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The junction conditions for the most general gravitational theory with a Lagrangian containing terms quadratic in the curvature are derived. We include the cases with a possible concentration of matter on the joining hypersurface -termed as thin shells, domain walls or braneworlds in the literature-as well as the proper matching conditions where only finite jumps of the energy-momentum tensor are allowed. In the latter case we prove that the matching conditions are more demanding than in General Relativity. In the former case, we show that generically the shells/domain walls are of a new kind because they possess, in addition to the standard energy-momentum tensor, a double layer energy-momentum contribution which actually induces an external energy flux vector and an external scalar pressure/tension on the shell. We prove that all these contributions are necessary to make the entire energy-momentum tensor divergence-free, and we present the field equations satisfied by these energy-momentum quantities. The consequences of all these results are briefly analyzed.

We explore the possibility of having a good description of classical signature change in the brane scenario.PACS Numbers: 04.50.+h, 98.80.Cq, 11.10.Kk, 04.20.Gz.The aim of this letter is to show, in simple terms, that a natural scenario for the change of signature in the physical spacetime is provided by the brane-world models [1][2][3] (see also [4][5][6] for an exhaustive list of references) or, in general, by every higher-dimensional theory [7] which may contain domain walls and/or branes.The main idea behind our proposal is that d-branes are nothing but timelike (d + 1)-surfaces in a higherdimensional spacetime (the bulk) [8]. However, nothing prevents the possibility of having perfectly regular branes which change its character from (say) spacelike to timelike, or which are partly null, or even more complicated possibilities. The first case corresponds to a signaturechanging brane. The interesting property is that both the bulk and the brane can be regular everywhere even though the change of signature may appear as a dramatical event when seen from within the brane. Notice that the signature in the bulk is left unchanged, so that our work differs significantly from other recent studies [9]. In our proposal, the study of the change of signature becomes the simple geometrical analysis of imbedded submanifolds in the bulk: a well-posed mathematical problem without pathologies. It is remarkable that many of the traditional ad hoc assumptions concerning signature change [10] are shown to become pure necessary conditions in the brane case, which indirectly proves the plausibility of our idea and makes it worth exploring it, possibly sheding some light into the "signature-change controversy" [10].Whether a signature change occurred in our effective spacetime is debatable, and several independent works have considered this possibility [12,11]. From a classical viewpoint, a signature change may serve to avoid the singularities of general relativity [13], such as the big-bang, which might be replaced by a Euclidean region prior to the birth of time. Signature change has also been vindicated as an effective classical description of both the no-boundary proposal [14] and the quantum tunneling [15] approach for the prescription of the Universe's wave function in quantum cosmology. In general, every process which can be studied by resorting to the "imaginary time", e.g. [14], can be also analyzed by means of change of the signature. All these possibilities could be naturally considered in our proposal.As a matter of concreteness we will focus on the recent models based on a single 3-brane embedded into a five-dimensional Lorentzian manifold with a noncompact fifth dimension [3] (a more geometrically focused review of this model can be found in [5]) and [16,17]. We can think of such a brane model as consisting of two Lorenztian regions joined together across corresponding smooth timelike boundaries. The matching between the two manifolds can be performed as long as the induced metrics on the two boundaries are isometric....

Hartle's model describes the equilibrium configuration of a rotating isolated compact body in perturbation theory up to second order in General Relativity. The interior of the body is a perfect fluid with a barotropic equation of state, no convective motions and rigid rotation. That interior is matched across its surface to an asymptotically flat vacuum exterior. Perturbations are taken to second order around a static and spherically symmetric background configuration. Apart from the explicit assumptions, the perturbed configuration is constructed upon some implicit premises, in particular the continuity of the functions describing the perturbation in terms of some background radial coordinate. In this work we revisit the model within a modern general and consistent theory of perturbative matchings to second order, which is independent of the coordinates and gauges used to describe the two regions to be joined. We explore the matching conditions up to second order in full. The main particular result we present is that the radial function m 0 (in the setting of the original work) of the second order perturbation tensor, contrary to the original assumption, presents a jump at the surface of the star, which is proportional to the value of the energy density of the background configuration there. As a consequence, the change in mass δM needed by the perturbed configuration to keep the value of the central energy density unchanged must be amended. We also discuss some subtleties that arise when studying the deformation of the star.

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General hypersurface layers are considered in order to describe brane-worlds and shell cosmologies. No restriction is placed on the causal character of the hypersurface which may thus have internal changes of signature. Strengthening the results in our previous letter [1], we confirm that a good, regular and consistent description of signature change is achieved in these brane/shells scenarios, while keeping the hypersurface and the bulk completely regular. Our formalism allows for a unified description of the traditional timelike branes/shells together with the signature-changing, or pure null, ones. This allows for a detailed comparison of the results in both situations. An application to the case of hypersurface layers in static bulks is presented, leading to the general RobertsonWalker geometry on the layer -with a possible signature change. Explicit examples on anti de Sitter bulks are then studied. The permitted behaviours in different settings (Z2-mirror branes, asymmetric shells, signature-changing branes) are analysed in detail. We show in particular that (i) in asymmetric shells there is an upper bound for the energy density, and (ii) that the energy density within the brane vanishes when approaching a change of signature. The description of a signature change as a 'singularity' seen from within the brane is considered. We also find new relations between the fundamental constants in the brane/shell, its tension, and the cosmological and gravitational constants of the bulk, independently of the existence or not of a change of signature.

In the literature, the matchings between spacetimes have been most of the times implicitly assumed to preserve some of the symmetries of the problem involved. But no definition for this kind of matching was given until recently. Loosely speaking, the matching hypersurface is restricted to be tangent to the orbits of a desired local group of symmetries admitted at both sides of the matching and thus admitted by the whole matched spacetime. This general definition is shown to lead to conditions on the properties of the preserved groups. First, the algebraic type of the preserved group must be kept at both sides of the matching hypersurface. Secondly, the orthogonal transivity of two-dimensional conformal (in particular isometry) groups is shown to be preserved (in a way made precise below) on the matching hypersurface. This result has in particular direct implications on the studies of axially symmetric isolated bodies in equilibrium in General Relativity, by making up the first condition that determines the suitability of convective interiors to be matched to vacuum exteriors. The definition and most of the results presented in this paper do not depend on the dimension of the manifolds involved nor the signature of the metric, and their applicability to other situations and other higher dimensional theories is manifest.

The standard definition of cylindrical symmetry in General Relativity is reviewed. Taking the view that axial symmetry is an essential pre-requisite for cylindrical symmetry, it is argued that the requirement of orthogonal transitivity of the isometry group should be dropped, this leading to a new, more general definition of cylindrical symmetry. Stationarity and staticity in cylindrically symmetric spacetimes are then defined, and these issues are analysed in connection with orthogonal transitivity, thus proving some new results on the structure of the isometry group for this class of spacetimes.