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A generalization of the relativistic equilibrium equations for a non-rotating star
Abstract
The problem of elastomechanical equilibrium for a static, spherically symmetric star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity theory is used, and the second order equilibrium equations are found. It is shown that the equilibrium conditions with anisotropic pressure introducedad hoc by some authors are in fact the dynamical conditions for a relativistic elastic material. The corresponding first order equations for the components of the metric and of the energy-momentum tensor reduce to the Tolman-Oppenheimer-Volkhoff equations if the material exhibits no shape-rigidity. Two interesting classes of solutions are discussed.
... To proceed further, we assume as a source an isotropic elastic medium. The energymomentum tensor of an elastic-solid sphere has been thoroughly discussed in Ref. [14]. In particular, it was shown that a spherically symmetric, relativistic body can be characterized by three quantities ǫ, p, Ω. ...
... The stresses may then be calculated as derivatives of the energy with respect to such invariants (in the case of a perfect fluid, this obviously amounts to say that the equation of state relates energy and specific volume v, and that the pressure may be obtained as the derivative of the energy with respect to v). In the particular case of spherical symmetry, it has been shown [14] that assigning the state equation of the body consists in giving the energy density as a function of only two invariants of the strain tensor, one of them being the specific volume (so that the derivative of ǫ with respect to it governs the isotropic response p) and the other one being the quadratic invariant of the deformation, the "response" associated to this invariant being Ω. ...
... It is, in fact, equivalent to a "constitutive" partial differential equation for ǫ. The solution of such equation identifies a well-defined class of elastic materials which depends on the choice of an arbitrary function of one variable only (we refer the reader to Ref. [14] for details). This arbitrariness may be used to assign the "on shell" value ǫ = ǫ(r). ...
New, simple models of ``black hole interiors'', namely spherically symmetric solutions of the Einstein field equations in matter matching the Schwarzschild vacuum at spacelike hypersurfaces ``R<2M'' are constructed. The models satisfy the weak energy condition and their matter content is specified by an equation of state of the elastic type.
... To proceed further, we assume as a source an isotropic elastic medium. The energymomentum tensor of an elastic-solid sphere has been thoroughly discussed in Ref. [14]. In particular, it was shown that a spherically symmetric, relativistic body can be characterized by three quantities ǫ, p, Ω. ...
... The stresses may then be calculated as derivatives of the energy with respect to such invariants (in the case of a perfect fluid, this obviously amounts to say that the equation of state relates energy and specific volume v, and that the pressure may be obtained as the derivative of the energy with respect to v). In the particular case of spherical symmetry, it has been shown [14] that assigning the state equation of the body consists in giving the energy density as a function of only two invariants of the strain tensor, one of them being the specific volume (so that the derivative of ǫ with respect to it governs the isotropic response p) and the other one being the quadratic invariant of the deformation, the "response" associated to this invariant being Ω. ...
... It is, in fact, equivalent to a "constitutive" partial differential equation for ǫ. The solution of such equation identifies a well-defined class of elastic materials which depends on the choice of an arbitrary function of one variable only (we refer the reader to Ref. [14] for details). This arbitrariness may be used to assign the "on shell" value ǫ = ǫ(r). ...
New, simple models of “black hole interiors”, namely spherically symmetric solutions of the Einstein field equations in matter matching the Schwarzschild vacuum at spacelike hypersurfaces “R < 2M” are constructed. The models satisfy the weak energy condition and their matter content is specified by an equation of state of the elastic type.
... Spherically symmetric models with elastic matter in general relativity have been studied by a number of authors: Magli and Kijowski [1] investigated the problem of elastomechanical equilibrium for a non-rotating star, Park [2] proved existence theorems for spherically symmetric elastic bodies, Magli [3] analysed the relativistic interior dynamics of a spherically symmetric non-rotating star composed of an elastic material, Frauendiener and Kabobel [4] discussed spherically symmetric solutions of the general relativistic elasticity equations with different stored energy functions; and Karlovini and Samuelsson [5] showed how physically prestressed stellar models, which serve as backgrounds in investigations of stellar perturbations, can be produced numerically and investigated radial and axial perturbations of static spherically symmetric elastic configurations [6], [7]; just to name a few. ...
... In this paper, after providing a short summary of some relevant issues concerning relativistic elasticity (the reader is referred to [13], [14], [5], [1] for further details), the definitions of sound wave front, propagation speed and characteristic equation are presented in Section 3. Based on these definitions, an expression for the propagation speed of the wave front in spherically symmetric spacetimes with elastic matter is derived in Section 4, which depends on the energy density, the radial pressure and the elasticity tensor. In Section 5, we consider shear-free static and non-static solutions obtained in [14], determine their radial propagation speeds and analyse if they satisfy the causality condition (i.e.: propagation speed less than or equal to the speed of light), showing that there are open regions where causality is preserved besides satisfying the Dominant Energy Condition and being singularity free. ...
