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J. L. Hernandez-Pastora

Publications and source records attributed to J. L. Hernandez-Pastora.

18 recordsLinked to original sources
arXiv2025

Axially symmetric ghost stars

We present static axially symmetric fluid distributions not producing gravitational field outside their boundaries (i.e. fluid sources which match smoothly on the boundary surface to Minkowski space-time). These solutions provide further examples of ghost stars. A specific model is fully described, and its physical and geometrical properties are analyzed in detail. This includes the multipole moment structure of the source and its complexity factors, both of which vanish for our solution.

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arXiv2022

Whom actually do multipole moments belong to?

Using an integral definition to calculate the relativistic multipole moments (RMM), and the ensuing generalized relativistic Gauss theorem, we prove that the evaluation of that volume integral in Erez-Rosen coordinates, leads to a specific link between the RMM and the source of the exterior space--time, provided we have a global static axisymmetric metric in that coordinate system for any Weyl exterior field. This result allows to establish a relationship between the RMM and certain volume integral expressions involving the material content of the source from its energy-momentum tensor as well as the interior metric. In particular the relativistic quadrupole moment for the Erez-Rosen space-time is obtained.

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arXiv2022

All analytic solutions for geodesic motion in axially symmetric space-times

Recent observations of the orbits of star clusters around Sgr $A^\star$, imaging of black holes and gravitational waveforms of merging compact objects require a detailed understanding of the general relativistic geodesic motion. We came up with a method to provide all the possible geodesics in an axially symmetric space-time. The Kerr metric is explicitly worked out, recovering the Schwarzschild geodesics in the static limit. We also found the most general Killing tensor and its associated constant of motion for an axisymmetric space-time. The relevance of these results is crucial to understanding the different scenarios and the fundamental nature of the compact object at the galactic center.

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arXiv2020

Connecting the exterior gravitational field with the energy momentum tensor of axially symmetric compact objects

A method to construct interior axially symmetric metrics that appropriately match with any vacuum solution of the Weyl family is developed in Hernandez-Pastora etal. (Class Quantum Gravity 33:235005, 2016). It was shown,for the case of some vacuum solutions, that the simplestsolution for the interior metric leads to sources with well behaved energy conditions. Now, we integrate the field equa-tions to obtain the interior metric functions in terms of theanisotropies and pressures of the source. As well, the compatible equations of state for these global models are calculated. The interior metric and the suitable energy momentum tensor describing the source are constructed in terms of the exterior metric functions. At the boundary of the compact object,the behaviour of a pressure Tm, defined from the energy momentum tensor, is shown to be related with the exterior gravitational field. This fact allows us to explore the differences arising at the matter distribution when the sphericalsymmetry of the global metric is dropped. Finally, an equation derived from the matching conditions is obtained whichallows us to calculate the Weyl coefficients of the exteriormetric as source integrals. Hence the Relativistic MultipoleMoments of the global model can be expressed in terms ofthe matter distribution of the source.

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arXiv2017

Physical properties of a source of the Kerr metric: Bound on the surface gravitational potential and conditions for the fragmentation

We investigate some important physical aspects of a recently presented interior solution for the Kerr metric. It is shown that, as in the spherically symmetric case, there is a specific limit for the maximal value of the surface potential (degree of compactness), beyond which, unacceptable physical anomalies appear. Such a bound is related to the appearance of negative (repulsive) gravitational acceleration, that is accompanied by the appearance of negative values of the pressure. A detailed discussion on this effect is presented. We also study the possibility of a fragmentation scenario, assuming that the source leaves the equilibrium, and we bring out the differences with the spherically symmetric case.

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arXiv2017

Interior solution for the Kerr metric

A, recently presented, general procedure to find static and axially symmetric, interior solutions to the Einstein equations, is extended to the stationary case, and applied to find an interior solution for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy conditions for a wide range of values of the parameters, and matches smoothly to the Kerr solution, thereby representing a globally regular model describing a non spherical and rotating source of gravitational field. In the spherically symmetric limit, our model converges to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. The physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments.

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arXiv2016

Axially symmetric static sources of gravitational field

A general procedure to find static and axially symmetric, interior solutions to the Einstein equations is presented. All the so obtained solutions, verify the energy conditions for a wide range of values of the parameters, and match smoothly to some exterior solution of the Weyl family, thereby representing globally regular models describing non spherical sources of gravitational field. In the spherically symmetric limit, all our models converge to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. Some particular sources are obtained, and the physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments. The total mass of different configurations is also calculated, it is shown to be equal to the monopole of the exterior solution.

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arXiv2016

Source integrals of multipole moments for static space-times

The definition of Komar for the mass of a relativistic source is used as a starting point to introduce volume integrals for Relativistic Multipole Moments (RMM). A certain generalization of the classical Gauss theorem is used to rewrite these multipole moments as integrals over a surface at the infinity. Therefore it is shown that the above generalization leads to Asymptotic Relativistic Multipole Moments (ARMM), recovering the multipoles of Geroch or Thorne, when the integrals are evaluated in asympotically cartesian harmonic coordinates. Relationships regarding the Thorne definition and the classical theory of moments are shown.

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arXiv2013

On the Solutions of infinite systems of linear equations

New theorems about the existence of solution for a system of infinite linear equations with a Vandermonde type matrix of coefficients are proved. Some examples and applications of these results are shown. In particular, a kind of these systems is solved and applied in the field of the General Relativity Theory of Gravitation. The solution of the system is used to construct a relevant physical representation of certain static and axisymmetric solution of the Einstein vacuum equations. In addition, a newtonian representation of these relativistic solutions is recovered. It is shown as well that there exists a relation between this application and the classical Haussdorff moment problem.

