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.     

On static spherically symmetric solutions of the vacuum Brans-Dicke theory

It is shown that among the four classes of the static spherically symmetric solutions of the vacuum Brans-Dicke theory of gravity only two are really independent. Further, by matching exterior and interior (due to physically reasonable spherically symmetric matter source) scalar fields it is found that only the Brans class I solution with a certain restriction on the solution parameters may represent an exterior metric for a nonsingular massive object. The physical viability of the black hole nature of the solution is investigated. It is concluded that no physical black hole solution different from the Schwarzschild black hole is available in the Brans-Dicke theory.     

Fluid spheres and R- and T-regions in general relativity

R- and T-regions of spacetime are first defined in a particular coordinate system and then with the aid of the Schwarzschild vacuum solution are shown to represent the outside and inside of a black hole respectively. A certain class of interior solutions, relating to a perfect fluid, are also considered and it is found that these R- and T-solutions have distinct physical properties. The R-solutions are static, spherically symmetric, permanent, and have a classical analogue, while the corresponding T-solutions, which are wholly time dependent, are cylindrical, temporary, and do not have a classical analogue. It is shown that these T-solutions cannot be generated from their R-region counterparts. Particular T-solutions are also presented in which the corresponding fluid occupies the whole of a T-region. The fluid would under certain circumstances be black body radiation while for other cases the internal pressure is always greater than the density.     

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.     

Cylindrically symmetric distributions of matter taking into account pressure

A cylindrically symmetric distribution of matter under pressure which evolves with time is considered. A new class of non-steady-state solutions to the Einstein equations with cylindrical symmetry is found (for cases when the matter is under zero, constant, and time-dependent pressure). A homogeneous, anisotropic universe and an analog of the Schwarzschild solution are considered as particular cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 33–36, December, 1985.     

A rotating three component perfect fluid source and its junction with empty space-time

The Kerr solution for empty space-time is presented in an ellipsoidally symmetric coordinate system and it is used to produce generalised ellipsoidal metrics appropriate for the generation of rotating interior solutions of Einstein’s equations. It is shown that these solutions are the familiar static perfect fluid cases commonly derived in curvature coordinates but now endowed with rotation. These are also shown to be potential fluid sources for not only Kerr but also Kerr-de Sitter empty space-time. The approach is further discussed in the context of T-solutions of Einstein’s equations and the vacuum T-solution outside a rotating source is presented. The interior source for these solutions is shown not to be a perfect fluid but rather an anisotropic three component perfect fluid for which the energy momentum tensor is derived. The Schwarzschild interior solution is given as an example of the approach.     

A new algorithm for anisotropic solutions

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.     

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     

A pure geometric approach to stellar structure

The present work represents a step in dealing with stellar structure using a pure geometric approach. Geometric field theory is used to construct a model for a spherically symmetric configuration. In this case, two solutions have been obtained for the field equations. The first represents an interior solution which may be considered as a pure geometric one in the sense that the tensor describing the material distributions is not a phenomenological object, but a part of the geometric structure used. A general equation of state for a perfect fluid, is obtained from, and not imposed on, the model. The second solution gives rise to Schwarzschild exterior field in its isotropic form. The two solutions are matched, at a certain boundary, to evaluate the constants of integration. The interior solution obtained shows that there are different zones characterizing the configuration: a central radiation dominant zone, a probable convection zone as a physical interpretation of the singularity of the model, and a corona like zone. The model may represent a type of main sequence stars. The present work shows that Einstein’s geometerization scheme can be extended to gain more physical information within material distribution, with some advantages.     

Are there any models with homogeneous energy density?

By applying a recent method—based on a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor—to the simple spherical static case, we found that the only static solution with homogeneous energy density is the Schwarzschild solution and that there are no spherically symmetric dynamic solutions consistent with the homogeneous energy density assumption. Finally, a circular equivalence is shown among the most frequent conditions considered in the spherical symmetric case: homogeneous density, isotropy in pressures, conformally flatness and shear-free conditions. We demonstrate that, due to the regularity conditions at the center of the matter distribution, the imposition of two conditions necessarily leads to the static case.     

Space-times containing perfect fluids and having a vanishing conformal divergence

The solutions of the Einstein field equations are studied under the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the divergence of the conformal (Weyl) tensor vanishes, and (3a) either an equation of state exists such thatp=p (w),p being the pressure andw the rest energy density, or (3b) the rest particle density is conserved. Under assumptions (1), (2), and (3a) it is shown that the space-time is conformally flat and the metric is a Robertson-Walker metric. The flow is irrotational, shear-free, and geodesic. Under assumptions (1), (2), and (3b) it is shown that either the line element is static or the fluid has a very special caloric equation of state. Conditions for a static solution to exist are examined, and it is shown that the Schwarzschild interior solution satisfies these conditions as does the Einstein universe. The Schwarzschild interior and the Einstein universe are the only conformally flat, static solutions obeying (1), (2), and (3b).The research reported herein was supported in part by the Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement No. 125.     

