Perfect fluid metrics conformal to the Schwarzschild exterior spacetime

We construct perfect fluid spacetimes by performing a conformal transformation on a non-conformally flat vacuum solution, namely the Schwarzschild exterior metric. It should be noted that conformally Ricci flat perfect fluid solutions, except those that are conformally flat, are rarely reported explicitly. In this article it is demonstrated that perfect fluid metrics conformal to the Schwarzschild exterior line element are necessarily static. The Einstein field equations for the static case reduce to a fully determined system of three differential equations in three unknowns and the conformal factor is uniquely determined in closed form. The solution is analysed for physical plausibility by establishing the positivity of the energy density and pressure profiles graphically. Additionally, the solution is observed to be causal in an appropriate limit and both the energy density and pressure is shown to be decreasing outwards towards the boundary. Finally, the weak, strong and dominant energy conditions are found to be satisfied in the region under investigation. Accordingly, the most common elementary physical conditions are met and the model is seen to be suitable for a core-envelope stellar model.     

A charged generalization of Florides' interior Schwarzschild solution

The new class of interior Schwarzschild solutions found by Florides is generalized to the charged case. A particular solution within this class is found, which represents an electromagnetic mass-model of a neutral spherically symmetric system. The pressure is isotropic, decreasing monotonously with increasing radius and vanishes at the surface of the matter distribution. The solution is regular everywhere inside a radiusR, and is joined continuously to the exterior Schwarzschild solution at this radius.     

Innere Reissner-Weyl-Lösung

With the aid of an invariance transformation of the Lagrangian we obtain a class of exact static solutions of the Einstein-Maxwell equations including perfect fluid. Application of the method to the interior and exterior Schwarzschild solution yields a corresponding solution with electromagnetic field (Reissner-Weyl solution). The boundary conditions of the resulting metric are automatically fulfilled.     

Generalized Einstein static universe

We consider the equilibrium configurations for static spherically symmetric self-gravitating perfect fluids in general relativity. The fluid obeys an equation of state which is the extreme limit allowed by Hawking's strong energy condition. The new solutions we have found are generalized Einstein static universes which have a Killing horizon similar to the one we are familiar with from the vacuum Schwarzschild solution.     

Bondian frames to couple matter with radiation

A study is presented for the non linear evolution of a self gravitating distribution of matter coupled to a massless scalar field. The characteristic formulation for numerical relativity is used to follow the evolution by a sequence of light cones open to the future. Bondian frames are used to endow physical meaning to the matter variables and to the massless scalar field. Asymptotic approaches to the origin and to infinity are achieved; at the boundary surface interior and exterior solutions are matched guaranteeing the Darmois–Lichnerowicz conditions. To show how the scheme works some numerical models are discussed. We exemplify evolving scalar waves on the following fixed backgrounds: (a) an atmosphere between the boundary surface of an incompressible mixtured fluid and infinity; (b) a polytropic distribution matched to a Schwarzschild exterior; (c) a Schwarzschild–Schwarzschild spacetime. The conservation of energy, the Newman–Penrose constant preservation and other expected features are observed.     

An improved singular perturbation solution for bound geodesic orbits in the Schwarzschild metric

A modification to the Lindstedt-Poincaré method of strained parameters is applied to the differential equation of the orbit of a test particle in the Schwarzschild exterior metric. A new perturbation solution for the equation of the bound orbit, which is completely free of secular terms in the angular coordinate, is derived. The precession of the orbit per revolution is calculated using this solution and it is found to give a more accurate result than existing perturbation solutions. The method should be applicable to similar orbital problems in general relativity.     

Static anisotropic fluid spheres in general relativity with nonuniform density

An Ansatz developed by Maharaj and Maartens is used to obtain solutions of Einstein's field equations for static anisotropic fluid spheres with nonuniform density. These solutions are matched with the Schwarzschild exterior solution.     

A theorem on the conformal structure of interior space-times

It is well known that the interior and exterior Schwarzschild solutions are of different conformal type. More examples of this phenomenon are easily found in the literature. A precise statement of this phenomenon is given together with a theorem stating that this will take place if the matter tensor of the interior solution satisfies some weak energy conditions and if some reasonable conditions are fulfilled.     

An interior solution for the gamma metric

Interior solutions for a static, axially symmetric family of solutions of Einstein's equations are described. The interior solutions correspond to spatially bound matter and are properly matched to an exterior vacuum solution. The family of solutions discussed include the Schwarzschild solution as a special case. A general method is exhibited for transforming any spherically symmetric interior solution to an interior for the other members of the family of solutions. The energy density remains positive for at least a finite range of the parameter that describes the family of solutions. Two solutions are explicitly exhibited. One is transformed from the constant density Schwarzschild interior solution and one from the Adler interior solution. The first solution would be expected to be unstable under adiabatic perturbations of the matter, the second would be expected to be stable.Supported in part by The National Science Foundation under Grant No. INT 782-5663.Supported in part by Consejo Nacional de Investigaciones Cientificas y Technologicas (CONICIT), Venezuela.     

Physical properties of an exact spherically symmetric solution with shear in general relativity

The properties of an exact spherically symmetric perfect fluid solution obtained in non-comoving coordinates are examined. This solution contains shear, and the pressure and the density are positive in the interior of the fluid. Their respective gradients with respect to comoving radial coordinate are equal and negative, and the speed of sound in this fluid is less than the speed of light in vacuum and is increasing outwards. There is a singularity at the center of the fluid since the pressure and the density become infinite there, though their ratio becomes unity. This singularity is naked, since there does not exist a trapped surface in the fluid outside this singularity. The circumference is an increasing function of radical comoving coordinate, and the mass function is positive and is increasing outwards. There are no tidal forces in radial direction, but the tidal forces normal to this direction are non-vanishing. We also give the kinematic quantities for this fluid. However, it is not possible to match this solution with an exterior vacuum Schwarzschild solution. Moreover, the dominant energy condition produces imaginary values for the sound speed.     

