Vacuum Einstein Field Equations Research Articles

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.

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.

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.

Imploding-exploding gravitational waves

A solution of Einstein's vacuum field equations is constructed describing an imploding spherical impulsive gravitational wave followed by an exploding similar wave. The two waves propagate in Minkowskian spacetime and the history of the process is the past and future sheets of the null-cone of an event (taken as origin) in the spacetime. The solution is a superposition of two of Penrose's impulsive wave solutions and is described in a single coordinate system in which the metric tensor components are continuous across the histories of the wave fronts.

Differential forms, spin-3/2 fields and Einstein's equations

The relationship between Einstein's vacuum field equations and classical spin-3/2 field equations is re-explored using two-component spinor valued differential forms. One-form and two-form expressions for spin-3/2 fields and their field equations are considered and a Lagrangian formulation is presented. Complex structures defined by spin-3/2 fields are considered.

Higher-dimensional vacuum solutions of Einstein's field equations

Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.

Relativity in three dimensions

We present a new formulation of relativistic kinematics and field dynamics in an entirely three-dimensional Riemannian space. Here the basic idea is to represent local physical observers by three-dimensional vector fields of bounded length and the dynamics of physical fields by the flows of vector fields, which are one-parameter groups of local smooth transformations of the 3-space. The flow parameter plays the role of local time for each physical observer, and the Lie derivative replaces the time derivative. In the kinematics part we show how the introduction of a relative velocity function in terms of the 3-metric of the space leads immediately to the Lorentz time-dilation formula for two inertially equivalent physical observers. A treatment of equivalence class of inertial observers leads to the definition of generalized Lorentz matrix functions with corresponding generalized properties, which reduce to the usual Lorentz matrices when the underlying space is Euclidean. In the dynamics part we treat the Einstein field equations in Gaussian normal coordinates by reformulating them in a three-dimensional set-up and show how certain solutions, including the de Sitter solution, can be derived from a flat-space 3-metric by subjecting it to a transformation by the flow of a three-dimensional vector field. In this way, we obtain a new solution of the vacuum Einstein field equations which is related to the complex form of (real) Kasner-type solutions. Our formalism can also be applied to other field equations, such as the Maxwell equations, and provide a corresponding set of three-dimensional equations to generate four-dimensional solutions.

Lorentz group and spherical impulsive gravity waves.

The method for constructing a spherical impulsive gravitational wave solution of Einstein's vacuum field equations has been described by Penrose. Recently the author gave an explicit transformation leading to coordinates in which the metric tensor is continuous across the history of the wave front in Minkowskian space-time. In this paper we exhibit the relationship between this transformation and proper, orthochronous Lorentz transformations. This provides a novel viewpoint on spherical impulsive gravity waves which facilitates a description of them in terms of new continuous coordinate systems.

Higher dimensional vacuum cosmologies

view Abstract Citations (33) References (66) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Higher Dimensional Vacuum Cosmologies Coley, A. A. Abstract Higher dimensional cosmological models are reviewed, and, in particular, vacuum solutions of the field equations of generalized Lagrangian theories are investigated. The possibility that the (four-dimensional) properties of matter may be geometric in origin is discussed by studying whether the higher dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with a nonzero energy-momentum tensor (taken to be a cosmological perfect fluid) constituting the material source. In a sense, this generalizes the work done in the five-dimensional Einstein theory by considering that Einstein's four-dimensional theory with matter is embedded in a higher dimensional generalized Lagrangian theory. It is known that the five-dimensional (D = 5) vacuum Einstein field equations (in which the metric is independent of the fifth dimension) give rise to the familiar radiation Friedmann-Robertson-Walker cosmological model (of Einstein's four-dimensional theory with perfect fluid source). However, it is of interest to study models with more general forms of matter, and so we are motivated to study vacuum solutions in higher dimensions (D greater than 5) and in theories more general than Einstein's theory of gravity. A variety of different higher dimensional vacuum solutions with a cosmological metric are found and discussed. In particular, theories in five dimensions (D = 5) with the addition of general quadratic curvature invariants to the usual Einstein-Hilbert action, Einstein's theory in higher dimensions (D greater than 5), and the general D-dimensional Gauss-Bonnet theory are investigated in detail. Many new higher dimensional vacuum cosmological solutions are found. In some cases of particular interest, the general solution is obtained. Approximate solutions are also investigated. The solutions that are obtained then give rise to models with a variety of different forms for the four-dimensional matter; indeed, it is found that models with a wide range of physically acceptable equations of state are possible. Publication: The Astrophysical Journal Pub Date: June 1994 DOI: 10.1086/174169 Bibcode: 1994ApJ...427..585C Keywords: Astronomical Models; Cosmology; Einstein Equations; Gravitational Fields; Lagrangian Function; Relativity; Space-Time Functions; Unified Field Theory; Vacuum; Curvature; Equations Of State; Matter (Physics); Particle Mass; String Theory; Supergravity; Supersymmetry; Tensors; Physics (General); COSMOLOGY: THEORY full text sources ADS |

A Finsler generalisation of Einstein's vacuum field equations

This paper gives a generalisation of Einstein's vacuum field equations for Finsler metrics. The given generalised field equation reproduces the Einstein equations for Riemannian metrics, and also admits non-Riemannian solutions. This is shown in detail by deriving a first order Finsler perturbation, solving the new field equation, of the Schwarzschild metric. This perturbation turns out to be time independent. The effects of the perturbation on the three Classical Tests of General Relativity are derived, and used to give limits on the size of the perturbation parameter involved.

