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In the study of rotating neutron stars the $I$-Love-$Q$ relations refer to the existence of various approximate, equation of state-independent relations involving the moment of inertia, the Love number and the quadrupole moment. These relations are relevant for observational astrophysics, since they allow (in theory) the inference of any two quantities within the $I$-Love-$Q$ triad out of the third one alone. However, the quantities involved in the relations are, in fact, normalized by a parameter $M_0$ that arises in the usual perturbative analytical approach as the mass of the background configuration. Since $M_0$ is not the mass of the rotating star $M_S$, it is not an observational quantity, which may affect the application of the relations to actual observations. This situation is usually ignored in most studies by taking $M_0$ to be the mass of the star, an approximation that can, in some cases, be inconsistent. In this paper we extract the value of $M_0$ using an $\textit{extended}$ version of the universal relations that involve a fourth parameter, $\delta M$, proportional to the difference $M_S-M_0$. We analyze to which degree this extended set of relations yields a more precise inference of compact star properties and equation of state parameters., Comment: 15 pages, 6 figures, 4 tables

The tidal problem is used to obtain the tidal deformability (or Love number) of stars. The semi-analytical study is usually treated in perturbation theory as a first order perturbation problem over a spherically symmetric background configuration consisting of a stellar interior region matched across a boundary to a vacuum exterior region that models the tidal field. The field equations for the metric and matter perturbations at the interior and exterior regions are complemented with corresponding boundary conditions. The data of the two problems at the common boundary are related by the so called matching conditions. These conditions for the tidal problem are known in the contexts of perfect fluid stars and superfluid stars modelled by a two-fluid. Here we review the obtaining of the matching conditions for the tidal problem starting from a purely geometrical setting, and present them so that they can be readily applied to more general contexts, such as other types of matter fields, different multiple layers or phase transitions. As a guide on how to use the matching conditions, we recover the known results for perfect fluid and superfluid neutron stars.

We study the tidal problem and the resulting $I$-Love-$Q$ approximate universal relations for rotating superfluid neutron stars in the Hartle-Thorne formalism. Superfluid stars are described in this work by means of a two-fluid model consisting of superfluid neutrons and all other charged constituents. We employ a stationary and axisymmetric perturbation scheme to second order around a static and spherically symmetric background. Recently, we used this scheme to study isolated rotating superfluid stars. In this paper it is applied to analyze the axially symmetric sector of the tidal problem in a binary system. We show that a consistent use of perturbative matching theory amends the original two-fluid formalism for the tidal problem to account for the possible non-zero value of the energy density at the boundary of the star. This is exemplified by building numerically different stellar models spanning three equations of state. Significant departures from universality are found when the correct matching relations are not taken into account. We also present an augmented set of universal relations for superfluid neutron stars which includes the contribution to the total mass of the star at second order, $\delta M$. Therefore, our results complete the set of universal relations for rotating superfluid stars, generalizing our previous findings in the perfect fluid case., Comment: 18 pages, 3 figures. Reference added, typos and minor corrections, and some discussion expanded, to agree with the published version

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Roberson-Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow., Comment: 29 pages, no figures. Added references and included some further discussion on the result. Version matches the published version

We study spherical charged black holes in the presence of a cosmological constant with corrections motivated by the theory of loop quantum gravity. The effective theory is constructed at the Hamiltonian level by introducing certain correction terms under the condition that the modified constraints form a closed algebra. The corresponding metric tensor is then carefully constructed ensuring that the covariance of the theory is respected, that is, in such a way that different gauge choices on phase space simply correspond to different charts of the same spacetime solution. The resulting geometry is characterized by four parameters: the three usual ones that appear in the general relativistic limit (describing the mass, the charge, and the cosmological constant), as well as a polymerization parameter, which encodes the quantum-gravity corrections. Contrary to general relativity, where this family of solutions is generically singular, in this effective model the presence of the singularity depends on the values of the parameters. The specific ranges of values that define the family of singularity-free spacetimes are explicitly found, and their global structure is analyzed. In particular, the mass and the cosmological constant need to be nonnegative to provide a nonsingular geometry, while there can only be a bounded, relatively small, amount of charge. These conditions are suited for any known spherical astrophysical black hole in the de Sitter cosmological background, and thus this model provides a globally regular description for them., Comment: Minor corrections performed. Version matching the published paper. 38 pages, 10 figures, 6 appendices

