Invariant solutions of Liouville's equation in Robertson-Walker space-times

We find all solutions of Liouville's equation in Robertson-Walker space-times that are either spatially homogeneous or isotropic or both. Some of these solutions depend on constants of motion that are not generated by Killing vectors. We indicate how these solutions may be used to find Einstein-Liouville solutions.     

On the existence of interior solutions in general relativity

The existence problem of matter sources for given stationary axisymmetric solutions of Einstein's vacuum field equations is investigated. The existence of sources of a differentially or rigidly rotating perfect fluid can be proved at least in the neighborhood of boundary surfaces if these can be chosen suitably (theorem). In particular, there exist such half-local perfect fluid sources for the Kerr metric. Hence the existence of a global regular Kerr-interior solution cannot be excluded by local considerations in an arbitrarily small neighborhood of a possible boundary.Work supported in part by the Deutsche Forschungsgemeinschaft.     

On stationary axially symmetric interior solutions in general relativity

This note presents the coordinate transformation by which the coordinate condition of a previous paper (Rawson-Harris, 1972) may be imposed.     

The Ernst equation in general relativity

Stationary axisymmetric gravitational fields are governed by the Ernst equation. Its internalSU(1, 1) symmetry gives rise to a linear problem and to Bäcklund transformations which map known solutions into new ones. The main features of the solution generating techniques are outlined, the generalization to Einstein-Maxwell fields is discussed and some applications are given.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.This paper summarizes some common work with G. Neugebauer. I would like to thank him for many stimulating discussions.     

On the uniqueness of static perfect-fluid solutions in general relativity

Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stellar models. Given a static, asymptotically flat perfectfluid spacetime with barotropic equation of state (p), and given another such spacetime which is spherically symmetric and has the same (p) and the same surface potential: we prove that both are identical provided (p) satisfies a certain differential inequality. This inequality is more natural and less restrictive than the conditions required by Masood-ul-Alam.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, project P-7197     

Stationary solutions in five-dimensional general relativity

Asymptotically Minkowskian stationary axisymmetric solutions of the five-dimensional Einstein equations are generated from particular classes of stationary axisymmetric solutions of the four-dimensional Einstein-Maxwell equations. First, the five-dimensional electrostatic and magnetostatic solutions are generated from the four-dimensional electrostatic solutions and a harmonic scalar field. Then, a new class of five-dimensional nonstatic solutions are generated from the four-dimensional class =+1. As an example, a three-parameter family of regular asymptotically flat rotating solutions is constructed. These solutions can be interpreted, after dimensional reduction to four dimensions, as extended elementary particles with mass, spin, electric charge, and magnetic dipole moment.     

Stationary solutions in three-dimensional general relativity

All the stationary solutions of the three-dimensional vacuum Einstein equations are obtained. These include a class of multicenter solutions representing systems of massive and spinning point particles. The geodesic motion of a test particle in the one-particle metric is discussed. A class of geodesics contain finite intervals where the particle moves back in coordinate time, without violation of causality.     

Kinetic equation in the general theory of relativity

An attempt is made to discover the physical content of the general-relativistic theory of gases. Under an invariant interpretation of the collision term, this theory does not satisfy the correspondence principle for classical theory. The collision term in the classical Boltzmann form is meaningful only in an isolated reference frame realized by a locally inertial frame with the origin at the collision point. With this formulation of the kinetic equations, the class of equilibrium states in GR expands considerably and also covers nonstationary distributions. A specific example of such a distribution that is locally Maxwellian is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 72–76, February, 1979.In conclusion, the author considers it his pleasant duty to thank Prof. N. A.Chernikov for useful discussion of certain problems touched on this paper, and A. V. Zakharov, who made this result of his article koown to the author.     

Spherically symmetric solutions in five-dimensional general relativity

The most general time-independent spherically symmetric (in the usual three space dimensions) solution to the five-dimensional vacuum Einstein equations is found, subject to the existence of a Killing vector in the fifth direction. The significance of these solutions is discussed within the context of a previously proposed extension of the Kaluza-Klein model in which the universe, although (4+1)-dimensional, has evolved over cosmic times into an effectively (3+l)-dimensional one.     

Shear free solutions in general relativity theory

The Goldberg-Sachs theorem is an exact result on shear-free null geodesics in a vacuum spacetime. It is compared and contrasted with an exact result for pressure-free matter: shear-free flows cannot both expand and rotate. In both cases, the shear-free condition restricts the way distant matter can influence the local gravitational field. This leads to intriguing discontinuities in the relation of the General Relativity solutions to Newtonian solutions in the timelike case, and of the full theory to the linearised theory in the null case.     

Static spherically symmetric solutions in general projective relativity

Static spherically symmetric solutions have been obtained for general projective relativity withn=0 andn0 both in isotropic and curvature coordinates. In curvature coordinates, only a restricted exact solution is possible. However, an approximate solution can always be obtained following a method similar to Vanden Bergh. In these spacetimes there is no horizon, but only a naked singularity atr=0. Thus there are no black holes. It is shown that there is no solution in static, spherically symmetric, conformally flat spacetime.     

Stationary axisymmetric perfect fluid solutions in general relativity

A new class of exact solutions of Einstein's field equations with the energy-momentum tensor of a perfect fluid is given. The class of solutions is invariantly characterized by means of the following properties: (i) The energy-momentum tensor describes a perfect fluid. (ii) There are two commuting Killing vectors and which form an abelian groupG ₂ of motion. (iii) There is a timelike Killing vector parallel to the four-velocity of the fluid (rigid rotation of the fluid). (iv) The four-vector of the angular velocity of the fluid is a gradient ⁱ=–(1/4_c)^irklU_l (U_r:k–U_k:r)= ⁱ. The last assumption is the reason that all solutions of this class can be found by solving an ordinary differential equation of the second order.     

New solutions for charged spheres in general relativity

Exact solutions of the Einstein-Maxwell field equations are obtained for the case of static and spherically symmetric distribution of charged matter. The solutions are obtained through the extension of a method originally used for neutral configurations and are conveniently matched to the Reissner-Nordstrom exterior metric. The physical plausability of the solutions is discussed and it is shown that some of them reduce in appropriate limits to known neutral or charged solutions.     

Axisymmetric regular multiwormhole solutions in five-dimensional general relativity

A class of regular, asymptotically flat solutions to the five-dimensional vacuum Einstein equations with a two-parameter Abelian isometry group is constructed, under the additional assumption of axial symmetry in three-dimensional space. The possibility of interpreting these multiwormhole solutions as multiparticle systems is discussed.     

On axially symmetric soliton solutions of the coupled scalar-vector-tensor equations in general relativity

The inverse scattering theory used by Belinsky and Zakharov to obtain soliton solutions of the of the Einstein equations is here applied to the case of a five-dimensional space and interpreted in the framework of the Jordan-Kaluza-Klein theory. For two solitons exact, stationary, axially symmetric and asymptotically flat solutions are obtained.     

On general and restricted covariance in general relativity

We reconsider the principle of general covariance and give a rigorous formulation of a principle ofrestricted covariance. We give a number of examples of preferred coordinate systems, considered in the literature, and in each case demonstrate the applicability of the notion of restricted covariance proposed.     

Some solutions of the Einstein-Maxwell equations in general relativity

A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.