Narrow Results

We look for "static" spherically symmetric solutions of Einstein's Equations for perfect fluid source with equation of state p = wρ, for 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 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.

It is proved that all thin-shell wormholes built from two identical regions of vacuum static, spherically symmetric spacetimes have a negative shell surface energy density in any scalar–tensor theory of gravity with a non-ghost massless scalar field and a non-ghost graviton.

In this paper, using the apparatus of the Clifford bundle formalism, we show how straightforwardly solve in Minkowski space–time the Dirac–Hestenes equation — which is an appropriate representative in the Clifford bundle of differential forms of the usual Dirac equation — by separation of variables for the case of a potential having spherical symmetry in the Cartesian and spherical gauges. We show that, contrary to what is expected at a first sight, the solution of the Dirac–Hestenes equation in both gauges has exactly the same mathematical difficulty.

The fluid models mentioned in the title are studied in a modified approach, based on two formulas for the mass function. All characteristics of the fluid are expressed through a master potential, satisfying an ordinary second-order differential equation. Different constraints are imposed on this core of relations, finding new solutions and deriving the classical results for perfect fluids as particular cases. All charged anisotropic solutions, all conformally flat and all uniform density solutions are found. A large class of solutions with linear equation among the two pressures is derived, including the case of vanishing tangential pressure. The mechanism responsible for the appearance of equation of state is elucidated.

In this paper, we study the initial boundary value problem for the isentropic Euler–Poisson system in an exterior domain with spherical symmetry. The initial data is supposed to be bounded and satisfy other suitable assumptions. Using a fractional step Godunov scheme, we construct the approximate solutions and prove the uniform $L^{\infty }$ estimates for the approximate solutions. Then the compensated compactness argument implies the convergence of the solutions. The weak entropy solution also satisfies the initial value and boundary value in the sense of trace.

We consider a graph link L in S³ which admits a symmetry of the exterior given by a boundary component preserving periodic homeomorphism f (orientation-preserving or not). This paper shows that L is obtained by a specific construction method if f is not a strong inversion. As an application, we provide a complete classification of 2-component graph links with symmetries of the exteriors which are not extendable to symmetries of S³.

We show that holography follows directly from the basic structure of spherically symmetric loop quantum gravity. The result is not dependent on detailed assumptions about the dynamics of the theory being considered. It ties strongly the amount of information contained in a region of space to the tight mathematical underpinnings of loop quantum geometry.

Modified Chaplygin gas (MCG) is a strong candidate for the unified model of dark matter and dark energy. The equation of state of this modified model is valid from the radiation era to the ΛCDM model. In the early epoch (when ρ was large), dark matter had the dominant role, while at later stages (when ρ is small), the MCG model behaves as dark energy. In this work, we have found an exact solution to static, spherically symmetric Einstein equations describing a wormhole for an inhomogeneous distribution of MCG. For the existence of a wormhole solution, there are some restrictions concerning the parameters in the equation of state for MCG and the throat radius of the wormhole. Physical properties and characteristics of such modified Chaplygin wormholes are analyzed in detail.

Considering the standard "static" spherically symmetric ansatz ds² = −B(r) dt² + A(r) dr² + r² dΩ² for Einstein's equations with perfect fluid source, we ask how we can interpret solutions where A(r)andB(r) are not positive, as they must be for the static matter source interpretation to be valid.

Noting that the requirement of Lorentzian signature implies A(r) B(r) > 0, we find two possible interpretations:

(i) The nonzero component of the source four-velocity does not have to be u⁰. This provides a connection from the above ansatz to the Kantowski–Sachs (KS) space–times.

(ii) Regions with negative A(r) and B(r) of "static" solutions in the literature must be interpreted corresponding to the tachyonic source.

The combinations of source type and four-velocity direction result in four possible cases. One is the standard case, one is identical to the KS case, and two are tachyonic. The dynamic tachyonic case was anticipated in the literature, but the static tachyonic case seems to be new. We derive Oppenheimer–Volkoff-like equations for each case, and find some simple solutions. We conclude that new "simple" black hole solutions of the above form, supported by a perfect fluid, do not exist.

The fluid models mentioned in the title are classified. All characteristics of the fluid are expressed through a master potential, satisfying an ordinary second-order differential equation. Different constraints are imposed on this core of relations, to find new solutions and to derive the classical results for perfect fluids and dust as particular cases. Many uncharged and charged anisotropic solutions, all conformally flat, and some uniform density solutions are found. A number of solutions with linear equation among the two pressures are derived, including the case of vanishing tangential pressure.

We derive the equations governing static, spherically symmetric vacuum solutions to the Einstein equations, as modified by the frame-dependent effective action (derived from trace dynamics) that gives an alternative explanation of the origin of "dark energy". We give analytic and numerical results for the solutions of these equations, first in polar coordinates, and then in isotropic coordinates. General features of the static case are that: (i) there is no horizon, since g₀₀ is nonvanishing for finite values of the polar radius, and only vanishes (in isotropic coordinates) at the internal singularity, (ii) the Ricci scalar R vanishes identically, and (iii) there is a physical singularity at cosmological distances. The large distance singularity may be an artifact of the static restriction, since we find that the behavior at large distances is altered in a time-dependent solution using the McVittie Ansatz.