We discuss slowly-rotating, general relativistic, superfluid neutron stars in the Hartle-Thorne formulation. The composition of the stars is described by a simple two-fluid model which accounts for superfluid neutrons and all other constituents. We apply a perturbed matching framework to derive a new formalism for slowly-rotating superfluid neutron stars, valid up to second-order perturbation theory, building on the original formulation reported by Andersson and Comer in 2001. The present study constitutes an extension of previous work in the single-fluid case where it was shown that the Hartle-Thorne formalism needs to be amended since it does not provide the correct results when the energy density does not vanish at the surface of the star. We discuss in detail the corrections that need to be applied to the original two-fluid formalism in order to account for non vanishing energy densities at the boundary. In the process, we also find a correction needed in the computation of the deformation of the stellar surface in the original two-fluid model in all cases (irrespective of the value of the energy density at the surface). The discrepancies found between the two formalisms are illustrated by building numerical stellar models, focusing on the comparison in the calculation of the stellar mass, the deformation of the star, and in the Kepler limit of rotation. In particular, using a toy-model equation of state for which the energy density does not vanish at the boundary of the star we demonstrate that the corrections to the formalism we find impact the structure of slowly-rotating superfluid neutron stars in a significant way., Comment: 24 pages, 16 figures: v2. minor corrections and changes to match the published version

We present a covariant model of a spherically symmetric black hole with corrections motivated by loop quantum gravity. The effective modifications, parametrized by a positive constant $\lambda$, are implemented through a canonical transformation and a linear combination of the constraints of general relativity, in such a way that the theory remains free of anomalies and general relativity is recovered for $\lambda=0$. In addition, the corresponding metric is constructed in a fully covariant way to ensure that gauge transformations on phase space correspond to coordinate changes. The solution for each gauge choice provides a chart and corresponding line element of a spacetime solution whose geometry is unambiguously determined in terms of the parameter $\lambda$ and a constant of motion $m$. For positive values of $m$, the solution is asymptotically flat and contains a globally hyperbolic black-hole/white-hole region with a minimal spacelike hypersurface that replaces the Schwarzschild singularity. The corresponding exterior regions are isometric and, in particular, allow the computation of the ADM mass. The procedure to obtain the global causal structure of the solution yields also its maximal analytic extension., Comment: Minor changes to match the published version

We present an effective theory to describe the quantization of spherically symmetric vacuum in loop quantum gravity. We include anomaly-free holonomy corrections through a canonical transformation of the Hamiltonian of general relativity, such that the modified constraint algebra closes. The system is then provided with a fully covariant and unambigous geometric description, independent of the gauge choice on the phase space. The resulting spacetime corresponds to a singularity-free (black-hole/white-hole) interior and two asymptotically flat exterior regions of equal mass. The interior region contains a minimal smooth spacelike surface that replaces the Schwarzschild singularity. We find the global causal structure and the maximal analytical extension. Both Minkowski and Schwarzschild spacetimes are directly recovered as particular limits of the model., Comment: 6 pages, one figure. Minor changes to match the published version, and one reference updated

Existence and uniqueness of rotating fluid bodies in equilibrium is still poorly understood in General Relativity (GR). Apart from the limiting case of infinitely thin disks, the only known global results in the stationary rotating case (Heilig [14] and Makino [21] [arXiv:1705.07392]) show existence in GR nearby a Newtonian configuration (under suitable additional restrictions). In this work we prove existence and uniqueness of rigidly (slowly) rotating fluid bodies in equilibrium to second order in perturbation theory in GR. The most widely used perturbation framework to describe slowly rigidly rotating stars in the strong field regime is the Hartle-Thorne model. The model involves a number of hypotheses, some explicit, like equatorial symmetry or that the perturbation parameter is proportional to the rotation, but some implicit, particularly on the structure and regularity of the perturbation tensors and the conditions of their matching at the surface. In this work, with basis on the gauge results obtained in [25], the Hartle-Thorne model is fully derived from first principles and only assuming that the perturbations describe a rigidly rotating finite perfect fluid ball (with no layer at the surface) with the same barotropic equation of state as the static ball. Rigidly rotating fluid balls are analyzed consistently in second order perturbation theory by imposing only basic differentiability requirements and boundedness. Our results prove in particular that, at this level of approximation, the spacetime must be indeed equatorially symmetric and is fully determined by two parameters, namely the central pressure and the uniform angular velocity of the fluid., Comment: 81 pages, no figures

