All "static" spherically symmetric perfect fluid solutions of Einstein's equations with constant equation of state parameter and finite-polynomial "mass function"
Ibrahim Semiz
备注：
24 standard LaTeX pages, 4 tables, no figures. -Title changed to avoid equation in title. -The set of solutions and their interpretation remains unchanged, but new classification of solutions (solution labels also changed), consolidation of appendix into a table (omitting calculation details) resulted in shorter paper. -Updated Publication info for preprints in References
We look for "static" spherically symmetric solutions of Einstein's Equations for perfect fluid source with equation of state $p=wρ$. In order to include the possibilities of recently popularized dark energy and phantom energy possibly pervading the spacetime, we put no constraints on the constant $w$. We consider all four cases compatible with the standard ansatz for the line element, discussed in previous work. For each case we derive the equation obeyed by the mass function or its analogs. For these equations, we find {\em all} finite-polynomial solutions, including possible negative powers.
For the standard case, we find no significantly new solutions, but show that one solution is a static phantom solution, another a black hole-like solution. For the dynamic and/or tachyonic cases we find, among others, dynamic and static tachyonic solutions, a Kantowski-Sachs (KS) class phantom solution, another KS-class solution for dark energy, and a second black hole-like solution.
The black hole-like solutions feature segregated normal and tachyonic matter, consistent with the assertion of previous work. In the first black hole-like solution, tachyonic matter is inside the horizon, in the second, outside.
The static phantom solution, a limit of an old one, is surprising at first, since phantom energy is usually associated with super-exponential expansion. The KS-phantom solution stands out since its "mass function" is a ninth order polynomial.
We investigate solutions of Einstein field equations for the non-static spherically symmetric perfect fluid case using different equations of state. The properties of an exact spherically symmetric perfect fluid solutions are obtained which contain shear. We obtain three different solutions out of these one turns out to be an incoherent dust solution and the other two are stiff matter solutions.
This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian Gravity and General Relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding perfect fluid solutions possess finite/infinite extent. In the Newtonian case, for the large class of monotonic equations of state, and in General Relativity we improve earlier results.
In this short note we comment about some criticisms - appeared in a recent paper by Iguchi et al - to our previous works on gravitational collapse of perfect fluids. We show that those criticisms are incorrect on their own.
Using an estimate, we prove that if solution of the spherically symmetric Einstein-Vlasov-Maxwell system develops a singularity at all time, then the first one has to appear at the center of symmetry.
We propose a new test of Einstein's theory of gravitation. It concerns the velocity distribution of low-energy particles in a spherically symmetric gravitational field.
We categorize the global structure of spherically symmetric static solutions of Einstein SU(2) Yang Mills equations with positive cosmological constant that are smooth at the center of spherical symmetry.
Using the iterative Scheme we prove the local existence and uniqueness of solutions of the spherically symmetric Einstein-Vlasov-Maxwell system with small initial data. We prove a continuation criterion to global in-time solutions.
We study the spherically symmetric collapse of a perfect fluid using area-radial coordinates. We show that analytic mass functions describe a static regular centre in these coordinates. In this case, a central singularity can not be realized without an infinite discontinuity in the central density. We construct mass functions involving fluid dynamics at the centre and investigate the relationship between those and the nature of the singularities.
The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given.
Using ODE techniques we prove the existence of large classes of initial data satisfying the constraints for the spherically symmetric Einstein-Vlasov-Maxwell system. These include data for which the ratio of total charge to total mass is arbitrarily large.
Einstein's field equations for timelike self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively using the theory of dynamical systems.
Einstein's field equations for spatially self-similar spherically symmetric perfect-fluid models are investigated. The field equations are rewritten as a first-order system of autonomous differential equations. Dimensionless variables are chosen in such a way that the number of equations in the coupled system is reduced as far as possible and so that the reduced phase space becomes compact and regular. The system is subsequently analysed qualitatively with the theory of dynamical systems.
