Empty spaces and perfect fluids with homothetic transformation

A class of Lorentzian metrics in one variable that admits homothetic transformation is introduced. Solutions of Einstein's empty space equations are presented. The case of perfect fluids is discussed.     

Homothetic motions in general relativity

Properties of homothetic or self-similar motions in general relativity are examined with particular reference to vacuum and perfect-fluid space-times. The role of the homothetic bivector with componentsH _[a;b] formed from the homothetic vectorH is discussed in some detail. It is proved that a vacuum space-time only admits a nontrivial homothetic motion if the homothetic vector field is non-null and is not hypersurface orthogonal. As a subcase of a more general result it is shown that a perfect-fluid space-time cannot admit a nontrivial homothetic vector which is orthogonal to the fluid velocity 4-vector.     

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

Classification of Teleparallel Homothetic Vector Fields in Cylindrically Symmetric Static Space-Times in Teleparallel Theory ofGravitation

In this paper we classify cylindrically symmetric static space-timesaccording to their teleparallel homothetic vector fields using directintegration technique. It turns out that the dimensions of the teleparallelhomothetic vector fields are 4, 5, 7 or 11, which are the same in numbers asin general relativity. In case of 4, 5 or 7 proper teleparallel homotheticvector fields exist for the special choice to the space-times. In the caseof 11 teleparallel homothetic vector fields the space-time becomes Minkowskiwith all the zero torsion components. Teleparallel homothetic vector fieldsin this case are exactly the same as in general relativity. It is importantto note that this classification also covers the plane symmetric static space-times.     

Homothetic motions and perfect fluid cosmologies

A theorem on homothetic (self-similar) motions in spacetimes with a perfect fluid is derived. The main result is that a perfect fluid cosmological model cannot have a non-trivial homothetic motion orthogonal to the fluid 4-velocity vector.     

Null charts and naked singularities in spherically symmetric,homothetic spacetimes

We give a unified derivation of a null chart for all spherically symmetric, homothetic spacetimes. These spacetimes contain an interesting class of naked singularities which we are also able to elucidate. Much use is made of graphical representation; in particular a chart of the spacetimes based on their homothetic group motions is introduced. Dust spacetimes, and two homogeneous examples with non-zero pressure (flat Robertson-Walker and a Kantowski-Sachs example) are studied in detail. We show the horizon structure in the null atlas, in comoving coordinates, in terms of the areal radius and comoving time, and in the homothetic diagrams. The critical delay between comoving observers for the onset of nakedness is interpreted in terms of a decreasing mass concentration in the spirit of Thorne's hoop conjecture. We also give a simple criterion for the existence of apparent horizons isolating the various singularities, and study in detail how this criterion is circumvented in the naked examples. We conclude that this type of naked singularity is a consequence of the imposed homothetic symmetry, by showing it to be generally present and timelike in the homothetic group chart even when it is not visible at comoving infinity (before the onset of criticality). It is the delayed final collapse of initially distant observers in inhomogeneous spacetimes that causes the initial singularity to become visible at comoving infinity. We conclude that these examples do not present an obstacle to the Event Horizon Conjecture as summarized by W. Israel (1984). That is, one can formulate criteria for the formation of apparent horizons that do not imply that all singularities are necessarily so enclosed. It is still possible that all singularities stronger than homothetic are isolated by an apparent horizon, in the spirit of Tipler's conjecture.On leave from Department of Physics, Queen's University, Kingston, Ontario, Canada     

Homothetic motions with null homothetic bivectors in general relativity

It is shown that if a nonflat vacuum space-time admits a homothetic vector field with a null homothetic bivector then that space-time is algebraically special. If that homothetic vector field is a nontrivial one (not a Killing one) then the space-time is Petrov type III orN.     

Projective differential geometry and geodesic conservation laws in general relativity,II: Conservation laws

We examine the geodesic conservation laws associated with the projective actions discussed in our earlier paper with the same overall title. Using the Cartan formalism, a one-to-one correspondence between a class of these actions and all geodesic conservation laws is possible. In particular there is a natural geometric interpretation of Killing tensors. Homothetic motions are shown to correspond to conserved quantities on all geodesies (not just null ones). The same approach identifies homothetic Killing tensors and a universal quadratic first integral which reduces to the conformai Killing tensor case on null geodesics.     

Homothetic transformations with fixed points in spacetime

A study is made of homothetic motions with fixed points in spacetime. Some general properties of such spacetimes are established, and a characterization of generalized plane wave spacetimes is proved. A general discussion of homothetic motions in Einstein's theory is given.This is in the sense that no open subset ofM is flat.     

