Static Vacuum Solutions Research Articles

A note on static metrics: the degenerate case

We give the necessary and sufficient conditions for a 3-metric to be the adapted spatial metric of a static vacuum solution. This work accomplishes for the degenerate cases the already known study for the regular ones (Bartnik and Tod 2006 Class. Quantum Grav. 23 569–71).

Vacuum spherically symmetric solutions in f(T) gravity

Spherically symmetric static vacuum solutions have been built in $f(T)$ models of gravity theory. We apply some conditions on the metric components; then the new vacuum spherically symmetric solutions are obtained. Also, by extracting metric coefficients we determine the analytical form of $f(T)$.

HIGHER DIMENSIONAL BLACK HOLES

This paper reviews black hole solutions of higher-dimensional General Relativity (GR). The focus is on stationary vacuum solutions and recent work on instabilities of such solutions.

Black holes and global structures of spherical spacetimes in Horava-Lifshitz theory

We systematically study black holes in the Horava-Lifshitz (HL) theory by following the kinematic approach, in which a horizon is defined as the surface at which massless test particles are infinitely redshifted. Because of the nonrelativistic dispersion relations, the speed of light is unlimited, and test particles do not follow geodesics. As a result, there are significant differences in causal structures and black holes between general relativity (GR) and the HL theory. In particular, the horizon radii generically depend on the energies of test particles. Applying them to the spherical static vacuum solutions found recently in the nonrelativistic general covariant theory of gravity, we find that, for test particles with sufficiently high energy, the radius of the horizon can be made as small as desired, although the singularities can be seen in principle only by observers with infinitely high energy. In these studies, we pay particular attention to the global structure of the solutions, and find that, because of the foliation-preserving-diffeomorphism symmetry, ${Diff}(M,{\cal{F}})$, they are quite different from the corresponding ones given in GR, even though the solutions are the same. In particular, the ${Diff}(M,{\cal{F}})$ does not allow Penrose diagrams. Among the vacuum solutions, some give rise to the structure of the Einstein-Rosen bridge, in which two asymptotically flat regions are connected by a throat with a finite non-zero radius. We also study slowly rotating solutions in such a setup, and obtain all the solutions characterized by an arbitrary function $A_{0}(r)$. The case $A_{0} = 0$ reduces to the slowly rotating Kerr solution obtained in GR.

Cylindrically and toroidally symmetric solutions with a cosmological constant

Cylindrical-like coordinates for constant-curvature 3-spaces are introduced and discussed. This helps to clarify the geometrical properties, the coordinate ranges and the meaning of free parameters in the static vacuum solution of Linet and Tian. In particular, when the cosmological constant is positive, the spacetimes have toroidal symmetry. One of the two curvature singularities can be removed by matching the Linet–Tian vacuum solution across a toroidal surface to a corresponding region of the dust-filled Einstein static universe. Some other properties and limiting cases of these space-times are also described, together with their generalisation to higher dimensions.

A NEW METHOD TO CONSTRUCT THE DEGENERATE STATIC VACUUM SOLUTIONS WITH COSMOLOGICAL CONSTANT

In this paper, we present a special Lorentzian metric to solve the vacuum Einstein's field equation with cosmological constant Λ and get a general form of solution. By some proper coordinate transformations, a degenerate static vacuum solution of class A with cosmological constant is obtained, and the corresponding one of class B can also be obtained by the complex transformation. We show that this is a constructive and effective new method to extend the degenerate static vacuum solution to the case with cosmological constant.

Classical tests of general relativity in brane world models

The classical tests of general relativity (perihelion precession, deflection of light and the radar echo delay) are considered for several spherically symmetric static vacuum solutions in brane world models. Generally, the spherically symmetric vacuum solutions of the brane gravitational field equations have properties quite distinct as compared to the standard black hole solutions of general relativity. As a first step a general formalism that facilitates the analysis of general relativistic Solar System tests for any given spherically symmetric metric is developed. It is shown that the existing observational Solar System data on the perihelion shift of Mercury, on the light bending around the Sun (obtained using long-baseline radio interferometry), and ranging to Mars using the Viking lander constrain the numerical values of the parameters of the specific models.

