A class of static perfect fluid solutions

We consider a two-parameter family of equations of state for perfect fluids which forms the limiting case of a condition employed in a uniqueness proof of static, asymptotically flat solutions of the field equations. We find a geometric interpretation of this family and determine, for each of its members, the one-parameter family of regular spherically symmetric solutions.     

On a static axisymmetric solution of the Einstein equations

A family of static, axisymmetric, asymptotically flat solutions of the Einstein equations is discussed. A source with an exterior described by a member of this family initially could have an area smaller than that of a n appropriately defined Schwarzchild surface. Intuition does not suggest the fate of the collapsing source.     

Neutral perfect fluids of Majumdar-type in general relativity

We consider the extension of the Majumdar-type class of static solutions for the Einstein-Maxwell equations proposed by Ida to include charged perfect fluid sources. We impose the equation of state ρ+3p = 0 and discuss spherically symmetric solutions for the linear potential equation satisfied by the metric. In this particular case the fluid charge density vanishes and we locate the arising neutral perfect fluid in the intermediate region defined by two thin shells with respective charges Q and −Q. With its innermost flat and external (Schwarzschild) asymptotically flat spacetime regions, the resultant condenser-like geometries resemble solutions discussed by Cohen and Cohen in a different context. We explore this relationship and point out an exotic gravitational property of our neutral perfect fluid. We mention possible continuations of this study to embrace non-spherically symmetric situations and higher dimensional spacetimes.     

Existence of Axially Symmetric Static Solutions of the Einstein-Vlasov System

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.     

Exact solution for the gravitational field of a charged,magnetized, spinning mass

A family of solutions of the Einstein-Maxwell field equations is presented, corresponding to the exterior of stationaryaxisymmetric sources with charge, mass, angular momentum, and magnetic dipole moment. The Riemann tensor vanishes asymptotically for each member of the family; some solutions are asymptotically flat and some have NUT-like behavior asymptotically. For the asymptotically flat solutions, the gyromagnetic ratio may vary from zero to one. The corresponding value for the Kerr-Newman solution is one. A method for generating infinite chains of families of solutions of the Einstein-Maxwell equations is described.     

Lie symmetries of difference equations

The discrete heat equation is worked out to illustrate the search of symmetries of difference equations. Special attention it is paid to the Lie structure of these symmetries, as well as to their dependence on the derivative’s discretization. The case ofq-symmetries for discrete equations in aq-lattice is briefly considered at the end. Talk delivered by J. Negro at the DI-CRM Workshop held in Prague, 18–21 June 2000. This work has been partially supported by DGES of the Ministerio de Educación y Cultura of Spain under Projects PB98-0360 and the Junta de Castilla y León (Spain).     

Uniqueness and nonuniqueness of static black holes in higher dimensions

We prove a uniqueness theorem for asymptotically flat static charged dilaton black-hole solutions in higher-dimensional space-times. We also construct infinitely many nonasymptotically flat regular static black holes on the same space-time manifold with the same spherical topology. An application to the uniqueness of a class of flat p-branes is also given.     

Hamilton-Jacobi separable,axisymmetric, perfect-fluid solutions of Einstein's equations

In this paper we examine the Einstein equations with a perfect fluid source under the assumptions of (i) axial symmetry and time-independence, (ii) uniform rotation of the fluid about the symmetry axis, and (iii) separability of the Hamilton-Jacobi equation for the null geodesics of the space. These assumptions are made in an attempt to generalize the results of a similar investigation by Carter for the source-free case.We first extend Carter's results by showing that his additional assumption of separability of the wave equation is unnecessary, it being a consequence of the field equations.When the density of the fluid is non-zero, we are led to a particular solution discovered by Wahlquist, or to more symmetrical interior solutions with spherical equipressure surfaces. Except for the case of no rotation, these solutions cannot be matched to asymptotically flat exteriors.     

Homothetic and conformal symmetries of solutions to Einstein's equations

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.     

Cosmological black holes are coloured too

A new solution of the Einstein-Yang-Mills system with axial symmetry and cosmological constant is given here. This new metric, which in the absence of matter becomes the de Sitter universe, is the natural generalization of both a recent result of Perry for stellar (asymptotically flat) black holes and a well-knwn family of solutions of the Einstein-Maxwell system with nonvanishing cosmological constant given some time ago by Carter. Calculations have been carried out making extensive use of the coordinates found by Plebanski.     

