Vacuum Einstein Field Equations Research Articles

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.

Spacetime Ehlers group: transformation law for the Weyl tensor

The spacetime Ehlers group, which is a symmetry of the Einstein vacuumfield equations for strictly stationary spacetimes, is defined and analysedin a purely spacetime context (without invoking the projectionformalism). In this setting, the Ehlers group finds its natural descriptionwithin an infinite-dimensional group of transformations that mapsLorentz metrics into Lorentz metrics and which may be of independentinterest. The Ehlers group is shown to be well defined independently of thecausal character of the Killing vector (which may become null on arbitraryregions). We analyse which global conditions are required on thespacetime for the existence of the Ehlers group. The transformation lawfor the Weyl tensor under Ehlers transformations is obtainedexplicitly.This allows us to study where, and under what circumstances, curvaturesingularities in the transformed spacetime will arise. The results of thepaper are applied to obtain a local characterization of the Kerr-NUTmetric.

The method of adjoint operators and a symmetry operator for the Maxwell equations in type-D vacuum backgrounds

Since in all the type-D solutions of the Einstein vacuum field equations with a cosmological term each maximal spin-weighted component of the electromagnetic spinor field satisfies a decoupled equation and there exists a two-index Killing spinor field; we show, via the adjoint operators method, that a symmetry operator for the Maxwell equations can be constructed.

Xanthopoulos Theorem in the Kaluza–Klein Theory

It is shown that if ĝAB is an exact solution of the Einstein vacuum field equations in 4 + 1 dimensions, R^AB = 0, and lA is a null vector field, then ĝAB + lAlB is also an exact solution of the Einstein equations R^AB = 0 if and only if the perturbation lAlB satisfies the Einstein equations linearized about ĝAB. Then, making use of the Kaluza–Klein approach, it is shown that this result allows us to obtain exact solutions of the Einstein–Maxwell equations (possibly coupled to a scalar field) by solving a system of linear equations.

Kerr and Schwarzschild black-holes in three-dimensional relativity

We consider the Kerr and Schwarzschild black-hole space–times in the framework of a three-dimensional formulation of relativistic kinematics and field dynamics, in which local physical observers are represented by non-singular vector fields of bounded length in a three-dimensional pseudo-Riemannian space. A space–time is represented by a pair of 3-metric and a fundamental 3-vector field satisfying a set of basic equations, each solution of which determines uniquely a solution of the vacuum Einstein field equations. It is shown that the only spherically symmetric solution of our basic equations leads to the Schwarzschild space–time, thereby proving a version of Birkoff’s theorem in this formalism. The Schwarzschild horizon and the Kerr stationary limit are both related to the upper bound of the length of the corresponding physical 3-vector fields. An example of a solution of the vacuum Einstein field equations shows that nonstationary space–times can also be formulated in this three-dimensional relativity.

Newman-Penrose Formalism and Gravitational Impulsive and Shock Waves**The project supported by the Scientific Foundation of the Chinese Academy of Sciences

Vacuum spacetimes endowed with two commuting spacelike Killing vector fields are considered. Subject to the hypothesis that there exists a shearfree null geodesic congruence orthogonal to the two-surface generated by the two commuting spacelike Killing vector fields,it is shown that, with a specific choice of null tetrad, the Newman-Penrose equations are reduced to an ordinary differential equation of Riccati type. fiom the consideration of this differential equation, exact solutions of the vacuum Einstein field equations with distribution valued Weyl curvature describing the propagation of gravitational impulsive and shock wave of variable polarization are then constructed.

An exact gravitational wave solution

An exact solution describing an axisymmetric gravitational wave propagating in z -direction is given in closed form. This solution to Einstein's vacuum field equations has no curvature singularities, in particular it is regular at the symmetry axis. The gravitational wave solution has the nice property that the curvature invariants tend to zero with increasing distance from the axis.

An exact gravitational wave solution

AbstractAn exact solution describing an axisymmetric gravitational wave propagating in z ‐direction is given in closed form. This solution to Einstein's vacuum field equations has no curvature singularities, in particular it is regular at the symmetry axis. The gravitational wave solution has the nice property that the curvature invariants tend to zero with increasing distance from the axis.