It is shown that exact spherically symmetric solutions to Einstein's Field Equations exist such that, over an open region of the spacetime, they are singularity free, satisfy the dominant energy condition, represent elastic matter with a well defined constitutive function, and are such that elastic perturbations propagate causally. Two toy-models are then built up in which a thick elastic, spherically symmetric shell with the above properties, separates two Robertson-Walker regions corresponding to different values of the curvature k. The junction conditions (continuity of the first and second fundamental forms) are shown to be exactly satisfied across the corresponding matching spherical surfaces.
... In spherical symmetry the subject started with [36] (see also [5,37,38]) and has meanwhile advanced much further, see the recent [39,40] and references therein. In these works there is no assumption of closeness to a stressfree configurations. ...
This paper is based on a series of talks given at the ESI program on 'Mathematical Perspectives of Gravitation Beyond the Vacuum Regime' in February 2022. It is meant to be an introduction to the field of relativistic elasticity for
readers with a good base in the mathematics of General Relativity with no necessary previous of knowledge of elasticity either in the classical or relativistic domain.
Despite its introductory purpose, the present work has new material, in particular related to the formal structure of the theory.
... Several works have reconsidered the general relativistic elasticity formalism such as Magli & Kijowski (1992); Beig & Schmidt (2005), compared by Frauendiener & Kabobel (2007) and Brito et al. (2011b,a) who extended the study by Magli & Kijowski. In particular, Karlovini & Samuelsson (2003) revisited the formalism by Carter & Quintana (1972) and applied it to the study of different types of perturbations in elastic neutron stars in a series of subsequent papers . ...
Neutron stars are the remnant of massive stars and are created during the core-collapse supernova that marks the end of their life. With a radius of a tenth of kilometers for a mass of one to two times the one of the Sun, they are very dense and relativistic objects sustained by the strong interaction. This thesis deals with the theoretical modeling of three aspects of the evolution and dynamics of neutron stars : the thermal evolution of isolated neutron stars and neutron stars accreting matter from a companion star, the influence of the elastic properties of their solid parts on the rotation of isolated neutron stars and the rotational evolution of accreting neutron stars. The confrontation with the observations enables us to probe the properties of dense matter.
... Anisotropic material balls also have been objects of study and many particular solutions of the corresponding Einstein equations have been derived (for example [3], [4], [5] and [6]) under various assumptions imposed on the stress-energy tensor. But, unlike in the case of the perfect fluid ball, there hasn't been any general statement concerning the existence of solutions for anisotropic balls. ...
The paper is concerned with the Einstein equations for a spherically symmetric static distribution of anisotropic matter. The equations are cast into a system of Fuchsian type ODE for certain scalar invariants of the strain. And then the existence and regularity of this ODE is studied under general constitutive relation. In the case the constitutive relation is given by a quadratic form of strain, it is also shown that the solutions stay regular up to the boundary of the material ball.
... ones not locally isometric to the Schwarzschild solution)? The equations for spherically symmetric elastic bodies have been studied in [9] but there is no existence theory for general constitutive relations known in that case. Problem Study the existence of spherically symmetric static elastic bodies in general relativity It can be seen from the above that quite a lot is known about asymptotically flat static solutions of the Einstein-matter equations. ...
Recent results on solutions of the Einstein equations with matter are surveyed and a number of open questions are stated. The first group of results presented concern asymptotically flat spacetimes, both stationary and dynamical. Then there is a discussion of solutions of the equations describing matter in special relativity and Newtonian gravitational theory and their relevance for general relativity. Next spatially compact solutions of the Einstein-matter equations are presented. Finally some remarks are made on the methods which have been used, and could be used in the future, to study solutions of the Einstein equations with matter.
... In [11] it is shown that coupling Einstein's equations with a Yang-Mills field can lead to static, singularity-free solutions with finite total mass. Static solutions of general-relativistic elasticity have been studied in [5]. For further references, especially on the astrophysical literature, we refer to [10]. ...
The Vlasov-Einstein system describes the evolution of an ensemble of particles (such as stars in a galaxy, galaxies in a galaxy cluster etc.) interacting only by the gravitational field which they create collectively and which obeys Einstein's field equations. The matter distribution is described by the Vlasov or Liouville equation for a collisionless gas. Recent investigations seem to indicate that such a matter model is particularly suited in a general relativistic setting and may avoid the formation of naked singularities, as opposed to other matter models. In the present note we consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular center and may have isotropic or anisotropic pressure, and solutions, which have a Schwarzschild-singularity at the center. The paper extends previous work, where smooth, globally defined solutions with regular center and isotropic pressure were considered.
... Other general results about solutions with a de Sitter core have appeared in the literature (see for instance [76], where an equation of state of the elastic type [77] is considered, as well as the analysis in [52]); extensions of the above results with gravity coupled to a nonlinear scalar field are also very interesting (see, e.g., [22,50] and references therein). This general analysis is of course important and preliminary for physical application of these ideas, which also have appeared (see, for instance, [1]). ...
We review, in a historical perspective, some results about black hole spacetimes with a regular center. We then see how their properties are realized in a specific solution that recently appeared; in particular we analyze in detail the (necessary) violation of the strong energy condition.