Cylindrically symmetric vacuum solutions in higher dimensional Brans–Dicke theory

Higher dimensional, static, cylindrically symmetric vacuum solutions with and without a cosmological constant in the Brans–Dicke theory are presented. We show that for a negative cosmological constant and for specific values of the parameters, a particular subclass of these solutions includes higher dimensional topological black hole-type solutions with a flat horizon topology. We briefly extend our discussion to stationary vacuum and Λ-vacuum solutions.

PLANE SYMMETRIC SOLUTIONS IN f(R) GRAVITY

The modified theories of gravity, especially the f(R) theory, have attracted much attention in recent years. In this context, we explore static plane symmetric vacuum solutions using the metric approach of this theory. The field equations are solved using the assumption of constant scalar curvature which may be zero or nonzero. We have found a total of three plane symmetric solutions. The correspondence of these solutions with the well-known solutions in General Relativity is given.

An anti-Schwarzshild solution: Wormholes and scalar-tensor solutions

We investigate a static solution with an hyperbolic nature, characterised by a pseudo-spherical foliation of space. This space-time metric can be perceived as an anti-Schwarzschild solution, and exhibits repulsive features. It belongs to the class of static vacuum solutions termed "a degenerate static solution of class A" (see [1]). In the present work we review its fundamental features, discuss the existence of generalised wormholes, and derive its extension to scalar-tensor gravity theories in general.

New horizons for black holes and branes

We initiate a systematic scan of the landscape of black holes in any spacetime dimension using the recently proposed blackfold effective worldvolume theory. We focus primarily on asymptotically flat stationary vacuum solutions, where we uncover large classes of new black holes. These include helical black strings and black rings, black odd-spheres, for which the horizon is a product of a large and a small sphere, and non-uniform black cylinders. More exotic possibilities are also outlined. The blackfold description recovers correctly the ultraspinning Myers-Perry black holes as ellipsoidal even-ball configurations where the velocity field approaches the speed of light at the boundary of the ball. Helical black ring solutions provide the first instance of asymptotically flat black holes in more than four dimensions with a single spatial U(1) isometry. They also imply infinite rational non-uniqueness in ultraspinning regimes, where they maximize the entropy among all stationary single-horizon solutions. Moreover, static blackfolds are possible with the geometry of minimal surfaces. The absence of compact embedded minimal surfaces in Euclidean space is consistent with the uniqueness theorem of static black holes.

Convergent Null Data Expansions at Space-Like Infinity of Stationary Vacuum Solutions

We present a characterization of the asymptotics of all asymptotically flat stationary vacuum solutions of Einstein's field equations. This characterization is given in terms of two sequences of symmetric trace free tensors (we call them the `null data'), which determine a formal expansion of the solution, and which are in a one to one correspondence to Hansen's multipoles. We obtain necessary and sufficient growth estimates on the null data to define an absolutely convergent series in a neighbourhood of spatial infinity. This provides a complete characterization of all asymptotically flat stationary vacuum solutions to the field equations.

Fundamental parameter-free solutions in modified gravity

Modified gravity (MOG) has been used successfully to explain the rotation curves of galaxies, the motion of galaxy clusters, the bullet cluster and cosmological observations without the use of dark matter or Einstein's cosmological constant. We now have the ability to demonstrate how these solutions can be obtained directly from the action principle, without resorting to the use of fitted parameters or empirical formulae. We obtain numerical solutions to the theory's field equations that are exact in the sense that no terms are omitted, in two important cases: the spherically symmetric, static vacuum solution and the cosmological case of a homogeneous, isotropic universe. We compare these results to selected astrophysical and cosmological observations.

One-parameter families of conformally related asymptotically flat, static vacuum data

Extending the results of Friedrich (2008 Class. Quantum Grav. 25 065012) we give a complete description of the asymptotically flat, conformally non-flat, static vacuum data which admit non-trivial, asymptotically smooth conformal mappings onto other such data. These data form a 3-parameter family which decomposes into 1-parameter families of data which are conformal to each other. The data and the associated static vacuum solutions are given explicitly in terms of elliptic and, in a special case, elementary functions.