NUMERICAL RELATIVITY: EVOLVING SPACETIME

The construction of numerical solutions of Einstein's General Relativity equations is formulated as an initial-value problem. The space-plus-time (3 + 1) decomposition of the spacetime metric tensor is used to discuss the structure of the field equations. The resulting evolution system is shown to depend in a crucial way on the coordinate gauge. The mandatory use of singularity avoiding coordinate conditions (like maximal slicing or similar gauges) is explained. A brief historical review of Numerical Relativity is included, showing the enormous effort in constructing codes based in these gauges, which lead to non-hyperbolic evolution systems, using "ad hoc" numerical techniques. A new family of first order hyperbolic evolution systems for the vacuum Einstein field equations in the harmonic slicing gauge is presented. This family depends on a symmetric 3 × 3 array of parameters which can be used to scale the dynamical variables in future numerical applications.

Exact treatment of time-dependent axisymmetric gravitation

This paper extends an earlier treatment of time-dependent gravitational fields that are axially symmetric and non-rotating. From a consideration of the canonical solution of the Einstein vacuum field equations previously obtained as an axial expansion, a new method has been found that now provides the exact solution, whenever a certain generative key function X ( t , z ) is known.

A spherical impulse gravity wave.

An exact solution of Einstein's vacuum field equations describing a spherical-fronted impulsive gravity wave is given in a coordinate system in which the metric tensor is continuous across the (future null cone) history of the wave front in Minkowskian space-time. A coordinate transformation is presented which reduces the line element to Minkowskian form in the future of the null cone (it is manifestly in Minkowskian form in the past of the null cone) and enables us to demonstrate how the solution has been constructed using the Penrose procedure of subdividing flat space-time into two halves with the future null cone as boundary and then reattaching the halves with a warp.

Boost symmetries in spatially compact spacetimes with a cosmological constant

The author characterizes those boost symmetric spacetimes which, in addition to de Sitter space, admit spatially compact, (locally) conformally flat, time symmetric Cauchy hypersurfaces and satisfy Einstein's vacuum field equations with a cosmological constant. The Cauchy surfaces all turn out to be (finite quotients of) Riemannian products and warped products of (S1,dy2) with (S2, gamma k) where gamma k is the canonical metric of constant positive sectional curvature k.

Solitonic solutions on a Bianchi II background generated by SHEEP algebraic manipulation

The exact solutions of Einstein's (vacuum) field equations corresponding to a (real-poles) three-solitonic perturbation of a Bianchi type II space-time are investigated by using computer algebra. Two-dimensional and three-dimensional plots of the relevant solitonic fields are obtained numerically.

Einstein vacuum field equations with a single non-null Killing vector

It is shown that, in the case where there is a single non-null Killing vector, the vacuum Einstein field equations imply that there is a Ricci collineation in the quotient 3-space. Using coordinates adapted to the collineation vector, we derive a fourth order partial differential equation involving the metric of the quotient 3-space and we show that if this equation is satisfied, the Ernst potential may be obtained by integrating a total Riccati equation and a straightforward set of total differential equations. We also show that if the collineation vector is null, the metric of the quotient 3-space may be expressed in terms of two real Clebsch potentials. Finally in the special case where the collineation vector is the generator of a timelike homothetic motion we reduce the field equations to a single second order partial differential equation of non-Painleve type in two independent variables and obtain Petrov type III solution of Robinson-Trautman type.

Vacuum solutions of Einstein's equations in double-null coordinates

For four-dimensional spacetimes with 2-spaces of constant curvature, the authors present a complete list of solutions of Einstein's vacuum field equations with nonzero cosmological constant. All solutions requiring a regular extension are given in manifestly regular double-null coordinates.

On null tratorial torsion in vacuum quadratic poincare gauge field theory

The field equations of vacuum quadratic Poincare gauge field theory (QPGFT) are solved for purely null tratorial torsion. Expressing the contortion tensor for this case is Klambda mu nu =-2(glambda mu anu -glambda nu amu ), where amu is null, and one can prove that any solution of vacuum QPGFT, whose V4 part is also a solution of Einstein's vacuum field equations with a cosmological constant, is necessarily of algebraic type N with respect to the generalised Weyl tensor. This class of exact solutions has an expansion-free, shear-free and twist-free autoparallel repeated principal null congruence. The relationship between this class and a similar class with null axial vector torsion is also discussed.

Anisotropic cosmology with extra dimensions

We present an exact solution of the n-dimensional (n > 4) vacuum Einstein field equations with a Bianchi type I metric. The solution may be interpreted as a four-dimensional anisotropic cosmological model. The extra dimensions are related to the energy density and pressures in the model. The physics of the results is discussed at the end of the paper.