In the scope of teleparallel equivalent to general relativity, we are interested in finding the energy–momentum and its suitable quantities of cylindrically axially symmetric rotating (CASR) space–time. We obtain the formulations for the torsion vector, torsion axial-vector, angular momentum and gravitational centre of mass in CASR solutions. We have investigated the gravitational energy–momentum, gravitational flux and matter flux of the space–time. The vector part connected with Dirac spin has been estimated as well. Finally, a special case of the solution are discussed.
We present a class of exact solutions of Einstein's gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solutions are obtained by assuming a particular form of the anisotropy factor. The energy density and both radial and tangential pressures are finite and positive inside the anisotropic star. Numerical results show that the basic physical parameters (mass and radius) of the model can describe realistic astrophysical objects like neutron stars.
It is shown that exact spherically symmetric solutions to Einstein’s field equations exist such that, over an open region of the spacetime, they are singularity free, satisfy the dominant energy condition, represent elastic matter with a well-defined constitutive function, and are such that elastic perturbations propagate causally. Two toy models are then built up in which a thick elastic, spherically symmetric shell with the above properties, separates two Robertson-Walker regions corresponding to different values of the curvature k in the first model and to the same value of k in the second model. The junction conditions (continuity of the first and second fundamental forms) are shown to be exactly satisfied across the corresponding matching spherical surfaces.
In the framework of teleparallel equivalent to general relativity, we study the energy–momentum and its relevant quantities of cylindrically axially symmetric rotating (CASR) space time. We calculate the gravitational energy–momentum and gravitation energy–momentum flux of the derived solution. We show that in specific cases, our results coincide with what the results derived before in general relativity.
We present a class of relativistic solutions for compact cold stars with strange matter in a pseudo-spheroidal space-time. Considering strange matter equation of state namely, , where , p and B are energy density, pressure and MIT Bag parameter respectively, stellar models are obtained. In the presence of anisotropy with a pseudo-spheroidal geometry described by Vaidya-Tikekar, metric stellar models are explored where the Bag parameter varies with the energy density () inside the compact object. We determine the density dependence of B at different anisotropy. It is noted that although B varies with anisotropy inside the star, finally at the surface it attains a value which is independent of the anisotropy. The Bag parameter B is found to increase with an increase in anisotropy for a given compactness factor (M/b) and spheroidicity . It is also noted that for a star with given mass and radius, the parameter B increases with the increase in and finally at large , it attains a constant. The equation of state (EoS) obtained here from geometrical consideration with allowed ‘B’ value is found same to that one obtains from micro-physics. The stability of the stellar models for compact stars with anisotropy in hydrostatic equilibrium obtained here is also studied.
Conformal transformations are applied in the context of general relativistic elasticity. Expressions relating relativistic elastic quantities and tensors are obtained for two conformal spacetimes, whose material metrics are also conformally related. Non-static shear-free spherically symmetric elastic solutions of the Einstein field equations are constructed by performing conformal transformations of spacetime and material metrics of a known non-static shear-free spherically symmetric elastic solution.
We develop here a procedure to obtain regular static configurations as
resulting from dynamical gravitational collapse of a massive matter cloud in
general relativity. Under certain general physical assumptions for the
collapsing cloud, we find the class of dynamical models that lead to an
equilibrium configuration. To illustrate this, we provide a class of perfect
fluid collapse models that lead to a static constant density object as limit.
We suggest that similar models might possibly constitute the basis for the
description of formation of compact objects in nature.
We present spherically symmetric black hole solutions for Einstein gravity coupled to anisotropic matter. We show that these black holes have arbitrarily short hair, and argue for stability by showing that they can arise from dynamical collapse. We also show that a recent "no short hair" theorem does not apply to these solutions.
We present spherically symmetric black hole solutions for Einstein gravity coupled to anisotropic matter. We show that these black holes have arbitrarily short hair, and argue for stability by showing that they can arise from dynamical collapse. We also show that a recent "no short hair" theorem does not apply to these solutions.
We present a class of exact solutions of the Einstein gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solution is represented in a closed integral form. The energy density and both radial and tangential pressure are finite and positive inside the anisotropic star. The energy density, radial pressure, pressure-density ratio and the adiabatic speed of sound are monotonically decreasing functions. Several stellar models with the anisotropy coefficient proportional to r2 are discussed, the values of the basic physical parameters of the star (radius, mass and red shift) and bound on anisotropy parameter is obtained.
We consider a static cylindrically symmetric spacetime with elastic
matter and study the matching problem of this spacetime with a suitable
exterior. For the exterior, we take the Levi-Civita spacetime and its
generalization including a cosmological constant, the Linet-Tian
spacetime. We show that the matching is only possible with the
Linet-Tian solution.
