Tolman Solution Research Articles

Magnetized Dusty Black Holes and Wormholes

We consider the generalized Tolman solution of general relativity, describing the evolution of a spherical dust cloud in the presence of an external electric or magnetic field. The solution contains three arbitrary functions f(R), F(R) and τ0(R), where R is a radial coordinate in the comoving reference frame. The solution splits into three branches corresponding to hyperbolic (f>0), parabolic (f=0) and elliptic (f<0) types of motion. In such models, we study the possible existence of wormhole throats defined as spheres of minimum radius at a fixed time instant, and prove the existence of throats in the elliptic branch under certain conditions imposed on the arbitrary functions. It is further shown that the normal to a throat is a timelike vector (except for the instant of maximum expansion, when this vector is null), hence a throat is in general located in a T-region of space-time. Thus, if such a dust cloud is placed between two empty (Reissner–Nordström or Schwarzschild) space-time regions, the whole configuration is a black hole rather than a wormhole. However, dust clouds with throats can be inscribed into closed isotropic cosmological models filled with dust to form wormholes which exist for a finite period of time and experience expansion and contraction together with the corresponding cosmology. Explicit examples and numerical estimates are presented. The possible traversability of wormhole-like evolving dust layers is established by a numerical study of radial null geodesics.

Decoupled anisotropic spheres in self-interacting Brans-Dicke gravity

The focus of this paper is to obtain anisotropic spherically symmetric solutions by means of gravitational decoupling in the background of self-interacting Brans-Dicke theory. We introduce minimal geometric deformation in the radial metric component to decouple the field equations into two arrays. The first set, governed by the seed source, is determined through metric functions of isotropic solution (Heintzmann/Tolman VII spacetimes) while the second set is solved by imposing two constraints on the anisotropic source. The unknown constants are evaluated via matching conditions at the stellar boundary. We investigate the effects of massive scalar field as well as decoupling parameter on the physical structure of anisotropic models and check them for viability through energy conditions. It is concluded that the anisotropic solutions obtained through constraint I are well-behaved for selected values of the decoupling parameter. For the second constraint, the extended Heintzmann solution is viable but anisotropic Tolman solution does not comply with dominant energy condition for higher values of the decoupling parameter.

Dynamical behavior of the Tolman metrics in f(R,T) gravity

We analyze the behavior of well-known stellar models within the context of $f(R,T)$ modified theory of gravity, in which the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$, namely $f(R,T)=R+2\ensuremath{\chi}T$ for some constant $\ensuremath{\chi}$. The equation of pressure isotropy in this theory is identical to that of the standard Einstein theory therefore all known metric potentials solving Einstein's equations are valid here. However, the pressure and energy density profiles are markedly different due to the presence of the term $2\ensuremath{\chi}T$. The exact solutions to the corresponding static spherically symmetric field equations with a perfect fluid source are the well known Tolman solutions [Phys. Rev. 55, 364 (1939)] in general relativity. To support the theoretical results, graphical representation are employed to investigate the physical viability of compact stars. Specifically we study the density and pressure profiles, the sound speed behavior as well as the energy conditions and mass behavior where appropriate. It is found that in some cases the $f(R,T)$ model displays more pleasing behavior than its Einstein counterpart while in other cases the behavior is similar. In no case does the $2\ensuremath{\chi}T$ addition negatively impact the model's behavior.

Erratum to: “Incompatibility of the Tolman Solution for Dust-like Matter with the Causality Principle”

Erratum to: “Incompatibility of the Tolman Solution for Dust-like Matter with the Causality Principle”

Incompatibility of the Tolman solution for dust-like matter with the causality principle

Incompatibility of the Tolman solution for dust-like matter with the causality principle

Effects of inhomogeneities on apparent cosmological observables: “fake” evolving dark energy

Using the exact Lemaitre–Bondi–Tolman solution with a non-vanishing cosmological constant Λ, we investigate how the presence of a local spherically symmetric inhomogeneity can affect apparent cosmological observables, such as the deceleration parameter or the effective equation of state of dark energy (DE), derived from the luminosity distance under the assumption that the real space-time is exactly homogeneous and isotropic. The presence of a local underdensity is found to produce apparent phantom behavior of DE, while a locally overdense region leads to apparent quintessence behavior. We consider relatively small large scale inhomogeneities which today are not linear and could be seeded by primordial curvature perturbations compatible with CMB bounds. Our study shows how observations in an inhomogeneous ΛCDM universe with initial conditions compatible with the inflationary beginning, if interpreted under the wrong assumption of homogeneity, can lead to the wrong conclusion about the presence of “fake” evolving dark energy instead of Λ.

