Solutions Of Vacuum Einstein Equations Research Articles

On the Uniqueness of Schwarzschild–de Sitter Spacetime

We establish a new uniqueness theorem for the three dimensional Schwarzschild–de Sitter metrics. For this, some new or improved tools are developed. These include a reverse Łojasiewicz inequality, which holds in a neighborhood of the extremal points of any smooth function. We further prove the smoothness of the set of maxima of the lapse, whenever this set contains a topological hypersurface. This leads to a new strategy for the classification of well behaved static solutions of vacuum Einstein equations with a positive cosmological constant, based on the geometry of the maximum-set of the lapse.

On uniqueness of static asymptotically anti-de Sitter black hole

In this paper, the asymptotically anti-de Sitter spacetime with a hypersurface orthogonal Killing vector field is discussed near the conformal boundary by using Newman–Penrose formalism. We first investigate the multipole structure of a static, asymptotically anti-de Sitter spacetime. We note that, different to asymptotically flat case in which the multipoles are contained in , the multipoles of a static, asymptotically anti-de Sitter spacetime are all contained in . We then show that the Schwarzschild anti-de Sitter spacetime is the only asymptotically anti-de Sitter, vacuum, static, axisymmetric, type-D analytic solution of vacuum Einstein equation with a negative cosmological constant in a Bondi coordinate neighborhood of the conformal boundary if the spherical harmonic expansion of is finite.

Quasinormal modes of static and spherically symmetric black holes with the derivative coupling

We investigate the quasinormal modes of a class of static and spherically symmetric black holes with the derivative coupling. The derivative coupling has rarely been paid attention to the study of black hole quasinormal modes. Specifically, we study the effect of derivative coupling on the quasinormal modes for four kinds of black holes. They are Reissner–Nordström black holes, Bardeen black holes, noncommunicative geometry inspired black holes and dilaton black holes. These black holes are not the solutions of vacuum Einstein equations which guarantees the effect of derivative coupling is not trivial. We find the influence of derivative coupling on the quasinormal modes roughly mimics the overtone numbers. In other words, there is a qualitative similarity in the trend of quasinormal modes frequencies due to increasing either the coupling constant and the overtone number.

Higher order perturbations of anti–de Sitter space and time-periodic solutions of vacuum Einstein equations

Motivated by the problem of stability of Anti-de Sitter (AdS) spacetime, we discuss nonlinear gravitational perturbations of maximally symmetric solutions of vacuum Einstein equations in general and the case of AdS in particular. We present the evidence that, similarly to the self-gravitating scalar field at spherical symmetry, the negative cosmological constant allows for the existence of globally regular, asymptotically AdS, time-periodic solutions of vacuum Einstein equations whose frequencies bifurcate from linear eigenfrequencies of AdS. Interestingly, our preliminary results indicate that the number of one parameter families of time-periodic solutions bifurcating from a given eigenfrequency equals the multiplicity of this eigenfrequency.

Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.

Petrov type I spacetime and dual relativistic fluids

The Petrov type I condition for the solutions of vacuum Einstein equations in both of the nonrelativistic and relativistic hydrodynamic expansions is checked. We show that it holds up to the third order of the nonrelativistic hydrodynamic expansion parameter, but it is violated at the fourth order even if we choose a general frame. On the other hand, it is found that the condition holds at least up to the second order of the derivative expansion parameter. Turn the logic around, through imposing the Petrov type I condition and Hamiltonian constraint on a finite cutoff surface, we show that the stress tensor of the relativistic fluid can be recovered with correct first order and second order transport coefficients dual to the solutions of vacuum Einstein equations.

Charged black rings from inverse scattering

The inverse scattering method of Belinsky and Zakharov is a powerful method to construct solutions of vacuum Einstein equations. In particular, in five dimensions this method has been successfully applied to construct a large variety of black hole solutions. Recent applications of this method to Einstein-Maxwell-dilaton (EMd) theory, for the special case of Kaluza-Klein dilaton coupling, has led to the construction of the most general black ring in this theory. In this contribution, we review the inverse scattering method and its application to the EMd theory. We illustrate the efficiency of these methods with a detailed construction of an electrically charged black ring.

Generating black strings

It is generated a five dimensional solution of vacuum Einstein equations, with three free parameters, by means of equipping the Zipoy-Voorhees solution with a scalar field then lifting it to a fifth dimension. This solution presents as particular cases the black string with arbitrary tension and the Gregory-Laflamme black string.The trapped surfaces, mass, tension and the condition of spherical symmetry of the solution are studied.

The local invariable solution of vacuum Einstein equation

In this paper, using the method of fibre bundle, we solve the coefficient problem in the operator of prolongation group with the aid of the conception of prolongation group for Lie transformation group. We obtain the permitted group of the vacuum Einstein equation, and introduce a doable program of getting local invariable solution of the equation.