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arXiv2013

Geodesics in the linearized multipole solution: Distinguishing black holes from naked singularities

We analyze the behaviour of geodesic motion of test particles in the spacetime of a specific class of axially symmetric static vacuum solutions to the Einstein equations, hereafter referred to as linearized multipole solution (LM). We discuss about its suitability to describe a quasi-spherical spacetime. The existence of an ISCO (innermost stable circular orbit) very close to the (singular) horizon of the source, is established. The existence of such stable orbit, inner than the one of the Schwarzschild metric, as well as the appearance of a splitting in the admissible region of circular orbits, is shown to be due to the multipole structure of the solution, thereby providing additional potential observational evidence for distinguishing Schwarzschild black holes from naked singularities.

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arXiv2013

Linearized multipole solutions and their representation

The monopole solution of the Einstein vacuum field equations (Schwarzschild`s solution) in Weyl coordinates involves a metric function that can be interpreted as the gravitational potential of a bar of length $2m$ with constant linear density. The question addressed in this work is whether similar representations can be constructed for Weyl solutions other than the spherically symmetric one. A new family of static solutions of the axisymmetric vacuum field equations generalizing the M-Q$^{(1)}$ solution is developed. These represent slight deviations from spherical symmetry in terms of the relativistic multipole moments (RMM) we wish the solution to contain. A Newtonian object referred to as a dumbbell can be used to describe these solutions in a simple form by means of the density of this object, since the physical properties of the relativistic solution are characterized by its behaviour. The density profile of the dumbbell, which is given in terms of the RMM of the solution, allows us to distinguish general multipole Weyl solutions from the constant-density Schwarzschild solution. The range of values of the multipole moments that generate positive-definite density profiles are also calculated. The bounds on the multipole moments that arise from this density condition are identical to those required for a well-behaved infinite-redshift surface $g_{00}=0$.

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arXiv2011

Event Horizon of the Monopole-Quadrupole solution: geometric and thermodynamic properties

We investigate the general geometric properties of the surface of infinite red-shift corresponding to the event horizon of the static and axisymmetric solution of the Einstein vacuum equations that only possesses mass $M$ and quadrupole moment $Q$. The deformation of the Schwarzschild surface $r=2M$ produced by the quadrupole moment is shown, and the range of values of this multipole moment is specified, which preserves a regular, closed, continuous and differentiable surface. Some thermodynamic consequences and speculations ensuing from our results are discussed.

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arXiv2010

From geodesics of the multipole solutions to the perturbed Kepler problem

A static and axisymmetric solution of the Einstein vacuum equations with a finite number of Relativistic Multipole Moments (RMM) is written in MSA coordinates up to certain order of approximation, and the structure of its metric components is explicitly shown. From the equation of equatorial geodesics we obtain the Binet equation for the orbits and it allows us to determine the gravitational potential that leads to the equivalent classical orbital equations of the perturbed Kepler problem. The relativistic corrections to Keplerian motion are provided by the different contributions of the RMM of the source starting from the Monopole (Schwarzschild correction). In particular, the perihelion precession of the orbit is calculated in terms of the quadrupole and 2$^4$-pole moments. Since the MSA coordinates generalize the Schwarzschild coordinates, the result obtained allows measurement of the relevance of the quadrupole moment in the first order correction to the perihelion frequency-shift.

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arXiv2010

New Representation of some Static and Axisymmetric Vacuum Solutions

We solve the Einstein vacuum-equations for the case of static and axisymmetric solutions in a system of coordinates different from the Weyl standard one. We prove that there exists a class of solutions with the appropriate asymptotical behaviour which can be written in a simple compact form, in terms of a function that must satisfies certain Cauchy-Newman problem. The relation between the choice of coordinates and the form of the metric functions that describe the solution is given by providing that analytic function which characterizes the metric as well as the gauge.

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arXiv2009

On the existence of MSA coordinates

The static solutions of the axially symmetric vacuum Einstein equations with a finite number of Relativistic Multipole Moments are described by means of a function that can be written in the same analytic form as the Newtonian gravitational multipole potential. A family of so-called MSA (Multipole-Symmetry Adapted) coordinates are introduced and calculated at any multipole order to perform the transformation of the Weyl solutions. In analogy with a previous result obtained in Newtonian gravity, the existence of a symmetry of a certain system of differential equations leading to the determination of that kind of multipole solutions in General Relativity is explored. The relationship between the existence of this kind of coordinate and the symmetries mentioned is proved for some cases, and the characterization of the MSA system of coordinates by means of this relationship is discussed.

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arXiv1998

Approaches to the Monopole-Dynamic Dipole Vacuum Solution Concerning the Structure of its Ernst's Potential on the Symmetry Axis

The FHP algorithm allows to obtain the relativistic multipole moments of a vacuum stationary axisymmetric solution in terms of coefficients which appear in the expansion of its Ernst's potential on the symmetry axis. First of all, we will use this result in order to determine, at a certain approximation degree, the Ernst's potential on the symmetry axis of the metric whose only multipole moments are mass and angular momentum. By using Sibgatullin's method we analyse a series of exacts solutions with the afore mentioned multipole characteristic. Besides, we present an approximate solution whose Ernst's potential is introduced as a power series of a dimensionless parameter. The calculation of its multipole moments allows us to understand the existing differences between both approximations to the proposed pure multipole solution.

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arXiv1997

Thermal Conduction in Systems out of Hydrostatic Equilibrium

We analyse the effects of thermal conduction in a relativistic fluid, just after its departure from hydrostatic equilibrium, on a time scale of the order of thermal relaxation time. It is obtained that the resulting evolution will critically depend on a parameter defined in terms of thermodynamic variables, which is constrained by causality requirements.

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