Static Vacuum Solutions Research Articles

On relativistic multipole moments of stationary space-times.

Among the known exact solutions of Einstein's vacuum field equations the Manko–Novikov and the Quevedo–Mashhoon metrics might be suitable ones for the description of the exterior gravitational field of some real non-collapsed body. A new proposal to represent such exterior field is the stationary q-metric. In this contribution, we computed by means of the Fodor–Hoenselaers–Perjés formalism the lowest 10 relativistic multipole moments of these metrics. Corresponding moments were derived for the static vacuum solutions of Gutsunayev–Manko and Hernández–Martín. A direct comparison between the multipole moments of these non-isometric space–times is given.

Static vacuum solutions on curved space–times with torsion

The Einstein–Cartan–Kibble–Sciama ( ECKS ) theory of gravity naturally extends Einstein’s general relativity ( GR ) to include intrinsic angular momentum (spin) of matter. The main feature of this theory consists of an algebraic relation between space–time torsion and spin of matter, which indeed deprives the torsion of its dynamical content. The Lagrangian of ECKS gravity is proportional to the Ricci curvature scalar constructed out of a general affine connection so that owing to the influence of matter energy–momentum and spin, curvature and torsion are produced and interact only through the space–time metric. In the absence of spin, the space–time torsion vanishes and the theory reduces to GR . It is however possible to have torsion propagation in vacuum by resorting to a model endowed with a nonminimal coupling between curvature and torsion. In the present work we try to investigate possible effects of the higher order terms that can be constructed from space–time curvature and torsion, as the two basic constituents of Riemann–Cartan geometry. We consider Lagrangians that include fourth-order scalar invariants from curvature and torsion and then investigate the resulting field equations. The solutions that we find show that there could exist, even in vacuum, nontrivial static space–times that admit both black holes and naked singularities.

Comparison of vacuum static quadrupolar metrics.

We investigate the properties of static and axisymmetric vacuum solutions of Einstein equations which generalize the Schwarzschild spherically symmetric solution to include a quadrupole parameter. We test all the solutions with respect to elementary and asymptotic flatness and curvature regularity. Analysing their multipole structure, according to the relativistic invariant Geroch definition, we show that all of them are equivalent up to the level of the quadrupole. We conclude that the q-metric, a variant of the Zipoy–Voorhees metric, is the simplest generalization of the Schwarzschild metric, containing a quadrupole parameter.

The rotation curve of a point particle in stringy gravity

Double Field Theory suggests to view the whole massless sector of closed strings as the gravitational unity. The fundamental symmetries therein, including the O(D,D) covariance, can determine unambiguously how the Standard Model as well as a relativistic point particle should couple to the closed string massless sector. The theory also refines the notion of singularity. We consider the most general, spherically symmetric, asymptotically flat, static vacuum solution to D=4 Double Field Theory, which contains three free parameters and consequently generalizes the Schwarzschild geometry. Analyzing the circular geodesic of a point particle in string frame, we obtain the orbital velocity as a function of R/(M∞G) which is the dimensionless radial variable normalized by mass. The rotation curve generically features a maximum and thus non-Keplerian over a finite range, while becoming asymptotically Keplerian at infinity, R/(M∞G)→ ∞. The adoption of the string frame rather than Einstein frame is the consequence of the fundamental symmetry principle. Our result opens up a new scheme to solve the dark matter/energy problems by modifying General Relativity at 'short' range of R/(M∞G).

Building cosmological frozen stars

Janis–Newman–Winicour (JNW) solutions generalize Schwarzschild to include a massless scalar field. While they share the familiar infinite redshift feature of Schwarzschild, they suffer from the presence of naked singularities. Cosmological versions of JNW spacetimes were discovered some years ago, in the most general case, by Fonarev. Fonarev solutions are also plagued by naked singularities, but have the virtue, unlike e.g. Schwarzschild–deSitter, of being dynamical. Given that exact dynamical cosmological black hole solutions are scarce, Fonarev solutions merit further study. We show how Fonarev solutions can be obtained via generalized dimensional reduction from simpler static vacuum solutions. These results may lead towards constructions of actual dynamical cosmological black holes. In particular, we note that cosmological versions of extremal charged dilaton black holes are known. JNW spacetimes represent a different limiting case of the family of charged dilaton black holes, which have been important in the context of string theory, and better understanding their cosmological versions of JNW spacetimes thus provides a second data point towards finding cosmological versions of the entire family.

Lee–Wick black holes

We derive and study an approximate static vacuum solution generated by a point-like source in a higher derivative gravitational theory with a pair of complex conjugate ghosts. The gravitational theory is local and characterized by a high derivative operator compatible with Lee-Wick unitarity. In particular, the tree-level two-point function only shows a pair of complex conjugate poles besides the massless spin two graviton. We show that singularity-free black holes exist when the mass of the source $M$ exceeds a critical value $M_{\rm crit}$. For $M > M_{\rm crit}$ the spacetime structure is characterized by an outer event horizon and an inner Cauchy horizon, while for $M = M_{\rm crit}$ we have an extremal black hole with vanishing Hawking temperature. The evaporation process leads to a remnant that approaches the zero-temperature extremal black hole state in an infinite amount of time.