The Einstein field equations are derived for a static cylindrically symmetric spacetime with elastic matter. The equations can be reduced to a system of two nonlinear ordinary differential equations and we present analytical and numerical solutions satisfying the dominant energy conditions. Furthermore, we show that the solutions can be matched at a finite radius to suitable Lambda-vacuum exteriors given by the Linet-Tian spacetime. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4769223]
In the context of relativistic elasticity it is interesting to study axially symmetric space-times due to their significance in modeling neutron stars and other astrophysical systems of interest. To approach this problem, here, a particular class of these space-times is considered. A cylindrically symmetric elastic space-time configuration is studied, where the material metric is taken to be flat. The components of the energy-momentum tensor for elastic matter are written in terms of the invariants of the strain tensor, here chosen to be the eigenvalues of the pulled-back material metric. The Einstein field equations are presented and a condition confirming the existence of a constitutive function is obtained. This condition leads to special cases, in one of which a new system for the metric functions and an expression for the constitutive function are deduced. The new system depends on a particular function, which builds up the constitutive equation.
Given a space-time and a continuous medium with elastic properties described by a 3-dimensional material space, one can ask whether they are compatible in the context of relativistic elasticity. Here a non-static, spherically symmetric spacetime metric is considered and we investigate the conditions for that metric to correspond to different 3-dimensional material metrics.KeywordsKilling VectorParticle Number DensitySymmetric SpacetimeMaterial SpaceProjection TensorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A new formulation of relativistic elastomechanics is presented. It is free of any assumption about the existence of a global relaxation state of the material. The strain, the stress and the energy-momentum tensors are expressed in terms of the first-order derivatives of a field describing the configuration of the material. Its elastic properties are completely determined by a scalar function describing the dependence of the mechanical energy accumulated in the deformed material upon the three invariants of the deformation. The stress-strain relations are generated in a canonical way by this function. Dynamical equations of the theory are derived from the variational principle. They form a hyperbolic system of second-order partial differential equations for the unknown field. Energy-momentum conservation laws are consequences of the Noether theorem. The Hamiltonian formulation of the dynamics and the linear version of the theory are also discussed.
The null tetrad formalism is used to investigate the structure of the Einstein field equations for Kerr–Schild gravitational fields in the presence of an elastic solid source. It is shown that such equations may be reduced to five nonlinear partial differential equations for five variables. It turns out that, when the interior solutions admit the same preferred null congruence of the vacuum ones and some compatibility conditions hold, it is possible to reduce them to a linear system and to develop a method of solution which closely resembles the ‘‘variation of the arbitrary constants’’ for ordinary differential equations. In the present paper, the above technical framework is developed in general and applied to two simple examples, deferring to future work the approach to the Kerr–interior problem.
We present a new variational framework for dissipative general relativistic
fluid dynamics. The model extends the convective variational principle for
multi-fluid systems to account for a range of dissipation channels. The key
ingredients in the construction are i) the use of a lower dimensional matter
space for each fluid component, and ii) an extended functional dependence for
the associated volume forms. In an effort to make the concepts clear, the
formalism is developed in steps with the model example of matter coupled to
heat considered at each level. Thus we discuss a model for heat flow, derive
the relativistic Navier-Stokes equations and discuss why the individual
dissipative stress tensors need not be spacetime symmetric. We argue that the
new formalism, which notably does not involve an expansion away from an assumed
equilibrium state, provides a conceptual breakthrough in this area of research
and provide an ambitious list of directions in which one may want to extend it
in the future. This involves an exciting set of problems, relating to both
applications and foundational issues.
The equations describing the hydrostatic equilibrium (mass continuity and Tolman–Oppenheimer–Volkoff) of a static anisotropic general relativistic fluid sphere are obtained in D (D ≥ 4) space–time dimensions in the presence of a cosmological constant. The formalism thus developed is used to study homogeneous anisotropic constant density charged fluid spheres and homogeneous anisotropic charged spheres with a neutral isotropic core in higher dimensions. For these configurations and with a particular choice of the proper charge density a complete solution of the coupled Einstein–Maxwell equations is obtained. © 2000 American Institute of Physics.
The field equations governing Bianchi type I space–times having elastic media as sources are constructed. Such equations turn out to be a system of three ordinary differential equations for the three independent components of the gravitational field. Plane symmetric and Robertson–Walker space–times are also discussed, and some examples of solutions are given. In the Robertson–Walker case, it turns out that it is possible to ‘‘tune’’ the elastic constitutive equations in such a way that any ‘‘required’’ behavior of the scale factor may be obtained. This feature closely resembles the scalar‐field dominated case. © 1995 American Institute of Physics.
Stationary Kerr–Schild gravitational fields in the interior of elastic–solid bodies are considered. It turns out that the Einstein field equations for such systems split into two parts: a ‘‘differential’’ part which may be written as a linear partial differential equation of the Maxwell type for a suitably defined, three‐dimensional Euclidean vector and an ‘‘algebraic’’ (generally nonlinear) part which allows the calculation of the stresses supporting the bodies. These results are applied to the problem of finding the sources of the Kerr space–time, and a new class of solutions of the Einstein field equations is found. Such solutions describe sources of the Kerr metric composed by an elastic body and a shell distribution of elastic stress energy located on the matching surface. Some explicit examples of ‘‘half local’’ (singular) Kerr–Schild sources of the Kerr space–time are also given. © 1995 American Institute of Physics.
