Static Vacuum Solutions Research Articles

Spatial infinity in higher dimensional spacetimes

Motivated by recent studies on the uniqueness or non-uniqueness of higher dimensional black hole spacetime, we investigate the asymptotic structure of spatial infinity in n-dimensional spacetimes($n \geq 4$). It turns out that the geometry of spatial infinity does not have maximal symmetry due to the non-trivial Weyl tensor {}^{(n-1)}C_{abcd} in general. We also address static spacetime and its multipole moments P_{a_1 a_2 ... a_s}. Contrasting with four dimensions, we stress that the local structure of spacetimes cannot be unique under fixed a multipole moments in static vacuum spacetimes. For example, we will consider the generalized Schwarzschild spacetimes which are deformed black hole spacetimes with the same multipole moments as spherical Schwarzschild black holes. To specify the local structure of static vacuum solution we need some additional information, at least, the Weyl tensor {}^{(n-2)}C_{abcd} at spatial infinity.

Global Monopole in Kalb–Ramond Background

In recent, S. SenGupta and S. Sur [Phys. Lett. B 502 (2001) 350] have obtained static vacuum solutions of the gravitational field equations in back ground space time with torsion. The torsion is identified with the field strength of a second-rank anti-symmetric tensor field, namely Kalb-Ramond field. In this work, we present the solutions for the metric outside a monopole resulting from the breaking of a global 0(3) symmetry in Kalb-Ramond background. A comparison is made with the corresponding results predicted by general relativity.

A new approach of the stationary axisymmetric vacuum S(A) solutions

We revisit axisymmetric stationary vacuum solutions of the Einstein equations, like we did for the cylindrical case [J. Math. Phys. 41, 7535 (2000)]. We explicitly formulate the simplest hypothesis under which the S(A) solutions, or axisymmetric Lewis solutions can be found and demonstrate that this hypothesis leads to a linear relation between the potentials. We show that the field equations still can be associated to the motion of a classical particle in a central field, where an arbitrary harmonic χ function plays the role of time. Three classes of solutions are obtained without the need of invoking the Papapetrou class. They depend on two real parameters, and the potentials are functions of χ only. The new approach exempts the need of complex parameters. We interpret one of the parameters as related to the vorticity of the source.

Static axisymmetric vacuum solutions and non-uniform black strings

We describe new numerical methods to solve the static axisymmetric vacuum Einstein equations in more than four dimensions. As an illustration, we study the compactified non-uniform black string phase connected to the uniform strings at the Gregory–Laflamme critical point. We compute solutions with a ratio of maximum to minimum horizon radius of up to nine. For a fixed compactification radius, the mass of these solutions is larger than the mass of the classically unstable uniform strings. Thus they cannot be the end state of the instability.

Constant scalar curvature and warped product globally null manifolds

This paper deals with the curvature properties of a class of globally null manifolds ( M, g) which admit a global null vector field and a complete Riemannian hypersurface. Using the warped product technique we study the fundamental problem of finding a warped function such that the degenerate metric g admits a constant scalar curvature on M. Our work has an interplay with the static vacuum solutions of the Einstein equations of general relativity.

On Stationary Vacuum Solutions to the Einstein Equations

It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum solutions away from the boundary or horizon.

Factor structure of the Tomimatsu‐Sato solutions and their recurrence relations

AbstractWe consider stationary axisymmetric vacuum solutions of Einstein's equations for which the Ernst potential is rational in prolate spheroidal coordinates. Extending an earlier study we show that there are several new expressions which are factorizable. In particular, we concentrate on the Tomimatsu‐Sato solutions and their recurrence relations. Various continuum limits of the recurrence relations will be discussed.

Existence of stationary vacuum solutions of Einstein's equations in an exterior domain

A proof is given for the existence and uniqueness of a stationary vacuum solution (M, g, ξ) of the boundary value problem consisting of Einstein's equations in an exterior domain M diffeomorphic to R × Σ (where Σ = R3\B(0, R)) and boundary data depending on the Killing field ξ on ∂Σ. The boundary data must be sufficiently close to that of a stationary, spatially conformally flat vacuum solution .

A New Solution of Einstein-Maxwell Equations

A new generation theorem for Einstein-Maxwell field is attained. According to it, a new solution of Einstein-Maxwell equations can be generated from a static vacuum solution. Starting with Schwarzschild solution, a new solution of the Einstein-Maxwell equations is obtained by means of this theorem.

An intrinsic characterization of the Schwarzschild metric

An intrinsic algorithm that exclusively involves conditions on the metric tensor and its differential concomitants is presented to identify every type-D static vacuum solution. In particular, the necessary and sufficient explicit and intrinsic conditions are given for a Lorentzian metric to be the Schwarzschild solution.

Taub’s plane-symmetric vacuum spacetime reexamined

The gravitational properties of the only static plane-symmetric vacuum solution of Einstein's field equations without a cosmological term (Taub's solution, for brevity) are presented: some already known properties (geodesics, weak field limit, and pertainment to the Schwarzschild family of spacetimes) are reviewed in a physically much more transparent way, as well as new results about its asymptotic structure, possible matchings, and the nature of the source are furnished. The main results point to the fact that the solution must be interpreted as representing the exterior gravitational field due to a negative mass distribution, confirming previous statements to that effect in the literature. Some analogies to Kasner's spatially homogeneous cosmological model are also mentioned.