Perturbation theory in geometric theories of gravitation is a gauge theory of symmetric tensors defined on a Lorentzian manifold (the background spacetime). The gauge freedom makes uniqueness problems in perturbation theory particularly hard as one needs to understand in depth the process of gauge fixing before attempting any uniqueness proof. This is the first paper of a series of two aimed at deriving an existence and uniqueness result for rigidly rotating stars to second order in perturbation theory in General Relativity. A necessary step is to show the existence of a suitable choice of gauge and to understand the differentiability and regularity properties of the resulting gauge tensors in some "canonical form", particularly at the centre of the star. With a wider range of applications in mind, in this paper we analyse the fixing and regularity problem in a more general setting. In particular we tackle the problem of the Hodge-type decomposition into scalar, vector and tensor components on spheres of symmetric and axially symmetric tensors with finite differentiability down to the origin, exploiting a strategy in which the loss of differentiability is as low as possible. Our primary interest, and main result, is to show that stationary and axially symmetric second order perturbations around static and spherically symmetric background configurations can indeed be rendered in the usual "canonical form" used in the literature while loosing only one degree of differentiability and keeping all relevant quantities bounded near the origin., Comment: 60 pages, no figures

In a recent paper we applied a rigorous perturbed matching framework to show the amendment of the mass of rotating stars in Hartle's model. Here, we apply this framework to the tidal problem in binary systems. Our approach fully accounts for the correction to the Love numbers needed to obtain the universal $I$-Love-$Q$ relations. We compute the corrected mass vs radius configurations of rotating quark stars, revisiting a classical paper on the subject. These corrections allow us to find a universal relation involving the second-order contribution to the mass $\delta M$. We thus complete the set of universal relations for the tidal problem in binary systems, involving four perturbation parameters, namely $I$, Love, $Q$, and $\delta M$. These relations can be used to obtain the perturbation parameters directly from observational data., Comment: 6 pages, 2 figures. Accepted for publication in MNRAS

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., Comment: 43 pages, 1 figure

We show how the correction to the calculation of the mass in the original relativistic model of a rotating star by Hartle [6], found recently [10], appears in the Newtonian limit, and that the correcting term is indeed present, albeit hidden, in the original Newtonian approach by Chandrasekhar [2]., Comment: 11 pages, uses iopart style and a custom bibtex style

Hartle's model provides the most widely used analytic framework to describe isolated compact bodies rotating slowly in equilibrium up to second order in perturbations in the context of General Relativity. Apart from some explicit assumptions, there are some implicit, like the "continuity" of the functions in the perturbed metric across the surface of the body. In this work we sketch the basics for the analysis of the second order problem using the modern theory of perturbed matchings. In particular, the result we present is that when the energy density of the fluid in the static configuration does not vanish at the boundary, one of the functions of the second order perturbation in the setting of the original work by Hartle is not continuous. This discrepancy affects the calculation of the change in mass of the rotating star with respect to the static configuration needed to keep the central energy density unchanged., Comment: 6 pages, no figures. To appear in the proceedings of the ERE2014 in Journal of Physics: Conference Series

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 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., Comment: 38 pages, two figures; two paragraphs improved with a clearer statement, typos corrected, added two references, updated one; matches published version

The Einstein-Straus model consists of a Schwarzschild spherical vacuole in a Friedman-Lema\^{\i}tre-Robertson-Walker (FLRW) dust spacetime (with or without \Lambda). It constitutes the most widely accepted model to answer the question of the influence of large scale (cosmological) dynamics on local systems. The conclusion drawn by the model is that there is no influence from the cosmic background, since the spherical vacuole is static. Spherical generalizations to other interior matter models are commonly used in the construction of lumpy inhomogeneous cosmological models. On the other hand, the model has proven to be reluctant to admit non-spherical generalizations. In this review, we summarize the known uniqueness results for this model. These seem to indicate that the only reasonable and realistic non-spherical deformations of the Einstein-Straus model require perturbing the FLRW background. We review results about linear perturbations of the Einstein-Straus model, where the perturbations in the vacuole are assumed to be stationary and axially symmetric so as to describe regions (voids in particular) in which the matter has reached an equilibrium regime., Comment: 29 pages, 1 figure