The link between the surface value of the potential and the highest
occurring ratio of the trace of the pressure tensor to the local density
is investigated on the basis of a study of static spheres. The
transverse pressure is permitted to differ from the radial one. Both
Newtonian and relativistic models are examined. Considerably larger
redshift values are found when anisotropic pressures are allowed than in
the isotropic case.
The gravitational dynamics of anisotropic elastic spheres supported only by tangential stresses and satisfying an equation of state is analysed, and a fairly large class of non-static, spherically symmetric solutions of the Einstein field equations is found by quadratures. The solutions contain three arbitrary functions. Two such functions are immediately recognized as the initial distributions of mass and energy, familiar from the Tolman - Bondi (dust) models, while the third is the elastic internal energy per unit volume. If this function is a constant, the energy density becomes proportional to the matter density and therefore the metric reduces to the Tolman - Bondi one. In the general case, however, the solutions contain oscillating models as well as finite-bouncing models.
The relativistic theory of elasticity is reviewed within the spherically symmetric context with a view towards the modelling of star interiors possessing elastic properties such as the ones expected in neutron stars. Emphasis is placed on generality in the main sections of the paper, and the results are then applied to specific examples. Along the way, a few general results for spacetimes admitting isometries are deduced, and their consequences are fully exploited in the case of spherical symmetry relating them next to the the case in which the material content of the spacetime is some elastic material. This paper extends and generalizes the pioneering work by Magli and Kijowski [1], Magli [2] and [3], and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers [4], [5] and [6].
Spherically symmetric spacetimes representing an anisotropic fluid and admitting a proper inheriting conformal Killing vector (ICKV) are studied and all such spacetimes are found. It is shown that the ICKV all lie in the (t,r)-plane except in the case of those solutions that are conformally flat, in which case there exist three angular-dependent ICKV. The physical properties of these solutions are investigated (with particular attention focused on static spacetimes), and, in the context of these solutions, examples are given which can be interpreted as stellar models (with reasonable physical properties), as models of magnetic fields in a plasma, and as models of viscous heat-conducting fluids. It is shown that global monopole solutions cannot admit a proper ICKV.
We present an exact analytical solution of the gravitational field equations describing a static spherically symmetric anisotropic quark matter distribution. The radial pressure inside the star is assumed to obey a linear equation of state, while the tangential pressure is a complicated function of the radial coordinate. In order to obtain the general solution of the field equations a particular density profile inside the star is also assumed. The anisotropic pressure distribution leads to an increase in the maximum radius and mass of the quark star, which in the present model is around three solar masses.
The relativistic interior dynamics of a spherically symmetric, non- rotating star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity is used, and the corresponding second order Einstein equations are found. The equations arise naturally in a non-comoving frame and reduce, in the static case, to the “generalized” Tolman-Oppenheimer-Volkhoff equations considered by J.Kijowski and the author in a previous paper [J. Geom. Phys. 9, No. 3, 207-223 (1992; Zbl 0771.73004)]. The charged case is also discussed.
The equilibrium problem for an axially symmetric, uniformly rotating, elastic solid neutron star is considered within the framework of a new formulation of relativistic elasticity. The analysis is fully relativistic and free of any assumption about the existence of a globally relaxed state of the star. Static configurations may exist only for rigid motions, and for such motions a set of six Einstein equations for six variables is proposed. Due both to the importance of elastic-solid lattices for the structure of neutron stars crusts and to the lack, up to now, of satisfactory axially symmetric perfect fluid interior solutions matching an asimpotically flat exterior, the results seem to indicate the possibility of constructing physically valid models of axially symmetric astrophysical objects in general relativity. The particular case of non-rotating axially symmetric elastic sources is also discussed.
The Einstein equations for a spherically symmetric static distribution of elastic matter are examined. The existence of regular solutions near the center is proven under a fairly mild hypothesis on the constitutive equation. These solutions are uniquely determined by the choice of central pressure and constitutive equation. It is also shown for a Hookean elastic material that these solutions can be integrated outward till the radial pressure vanishes, thus one can join an exterior Schwarzschild metric to obtain a maximal solution of the Einstein equations.
Using the Einstein field equations in the presence of a cosmological constant, the equations which describe the hydrostatic
equilibrium of a static anisotropic fluid sphere are obtained inD (D≥4) space-time dimensions. With suitable transformations, the equation of mass-continuity and of hydrostatic equilibrium are
given in a non-dimensional form. The formalism thus developed is used to study homogeneous charged fluid spheres in higher
dimensions and for these configurations a complete solution is obtained.