All static black holes inAdS3

In this work we find the general static vacuum solution of three dimensional gravity with negative cosmological constant. Even though all solutions are locally diffeomorphic to pure $AdS_{3}$, solutions that differ globally from the latter space exist. New solutions with black holes on the $AdS_3$ boundary are found in both global and Poincare coordinates. In the Poincare coordinates such solutions are known as black funnels and black droplets. The black funnel provided by our general static metric is dual to the Hartle-Hawking state in the $1+1$ boundary theory.

Exploring plane-symmetric solutions in f(R) gravity

The modified theories of gravity, especially the $f(R)$ gravity, have attracted much attention in the last decade. This paper is devoted to exploring plane symmetric solutions in the context of metric $f(R)$ gravity. We extend the work on static plane symmetric vacuum solutions in $f(R)$ gravity already available in literature [1, 2]. The modified field equations are solved using the assumption of both constant and non-constant scalar curvature. Some well known solutions have been recovered with power law and logarithmic forms of $f(R)$ models.

(2+1)-dimensional gravity in Weyl integrable spacetime

We investigate (2+1)-dimensional gravity in a Weyl integrable spacetime (WIST). We show that, unlike general relativity, this scalar-tensor theory has a Newtonian limit for any dimension and that in three dimensions the congruence of world lines of particles of a pressureless fluid has a non-vanishing geodesic deviation. We present and discuss a class of static vacuum solutions generated by a circularly symmetric matter distribution that for certain values of the parameter ω corresponds to a spacetime with a naked singularity at the center of the matter distribution. We interpret all these results as being a direct consequence of the spacetime geometry.

Reciprocal NUT spacetimes

In this paper, we study the Ehlers' transformation (sometimes called gravitational duality rotation) for reciprocal static metrics. First, we introduce the concept of reciprocal metric. We prove a theorem which shows how we can construct a certain new static solution of Einstein field equations using a seed metric. Later, we investigate the family of stationary spacetimes of such reciprocal metrics. The key here is a theorem from Ehlers', which relates any static vacuum solution to a unique stationary metric. The stationary metric has a magnetic charge. The spacetime represents Newman-Unti-Tamburino (NUT) solutions. Since any stationary spacetime can be decomposed into a 1 + 3 time-space decomposition, Einstein field equations for any stationary spacetime can be written in the form of Maxwell's equations for gravitoelectromagnetic fields. Further, we show that this set of equations is invariant under reciprocal transformations. An additional point is that the NUT charge changes the sign. As an instructive example, by starting from the reciprocal Schwarzschild as a spherically symmetric solution and reciprocal Morgan–Morgan disk model as seed metrics we find their corresponding stationary spacetimes. Starting from any static seed metric, performing the reciprocal transformation and by applying an additional Ehlers' transformation we obtain a family of NUT spaces with negative NUT factor (reciprocal NUT factors).

Spherically symmetric static vacuum solutions in Mimetic gravity

In this paper we analyze spherically symmetric static vacuum solutions with various topologies in mimetic gravity. When the Einstein's tensor is different from zero, a new class of solutions different from the Schwarzschild one emerges from the theory. We analyze the feature of the new solutions and we study the planar motion for the spherical case.

Reflection symmetry in higher dimensional black hole spacetimes

In four spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein’s equation there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in spacetime dimensions such that the action of the isometry group is trivial. (Here ‘trivial’ means that if the ‘axes’—i.e., the points where the axial Killing fields are linearly dependent—are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.

High-dimensional Lifshitz-type spacetimes, universal horizons, and black holes in Hořava-Lifshitz gravity

In this paper, we present all $[(d+1)+1]$-dimensional static diagonal vacuum solutions of the non-projectable Ho\v{r}ava-Lifshitz gravity in the IR limit, and show that they give rise to very rich Lifshitz-type structures, depending on the choice of the free parameters of the solutions. These include the Lifshitz spacetimes with or without hyperscaling violation, Lifshitz solitons, and black holes. Remarkably, even the theory breaks explicitly the Lorentz symmetry and allows generically instantaneous propagations, universal horizons still exist, which serve as one-way membranes for signals with any large velocities. In particular, particles even with infinitely large velocities would just move around on these boundaries and cannot escape to infinity. Another remarkable feature appearing in the Lifshitz-type spacetimes is that the dynamical exponent $z$ can take its values only in the ranges $1 \le z < 2$ for $d \ge 3$ and $1 \le z <\infty$ for $d = 2$, due to the stability and ghost-free conditions of the theory.