We critically examine the Roberts homothetic solution for the spherically symmetric Einstein-scalar field equations in double null coordinates, and show that the Roberts solution indeed solves the field equations only for one non-trivial case. We generalize this solution and discuss its relations with other known exact solutions.
We consider static spherically symmetric solutions of the Einstein equations with cosmological constant coupled to the SU(2) Yang Mills equations. We prove that most solutions can be continued back to the origin of spherical symmetry and that the qualitative behavior of solutions near this origin does not depend on the value of the cosmological constant.
The Proca wave equation describes a classical massive spin 1 particle. We analyze the gravitational interaction of this vector field. In particular, the spherically symmetric solutions of the Einstein-Proca coupled system are obtained numerically. Although at infinity the metric field approaches the usual Schwarzschild (Reissner-Nordström) limit, we demonstrate the absence of black hole type configurations.
We use Moeller's energy-momentum complex in order to explicitly compute the energy and momentum density distributions for an exact solution of Einstein's field equations with a negative cosmological constant minimally coupled to a static massless scalar field in a static, spherically symmetric background in (2+1)-dimensions.
The present work investigates the gravitational collapse of a perfect fluid in $f(R)$ gravity models. For a general $f(R)$ theory, it is shown analytically that a collapse is quite possible. The singularity formed as a result of the collapse is found to be a curvature singularity of shell focusing type. The possibility of the formation of an apparent horizon hiding the central singularity depends on the initial conditions.
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial spacelike hypersurface with a timelike boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
We prove a new global existence result for the asymptotically flat, spherically symmetric Einstein-Vlasov system which describes in the framework of general relativity an ensemble of particles which interact by gravity. The data are such that initially all the particles are moving radially outward and that this property can be bootstrapped. The resulting non-vacuum spacetime is future geodesically complete.
We consider static spherically symmetric solutions of Einstein's equations coupled to an SU(2) Yang Mills field that are smooth at the center of spherical symmetry. We prove that with small cosmological constant there exist solutions that possess a coordinate singularity at some r that is not maximum. The singularity can be removed with a Kruskal-like change of coordinates.
It is shown that for the spherically-symmetric and static systems the hypotheses posed by Yang and Radinschi and by Vagenas can be related to the particular distribution of the source. Simple proofs are given and a number of examples are discussed with the special emphasis put on the quantum corrected Schwarzschild black hole.
We consider the self-similar solutions associated with the critical behavior observed in the gravitational collapse of spherically symmetric perfect fluids with equation of state $p=αμ$. We identify for the first time the global nature of these solutions and show that it is sensitive to the value of $α$. In particular, for $α>0.28$, we show that the critical solution is associated with a new class of asymptotically Minkowski self-similar spacetimes. We discuss some of the implications of this for critical phenomena.
We classify all spherically symmetric and homothetic spacetimes that are allowed kinematically by constructing them from a small number of building blocks. We then restrict attention to a particular dynamics, namely perfect fluid matter with the scale-free barotropic equation of state p = alpha mu where 0<alpha<1 is a constant. We assign conformal diagrams to all solutions in the complete classification of Carr and Coley, and so establish which of the kinematic possibilities are realized for these dynamics. We pay particular attention to those solutions which arise as critical solutions during…
According to Birkhoff's theorem the only spherically symmetric solution of the vacuum Einstein field equations is the Schwarzschild solution. Inspite of imposing asymptotically flatness and staticness as initial conditions we obtain that these equations have general solutions with the Schwarzschild metric as merely a special and simplest form of them. It is possible to have perfect and smooth metrics with the same Newtonian and post-Newtonian limits of Schwarzschild by a convenient and correct selection.
The Einstein equations with small positive cosmological constant coupled to an SU(2) Yang Mills field admits solutions that possess a coordinate singularity at a noncritical radius. Here, we prove that these solutions are otherwise globally smooth and that they asymptotically approach Schwarzschild deSitter space with a vanishing Yang Mills field.