Static Vacuum Solutions from Convergent Null Data Expansions at Space-Like Infinity

We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1:1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations.

The asymptotics of stationary electro-vacuum metrics in odd spacetime dimensions

We show that for stationary, asymptotically flat solutions of the electro-vacuum Einstein equations, the Riemannian metric on the quotient space of the timelike Killing vector is analytic at i0, for a large family of gauges, in odd spacetime dimensions larger than 7. The same is true in spacetime dimension 5 for static vacuum solutions with non-vanishing mass.

Conformal Continuations in Gravitation Theory with Lagrangian F(R)

Static spherically-symmetric vacuum solutions of gravitation theory equations with Lagrangian f(R) are examined, where R is a scalar curvature and f is an arbitrary function. Equations of f(R)-theories are reduced to the Einstein scenario — general relativity theory (GRT) equations with a source in the form of a scalar field with potential — with the use of the well-known conformal transformation. The necessary and sufficient conditions of existence of solutions admitting conformal continuations are formulated. This means that the central singularity of the Einstein scenario is mapped into a regular sphere Strans of the Jordan scenario (that is, into the manifold corresponding to the initial formulation of the theory), and a solution of the field equations can be smoothly continued through it. The value of curvature R on the sphere Strans corresponds to an extremum of the function f(R). Concrete examples are considered.

Five dimensional cosmology in Horava-Witten M theory

The cosmology in the Hubble expansion era of the Horava-Witten M theory compactified on a Calabi-Yau threefold is studied in the reduction to five-dimensions where the effects of the Calabi-Yau manifold are summarized by the volume modulus, and all perturbative potentials are included. Matter on the branes are treated as first order perturbations of the static vacuum solution, and all equations in the bulk and all boundary conditions on both end branes are imposed. It is found that for a static volume modulus and a static fifth dimension, $y$, one can recover the four-dimensional Robertson Friedmann Walker cosmology for relativistic matter on the branes, but not for nonrelativistic matter. In this case, the Hubble parameter $H$ becomes independent of $y$ to first order in matter density. This result holds also when an arbitrary number of 5-branes are included in the bulk. The five-dimensional Horava-Witten model is compared with the Randall-Sundrum phenomenology with a scalar field in the bulk where a bulk and brane potential are used so that the vacuum solutions can be rigorously obtained. (In the Appendix, the difficulty of obtaining approximate vacuum solutions for other potentials is discussed.) In this case nonrelativistic matter is accommodated by allowing the distance between the branes to vary. It is suggested that nonperturbative potentials for the vacuum solution of Horava-Witten theory are needed to remove the inconsistency that nonrelativistic matter creates.

How can exact and approximate solutions of Einstein's field equations be compared?

The problem of comparison of the stationary axisymmetric vacuum solutions obtained within the framework of exact and approximate approaches for the description of the same general relativistic systems is considered. We suggest two ways of carrying out such comparison: (i) through the calculation of the Ernst complex potential associated with the approximate solution whose form on the symmetry axis is subsequently used for the identification of the exact solution possessing the same multipole structure, and (ii) the generation of approximate solutions from exact ones by expanding the latter in series of powers of small parameters. The central result of our paper is the derivation of the correct approximate analogues of the double-Kerr solution possessing the physically meaningful equilibrium configurations. We also show that the interpretation of an approximate solution originally attributed to it on the basis of some general physical suppositions may not coincide with its true nature established with the aid of a more accurate technique.

Low energy branes, effective theory, and cosmology

The low energy regime of cosmological BPS-brane configurations with a bulk scalar field is studied. We construct a systematic method to obtain five-dimensional solutions to the full system of equations governing the geometry and dynamics of the bulk. This is done for an arbitrary bulk scalar field potential and taking into account the presence of matter on the branes. The method, valid in the low energy regime, is a linear expansion of the system about the static vacuum solution. Additionally, we develop a four-dimensional effective theory describing the evolution of the system. At the lowest order in the expansion, the effective theory is a bi-scalar tensor theory of gravity. One of the main features of this theory is that the scalar fields can be stabilized naturally without the introduction of additional mechanisms, allowing satisfactory agreement between the model and current observational constraints. The special case of the Randall-Sundrum model is discussed.