Formation of Gyrs old black holes in the centers of galaxies within the Lemaître–Tolman model

In this article we present a model of formation of a galaxy with a black hole in the center. It is based on the Lemaître–Tolman solution and is a refinement of an earlier model. The most important improvement is the choice of the interior geometry of the black hole allowing for the formation of Gyrs old black holes. Other refinements are the use of an arbitrary Friedmann model as the background (unperturbed) initial state and the adaptation of the model to an arbitrary density profile of the galaxy. Our main interest was the M87 galaxy (NGC 4486), which hosts a supermassive black hole of mass 3.2 × 109 M⊙. It is shown that for this particular galaxy, within the framework of our model and for the initial state being a perturbation of the ΛCDM model, the age of the black hole can be up to 12.7 Gyrs. The dependence of the model on the chosen parameters at the time of last scattering was also studied. The maximal age of the black hole as a function of the Ω m and ΩΛ parameters for the M87 galaxy can be 3.717 or 12.708 Gyr.

Higher powers in gravitation

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalize the Einstein-Hilbert action by replacing the Ricci scalar R with some function f(R). The general asymptotic behavior of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of R to the Einstein-Hilbert action is investigated. The assumption that the highest powers of R should dominate the Universe's early history, and that the lowest powers should dominate its future is found to be inaccurate. The behavior of the general solution is complicated, and while it can be the case that single powers of R dominate the dynamics at late times, it can be either the higher or lower powers that do so. It is also shown that it is often the lowest powers of R that dominate at early times, when approach to a bounce or a Tolman solution are generic possibilities. Various examples are considered, and both vacuum and perfect fluid solutions are investigated.

Non-Abelian Born-Infeld cosmology

We investigate homogeneous and isotropic cosmological solutions supported by the SU(2) gauge field governed by the Born-Infeld Lagrangian. In the framework of the Friedmann-Robertson-Walker cosmology with or without the cosmological constant $\ensuremath{\lambda},$ we derive dynamical systems that give a rather complete description of the space of solutions. For $\ensuremath{\lambda}=0$ the effective equation of state $\ensuremath{\varepsilon}(p)$ is shown to interpolate between $p=\ensuremath{-}\ensuremath{\varepsilon}/3$ in the regime of the strong field and $p=\ensuremath{\varepsilon}/3$ for the weak field. Correspondingly, the universe starts with zero acceleration and gradually enters the decelerating regime, asymptotically approaching the Tolman solution.

Integrable cases of gravitating static isothermal fluid spheres

It is shown that different approaches toward the solution of the Einstein equations for a static spherically symmetric perfect fluid with a γ-law equation of state lead to an Abel differential equation of the second kind. Its only integrable cases at present are flat space–time, de Sitter solution and its Buchdahl transform, Einstein static universe, and the Klein–Tolman solution.

A new class of relativistic stellar models

Einstein field equations for a static and spherically symmetric perfect fluid are considered. A formulation given by Patino and Rago is used to obtain a class of nine solutions, two of them are Tolman solutions I, IV and the remaining seven are new. The solutions are the correct ones corresponding to expressions derived by Patino and Rago which have been shown by Knutsen to be incorrect. Similar to Tolman solution IV each of the new solutions satisfies energy conditions inside a sphere in some range of two independent parameters. Besides, each solution could be matched to the exterior Schwarzschild solution at a boundary where the pressure vanishes and thus the solutions constitute a class of new physically reasonable stellar models.

Anisotropic Tolman-like solutions of Einstein's equations

In this work we present new solutions of Einstein's equations assuming a spherically symmetric anisotropic fluid with shear in comoving coordinates. We study two limit cases: when the tangential pressure vanishes and when the radial pressure vanishes. The new solutions are based on the Tolman's solutions.

On Modeling a Relativistic Hierarchical (Fractal) Cosmology by Tolman's Spacetime. III. Numerical Results

view Abstract Citations (34) References (38) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On Modeling a Relativistic Hierarchical (Fractal) Cosmology by Tolman's Spacetime. III. Numerical Results Ribeiro, Marcelo B. Abstract This paper presents numerical solutions of particular Tolman models approximating a fractal behavior along the past light cone. The initial conditions of the numerical problem are discussed and the algorithm used to carry out the numerical integrations is presented. It was found that the numerical solutions are stiff across the flat-curved interface necessary to obtain the initial conditions of the problem. The spatially homogeneous Friedmann models are treated as special cases of the Tolman solution and solved numerically. Extending the results of Paper II on the Einstein e Sitter model, to the K = ±1 models, it was found that the open and closed Friedmann models also do not appear to remain homogeneous along the backward null cone, with a vanishing volume (average) density as one approaches the big bang singularity hypersurface Fractal solutions, that is, solutions representing an averaged and smoothed-out single fractal, were obtained in all three classes of the Tolman metric, but only the hyperbolic ones were found to be in agreement with observations, meaning that a possible Friedmann background universe would have to be an open one. The best fractal metric obtained through numerical simulations is also analyzed in terms of evolution, homothetic self-similarity, comparison with the respective spatially homogeneous case and the fitting problem in cosmology. The paper finishes with a discussion on some objections raised by some authors against a fractal cosmology. Publication: The Astrophysical Journal Pub Date: October 1993 DOI: 10.1086/173179 arXiv: arXiv:0807.1021 Bibcode: 1993ApJ...415..469R Keywords: COSMOLOGY: THEORY; Astrophysics; General Relativity and Quantum Cosmology E-Print: 57 pages. 22 figures. LaTeX. Paper published in 1993. The numerical code used in this paper is discussed in gr-qc/0205095 full text sources arXiv | ADS | Related Materials (2) Part 1: 1992ApJ...388....1R Part 2: 1992ApJ...395...29R