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.
It is shown analytically that every static, spherically symmetric solution to the Einstein Yang Mills equations with SU(2) gauge group that is defined in the far field has finite ADM mass. Moreover, there can be at most two horizons for such solutions. The three types of solutions possible, Bartnik-McKinnon particle-like solutions, Reissner-Nordstrom-like solutions, and black hole solutions having only one horizon are distinguished by the behavior of the metric coefficients at the origin.
By a choice of new variables the pressure isotropy condition for spherically symmetric static perfect fluid spacetimes can be made a quadratic algebraic equation in one of the two functions appearing in it. Using the other variable as a generating function, the pressure and the density of the fluid can be expressed algebraically by the function and its derivatives. One of the functions in the metric can also be expressed similarly, but to obtain the other function, related to the redshift factor, one has to perform an integral. Conditions on the generating function ensuring regularity and phys…
For field equations of 4th order, following from a Lagrangian `Ricci scalar plus Weyl scalar', it is shown (using methods of non-standard analysis) that in a neighbourhood of Minkowski space there do not exist regular static spherically symmetric solutions. With that (besides the known local expansions about r = 0 and r = infinity resp.) for the first time a global statement on the existence of such solutions is given. Finally, this result will be discussed in connection with Einstein's particle programme.
The aim of this paper is to continue the research of JMP 46, 042501 (2005) of regular static spherically symmetric spacetimes in Einstein-Born-Infeld theories from the point of view of the spacetime geometry and the electromagnetic structure. The energy conditions, geodesic completeness and the main features of the horizons of this spacetime are explicitly shown. A new static spherically symmetric dyonic solution in Einstein-Born-Infeld theory with similar good properties as in the regular pure electric and magnetic cases of our previous work, is presented and analyzed. Also, the circumvention…
A new class of spacetime defect solutions of Einstein Field equations of Edelen's direct Poincaré Gauge Field theory without biaxial symmetry is presented. The interior solution describes a core of defects where curvature vanishes and Cartan torsion is nonvanishing. Outside the core (in vacuum) the solution represents a spacetime with vanishing curvature and torsion describing a nontrivial topological defect solution of Einstein equations of gravity. Our solution corresponds to a very weak strenght of Tachyons can be found far away from the core defect.
Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy distribution, and its time rate of change on an initial spacelike hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface, but …
In this work a class of interior solution for Einstein field equations corresponding to a spherically symmetric anisotropic fluid sphere has been obtained under the assumption that the cosmological constant is spatially variable. The solution obtained has the characteristics that the pressure and the cosmological parameter vanish at the centre and at the boundary with a maximum value somewhere inside the body. It has been argued that a variable $ Λ$ is as much important physically in Astrophysics as in Cosmology.
In computational relativity, critical behaviour near the black hole threshold has been studied numerically for several models in the last decade. In this paper we present a spatial Galerkin method, suitable for finding numerical solutions of the Einstein-Dirac equations in spherically symmetric spacetime (in polar/areal coordinates). The method features exact conservation of the total electric charge and allows for a spatial mesh adaption based on physical arclength. Numerical experiments confirm excellent robustness and convergence properties of our approach. Hence, this new algorithm is well…
Static spherically symmetric distributions of electrically counterpoised dust (ECD) are used to construct solutions to Einstein-Maxwell equations in Majumdar--Papapetrou formalism. Unexpected bifurcating behaviour of solutions with regard to source strength is found for localized, as well as for the delta-function ECD distributions. Unified treatment of general ECD distributions is accomplished and it is shown that for certain source strengths one class of regular solutions approaches Minkowski spacetime, while the other comes arbitrarily close to black hole solutions.