Dilatonic AdS-Kerr solution to the AdS/CFT correspondence

We consider the $AdS_5$ solution deformed by a non-constant dilaton interpolating between the standard AdS (UV region) and flat boundary background (IR region). We show that this dilatonic solution can be generalized to the case of a non-flat boundaries provided that the metric of the boundaries satisfies the vacuum Einstein field equations. As an example, we describe the case when the four-dimensional boundaries represent the Kerr space-time.

Colliding Plane Impulsive Gravitational Waves

When two non-interacting plane impulsive gravitational waves undergo a head-on collision, the vacuum interaction region between the waves after the collision contains backscattered gravitational radiation from both waves. The two systems of backscattered waves have each got a family of rays (null geodesics) associated with them. We demonstrate that if it is assumed that a parameter exists along each of these families of rays such that the modulus of the complex shear of each is equal then Einstein's vacuum field equations, with the appropriate boundary conditions, can be integrated systematically to reveal the well-known solutions in the interaction region. In so doing the mystery behind the origin of such solutions is removed. With the use of the field equations it is suggested that the assumption leading to their integration may be interpreted physically as implying that the energy densities of the two backscattered radiation fields are equal. With the use of different boundary conditions this approach can lead to new collision solutions.

Gravitational waves in vacuum spacetimes with cosmological constant. I. Classification and geometrical properties of nontwisting type N solutions

All nontwisting Petrov-type N solutions of vacuum Einstein field equations with cosmological constant Λ are summarized. They are shown to belong either to the nonexpanding Kundt class or to the expanding Robinson–Trautman class. Invariant subclasses of each class are defined and the corresponding metrics are given explicitly in suitable canonical coordinates. Relations between the subclasses and their geometrical properties are analyzed. In the subsequent paper these solutions are interpreted as exact gravitational waves propagating in de Sitter or anti-de Sitter spacetimes.

Self-duality and gravitational perturbations

It is shown that there exists a one-to-one correspondence between the real solutions of the Einstein vacuum field equations linearized about the Minkowski metric and the (complex) metric perturbations whose curvature to first order in the metric perturbation is self-dual. It is also shown that the self-duality condition of the curvature to first order in the metric perturbation is equivalent to a set of first-order equations for the metric perturbation whose solution is given by a scalar potential that obeys the wave equation.

Non-static spherically symmetric solution of Einstein vacuum field equations with Λ

The Schwarzschild-deSitter metric is the known solution of Einstein field equations with cosmological constant term for vacuum spherically symmetric space around a point mass M. Recently it has been reported that in a $Lamda$-dominant world the Schwarzschild type coordinate systems are disqualified by redshift-magnitude test as a proper frame of reference(gr-qc/9812092). We derive the solution in a FRW type coordinate system which is qualified according to the mentioned test. Asymptotically it approachs to the non-static form of deSitter metric. The obtained metric is transformable to Schwarzschild-deSitter metric. It is an analytic function of $r$ for all values except $r=0$which is singular. This is carried out with no making use of Eddington-Finkelstein coordinates and without entering any cross term in the metric.

The Initial Boundary Value Problem for Einstein's Vacuum Field Equation

We study the initial boundary value problem for Einstein's vacuum field equation. We prescribe initial data on an orientable, compact, 3-dimensional manifold S with boundary Σ≠? and boundary conditions on the manifold T= Re+ 0×Σ. We assume the boundaries Σ and { 0 }×, Σ of S and T to be identified in the natural way. Furthermore, we prescribe certain gauge source functions which determine the evolution of the fields. Provided that all data are smooth and certain consistency conditions are met on Σ, we show that there exists a smooth solution to Einstein's equation Ric[g] = 0 on a manifold which has (after an identification) a neighbourhood of S in T∪S as a boundary. The solution is such that S is space-like, the initial data are induced by the solution on S, and, in the region of T where the solution is defined, T is time-like and the boundary conditions are satisfied.

Debye Potentials for Self-Dual Fields

It is shown that in a space-time that admits ageodetic and shear-free null vector field which is aprincipal direction of the conformal curvature(therefore, in any algebraically special solution of the Einstein vacuum field equations), any self-dualelectromagnetic field is locally given by a scalar(Debye) potential which obeys a second-orderdifferential equation and, similarly, that any self-dualYang-Mills field is locally given by a matrix-valuedpotential governed by a nonlinear second-orderdifferential equation. Using the fact that any self-dualelectromagnetic field is the self-dual part of a realsolution of the source-free Maxwell equations, it isshown that in any space-time of this class, the solutionof the source-free Maxwell equations is locally given bya Debye potential.

