Vacuum Field Equations Research Articles

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.

A Brief Guide to Variations in Teleparallel Gauge Theories of Gravity and the Kaniel-Itin Model

Recently Kaniel & Itin proposed a gravitational model with the wave type equation [\square+\lambda(x)]\vartheta^\alpha=0 as vacuum field equation, where \vartheta^\alpha denotes the coframe of spacetime. They found that the viable Yilmaz-Rosen metric is an exact solution of the tracefree part of their field equation. This model belongs to the teleparallelism class of gravitational gauge theories. Of decisive importance for the evaluation of the Kaniel-Itin model is the question whether the variation of the coframe commutes with the Hodge star. We find a master formula for this commutator and rectify some corresponding mistakes in the literature. Then we turn to a detailed discussion of the Kaniel-Itin model.

Energy and Momentum in the Tetrad Theory of Gravitation

We study the energy and momentum of an isolated system in the tetrad theory of gravitation, starting from the most general Lagrangian quadratic in torsion, which involves four unknown parameters. When applied to the static spherically symmetric case, the parallel vector fields take a diagonal form, and the field equation has an exact solution. We analyze the linearized field equation in vacuum at distances far from the isolated system without assuming any symmetry property of the system. The linearized equation is a set of coupled equations for a symmetric and skew-symmetric tensor fields, but it is possible to solve it up to $O(1/r)$ for the stationary case. It is found that the general solution contains two constants, one being the gravitational mass of the source and the other a constant vector ${\grave B_\alpha}$. The total energy is calculated from this solution and is found to be equal to the gravitational mass of the source. We also calculate the spatial momentum and find that its value coincides with the constant vector ${\grave B_\alpha}$. The linearized field equation in vacuum, which is valid at distances far from the source, does not give any information about whether the constant vector ${\grave B_\alpha}$ is vanishing or not. For a weakly gravitating source for which the field is weak everywhere, we find that the constant vector ${\grave B_\alpha}$ vanishes.

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.

Combining Cauchy and characteristic codes. IV. The characteristic field equations in axial symmetry

This paper is part of a long term program to develop combined Cauchy and characteristic codes as investigative tools in numerical relativity. In the third stage attention is devoted to axisymmetric systems possessing two spatial degrees of freedom. In the second of two preliminary theoretical papers, the vacuum field equations for the characteristic region are obtained and the compactified equations are regularized.

Extended double Riemann–Hilbert problem and multiple representations of the infinite-dimensional symmetric group for the stationary axisymmetric vacuum field equations

The double-complex function method suggested in previous papers is extended and the extended method is used to establish two mutually dual extended double Hauser–Ernst equations and related Riemann–Hilbert problems, then quadruple representations of the semidirect product of the Kac–Moody and Virasoro groups are given. These show that, for the stationary axisymmetric vacuum space–times, there are more and richer symmetry structures than previously expected. The multiple forms of some well-known transformations can be obtained as some special cases.

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.

CLASSICAL SOLUTIONS IN FIVE-DIMENSIONAL-INDUCED MATTER THEORY AND ITS RELATION TO AN IMPERFECT FLUID

We study five-dimensional cosmological models with four-dimensional hypersurfaces of the Bianchi types I and V. In this way the five-dimensional vacuum field equations GAB=0 led us to four-dimensional matter equations Gμv=Tμv and the matter is interpreted as a purely geometrical property of a fifth dimension. Also, we find that the energy-momentum tensor induced from the fifth dimension has the structure of an imperfect fluid that has dissipative terms.

Vacuum spacetimes with toroidal null infinities

It is shown that some known solutions of Einstein's vacuum field equations admit null infinities such that the sections of the null hypersurface at infinity are toroidal.

Equivalence Principle in the New General Relativity

We study the problem of whether the active gravitational mass of an isolated system is equal to the total energy in the tetrad theory of gravitation. The superpotential is derived using the gravitational Lagrangian which is invariant under parity operation, and applied to an exact spherically symmetric solution. Its associated energy is found equal to the gravitational mass. The field equation in vacuum is also solved at far distances under the assumption of spherical symmetry. Using the most general expression for parallel vector fields with spherical symmetry, we find that the equality between the gravitational mass and the energy is always true if the parameters of the theory $a_1$, $a_2$ and $a_3$ satisfy the condition, $(a_1+ a_2) (a_1-4a_3/9)\neq0$. In the two special cases where either $(a_1+a_2)$ or $(a_1-4a_3/9)$ is vanishing, however, this equality is not satisfied for the solutions when some components of the parallel vector fields tend to zero as $1/\sqrt{r}$ for large $r$.

A class of Kaluza-Klein soliton solutions

We give a class of exact solutions of the 5D field equations in vacuum that has 3 independent parameters, thereby extending the 2-parameter class of Kaluza-Klein solitons.

On the existence of C ∞ solutions to the asymptotic characteristic initial value problem in general relativity

The asymptotic characteristic initial value problem for Einstein's vacuum field equations is treated. It is shown that C$^{\infty}$ solutions exist and are unique for C$^{\infty}$ initial values. The proof is based on Friedrich's regular conformal vacuum field equations and Rendall's method of reducing the characteristic to an ordinary initial value problem.

A cosmological approach to a high-order effective field theory

A class of cosmological solutions is found to the vacuum field equations of an effective field theory derived from string theory in a -dimensional spacetime. This effective field theory concerns only the massless sector of the parent string theory. In order to make it operationally accessible, it has been proposed that the notion of the dimension of spacetime be based on measure-theoretic concepts, thus admitting the possibility of non-integer dimensions. As a consequence of the dimension of actual spacetime less than four , a relativistic quantum field theory becomes finite. Also, the radiative corrections of perturbation theory are sensitive to the actual value of the dimension . Present experimental results and standard theoretical predictions for the electromagnetic moment of the electron seem to suggest a non-vanishing value for . We take . In this paper, it has been proved that the coupling coefficients of high-order terms of the equations of the effective field theory depend on the value of . The results have been compared with the ones in the literature.

Induced matter theory and embeddings in Riemann flat space–times

A class of five-dimensional space-times that contain four-dimensional hypersurfaces whose intrinsic metrics are of cosmological interest is investigated. First, the five-dimensional space–time is assumed to be Riemann flat—the problem of determining the intrinsic metrics of the four-dimensional hypersurfaces then becomes a problem of embedding in flat space–time. Second, the Riemann flat solutions are used as a starting point to find solutions to Einstein’s vacuum field equations in five dimensions that are not Riemann flat. In particular, a new general class of five-dimensional vacuum solutions is found.

Arbitrariness of the general solution and symmetries

The computation of a number of arbitrary functions in the general solution is briefly reviewed. The results are used to study normal systems and their symmetry reduction. We discuss the treatment of gauge systems, especially the analysis of gauge fixing conditions. As examples, the Yang-Mills equations with the Lorentz gauge and Einstein's vacuum field equations with harmonic coordinates are considered.

Killing spinors and separability of Rarita–Schwinger’s equation in type {2,2} backgrounds

It is shown that in a type {2,2} solution of the Einstein vacuum field equations, which admits a two-index Killing spinor, a differential operator can be constructed that maps a solution of the Rarita–Schwinger equation into a solution of its complex conjugate. Furthermore, by considering the Plebański–Demiański metric, which contains all the vacuum type {2,2} metrics, it is shown that the separable solutions of the Rarita–Schwinger equation are eigenfunctions of a certain differential operator with the Starobinsky constant as the eigenvalue.