Spherically Symmetric Static Solutions of the Einstein Equations with Elastic Matter Source

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.     

Static spherically symmetric star in Gauss-Bonnet gravity

We explore static spherically symmetric stars in Gauss-Bonnet gravity without a cosmological constant, and present an exact internal solution which attaches to the exterior vacuum solution outside stars. It turns out that the presence of the Gauss-Bonnet term with a positive coupling constant completely changes thermal and gravitational energies, and the upper bound of the red shift of spectral lines from the surface of stars. Unlike in general relativity, the upper bound of the red shift is dependent on the density of stars in our case. Moreover, we have proven that two theorems for judging the stability of equilibrium of stars in general relativity can hold in Gauss-Bonnet gravity.     

Five-dimensional teleparallel theory equivalent to general relativity, the axially symmetric solution, energy and spatial momentum

A theory of (4+1)-dimensional gravity is developed on the basis of the teleparallel theory equivalent to general relativity. The fundamental gravitational field variables are the five-dimensional vector fields (pentad), defined globally on a manifold M, and gravity is attributed to the torsion. The Lagrangian density is quadratic in the torsion tensor. We then give the exact five-dimensional solution. The solution is a generalization of the familiar Schwarzschild and Kerr solutions of the four-dimensional teleparallel equivalent of general relativity. We also use the definition of the gravitational energy to calculate the energy and the spatial momentum.     

A two-mass expanding exact space-time solution

In order to understand how locally static configurations around gravitationally bound bodies can be embedded in an expanding universe, we investigate the solutions of general relativity describing a space–time whose spatial sections have the topology of a 3-sphere with two identical masses at the poles. We show that Israel junction conditions imply that two spherically symmetric static regions around the masses cannot be glued together. If one is interested in an exterior solution, this prevents the geometry around the masses to be of the Schwarzschild type and leads to the introduction of a cosmological constant. The study of the extension of the Kottler space–time shows that there exists a non-static solution consisting of two static regions surrounding the masses that match a Kantowski–Sachs expanding region on the cosmological horizon. The comparison with a Swiss-Cheese construction is also discussed.     

均匀密度零压星的完整的引力解

证明了Oppenheimer和Snyder关于均匀密度零压星的引力塌缩的经典解是不完整的,它并不能正确地连接作为内解和外解的Friedmann度规和Schwarzschild度规;通过在离散时空上拓展解参数而构成了一个完整的引力解,它实现了Friedmann度规和Schwarzschild度规之间的等价连接,并可以证明是奇性自由的;这个完整的引力解显示了物质,引力和离散时空结构之间的关联性 关键词： 均匀密度零压星 Friedmann度规 Schwarzschild度规 离散时空     

Horizon-less Spherically Symmetric Vacuum-Solutions in a Higgs Scalar-Tensor Theory of Gravity

The exact static and spherically symmetric solutions of the vacuum field equations for a Higgs Scalar-Tensor theory (HSTT) are derived in Schwarzschild coordinates. It is shown that in general there exists no Schwarzschild horizon and that the fields are only singular (as naked singularity) at the center (i.e. for the case of a point-particle). However, the Schwarzschild solution as in usual general relativity (GR) is obtained for the vanishing limit of Higgs field excitations.     

The energy distribution for a spherically symmetric isolated system in general relativity

The problems of the tolal energy and quasilocalenergy density or an isolated spherically symmetric static system in general relativity (GR) are considered with examples of some exact suintions. The field formulation of GR dereloped earlier hy L. P. Grishchuk. el al. (1984). in ihe framework of which all the dynamical fields, including the gravitation field, are considered in a fixed background spacetime is used intensively. The exact Schwarzschild and Reissner Nordstrom solutions are investigated in detail, and the results are compared with those in the recent work by J. D. Brown and J. W. York. Jr. (1993) as well as discussed with respect to the principle of nonlocalization of the gravitational energy in GR. Those examples are illustrative and simple because the background is selected as Minkowski spacetime and, in fact, the field configurations are studied in the framework of special relativity. It is shown that some problems of the Schwarzschild solution which are difficult to resolve in the standard geometrical framework of GR are resolved in the framework of the field formulation.     

General relativistic electromagnetic mass models of neutral spherically symmetric systems

The recent work of Grøn [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.     

Interior of a Schwarzschild Black Hole Revisited

The Schwarzschild solution has played a fundamental conceptual role in general relativity, and beyond, for instance, regarding event horizons, spacetime singularities and aspects of quantum field theory in curved spacetimes. However, one still encounters the existence of misconceptions and a certain ambiguity inherent in the Schwarzschild solution in the literature. By taking into account the point of view of an observer in the interior of the event horizon, one verifies that new conceptual difficulties arise. In this work, besides providing a very brief pedagogical review, we further analyze the interior Schwarzschild black hole solution. Firstly, by deducing the interior metric by considering time-dependent metric coefficients, the interior region is analyzed without the prejudices inherited from the exterior geometry. We also pay close attention to several respective cosmological interpretations, and briefly address some of the difficulties associated to spacetime singularities. Secondly, we deduce the conserved quantities of null and timelike geodesics, and discuss several particular cases in some detail. Thirdly, we examine the Eddington–Finkelstein and Kruskal coordinates directly from the interior solution. In concluding, it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined, and is still open to considerable debate.