The sources of the A and B degenerate static vacuum fields

We study some known static vacuum solutions of Einstein's equations, namely, the A and B metrics of Ehlers and Kundt. All of them have axial symmetry, so they can be transformed to the Weyl form. In Weyl coordinates log √−g 44 obeys a Laplace equation, and from this a source, calledthe Newtonian image source, can be identified. We use the image sources to give a possible interpretation to the metrics. The procedure is successful in some cases. In others it fails because the Weyl transform does not have reasonable properties at infinity.

New sources for Kerr and other metrics: rotating relativistic discs with pressure support

Complete sequences of new analytic solutions of Einstein's equations which describe thin super massive disks are constructed. These solutions are derived geometrically. The identification of points across two symmetrical cuts through a vacuum solution of Einstein's equations defines the gradient discontinuity from which the properties of the disk can be deduced. The subset of possible cuts which lead to physical solutions is presented. At large distances, all these disks become Newtonian, but in their central regions they exhibit relativistic features such as velocities close that of light, and large redshifts. Sections with zero extrinsic curvature yield cold disks. Curved sections may induce disks which are stable against radial instability. The general counter rotating flat disk with planar pressure tensor is found. Owing to gravomagnetic forces, there is no systematic method of constructing vacuum stationary fields for which the non-diagonal component of the metric is a free parameter. However, all static vacuum solutions may be extended to fully stationary fields via simple algebraic transformations. Such disks can generate a great variety of different metrics including Kerr's metric with any ratio of a to m. A simple inversion formula is given which yields all distribution functions compatible with the characteristics of the flow, providing formally a complete description of the stellar dynamics of flattened relativistic disks. It is illustrated for the Kerr disk.

Exact static vacuum solution of four-dimensional metric-affine gravity with nontrivial torsion

We present an exact spherically symmetric vacuum solution for a metric-affine quadratic Lagrangian. The metric part of the solution is of Reissner-Nordström type. The connection part includes nonmetricity depending on a dilatation charge and a shear charge. Furthermore, a nontrivial torsion depending on a certain torsion mass and with intrinsic contributions is present.

Torsion singularity in Weitzenböck space-time

In the framework of gravitational gauge theory with torsion in Weitzenbock space-time, scalar polynomial torsion singularity and scalar Riemann-Christoffel curvature singularity are studied by a static spherical symmetric vacuum solutions. Although the parameter e = 0, general relativity is recovered in a limit case and there exists torsion singularity at the Schwarzshild radius. Generally, there exists torsion and Riemann-Christoffel curvature singularity without real event horizon.

Construction of exact solutions of the Einstein-Maxwell equations corresponding to a given behaviour of the Ernst potentials on the symmetry axis

The method developed by one of the authors for the construction of exact solutions of the Einstein-Maxwell equations by setting the behaviour of the Ernst potentials on the symmetry axis is discussed in full detail. Some new results regarding the solution of the integral equations and the construction of the 3*3 matrix potential H are presented. Two new examples of the application of the method are considered, one of which is the stationary vacuum solution for a rotating mass, and the second one is the four-parameter metric which can be used for the description of the exterior field of a charged magnetized spinning source.

Astrophysically significant solutions of the Einstein and Einstein-Maxwell equations from the Laplace’s seed

The stationary solutions given by Amenedo and Manko generated from known solutions of Laplace’s equation as seed have been generalised to include the electromagnetic field. Further, the exterior solution of an axially symmetric rotating body with higher multipole moments and a solution corresponding to a Kerr object embedded in a gravitational field are given. We also give a method for constructing stationary vacuum solutions from static magnetovac solutions and vice versa and discuss a specific application of this method.

Some notes on stationary vacuum solutions of the Einstein equations with shearing nongeodesic eigenrays

The stationary vacuum solutions of the Einstein equations of General Relativity give the external space-time around stationary mass distributions, as e.g. final states of stellar evolution. The Kerr solution has shear-free geodesic eigenrays and describes all black hole configurations with good asymptotic behaviour at infinity. Other solutions of this class are unphysical. Classes with shearing geodesic or shearfree nongeodesic eigenrays do not contain physical solutions at all, so for other physical configurations one must turn to the generic case of shearing nongeodesic eigenrays. For the stationary axisymmetric case Ansatze for solutions can be formulated in form of a specific functional dependence between the parameters of shear and nongeodesicity, unless they both are constants. Here we investigate the second subcase. Our result is that there is no solution of the vacuum Einstein equations in this subclass, except when both parameters vanish.

Wilson line breaking and vacuum stability in Kaluza-Klein cosmology.

We attempt to apply the Wilson-line-breaking mechanism to Kaluza-Klein cosmology. Vacuum stability is considered in a simple SU(2) Einstein-Yang-Mills theory defined on space-time {ital M}{sup 4}{times}{ital S}{sup 1}. It is shown that static vacuum solutions with stable compactification exist.

Factor structure of rational vacuum metrics

The authors consider axisymmetric stationary vacuum solutions of the Einstein equations for which the metric functions are rational in certain coordinate systems. It is shown that various combinations of the polynomials in the Ernst potential and their derivatives are factorizable.