Approximate solutions of Einstein's vacuum field equations in the type N, twisting and diverging case

Approximate solutions of Einstein's vacuum field equations in the type N, twisting and diverging case are investigated. In this context, it is shown that the field equation determines a function of integration at any order and so can be satisfied. The field equation and other conditions for solutions to be type N are explicitly shown to have a first- and second-order approximate solution which is non-singular. It is argued that an approximate solution at any finite order can be calculated without the occurrence of singularities.

Spherically Symmetric Solutions of Einstein's Vacuum Equations in Five Dimensions

Multi-dimensional spherically symmetric spacetimes are of interest in the study of higher-dimensional black holes (and solitons) and higher-dimensional cosmological models. In this paper we shall present a comprehensive investigation of solutions of the five-dimensional spherically symmetric vacuum Einstein field equations subject only to the condition of separability in the radial coordinate (but not necessarily in the remaining two coordinates). A variety of new solutions are found which generalize a number of previous results. The properties of these solutions are discussed with particular attention being paid to their possible astrophysical and cosmological applications. In addition, the four-dimensional properties of matter can be regarded as geometrical in origin by a reduction of the five-dimensional vacuum field equations to Einstein's four-dimensional theory with a non-zero energy-momentum tensor constituting the material source; we shall also be interested in the induced matter associated with the new five-dimensional solutions obtained.

Quadrupole-quadrupole gravitational waves

This paper investigates the nonlinear self-interaction of quadrupole gravitational waves generated by an isolated system. The vacuum Einstein field equations are integrated in the region exterior to the system by means of a post-Minkowskian algorithm. Specializing to the quadrupole-quadrupole interaction (at the quadratic nonlinear order), we recover the known results concerning the non-local modification of the ADM mass-energy of the system accounting for the emission of quadrupole waves, and the non-local memory effect due to the re-radiation by the stress-energy distribution of linear waves. We then compute all the local (instantaneous) terms which are associated, in the quadrupole-quadrupole metric, with the latter non-local effects. Expanding the metric at large distances from the system, we obtain the corresponding radiation-field observables, including all non-local and transient contributions. This permits, notably, the completion of the observable quadrupole moment at the post-Newtonian order.

Hamiltonian reduction of Einstein's equations of general relativity

A program is outlined which resolves the problem of the Hamiltonian reduction of Einstein's vacuum field equations in (3 + 1)-dimensions. The problem involves writing Einstein's vacuum field equations as an unconstrained Hamiltonian dynamical system where the variables of the unconstrained system are the true degrees of freedom of the gravitational field. Our analysis is applicable to vacuum spacetimes that admit constant mean curvature compact spacelike hypersurfaces M that satisfy certain topological restrictions. We find that for these spacetimes (3 + 1)-reduction can be completed much as in the (2 + 1)-dimensional case. In both cases, one gets as the reduced phase space the contangent bundle T ast;T M of the Teichm üller space T M= M P D 0 of conformal structures on M and one gets reduction of the full classical Hamiltonian system with constraints to a non-local time-dependent reduced Hamiltonian system without constraints on the contact manifold R × T ∗τ M. For this reduced system, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact spacelike hypersurfaces in a neighborhood of the given initial one and the Hamiltonian is the volume functional of these hypersurfaces expressed in terms of the canonical variables of the hypersurface.

Killing spinors and spin- fields in type-D spacetimes

It is shown that if the background spacetime is a type-D solution of the Einstein vacuum-field equations with cosmological constant or of the Einstein - Maxwell field equations without cosmological constant, an operator can be constructed that maps a solution of the Rarita - Schwinger equation into another solution. In the case of the spin- fields in a type-D electrovac spacetime, we show that the maximal spin weight components, which obey separable decoupled equations, satisfy certain differential relations that are equivalent to the Teukolsky - Starobinsky identities fulfilled by the separated functions. It is also shown that the real and imaginary parts of the Starobinsky constants can be obtained from these relations.