We obtain a purely local characterisation that singles out the Majumdar-Papapetrou class, the near-horizon Bertotti-Robinson geometry and the Reissner-Nordstr\"om exterior solution, together with its plane and hyperbolic counterparts, among the static electrovacuum spacetimes. These five classes are found to form the whole set of static Einstein-Maxwell fields without sources and conformally flat space of orbits, this is, the conformastat electrovacuum spacetimes. The main part of the proof consists in showing that a functional relationship between the gravitational and electromagnetic potentials must always exist. The classification procedure provides also an improved characterisation of Majumdar-Papapetrou, by only requiring a conformally flat space of orbits with a vanishing Ricci scalar of the usual conveniently rescaled 3-metric. A simple global consideration allows us to state that the asymptotically flat subset of the Majumdar-Papapetrou class and the Reissner-Nordstr\"om exterior solution are the only asymptotically flat conformastat electrovacuum spacetimes., Comment: LaTeX; 31 pages. Uses iopart style files

We derive the linearly perturbed matching conditions between a Schwarzschild spacetime region with stationary and axially symmetric perturbations and a FLRW spacetime with arbitrary perturbations. The matching hypersurface is also perturbed arbitrarily and, in all cases, the perturbations are decomposed into scalars using the Hodge operator on the sphere. This allows us to write down the matching conditions in a compact way. In particular, we find that the existence of a perturbed (rotating, stationary and vacuum) Schwarzschild cavity in a perturbed FLRW universe forces the cosmological perturbations to satisfy constraints that link rotational and gravitational wave perturbations. We also prove that if the perturbation on the FLRW side vanishes identically, then the vacuole must be perturbatively static and hence Schwarzschild. By the dual nature of the problem, the first result translates into links between rotational and gravitational wave perturbations on a perturbed Oppenheimer-Snyder model, where the perturbed FLRW dust collapses in a perturbed Schwarzschild environment which rotates in equilibrium. The second result implies in particular that no region described by FLRW can be a source of the Kerr metric., Comment: LaTeX; 29 pages

We show that some types of sudden singularities admit a natural explanation in terms of regular changes of signature on brane-worlds in AdS$_{5}$. The present accelerated expansion of the Universe and its possible ending at a sudden singularity may therefore simply be an indication that our braneworld is about to change its Lorentzian signature to an Euclidean one, while remaining fully regular. An explicit example of this behaviour satisfying the weak and strong energy conditions is presented., Comment: LaTeX, 4 pages. Uses the eas.cls class. To appear in the proceedings of the Spanish Relativity Meeting ERE'07

We show that regular changes of signature on brane-worlds in AdS bulks may account for some types of the recently fashionable sudden singularities. Therefore, the possibility that the Universe seems to approach a future sudden singularity at an accelerated rate of expansion might simply be an indication that our braneworld is about to change from Lorentzian to Euclidean signature. Both the brane and the bulk remain fully regular everywhere. We present a model in which the weak and strong energy conditions hold on the brane, in contrast with the standard cosmologies leading to the analogous kinematical behaviour (that is, with a diverging Hubble factor)., Comment: 5 pages, 1 figure. Minor improvements in abstract and main text. New title and new reference added. To be published in PRD

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 Robertson-Walker 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 ($Z_{2}$-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., Comment: 23 pages, 2 figures

Perturbed stationary axisymmetric isolated bodies, e.g. stars, represented by a matter-filled interior and an asymptotically flat vacuum exterior joined at a surface where the Darmois matching conditions are satisfied, are considered. The initial state is assumed to be static. The perturbations of the matching conditions are derived and used as boundary conditions for the perturbed Ernst equations in the exterior region. The perturbations are calculated to second order. The boundary conditions are overdetermined: necessary and sufficient conditions for their compatibility are derived. The special case of perturbations of spherical bodies is given in detail., Comment: RevTeX; 32 pp. Accepted by Phys. Rev. D. Added references and extra comments in introduction