Vacuum type N space-times admitting homothetic vector fields with isolated fixed points

We consider vacuum space-times (M, g) which are of Petrov type N on an open dense subset ofM, and which admit (proper) homothetic vector fields with isolated fixed points. We prove that if such is the case then, at the fixed point, (M,g) is flat and the homothetic bivector,X _[a;b] , is necessarily simple-timelike. Furthermore, we prove that if the homothetic bivector remains simple-timelike in some neighbourhood of the fixed point then, around the fixed point, the space-time in question is a pp-wave. The paper ends with a local characterization and some examples of space-tunes satisfying these conditions.     

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.     

Vacuum spacetimes with a two-dimensional orthogonally transitive group of homothetic motions

Vacuum spacetimes with a two-dimensional orthogonally transitive groupH ₂ of proper homothetic motions acting on nonnull orbits are investigated with the aid of the Geroch-Held-Penrose formalism. It is found that these spacetimes admit in general anH ₃ of homothetic motions containing two commuting and hypersurface orthogonal Killing vector fields. The metric equations are integrated, and the line elements of the spacetimes in question are explicitly given in a diagonal form.     

Aligned Electric and Magnetic Weyl Fields

The results on the non-existence of purely magnetic solutions are extended to the wider class of spacetimes which have homothetic electric and magnetic Weyl fields. This class is a particularization of the spacetimes admitting a direction for which the relative electric and magnetic Weyl fields are aligned. We give an invariant characterization of these metrics and study the properties of their Debever null vectors. The directions observing aligned electric and magnetic Weyl fields are obtained for every Petrov-Bel type.     

Spatially homothetic cosmological models

Spatially homothetic cosmological models are defined as space-time manifolds acted on by a 3-parameter group of transformations transitive over spacelike hypersurfaces, whose effect is to multiply the metric by a constant conformal factor. Previous work on these models is reviewed briefly and the algebraic classification scheme of Eardley is described. Explicit forms of the metric and group generators are given for each class in terms of a conformally synchronous coordinate system using an invariant orthogonal basis of 1-forms. It is shown that certain subclasses are necessarily incomplete in the sense that a singularity of the conformally synchronous system must develop within a finite time.This article is Dedicated to Achille Papapetrou on the occasion of his retirement and forms a part of the Papapetrou Festschrift.     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

Construction of Sources for Majumdar-Papapetrou Spacetimes

We study Majumdar-Papapetrou solutions for the 3 + 1 Einstein-Maxwell equations, with charged dust acting as the external source for the fields. The spherically symmetric solution of Gürses is considered in detail. We introduce new parameters that simplify the construction of class C ¹, singularity-free geometries. The arising sources are bounded or unbounded, and the redshift of light signals allows an observer at spatial infinity to distinguish these cases. We find out an interesting affinity between the conformastatic metric and some homothetic and matter collineations. The associated geometric symmetries provide us with distinctive solutions that can be used to construct non-singular sources for Majumdar-Papapetrou spacetimes.     

Horizons in weyl metrics exhibiting extra symmetries

Static, axisymmetric, vacuum metrics (commonly called Weyl metrics) are classified according to the homothetic motions they admit, with all having more than two homothetic motions explicitly computed. Within the two new families of spacetimes so determined, none of the metrics is asymptotically flat. Although most of the horizons are unlike that of the Schwarzschild metric, it is shown that they nonetheless all fall within a classification scheme previously developed for two-dimensional static metrics. TheC — metric and the question of directional singularities are also briefly considered.Based on part of the author's doctoral dissertation submitted to Princeton University, 1970. This work has been assisted in part by NSF Grant No. GP7669.     

Symmetry mappings in Einstein-Maxwell space-times

General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.On leave from Mathematics Department, Monash University, Clayton, Victoria, 3168, Australia.     

Reduction of Einstein's field equations for twisting typeN-vacuum fields with anH 2

The problem of vacuum typeN solutions of Einstein's field equations with Killing vector and homothetic group can be reduced to a single ordinary differential equation of the third order for a single real function. Hauser's solution [4] is an exceptional case. If the homothetic parameterN takes the valueN=2 the third order differential equation becomes surprisingly simple. This caseN=2 is therefore the most promising one for the search of exact solutions.     

Homothetic self-similar solutions of three-dimensional Brans-Dicke gravity

All homothetic self-similar solutions of the Brans-Dicke scalar field in three-dimensional spacetime with circular symmetry are found in closed form.