We consider spherically symmetric configurations in general relativity (GR), supported by nonlinear electromagnetic fields with gauge-invariant Lagrangians depending on the single invariant $f=F_{\mu \nu }{F}^{\mu \nu }$. Static black hole (BH) and solitonic solutions are briefly described, both with only an electric or magnetic charge and with both nonzero charges (the dyonic ones). It is stressed that only pure magnetic solutions can be completely nonsingular. For dyonic systems, apart from a general scheme of obtaining solutions in quadratures for an arbitrary Lagrangian function $L(f)$, an analytic solution is found for the truncated Born–Infeld theory (depending on the invariant $f$ only). Furthermore, considering spherically symmetric metrics with two independent functions of time, we find a natural generalization of the class of wormholes found previously by Arellano and Lobo with a time-dependent conformal factor. Such wormholes are shown to be only possible for some particular choices of the function $L(f)$, having no Maxwell weak-field limit.

This paper formulates the exact static anisotropic spherically symmetric solution of the field equations through gravitational decoupling. To accomplish this work, we add a new gravitational source in the energy–momentum tensor of a perfect fluid. The corresponding field equations, hydrostatic equilibrium equation as well as matching conditions are evaluated. We obtain the anisotropic model by extending the known Durgapal and Gehlot isotropic solution and examined the physical viability as well as the stability of the developed model. It is found that the system exhibits viable behavior for all fluid variables as well as energy conditions and the stability criterion is fulfilled.

We study the stability properties of static, spherically symmetric configurations in $k$-essence theories with the Lagrangians of the form $F(X)$, $X\equiv {\varphi }_{,\alpha }{\varphi }^{,\alpha }$. The instability under spherically symmetric perturbations is proved for the two recently obtained exact solutions for $F(X)=F_0{X}^{1/3}$ and for $F(X)=F_0{X}^{1/2}-2\mathrm{\Lambda }$, where $F_0$ and $\mathrm{\Lambda }$ are constants. The first solution describes a black hole in an asymptotically singular spacetime, the second one contains two horizons of infinite area connected by a wormhole. It is argued that spherically symmetric $k$-essence configurations with $n<1/2$ are generically unstable because the perturbation equation is not of hyperbolic type.

In this research, core envelope model of a super dense spherically symmetric compact star is developed by considering anisotropic matter configuration. The core is represented by a linear equation of state (EOS), whereas the Van der Waals EOS is used in the envelope region. In the core and envelope of the star, all geometrical and physical factors are viable. The three regions, i.e. the core, envelope and outer space satisfy the junction conditions. The proposed model validates with the properties of Vela X-1, Her X-1 and SMC X-1. It is concluded that in the model presented, the core of the star compresses as the mass increases justifying the domination of gravitational effects on massive astronomical objects.

In this paper, we construct a class of spherically symmetric and self-similar analytical solutions to the pressureless Navier–Stokes equations with density-dependent viscosity coefficients satisfying $h(\rho )={\rho }^{𝜃}$, $g(\rho )=(𝜃-1){\rho }^{𝜃}$ for all $𝜃>0$. Under the continuous density free boundary conditions imposed on the free surface, we investigate the large-time behavior of the solutions according to various $𝜃>1$ and $0<𝜃<1$. When the time grows up, such solutions exhibit interesting information: Case (i) If the free surface initially moves inward, then the free surface infinitely approaches to the symmetry center and the fluid density blows up at the symmetry center, or the free surface tends to an equilibrium state; Case (ii) If the free surface initially moves outward, then the free surface infinitely expands outward and the fluid density decays and tends to zero almost everywhere away from the symmetry center, or the free surface tends to an equilibrium state. We also study the large-time behavior of the solutions for $𝜃=1$ without any boundary conditions.

In this paper, we consider a system of classical model equations describing a motion of gaseous star, spherically symmetric, with a free-surface and a rigid core in the center under the influence of gaseous self-gravitation and potential force of the core. In addition to it, we investigate the above model equations in the physically more realistic situation: the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the one in a fixed domain. Based on the fundamental local existence result and a priori estimates, we can construct a unique global classical solution.

Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group $\mathrm{\text{SO}}(3)$. The aim of this work is to make use of this tool also in the setting of Finsler geometry, or more general geometries on the tangent bundle, where the objects of relevance are d-tensors on the tangent bundle, or tensors in a pullback bundle, instead of ordinary tensors. For this purpose, we construct a set of d-tensor harmonics for spherical symmetry and show how these can be used for calculations in Finsler geometry.

We propose an analytic $C^1$ solution of the hydrodynamic equations for a main-sequence star in stationary and spherically symmetric conditions. We reach this result starting from convenient choices of mass density, specific function of the perfect gas, and luminosity function. The theoretical results, when applied to the Sun, are compared with the standard solar model data. A more accurate proof of Schwarzshild’s inequality and equation is given.

We critically analyze the models of a main-sequence star based on polytropic gases. First, we put in evidence that the hypothesis of polytropic gas is compatible with the constitutive equation of an ideal gas if the transformations inside the star are adiabatic. Then, neither the mono- or the polyphasic models of an ideal gas can be applied inside the stellar core for the existence in this region of nuclear reactions. Moreover, we prove that polyphasic models doesn’t allow the existence of C¹ solutions of the stationary hydrodynamic equations.