Cosmic background radiation anisotropies from cosmic structures - Models based on the Tolman solution

view Abstract Citations (60) References (24) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Cosmic Background Radiation Anisotropies from Cosmic Structures: Models Based on the Tolman Solution Panek, Miroslaw Abstract We investigate anisotropies of the Cosmic Background Radiation (CBR) resulting from second-order gravitational effects in structures like voids, the Great Attractor, and clusters of galaxies. Spherically symmetric models of the structures are based on the Tolman solution of Einstein equations. Predicted anisotropies of the order of 10^-7^-10^-6^ are below the sensitivity of present observations. Publication: The Astrophysical Journal Pub Date: April 1992 DOI: 10.1086/171146 Bibcode: 1992ApJ...388..225P Keywords: Anisotropy; Astronomical Models; Galactic Clusters; Relic Radiation; Einstein Equations; Gravitational Effects; Red Shift; Astrophysics; COSMOLOGY: COSMIC MICROWAVE BACKGROUND; COSMOLOGY: THEORY; GALAXIES: CLUSTERING full text sources ADS | data products SIMBAD (1)

On modeling a relativistic hierarchical (fractal) cosmology by Tolman's spacetime. I - Theory

This work examines a relativistic model for the observed inhomogeneities of the large scale structure where the hypothesis that this structure can be described as being a self-similar fractal system is advanced. The concept of hierarchical clustering is identified with a fractal distribution and the problems raised by the use of fractal ideas in a relativistic model are discussed, as well as their relations to the Copernican and Cosmological Principles. Voids, clusters and superclusters of galaxies are assumed to be part of a smoothed-out fractal structure described by a Lemaitre-Tolman solution. The basic concepts of the Newtonian model presented by Pietronero (1987) are reinterpreted and applied to this inhomogeneous curved spacetime. This fractal system is also assumed to have a crossover to homogeneity which leads to a "Swiss cheese" type model, composed by an interior Lemaitre-Tolman metric and an exterior dust Friedmann solution. The Darmois junction conditions between the two spacetimes are calculated, and the observational relations necessary to compare the model with observations are obtained for the interior region. The differential equations of the interior spacetime are set up and a numerical strategy is devised for finding particular Tolman solutions representing a fractal behaviour along the past light cone.

Spherically symmetric, self-gravitating dust is always 'self-similar'

This paper discusses spherically symmetric, self-gravitating dust from both the Newtonian and Einsteinian perspectives. A unified treatment in terms of comoving or Lagrangian coordinates is advocated, The most general type of self-similarity can be elucidated with this problem. It is argued that all Tolman solutions are in fact self-similar in the Lie sense in which an integral exists along a certain curve in space-time. These do not have a simple covariant expression in general at present. The limits in which more restrictive definitions apply are also explored. Finally, I treat the problem of classical spherical collapse with pressure in the same way. It is shown that previous discussions may thereby be simplified.

Perfect fluid spheres admitting flat 3-dimensional slices

We give the general spherically symmetric perfect fluid or dust solution with flat three-spaces orthogonal to the four-velocity. The case of dust corresponds to a subset of the Tolman solution. The shear-free case corresponds to the Friedmann line element.

Nonstatic radiating spheres in general relativity.

The method of Herrera, Jimenez, and Ruggeri of obtaining nonstatic solutions of Einstein's field equations to study the evolution of stellar bodies is applied to obtain two models of nonstatic radiating spheres from two well-known static solutions of field equations, viz., Tolman's solutions IV and V. Whereas Tolman's type-IV model is found to be contracting for the period under investigation, Tolman's type-V model shows a bounce after attaining a minimum radius.

Some exact anisotropic solutions in general relativity

Some exact solutions for anisotropic matter are worked out in the framework of general relativity. Four such solutions are obtained by a suitable modification of four well-known solutions by Tolman, viz., Tolman's solutions III, IV, V, and VI. The degree of anisotropy is determined by a parameter, and the range of values this parameter will have under realistic situations is calculated for all four classes. A singularity-free new solution for anisotropic matter is also presented.

Static configuration with an ultrarelativistic equation of state at the center

A class of solutions is obtained for the Einstein equations for liquid spheres with finite values of the energy density and pressure at the center. The fourth Tolman solution and the Adler solution belong to this class. A new solution is presented for which a merger is made with external empty space. Configuration parameters are calculated for an ultrarelativistic equation of state at the center.