On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations

We show existence and uniqueness of asymptotically flat solutions to the stationary Einstein equations inS=³–B _r , whereB _r is a ball of radiousr>0, when a small enough continuous complex function û on S is given. Regularity and decay estimates imply that these solutions are analytic in the interior ofS and also at infinity, when suitably conformally rescaled.     

The harmonic-map structure of the axially symmetric stationary Einstein equations

We systematically review the solutions of the vacuum Einstein equations for the axially symmetric stationary case which are harmonic maps. In particular, we show that the interesting part of the Kerr solution is a composition of a harmonic map intoH ₁ ² with a totally geodesic map fromH ₁ ² into SS(1,1). We also point out, relying on Sanchez' results, that there is an analogous structure for the Lorentz-domain cases involving cylindrical gravitational waves and colliding plane waves.     

Bäcklund transformations and exact solutions of the stationary axially symmetric Einstein equations

The Bäcklund transformations of the self-dual Yang-Mills fields are used to derive exact solutions of the stationary axially symmetric Einstein equations. Three solutions which belong to an infinite series of such solutions are explicitly calculated, the first of which is the Lewis solution. Also the first of another infinite series of solutions is calculated, which for some value of a parameter becomes the Papapetrou solution.     

Einstein equations and inertial forces in axially symmetric stationary spacetimes

We investigate the relations between the inertial forces and the Einstein equations in axially symmetric stationary spacetimes. For the vacuum stationary axially symmetric spacetimes, we use the Geroch formalism to express the Einstein equations in terms of inertial forces. For the spacetimes with the perfect fluid sources, we use the formalism developed by Hansen and Winicour for establishing the relations. As expected intuitively, the gradients of inertial forces represent the field equations.     

Non-existence of stationary,axisymmetric dust solutions of Einstein's equations on spatially compact manifolds

By using a theorem on the classifacation of SO(2)-actions on three-dimensional manifolds, it is shown that axisymmetric, stationary solutions of Einstein's equations do not exist on spacetimes with compact, space-like hypersurfaces, provided that the energy-momentum tensor is trace-free in the two-dimensional space of trajectories.     

New axially symmetric solutions of the Einstein-Maxwell equations

New exact solutions of the algebraic form for the static Einstein-Maxwell equations representing the exterior gravitational field of a massive magnetic dipole are derived. They are then used for construction of the stationary electrovacuum solutions reducing to the Schwarzschild metric in a pure vacuum limit.     

Non-existence of axially symmetric massive scalar fields

It has been observed that for the axially symmetric Einstein-Rosen metric, the stress-tensor of a scalar meson field associated with meson of rest mass cannot be the source term for generating gravitation. The above result also holds even when this meson field is coupled with an electromagnetic field.     

New series of asymptotically flat axisymmetric and stationary gravitational solutions

The equations for the pseudopotential of Harrison's Bäcklund transformations with seed solution an asymptotically flat solution of the Ernst equation are solved. A new series of asymptotically flat solutions of the equations of general relativity is obtained from the determinants of matrices, whose elements are known. The first solution of the series is calculated as a ratio of two sums of monomials. This solution has five arbitrary real constants. For a particular choice of the constants it becomes Kerr's solution.     

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 interior solution of axially symmetric,stationary and rigidly rotating dust configurations

It is shown that the interior solution of axially symmetric, stationary and rigidly rotating dust configurations is completely determined by the mass density along the axis of rotation. The particularly interesting case of a mass density, which is cylindrical symmetric in the interior of the dust configuration, is presented. Among other things, this proves the non-existence of homogeneous dust configurations.     

Exact conformally flat solutions of Einstein equations for neutral superfluids

A family of exact conformally flat solutions of Einstein equations (with Λ term) is presented. They correspond to a class of relativistic neutral superfluids. The particular velocity field chosen and the field equations determine the thermodynamical variables of the superfluid and restrict the form of the equation of state. One of the solution provides a geometrical interpretation for merons. In principle, a measurement of the time change of the circulation quantum, would distinguish among the various cosmologies of the Dirac-Canuto type.     

On axially symmetric solutions to the Higgs-Kibble-Yang-Mills field equations

The field equations for axially symmetric generalizations of the 't Hooft-Polyakov magnetic monopole are written out in a preferred coordinate system. It is argued that no soliton solutions exist when the product 2eg of the electric chargee with the magnetic chargeg is an even integer.Dedicated to Achille Papapetrou on the occasion of his retirement.     

Axially symmetric stationary solutions of Einstein-Maxwell equations

A scheme is presented by which we can obtain static electrovac exterior solutions from the stationary gravitational fields given by Kerr and Tomimatsu and Sato. A five-parameter solution of the combined Einstein-Maxwell equations is given that describes a source containing mass, electric charge, magnetic dipole, higher multipole moments of all three kinds and angular momentum. These are generated by using Kinnersley's method from Wang's electrovac solutions, which, in turn, were generated from the gravitational fields of Tomimatsu and Sato for =2–3. All the solutions are asymptotically flat.     

The axially symmetric non-rotating vacuum solutions of Rosen's equations

It is shown that all axially symmetric non-rotating solutions of Rosen's field equations can be expressed in terms of two harmonic functions as well as that the total energy of Rosen's metric isMc ².     

Bäcklund transformations of the axially symmetric stationary Einstein-Maxwell equations II

We obtained a Bäcklund transformation of the axially symmetric stationary Einstein-Maxwell equation which is a generalization of Neugebauer's I₁ transformation in the case of the Ernst equations. This transformation is shown to generate the generalized Ehlers transformation and the Harrison transformation.     

Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid

Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vector such that _;v + _v; =2g _v are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.This work was supported in part by the United States Atomic Energy Commission under contract number AT(04-3)-34 Project Agreement No. 125. It was completed when one of the authors (A.H.T.) was on sabbatical leave from the University of California, Berkeley and in residence at the Department of Applied Mathematics and Theoretical Physics, Cambridge University.     

Singularities in solutions of the Einstein equations for a static spherically symmetric field

Russian Physics Journal - The nature of the singularities in the solutions of Einstein's equations for a static spherically symmetric field is investigated as a function of the choice of...