On the spherical symmetry of static perfect fluids in general relativity

We present a theorem which establishes uniqueness, in particular spherical symmetry, of a wide class of general relativistic, static perfect-fluid models provided there exists a spherically symmetric model with the same equation of state and surface potential. The method of proof, which is inspired by recent work of Masood-ul-Alam, is illustrated by demonstrating uniqueness of a class of solutions due to Buchdahl which correspond to an extreme case of the inequality on the equation of state required by our theorem.     

Black holes in static vacuum space-times

An extension of Israel's theorem on the regularity of Killing horizons is proven. Well behaved asymptotically flat vacuum solutions of the Einstein equations which represent the exterior of a non-rotating black hole are considered. It is shown that the black hole has spherical topology and that the equipotential surfaces g ₀₀=constant are non-intersecting two-spheres. The solutions must therefore be members of the one-parameter family of spherically symmetric Schwarzschild solutions.This work has been carried out under a NATO Research Fellowship.     

Non-abelian solutions of <Emphasis Type="Italic">d</Emphasis>= 4 + 1 Einstein-Yang-Mills and Yang-Mills-dilaton theories

We construct static, asymptotically flat solutions of SU(2) Einstein-Yang-Mills theory in 4 + 1 dimensions, subject to bi-azimuthal symmetry. The results are compared with similar solutions of the SU(2) Yang-Mills-dilaton model. Both particle-like and black hole solutions are considered. The text was submitted by the authors in English.     

Static spherically symmetric solutions of the Einstein-Yang-Mills equations

We study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge groupSU(2). Our analysis results in three disjoint classes of solutions with a regular origin or a horizon. The 3-spaces (t=const.) of the first, generic class are compact and singular. The second class consists of an infinite family of globally regular, resp. black hole solutions. The third type is an oscillating solution, which although regular is not asymptotically flat.This article was processed by the author using the Springer-Verlag TEX CoMaPhy macro package 1991.     

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.     

Static axially symmetric solutions in Nordtvedt's theory

Static axially symmetric solutions in vacuum are obtained in the general scalar-tensor theory proposed by Nordtvedt. The solutions are asymptotically flat and under certain conditions give very large red shift. The asymptotic behavior and singularity are studied and a comparison is made with a corresponding solution with spherical symmetry. It is also observed that with a conformal transformation the Nordtvedt metric appears to reduce to the Brans-Dicke one.     

On the spherical symmetry of static stellar models

This paper completes the proof of the necessity of spherical symmetry in the static general-relativistic stellar models that have equations of state satisfying certain inequalities. The technical assumption — that there exists a reference spherical stellar model — that was essential in the previous discussions of this problem is removed. This paper also extends beyond previous discussions the class of equations of state included in the proof. The analysis of the equations for spherical stellar models, used here to demonstrate the existence of a reference spherical model, may also be of independent interest.     

On the physical interpretation of asymptotically flat gravitational fields

A problem in general relativity is how to extract physical information from solutions to the Einstein equations. Most often information is found from special conditions, e.g., special vector fields, symmetries or approximate symmetries. Our concern is with asymptotically flat space–times with approximate symmetry: the BMS group. For these spaces the Bondi four-momentum vector and its evolution, found at infinity, describes the total energy–momentum and the energy–momentum radiated. By generalizing the simple idea of the transformation of (electromagnetic) dipoles under a translation, we define (analogous to center of charge) the center of mass for asymptotically flat Einstein–Maxwell fields. This gives kinematical meaning to the Bondi four-momentum, i.e., the four-momentum and its evolution which is described in terms of a center of mass position vector, its velocity and spin-vector. From dynamical arguments, a unique (for our approximation) total angular momentum and evolution equation in the form of a conservation law is found. Third Award in the 2008 Essay Competition of the Gravity Research Foundation.     

Zero-mass scalar and electrostatic fields with the central symmetry in the general relativity

All asymptotically flat space solutions of Einstein equations with energy-momentum tensor of electrostatic and zero-mass scalar static central symmetric fields as a source were found. There are five branches of general solution; only two of them are contained in previous Penney's solution. In a limit of pure electrostatic field and pure scalar field our solutions become identical with corresponding solutions known previously.