The studies in general relativity of rotating finite objects in equilibrium have usually focused on the case when they are truly isolated, this is, the models to describe finite objects are embedded in an asymptotically flat exterior vacuum. Known results ensure the uniqueness of the vacuum exterior field by using the boundary data for the exterior field given at the surface of the object plus the decay of the exterior field at infinity. The final aim of the present work is to study the consequences on the interior models by changing the boundary condition at infinity to one accounting for the embedding of the object in a cosmological background. Considering first the FLRW standard cosmological backgrounds, we are studying the general matching of FLRW with stationary axisymmetric spacetimes in order to find the new boundary condition for the vacuum region. Here we present the first results., Comment: LaTeX, 6 pages, uses ere04.cls style, to appear in the proceedings of the Spanish Relativity Meeting ERE'04

A relatively recent study by Mars and Senovilla provided us with a uniqueness result for the exterior vacuum gravitational field of global models describing finite isolated rotating bodies in equilibrium in General Relativity (GR). The generalisation to exterior electrovacuum gravitational fields, to include charged rotating objects, is presented here., Comment: LaTeX, 6pages, uses indentfirst package. To appear in the proceedings of the Spanish Relativity Meeting ERE'03

The problem of describing isolated rotating bodies in equilibrium in General Relativity has so far been treated under the assumption of the circularity condition in the interior of the body. For a fluid without energy flux, this condition implies that the fluid flow moves only along the angular direction, i.e. there is no convection. Using this simplification, some recent studies have provided us with uniqueness and existence results for asymptotically flat vacuum exterior fields given the interior sources. Here, the generalisation of the problem to include general sources is studied. It is proven that the convective motions have no direct influence on the exterior field, and hence, that the aforementioned results on uniqueness and existence of exterior fields apply equally in the general case., Comment: 8 pages, LaTex, uses iopart style files. To appear in Class. Quatum Grav

We exploit once again the analogy between the energy-momentum tensor and the so-called ``superenergy'' tensors in order to build conserved currents in the presence of Killing vectors. First of all, we derive the divergence-free property of the gravitational superenergy currents under very general circumstances, even if the superenergy tensor is not divergence-free itself. The associated conserved quantities are explicitly computed for the Reissner-Nordstrom and Schwarzschild solutions. The remaining cases, when the above currents are not conserved, lead to the possibility of an interchange of some superenergy quantities between the gravitational and other physical fields in such a manner that the total, mixed, current may be conserved. Actually, this possibility has been recently proved to hold for the Einstein-Klein-Gordon system of field equations. By using an adequate family of known exact solutions, we present explicit and completely non-obvious examples of such mixed conserved currents., Comment: LaTeX, 19 pages; improved version adding new content to the second section and some minor corrections

We study the possibility of generalising the Einstein--Straus model to anisotropic settings, by considering the matching of locally cylindrically symmetric static regions to the set of $G_4$ on $S_3$ locally rotationally symmetric (LRS) spacetimes. We show that such matchings preserving the symmetry are only possible for a restricted subset of the LRS models in which there is no evolution in one spacelike direction. These results are applied to spatially homogeneous (Bianchi) exteriors where the static part represents a finite bounded interior region without holes. We find that it is impossible to embed finite static strings or other locally cylindrically symmetric static objects (such as bottle or coin-shaped objects) in reasonable Bianchi cosmological models, irrespective of the matter content. Furthermore, we find that if the exterior spacetime is assumed to have a perfect fluid source satisfying the dominant energy condition, then only a very particular family of LRS stiff fluid solutions are compatible with this model. Finally, given the interior/exterior duality in the matching procedure, our results have the interesting consequence that the Oppenheimer-Snyder model of collapse cannot be generalised to such anisotropic cases., Comment: LaTeX, 24 pages. Text unchanged. Labels removed from the equations. Submitted for publication

We study new separable orthogonally transitive abelian G_2 on S_2 models with two mutually orthogonal integrable Killing vector fields. For this purpose we consider separability of the metric functions in a coordinate system in which the velocity vector field of the perfect fluid does not take its canonical form, providing thereby solutions which are non-separable in comoving coordinates in general. Some interesting general features concerning this class of solutions are given. We provide a full classification for these models and present several families of explicit solutions with their properties., Comment: latex, 26 pages, accepted for publication in Class. Quantum Grav