while the curved brane solution is singular, the flat brane is not, opening the possibility for a revival of the self-tuning proposal.

We show that a general solution of the Einstein equations that describes approach to an inhomogeneous and anisotropic sudden spacetime singularity does not experience geodesic incompleteness. This generalises the result established for isotropic and homogeneous universes. Further discussion of the weakness of the singularity is also included. PACS number: 98.80.-k 1 Introduction There has been strong interest in the structure and ubiquity of finite-time singularities in general-relativistic cosmological models since they were first introduced by Barrow et al [1], as a counter-example to the belief [2] that closed Friedmann universes obeying the strong energy condition must collapse to a future singularity. They were characterised in detail as sudden singularities in refs. [3, 4, 5] and are 'weak' singularities in the senses defined by Tipler [6] and Krolak [7]. A sudden future singularity at t s is defined informally in terms of the metric expansion scale factor, a(t) with t s > 0, by 0 < a(t s ) < ∞, 0 <ȧ(t s ) < ∞,ä(t → t s ) → −∞. These archetypal examples have finite values of the metric scale