The conformal field equations are used to discuss the local existence of spherically symmetric solutions to the Einstein-Yang-Mills system which behave asymptotically like the anti-de Sitter spacetime. By using a gauge based on conformally privileged curves we obtain a formulation of the problem in terms of an initial boundary value problem on which a general class of maximally dissipative boundary conditions can be discussed. The relation between these boundary conditions and the notion of mass on asymptotically anti-de Sitter spacetimes is analysed.
We give a complete classification of all static, spherically symmetric solutions of the SU(2) Einstein-Yang-Mills theory with a positive cosmological constant. Our classification proceeds in two steps. We first extend solutions of the radial field equations to their maximal interval of existence. In a second step we determine the Carter-Penrose diagrams of all 4-dimensional space-times constructible from such radial pieces. Based on numerical studies we sketch a complete phase space picture of all solutions with a regular origin.
The Riemann, Ricci and Einstein tensors for N-dimensional spherically symmetric spacetimes in various systems of coordinates are studied, and the general metric for conformally flat spacetimes is given. As an application, all the Friedmann-Robertson-Walker-like solutions for a perfect fluid with an equation of state $p = k ρ$ are found. Then, these solutions are used to model the gravitational collapse of a compact ball, by first cutting them along a timelike hypersurface and then joining them with asymptotically flat Vaidya solutions. It is found that when the collapse has continuous self-sim…
A static spherically symmetric metric in Einstein-scalar-tensor gravity theory with a scalar field potential $V[ϕ]$ is non-singular for all real values of the coordinates. It does not have a black hole event horizon and there is no essential singularity at the origin of coordinates. The weak energy condition $ρ_ϕ> 0$ fails to be satisfied for $r\lesssim 1.3r_S$ (where $r_S$ is the Schwarzschild radius) but the strong energy condition $ρ_ϕ+3p_ϕ> 0$ is satisfied. The classical Einstein-scalar-tensor solution is regular everywhere in spacetime without a black hole event horizon. However, the viol…
In a previous paper two of the authors (G. R. and A. D. R.) showed that there exist global, classical solutions of the spherically symmetric Vlasov-Einstein system for small initial data. The present paper continues this investigation and allows also large initial data. It is shown that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The result adds weight to the conjecture that cosmic censorship holds if one replaces dust as matter model for which naked singularities do form by …
We construct the general spherically symmetric and self-similar solution of the Einstein-Vlasov system (collisionless matter coupled to general relativity) with massless particles, under certain regularity conditions. Such solutions have a curvature singularity by construction, and their initial data on a Cauchy surface to the past of the singularity can be chosen to have compact support in momentum space. They can also be truncated at large radius so that they have compact support in space, while retaining self-similarity in a central region that includes the singularity. However, the Vlasov …
We classify all spherically symmetric spacetimes admitting a kinematic self-similar vector of the second, zeroth or infinite kind. We assume that the perfect fluid obeys either a polytropic equation of state or an equation of state of the form $p=Kμ$, where $p$ and $μ$ are the pressure and the energy density, respectively, and $K$ is a constant. We study the cases in which the kinematic self-similar vector is not only ``tilted'' but also parallel or orthogonal to the fluid flow. We find that, in contrast to Newtonian gravity, the polytropic perfect-fluid solutions compatible with the kinematic…
We review results on the spherically symmetric, asymptotically flat Einstein-Vlasov system. We focus on a recent result where we found explicit conditions on the initial data which guarantee the formation of a black hole in the evolution. Among these data there are data such that the corresponding solutions exist globally in Schwarzschild coordinates. We put these results into a more general context, and we include arguments which show that the spacetimes we obtain satisfy the weak cosmic censorship conjecture and contain a black hole in the sense of suitable mathematical definitions of these …
By using of the Euler-Lagrange equations, we find a static spherically symmetric solution in the Einstein-aether theory with the coupling constants restricted. The solution is similar to the Reissner-Nordstrom solution in that it has an inner Cauchy horizon and an outer black hole event horizon. But a remarkable difference from the Reissner-Nordstrom solution is that it is not asymptotically flat but approaches a two dimensional sphere. The resulting electric potential is regular in the whole spacetime except for the curvature singularity. On the other hand, the magnetic potential is divergent…
