Empty Space-time Research Articles

Nonexistence of static conformally-flat solutions in self-creation cosmology

It is shown that there do not exist spherically-symmetric conformaly-flat static solutions, in the self-creation cosmology proposed by Barber, representing disordered radiation and in the presence of source-free electromagnetic field except for the trivial empty flat space-time of Einstein's theory. The solution in the vacuum case is also only a flat space time of Einstein's theory.

Plane-symmetric vacuum in self-creation cosmology

Vacuum field equations for the static and non-static plane-symmetric metric are obtained in self-creation theory of gravitation proposed by Barber (1982). It is shown that, in both static and non-static cases, the only plane-symmetric solution in vacuum is the empty flat space-time of Einstein's theory. It is observed that this result is quite different from that of the Brans-Dicke and other scalar-tensor theories of gravitation.

Strong cosmic censorship and the strong curvature singularities

Conditions are given under which any asymptotically simple and empty space-time that has a partial Cauchy surface with an asymptotically simple past is globally hyperbolic. It is shown that this result suggests that the Cauchy horizons of the type occurring in Reissner–Nordström and Kerr space-times are unstable. This in turn gives support for the validity of the strong cosmic censorship hypothesis.

Anisotropic fluid with SU(2)-type structure in general relativity: A model of localized matter

A model of anisotropic fluid with three perfect fluid components in interaction is studied. Each fluid component obeys the stiff matter equation of state and is irrotational. The interaction is chosen to reproduce an integrable system of equations similar to the one associated to self-dual SU(2) gauge fields. An extension of the Belinsky–Zakharov version of the inverse scattering transform is presented and used to find soliton solutions to the coupled Einstein equations. A particular class of solutions that can be interpreted as lumps of matter propagating in empty space-time is examined.

The gravitational field in the wave zone II. Consequences of asymptotic structure

We consider an asymptotically flat and empty space-time generated by a bounded source of perfect fluid. The vanishing of the conformal Weyl tensor onI+ and of the Ricci tensor nearI+ are used to simplify the expression obtained in the previous paper for the coefficient ofr− of the metric tensor after an expansion in powers ofc−1. The result is a very simple expression for the dominant term ofgαβ,0 in the radiation zone in terms of the quadrupole moment of the source. Using this expression and an invariant definition of the total energy, we calculate in the framework of full general relativity the radiated energy per unit time and prove that the first term is identical with the quadrupole radiation as given by the linearized version of general relativity.

Nonexistence of static conformally flat solutions in a new scalar–tensor theory

For the special case when the scalar field is massless and conformally invariant, it is shown that there do not exist spherically symmetric conformally flat solutions in a new scalar–tensor theory proposed by Schmidt et al. [Phys. Rev. D 24, 1484 (1981)] representing disordered radiation and in the presence of a source-free electromagnetic field except for the trivial empty flat space-time of Einstein’s theory. The solution in the vacuum case is also only a flat space-time of Einstein’s theory.

Plane symmetric cosmological solutions of the Lyttleton-Bondi Universe

The exterior field of the plane symmetric in the Lyttleton-Bondi Universe is considered and exact solutions are obtained in some specific situations. Our solutions generalise the empty space-time solutions, with plane symmetry, previously obtained by Taub and Bera to the situation when Lyttleton-Bondi field is present. Various physical aspects of the solutions are discussed.

An R4 spacetime with a Cauchy surface which is not R3

There exists a class of contractible open 3-manifolds Wp which are topologically distinct from R3. They have the property that Wp*R is diffeomorphic to R4. A spacetime with a cauchy surface diffeomorphic to some Wp therefore has a spacetime manifold diffeomorphic to R4. A theorem is established to the effect that any asymptotically simple and empty spacetime has a spacetime manifold diffeomorphic to R4, but can only have R3 Cauchy surfaces. Also J+ and J- must be diffeomorphic to S*R. These results supersede an argument of Geroch (1971) in support of similar conclusions.

Self-gravitating fluid in a conformally-flat space-time

The Einstein field equations for an irrotational perfect fluid with pressurep, equal to energy density are studied when the space-time is conformally flat. The coordinate transformation to co-moving coordinates is discussed. The energy and Hawking-Penrose inequalities are studied. Static and non-static solutions of the field equations are obtained. It is interesting to note that in the static case the only spherically-symmetric conformally flat solution for self-gravitating fluid is simply the empty flat space-time of general relativity.