We review and discuss possible causes for the appearance of local anisotropy (principal stresses unequal) in self-gravitating systems and present its main consequences. We consider both Newtonian and general relativistic examples. The results emerging from the stability analysis hint at the potential relevance of local anisotropy in the evolution of self-gravitating objects. In this respect particular attention is devoted to the Jeans instability criterion and to the occurrence of cracking in anisotropic spheres. A selection of solutions to Einstein equations for anisotropic matter is analyzed. The specific consequences derived from local anisotropy in these solutions, are exhibited. The differences between two different definitions of energy, within a slowly evolving distribution of anisotropic fluid, are discussed in detail. The conspicuous role played by the Weyl and shear tensors and their relationship with anisotropy of the fluid are brought out.
A description of relativistic elasticity based on a parametrization of material configurations in terms of unconstrained degrees of freedom is proposed. In this description elasticity may be treated as a gauge-type theory, where the role of gauge transformations is played by the diffeomorphisms of the material space. It is shown that the dynamics of the theory may be formulated in terms of three independent, hyperbolic, second-order partial differential equations imposed on three independent gauge potentials. The corresponding unconstrained variational principle is given. The equality between the canonical and the symmetric energy-momentum tensors (Rosenfeld-Belinfante theorem) is proved and the equivalence with existing formulations of the theory is discussed. The canonical (unconstrained) momenta conjugate to the three configuration variables and the Hamiltonian of the system are found.
The relativistic theory of elasticity is reviewed within the spherically symmetric context with a view towards the modeling of star interiors possessing elastic properties such as theones expected in neutron stars. Emphasis is placed on generality in the main sections of the paper, and the results are then applied to specific examples. Along the way, a few general results for spacetimes admitting isometries are deduced, and their consequences are fully exploited in the case of spherical symmetry relating them next to the the case in which the material content of the spacetime is some elastic material. This paper extends and generalizes the pioneering work by Magli and Kijowski [1], Magli [2] and [3], and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers [4], [5] and [6]. Comment: 23 pages
A new approach to relativistic elasticity theory is proposed. In this approach the theory becomes a gauge--type theory, with the diffeomorphisms of the material space playing the role of gauge transformations. The dynamics of the elastic material is expressed in terms of three independent, hyperbolic, second order partial differential equations imposed on three (independent) gauge potentials. The relationship with the Carter-Quintana approach is discussed.
We present an equation of state for elastic matter which allows for purely longitudinal elastic waves in all propagation directions, not just principal directions. The speed of these waves is equal to the speed of light whereas the transversal type speeds are also very high, comparable to but always strictly less than that of light. Clearly such an equation of state does not give a reasonable matter description for the crust of a neutron star, but it does provide a nice causal toy model for an extremely rigid phase in a neutron star core, should such a phase exist. Another reason for focusing on this particular equation of state is simply that it leads to a very simple recipe for finding stationary rigid motion exact solutions to the Einstein equations. In fact, we show that a very large class of stationary spacetimes with constant Ricci scalar can be interpreted as rigid motion solutions with this matter source. We use the recipe to derive a static spherically symmetric exact solution with constant energy density, regular centre and finite radius, having a nontrivial parameter that can be varied to yield a mass-radius curve from which stability can be read off. It turns out that the solution is stable down to a tenuity R/M slightly less than 3. The result of this static approach to stability is confirmed by a numerical determination of the fundamental radial oscillation mode frequency. We also present another solution with outwards decreasing energy density. Unfortunately, this solution only has a trivial scaling parameter and is found to be unstable. Comment: 20 pages, 6 figures. Added remarks on our numerical method as well as the relation to the Ernst equation
We present a class of exact solutions of Einstein's gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations. The solutions are obtained by assuming a particular form of the anisotropy factor. The energy density and both radial and tangential pressures are finite and positive inside the anisotropic star. Numerical results show that the basic physical parameters (mass and radius) of the model can describe realistic astrophysical objects like neutron stars. Comment: 10 pages, 5 figures
The aim of this paper is twofold. First, we set up the theory of elastic matter sources within the framework of general relativity in a self-contained manner. The discussion is primarily based on the presentation of Carter and Quintana but also includes new methods and results as well as some modifications that in our opinion make the theory more modern and transparent. For instance, the equations of motion for the matter are shown to take a neat form when expressed in terms of the relativistic Hadamard elasticity tensor. Using this formulation we obtain simple formulae for the speeds of elastic wave propagation along eigendirections of the pressure tensor. Secondly, we apply the theory to static spherically symmetric configurations using a specific equation of state and consider models either having an elastic crust or core. Comment: 33 pages, 5 eps-figures, referencenses and comments added
Black hole interiors (the T-domain) are studied here in great detail. Both the {\em general} and particular T-domain solutions are presented including non-singular ones. Infinitely many local T-domain solutions may be modeled with this scheme. The duality between the T and R domains is presented. It is demonstrated how generally well behaved R-domain solutions will give rise to exotic phases of matter when collapsed inside the event horizon. However, as seen by an external observer, the field is simply that of the Schwarzschild vacuum with well behaved mass term and no evidence of this behaviour may be observed. A singularity theorem is also presented which is independent of energy conditions.