New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness

New axisymmetric stationary solutions of the vacuum Einstein equations in five-dimensional asymptotically flat spacetimes are obtained by using solitonic solution-generating techniques. The new solutions are shown to be equivalent to the four-dimensional multi-solitonic solutions derived from particular class of four-dimensional Weyl solutions and to include different black rings from those obtained by Emparan and Reall.

Diverging and twisting type N solutions of vacuum Einstein equations: an approximative approach

An approximative method to find diverging and twisting type N solutions to the vacuum Einstein equations is considered. The general solution of iteration equations is found. The general metric of a regular spacetime in the first-order approximation is given. It is shown that there exists no regular solution in the second-order approximation. This leads us to the conclusion that there are severe difficulties in considering the vacuum type N solutions as those describing the gravitational radiation from a bounded source. In this paper, the Debney–Kerr–Schild coordinates are employed and the zero-order approximation is taken to have a particular simple form.

Generalized Weyl solutions

It was shown by Weyl that the general static axisymmetric solution of the vacuum Einstein equations in four dimensions is given in terms of a single axisymmetric solution of the Laplace equation in three-dimensional flat space. Weyl's construction is generalized here to arbitrary dimension $D\ge 4$. The general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal commuting non-null Killing vector fields is given either in terms of D-3 independent axisymmetric solutions of Laplace's equation in three-dimensional flat space or by D-4 independent solutions of Laplace's equation in two-dimensional flat space. Explicit examples of new solutions are given. These include a five-dimensional asymptotically flat ``black ring'' with an event horizon of topology S^1 x S^2 held in equilibrium by a conical singularity in the form of a disc.

Non-expanding impulsive gravitational waves

We investigate a class of impulsive gravitational waves which propagate either in Minkowski or in the (anti-) de Sitter background. These waves are constructed as impulsive members of the Kundt class of non-twisting, non-expanding type-N solutions of vacuum Einstein equations with a cosmological constant . We show that the only non-trivial waves of this type in Minkowski spacetime are impulsive pp-waves. For we demonstrate that the canonical subclasses of , which are invariantly different for smooth profiles, are all locally equivalent for impulsive profiles. Also, we present a coordinate system for these impulsive solutions which is explicitly continuous.

Structure of the space of solutions of Einstein’s equations: A new variables approach

Following Ashtekar’s new Hamiltonian formulation of general relativity [‘‘A new Hamiltonian formulation of general relativity,’’ Syracuse University preprint, 1986; Phys. Rev. Lett. 57, 2244 (1986); Proceedings of the Florence Conference on ‘‘Constrained Systems’’ (World Scientific, Singapore, 1986)], the results of Arms, Fischer, Marsden, and Moncrief [Ann. Inst. H. Poincaré 33, 147 (1980); Ann. Phys. (NY) 144, 81 (1982)] on the structure of the space of solutions of vacuum Einstein equations in the case of space-times admitting a compact Cauchy hypersurface are rederived and extended.

Soliton transformations for axially symmetric higher-dimensional gravity. II. Belinskii–Zakharov N-soliton transformations

The solutions of vacuum Einstein equations in 4+K dimensions with 2+K commuting Killing vectors under the Abelian Kaluza–Klein ansatz are considered. This system admits Belinskii–Zakharov-type soliton transformations. The explicit formulas for general N-soliton transformations are obtained by iterating the formulas for one-soliton transformation.

Generalized Neugebauer-Kramer transformations for non-linear sigma models

We consider the solutions of vacuum Einstein equations in 4 + K dimensions with 2 + K commuting Killing vectors and show that this system possesses a series of discrete symmetries I (1) generalizing the Neugebauer-Kramer transformation which corresponds to the K = 0 case. When conjugated with the dual symmetry, we obtain a series of continuous symmetries generalizing the I 1 transformation of Neugebauer. We argue that the discrete symmetries are in fact symmetries for any generalized non-linear sigma models.

Numerical analysis of one-soliton solutions on a Bianchi type-II background

The asymptotic behavior of a solution of vacuum Einstein equations describing the propagation of a single “soliton wave” on a Bianchi type-II background is investigated numerically. In the framework of the oscillatory approach to a cosmological singularity, the “transitional behavior” corresponding to a critical value of a fundamental integration constant is analyzed.

A classification theorem for static vacuum black holes, Part I: the study of the lapse

The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we extend this result by proving a classification theorem for all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without making any further assumption on the topology or the asymptotic. It is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In a broad sense, the theorem classifies all the static vacuum black holes in 3+1-dimensions. In this Part I we use introduce techniques in conformal geometry and comparison geometry a la Bakry-Emery to prove that vacuum static black holes have only one end, and, furthermore, that the lapse is bounded away from zero at infinity. The techniques have interest in themselves and could be applied in other contexts as well, for instance to study higher-dimensional static black holes.