Static vacuum solutions in non-Riemannian gravity

In the framework of non-riemannian geometry, we derive exact static vacuum solutions of the field equations obtained from the full equivalent version of the Einstein-Hilbert action when torsion degrees of freedom are taken into account. By imposing spherical symmetry and a suitable choice for the contorsion degrees of freedom, the static geometry provides deviations on the predictions of the observational tests predicted by General Relativity -- namely on the advance of planetary perihelia and the bending of light rays -- which we infer. The analytical extension is built in two particular domains of the parameter space. In the first domain we obtain a solution exhibiting an event horizon analogous to that of the Schwarzschild geometry. For the second domain, we show that the metric furnishes an exterior event horizon, and two interior horizons which enclose the singularity. For both branches we examine the effects of torsion corrections on the Hawking radiation. In this scenario the model extends Bekenstein's black hole geometrical thermodynamics, with an extra work term connected to a torsion parameter.

On generalized Einstein-Rosen waves in Brans-Dicke theory

In this paper cylindrically symmetric vacuum solutions corresponding to generalized Einstein-Rosen-type gravitational waves are considered in the framework of Brans-Dicke (BD) scalar-tensor theory. We see that, similar to axially symmetric vacuum solutions, under some assumptions, it is possible to generate BD-vacuum solutions from corresponding solutions in general relativity. Using this generating technique, we present several exact solutions corresponding to soliton, standing or pulse waves in BD theory. We also present a Kasner-type time-dependent generalization of cylindrically symmetric static vacuum solution. Some physical implications of these solutions are discussed in some detail. The effect of pulse type waves on test particle motion is briefly discussed and compared with static vacuum background. We also see that, similar to general relativity, there are no trapped cylinders in Einstein-Rosen-type solutions in BD theory.

Lifshitz spacetimes, solitons, and generalized BTZ black holes in quantum gravity at a Lifshitz point

In this paper, we study static vacuum solutions of quantum gravity at a fixed Lifshitz point in (2+1) dimensions, and present all the diagonal solutions in closed forms in the infrared limit. The exact solutions represent spacetimes with very rich structures: they can represent generalized BTZ black holes, Lifshitz space-times or Lifshitz solitons, in which the spacetimes are free of any kind of space-time singularities, depending on the choices of the free parameters of the solutions. We also find several classes of exact static non-diagonal solutions, which represent similar space-time structures as those given in the diagonal case. The relevance of these solutions to the non-relativistic Lifshitz-type gauge/gravity duality is discussed.

Geodesics in the linearized multipole solution: Distinguishing black holes from naked singularities

We analyze the behavior of the geodesic motion of test particles in the spacetime of a specific class of axially symmetric static vacuum solutions to the Einstein equations, hereafter referred to as the linearized multipole solution. We discuss its suitability to describe a quasispherical spacetime. The existence of an innermost stable circular orbit very close to the (singular) horizon of the source is established. The existence of such a stable orbit, closer than that of the Schwarzschild metric, as well as the appearance of a splitting in the admissible region of circular orbits, is shown to be due to the multipole structure of the solution, thereby providing additional potential observational evidence for distinguishing Schwarzschild black holes from naked singularities.

General-relativistic electromagnetic fields around a slowly rotating neutron star: stationary vacuum solutions

Pulsars are thought to be highly magnetized rotating neutron stars accelerating charged particles along magnetic field lines in their magnetosphere and visible as pulsed emission from the radio wavelength up to high energy X-rays and gamma-rays. Being highly compact objects with compactness close to $\Xi = R_s/R\approx0.5$, where $\Rs=2\,G\,M/c^2$ is the Schwarzschild radius and $\{M,R\}$ the mass and radius of the neutron star, general-relativistic effects become important close to their surface. This is especially true for the polar caps where radio emission is supposed to emanate from, leading to well defined signatures such as linear and circular polarization. In this paper, we derive a general formalism to extend to general relativity the Deutsch field solution valid in vacuum space. Thanks to a vector spherical harmonic expansion of the electromagnetic field, we are able to express the solution to any order in the spin parameter $\Omega$ of the compact object. We hope this analysis to serve as a benchmark to test numerical codes used to compute black hole and neutron star magnetospheres.

Stationary Axisymmetric and Slowly Rotating Spacetimes in Hořava-Lifshitz Gravity

Stationary, axisymmetric, and slowly rotating vacuum spacetimes in the Hořava-Lifshitz (HL) gravity are studied, and it is shown that, for any given spherical static vacuum solution of the HL theory (of any model, including the ones with an additional U(1) symmetry), there always exists a corresponding slowly rotating, stationary, and axisymmetric vacuum solution, which reduces to the former, when the rotation is switched off. The rotation is universal and only implicitly depends on the models of the HL theory and their coupling constants through the spherical seed solution. As a result, all asymptotically flat slowly rotating vacuum solutions are asymptotically identical to the slowly rotating Kerr solution. This is in contrast to the claim of Barausse and Sotiriou [Phys. Rev. Lett. 109, 181101 (2012)], in which slowly rotating black holes were reported (incorrectly) not to exist in the infrared limit of the nonprojectable HL theory.

A Note on Plane Symmetric Solutions in f(R) Gravity

The static plane symmetric vacuum solutions (Sharif and Shamir in Mod. Phys. Lett. A 25:1281, 2010) for n+1 dimension are reported. For this purpose, the generalized field equations are solved using the assumption of constant scalar curvature in metric f(R) gravity.