Some exact analytical solutions of the static Einstein–Maxwell equations for perfect and anisotropic fluids were found under the assumption of spherical symmetry and the existence of a one‐parameter group of conformal motions. All solutions are matched to the Reissner‘xnNordström metric and possess positive energy density larger than the stresses, everywhere within the sphere.
A family of solutions of the Einstein equations for a spherically symmetric distribution of anisotropic matter is presented, which can be matched with the flat (Minkowskian) space‐time on the boundary of the matter, although the energy density and stresses are nonvanishing within the sphere.
A heuristic procedure is developed to obtain interior solutions of Einstein's equations for anisotropic matter from known solutions for isotropic matter. Five known solutions are generalized to give solutions with anisotropic sources.
For specific equations of state, the paper studies the adiabatic
contraction of a sphere with anisotropic stresses for different degrees
of anisotropy. Attention is given to the equation of motion of particles
which is integrated numerically for different initial conditions. The
local pressure-density law for equilibrium is found to be different for
different regions in the material. Finally, it is noted that in some
examples the models are more stable than the isotropic model; in others
they are less stable.
A new interior Schwarzschild solution is presented. It is static,
spherically symmetric, regular everywhere inside a sphere of coordinate
radius a, and across the surface of this sphere it is joined smoothly to
the exterior Schwarzschild solution. There are no radial stresses inside
the sphere. The radius a is subject to the inequality a > 2m, where m
is the gravitational mass of the sphere. Under certain conditions the
new solution may be interpreted as the field inside an Einstein cluster.
A new formulation of relativistic elastomechanics is presented. It is free of any assumption about the existence of a global relaxation state of the material. The strain, the stress and the energy-momentum tensors are expressed in terms of the first-order derivatives of a field describing the configuration of the material. Its elastic properties are completely determined by a scalar function describing the dependence of the mechanical energy accumulated in the deformed material upon the three invariants of the deformation. The stress-strain relations are generated in a canonical way by this function. Dynamical equations of the theory are derived from the variational principle. They form a hyperbolic system of second-order partial differential equations for the unknown field. Energy-momentum conservation laws are consequences of the Noether theorem. The Hamiltonian formulation of the dynamics and the linear version of the theory are also discussed.
We present various analytic solutions for anisotropic fluid spheres in general relativity. First we consider genralizations of the P=αρ solution to the case where pressure is anisotropic, and study the effects of anisotropy on the structure of neutron stars. Next we study radiating anisotropic fluid spheres and present three classes of analytic solutions. We also study slowly rotating anisotropic fluid spheres and present two analytic solutions corresponding to the nonradiating case. One of these solutions corresponds to uniform rotation, while the other corresponds to differential rotation. We also present differential equations to be solved for slowly rotating and radiating anisotropic fluid spheres.
A two-perfect-fluid model of an anisotropic fluid is presented. The energy-momentum tensor associated with the sum of two perfect fluids, one perfect and one null fluid, and two null fluids is examined. Special attention is devoted to the study of the stress tensor. The special case wherein the two perfect fluids are irrotational is studied. A relation between the Einstein equations for this particular case and the Einstein equations for a massless complex scalar field is found. The general solution of Einstein equations for an anisotropic fluid constructed with two-null-fluid components in the plane-symmetric case is discussed. The energy-momentum tensor of a cloud of strings and the energy-momentum tensor of an anisotropic fluid formed by two null fluids are compared.
This book is intended to provide a thorough introduction to the theory
of general relativity. It is intended to serve as both a text for
graduate students and a reference book for researchers. According to
general relativity, as formulated by Einstein in 1915, the intrinsic,
observer-independent, properties of spacetime are described by a
spacetime metric, as in special relativity. The structure of spacetime
is related to the matter content of spacetime. Manifolds and tensor
fields are considered along with curvature, Einstein's equation,
isotropic cosmology, the Schwarzschild solution, methods for solving
Einstein's equation, causal structure, singularities, the initial value
formulation, asymptotic flatness, black holes, spinors, and quantum
effects in strong gravitational fields. Attention is also given to
topological spaces, maps of manifolds, Lie derivatives, Killing fields,
conformal transformations, and Lagrangian and Hamiltonian formulations
of Einstein's equation.
Two new exact analytical solutions to Einstein’s field equations representing static fluid spheres with anisotropic pressures are presented. One solution has a maximum value of mass of about 0.42 times the radius of the fluid sphere, dictated by causality and the corresponding values for the surface red shift and the central red shift are 1.58 and 9.03, respectively. The other solution has a maximum mass of about 0.435 times the radius of the fluid sphere and the corresponding red shifts from the surface and from the center are 1.77 and 16.28, respectively. In the low mass limit both solutions reduce to the constant density Schwarzschild interior solution.