The Lynden-Bell and Katz definition of gravitational energy: Applications to singular solutions

We apply the Lynden-Bell and Katz (LK) definition of gravitational energy to static and spherically symmetric space-times which admit a curvature singularity. These are the Tolman V, Tolman VI and the interior Schwarzschild solutions, the latter with the boundary limit of 9/8th of the gravitational radius. We show that the LK definition can still be applied to these solutions despite the presence of a singularity which nonetheless appears to carry no energy in the LK sense. While in the solutions that we mentioned the KL gravitational energy is positive definite everywhere in space time, this is not the case for the overcharged Reissner-Nordström space-time. In the latter case in fact the LK energy density becomes negative sufficiently close to the singularity hence we use the positivity criterion to impose a more stringent limit of validity to the Reissner-Nordström 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.     

Interior Schwarzschild Problem and Its Integration

The interior Schwarzschild metric for a static,spherically symmetric perfect fluid can be parametrizedwith two independent functions of the radial coordinate.These functions are easily expressed in terms of (radial) integrals involving the fluidenergy density and pressure. The pressure is, however,not independent, but is determined in terms of thedensity by one of Einstein's equations, theOppenheimer–Volkov (OV) equation. An approximate integral to theOV equation is presented which is accurate for slowlyvarying, realistic, densities, and exact in theconstant-density limit. It makes it possible to findcompletely integrated accurate solutions to the interiorSchwarzschild metric in terms of the density only. Somepost-Newtonian consequences of the solution are given aswell as the resulting general relativistic pressure for an energy densityr^-1/2.     

On ``Time-Periodic' Black-Hole Solutions to Certain Spherically Symmetric Einstein-Matter Systems

This paper explores black hole solutions of various Einstein-wave matter systems admitting a time-orientation preserving isometry of their domain of outer communications taking some point to its future. In the first two parts, it is shown that such solutions, assuming in addition that they are spherically symmetric and the matter has a certain structure, must be Schwarzschild or Reissner-Nordström. Non-trivial examples of matter for which the result applies are a wave map and a massive charged scalar field interacting with an electromagnetic field. The results thus generalize work of Bekenstein [1] and Heusler [13] from the static to the periodic case. In the third part, which is independent of the first two, it is shown that Dirac fields preserved by an isometry of a spherically symmetric domain of outer communications of the type described above must vanish. It can be applied in particular to the Einstein-Dirac-Maxwell equations or the Einstein-Dirac-Yang/Mills equations, generalizing work of Finster, Smoller and Yau [10, 8, 9 and also 7].     

Compact anisotropic spheres with prescribed energy density

Abstract New exact interior solutions to the Einstein field equations for anisotropic spheres are found. We utilise a procedure that necessitates a choice for the energy density and the radial pressure. This class contains the constant density model of Maharaj and Maartens (Gen. Rel. Grav. 21, 899–905 (1989)), and the variable density model of Gokhroo and Mehra (Gen. Rel. Grav. 26, 75–84 (1994)), as special cases. These anisotropic spheres match smoothly to the Schwarzschild exterior and gravitational potentials are well behaved in the interior. A graphical analysis of the matter variables is performed which points to a physically reasonable matter distribution.     

Sources for the Reissner-Nordström metric

New, physically motivated static sources for the Reissner-Nordström metric are found. One is a generalization of the Schwarzschild interior solution representing a sphere of constant nongravitational energy density. The other is a family of solutions for which the mass is electromagnetic in origin. Some general results are found. For a charged fluid sphere in equilibrium with pressure,m ² >q ². For a charged body with equation of state = (p), where (0)=0, the body is under tension at every point when the charge density has the same sign throughout.Supported by the National Research Council of Canada, Grant No. A5340.     

Letter: Slowly Rotating, Compact Fluid Sources Embedded in Kerr Empty Space-Time

Spherically symmetric static fluid sources are endowed with rotation and embedded in Kerr empty space-time up to and including quadratic terms in an angular velocity parameter using Darmois junction conditions. The boundary behaviour of the metric tensor and partial derivatives is used to develop a series solution of Einstein's equation's for the rotating fluid. The boundary of the rotating source is expressed explicitly in terms of sinusoidal functions of the polar angle. As an example of the analysis the Schwarzschild interior solution is endowed with rotation and the equation of the fluid boundary is generated together with surface behaviour of the fluid density and angular velocity.     

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.     

On a new interior Schwarzschild solution

It is shown explicitly that a new interior Schwarzschild solution satisfies a set of necessary and sufficient conditions for a spherically symmetric metric to join smoothly onto the vacuum field at a nonnull boundary surface. Moreover, the conditions do not prevent the radius of a spherical distribution from assuming values arbitrarily close to the Schwarzschild radius.