The stability features of steady states of the spherically symmetric Einstein-Vlasov system are investigated numerically. We find support for the conjecture by Zeldovich and Novikov that the binding energy maximum along a steady state sequence signals the onset of instability, a conjecture which we extend to and confirm for non-isotropic states. The sign of the binding energy of a solution turns out to be relevant for its time evolution in general. We relate the stability properties to the question of universality in critical collapse and find that for Vlasov matter universality does not seem …
In this paper we present a detailed calculation of an Ansatz that allows to obtain spherically symmetric Einstein-Dirac configurations in $d$-dimensions. We show that this is possible by combining $2^{\lfloor \frac{d-2}{2} \rfloor}$ Dirac fields, making use of the properties of the angular dependence of the spinors in a spherical background. By applying this Ansatz, we investigate some simple analytical solutions. One of them is a regular wormhole supported by the Dirac fields. Other solutions include a pathological black hole and a naked singularity. We analyze the domain of existence and pro…
We extend here the canonical treatment of spherically symmetric (quantum) gravity to the most simple matter coupling, namely spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored. The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We investigate several conceptual and technical details that might be…
We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is a…
This paper explores ``black hole'' solutions of various Einstein-wave matter systems admitting an isometry of their domain of outer communications taking every 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-Nordstrom. 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 [12]…
We review recent results concerning the spherically symmetric Einstein-scalar field system with positive cosmological constant. We do so by comparing with the classical results of Christodoulou concerning the asymptotically flat case (vanishing cosmological constant) and by discussing some of the issues which have emerged since the publication of our main results. Concerning the positive cosmological constant case, we also sketch an alternative proof of global in (Bondi) time existence, based on energy estimates, which is presumably more flexible and, consequently, amenable to generalizations;…
We study spherically-symmetric solutions in Massive Gravity generated by matter sources with polytropic equation of state. We concentrate in the non-perturbative regime where the mass term non-linearities are important, and present the main features of the solutions.
We present here the canonical treatment of spherically symmetric (quantum) gravity coupled to spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored. The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity. We also compare the reduced phase space quantization to the algebraic quantization. Altogether, we o…
We study a static spherically symmetric problem with a black hole and radially directed geodesic flows of dark matter. The obtained solutions have the following properties. At large distances, the gravitational field produces constant velocities of circular motion, i.e., flat rotation curves. At smaller distances, the field switches to Newtonian regime, then to Schwarzschild regime. Deviations from Schwarzschild regime start below the gravitational radius. The dark matter prevents the creation of event horizon, instead, a spherical region possessing extremely large redshift is created. The str…
Using both numerical and analytical tools we study various features of static, spherically symmetric solutions of the Einstein-Vlasov system. In particular, we investigate the possible shapes of their mass-energy density and find that they can be multi-peaked, we give numerical evidence and a partial proof for the conjecture that the Buchdahl inequality $\sup_{r > 0} 2 m(r)/r < 8/9$, $m(r)$ the quasi-local mass, holds for all such steady states--both isotropic {\em and} anisotropic--, and we give numerical evidence and a partial proof for the conjecture that for any given microscopic equation …
The Einstein-Vlasov-Maxwell (EVM) system can be viewed as a relativistic generalization of the Vlasov-Poisson (VP) system. As it is proved below, one of nice property obeys by the first system is that the strong energy condition holds and this allows to conclude that the above system is physically viable. We show in this paper that in the context of spherical symmetry, solutions of the perturbed (EVM) system by $γ:= 1/c^{2}$, $c$ being the speed of light, exist and converge uniformly in $L^{\infty}$-norm, as $c$ goes to infinity on compact time intervals to solutions of the non-relativistic (V…