Robertson-Walker Lyttleton-Bondi universe with cosmological constant

The exterior field of the Robertson-Walker metric in the Lyttleton-Bondi universe with cosmological constant is considered. It is shown that in the presence of cosmological constant, the exterior solution of this Universe is simply the empty space-time of general relativity.

Bianchi types of a three-parameter group of curvature collineations

The Bianchi types of the three-parameter group of curvature collineations admitted by a previously discussed family of typeN Robinson-Trautman empty space-times are obtained.

Towards the proof of the cosmic censorship hypothesis

An attempt is made to formulate the cosmic censorship hypotheses put forward by Penrose (1979) as a theorem which could be subject to a mathematical proof. It is proved that a weakly asymptotically simple and empty spacetime must be future asymptotically predictable if the energy and the strong causality conditions hold and either all singularities are of Tipler's strong curvature type and once singularity occurs there exists a marginally outgoing null geodesic or each singularity is preceded by the occurrence of a closed trapped surface. The marginally outgoing null geodesics may not be admitted by general naked singularities. However, it is shown that they occur if on the Cauchy horizon the global hyperbolicity is violated in such a way that causal simplicity also does not hold. This means that a wide class of nakedly singular spacetimes is considered. This result gives some support to the validity of Penrose's hypothesis.

Empty space-times of embedding class two

Using the canonical forms for symmetric tensors in Minkowski space and the Petrov forms of the Weyl tensor, algebraic relationships between the two kinds of classification are developed for empty space-times of embedding class two.

Conditions of existence for a class of self-dual solutions of the Einstein equations

We study the integrability of the Einstein equations for a class of empty homogeneous space-times (Bianchi class A), once the self-duality constraints are imposed on the space-time itself. This system is integrable in the case of Bianchi type I and is a subset of Bianchi types VI0 and VII0. Bianchi type II space-times do not admit self-dual solutions and for the case of Bianchi VIII and IX we were unable to find a general integrability condition.

Static plane-symmetric solution of a scalar-tensor theory of gravitation

AbstractVacuum field equations in a scalar-tensor theory of gravitation, proposed by Ross, are obtained with the aid of a static plane-symmetric metric. A closed form exact solution to the field equations in this theory is presented which can be considered as an analogue of Taub's empty space-time in Einstein's theory.

Static conformally flat solutions in a general scalar-tensor theory of gravitation

Explicit field equations in the general scalar-tensor theory of gravitation proposed by Nordtvedt are obtained with the aid of a static spherically symmetric conformally flat metric. Exact static solutions of Nordtvedt-Barker field equations both in vacuum and in the presence of a source-free electromagnetic field are presented and studied. It is shown that there are no spherically symmetric static conformally flat solutions of Nordtvedt-Barker field equations representing perfect fluid distribution with disordered radiation obeying the equation of stateρ=3p, except for the trivial empty flat space-time of Einstein's theory.

Mappings of empty space-times leaving the curvature tensor invariant

It is shown that if in some local coordinate system the componentsRijkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorRijkl invariant are found for each class of these space-times.

Imparting small vorticity to a Bianchi type-VIhempty spacetime

We present and briefly discuss a Bianchi type-$\mathrm{V}{\mathrm{I}}_{h}$ empty spacetime. The field equations have been solved after being linearized with respect to a parameter which imparts vorticity to the model. The limit of zero vorticity is an already known solution.

Instabilities of certain empty Robertson-Walker space-times

Results are established concerning perturbations of each empty Robertson-Walker space-time (M, g) with a nonvanishing cosmological constant. The perturbed space-times have the general form (\(\mathop {M,}\limits^ \sim \mathop g\limits^ \sim \)) with\(M\) an extension ofM, and\(g\) lying in an open neighborhood of g in a type ofWm topology. These results indicate that large classes of such perturbations give rise to space-times which suffer from one of two types of incompleteness.

Cylindrically symmetric empty space-time solutions of SEN — DUNN theory

A class of exact solutions is obtained for non-static cylindrically symmetric space-time ofMarder corresponding to vacuum field equations of a scalar-tensor theory of gravitation as proposed bySen andDunn. We observe that, given any solution of the vacuum field equations of Einstein’s theory, it is possible to generate the solution of the corresponding field equations of theSen—Dunn theory. The presence of the scalar-interaction functionx 0 influences one of the parameters (viz. α) of theMarder’s metric. The solutions preserve the wave character as in Einstein’s theory.