Based on the thermodynamical equations of a previous work, which synthesized the axiomatic approach to relativistic continuum physics with the acceptance of the relativistically invariant ponderomotive force and couple and electromagnetic power derived by de Groot and Suttorp from a microscopical treatment, this paper develops a constitutive theory for general relativistic deformable solids. After a review of the essentials of deformation theory of general relativistic continua, particular attention is paid to nonlinear elastic electromagnetic solids in normal conditions of pressure and temperature, then to the linear piezoelectric scheme which can be used for the theoretical analysis of gravitational‐wave detectors that use a piezoelectric device, and, finally, to magnetoelasticity under high pressure which describes the magnetomechanical behavior of matter in certain astrophysical objects (neutron stars).
Low-power nonlinear frequency conversion techniques such as second
harmonic generation (SHG) are being used to develop compact sources of
coherent green or blue visible radiation. Innovative methods must be
employed to achieve the conversion efficiencies required for practical
applications. The possibility of enhancing nonlinear interactions by
confinement in a waveguide led us to explore the production of single
-crystal lithium niobate (LiNbO_3) fibers. All the processing steps
necessary to make devices suitable for SHG were studied: (1)
laser-heated pedestal growth of single-crystal fibers (including
feedback control for maintaining a constant diameter); (2) control of
ferroelectric domain structure ("poling") for phase-matching; (3)
annealing for transparency and control of composition; (4) cladding with
a lowered refractive index for low-loss waveguiding; and (5) handling,
mounting, and polishing for fiber preparation and use. A diameter
measurement and control system for single-crystal fiber growth was
developed which produced fibers having diameter fluctuations well under
0.1% rms, and magnesium diffusion into LiNbO_3 for application to
in-diffused claddings for LiNbO _3 fibers was investigated. Methods for
producing LiNbO_3 fibers with rapidly-alternating sign of ferroelectric
domains were invented, and a model to explain the basic behavior of the
in-growth poling process was postulated. Periodically -poled LiNbO_3
produced using these methods was applied to quasi-phase-matched SHG of
blue light with wavelengths and nonlinear coefficients which would have
been impossible to phase-match using conventional birefringent
phase-matching techniques. In quasi-phase -matching (QPM), the sign of
the nonlinear coefficient is reversed at intervals spaced such that the
proper phase relationship between the fundamental and second-harmonic
waves is maintained on average, enabling the second-harmonic wave to
build up along the entire interaction length. The effect of departures
from ideal QPM were analyzed, including the tuning properties and
acceptance bandwidths of periodic structures, and deviations from
perfect periodicity. A technique for the synthesis of QPM tuning curves
using only sign-reversals in the coupling coefficient was devised.
Engineering the optical properties of an existing, well-studied material
provides an attractive alternative to the search for entirely new
materials, and the demonstration of this capability represents a
significant contribution to the technology of nonlinear optics.
An ansatz is developed to obtain interior solutions of the Einstein field equations for anisotropic spheres. This procedure necessitates a choice for the energy-density and the radial pressure. A class of solutions for a uniform energy-density source is presented. These anisotropic spheres match smoothly to the Schwarzschild exterior and are well-behaved in the interior of the sphere.
The recent work of Grn [1] concerning charged analogues of Florides' class of solutions is discussed and generalized. The properties of this kind of model are investigated. In particular it is shown that the ratiom/r as well as the acceleration of gravity are maximum inside the body rather than at the boundary. Some exact solutions of the Einstein-Maxwell equations illustrating these properties are presented. The solutions are matched continuously to the exterior Schwarzschild solution and they represent electromagnetic mass models of neutral systems. All physical quantities are finite inside the distributions. The energy density is positive and decreases monotonically from its maximum value at the center to zero at the boundary.
The Hamiltonian formulation of the theory of a gravitational field interacting with a perfect fluid is considered. There is a natural gauge related to the mechanical and thermodynamical properties of the fluid, which enables us to describe 2 degrees of freedom of the gravitational field and 4 degrees of freedom of the fluid (together with 6 conjugate momenta) by nonconstrained data (g,P) where g is a 3-dimensional metric and P is the corresponding Arnowitt-Deser-Misner momentum. The Hamiltonian of the theory, numerically equal to the entropy of the fluid, generates uniquely the evolution of the data. The Hamiltonian vanishes on the data satisfying the vacuum constraint equations and tends to infinity elsewhere as the amount of the matter tends to zero. In this way the vacuum theory with constraints is obtained as a limiting case of a ``deep potential well'' theory.
InProc. of IX Italian Conf. on General Relativity and Gravitation
- J Kijowski
- G Magli
- R Cianci
- R De
- M Ritis
- G Francaviglia
- C Marmo
- P Rubano
- Scudellaro
InProc. of IX Italian Conf
- J Kijowski
- G Magli
InHamittonian Thermodynamics
- J Jezierski
- J Kijowski
InSolvay conference on Astrophysics and Gravitation (Brussels)
- V Canuto
Elasticità finita e relativistica: introduzione ai metodi geometrici della teoria dei campi
- J Kijowski
Elasticit� finita e relativistica: introduzione ai metodi geometrici della teoria dei campi, Quad. Unione Matematica Italiana
- J Kijowski
Rendiconti di Matematica 8
- C Cattaneo
- A Gerardi
- C. Cattaneo