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.     

Special conformal symmetries in general relativity

A geometrical discussion of special conformal vector fields in space-time is given. In particular, it is shown that if such a vector field is admitted, it is unique up to a constant scaling and the addition of a homothetic or a Killing vector field. In the case when the gradient of the conformal scalar associated with is non-null it is shown that other homothetic and affine symmetries are necessarily admitted by the space-time, that an intrinsic family of 2-dimensional flat submanifolds is determined in the space-time, that is, in general, hypersurface orthogonal and that the space-time, if non-flat, is necessarily (geodesically) incomplete. Other geometrical features of such space-times are also considered.     

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.     

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.     

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.     

Inheriting geodesic flows

We investigate the propagation equations for the expansion, vorticity and shear for perfect fluid space-times which are geodesic. It is assumed that space-time admits a conformal Killing vector which is inheriting so that fluid flow lines are mapped conformally. Simple constraints on the electric and magnetic parts of the Weyl tensor are found for conformal symmetry. For homothetic vectors the vorticity and shear are free; they vanish for nonhomothetic vectors. We prove a conjecture for conformal symmetries in the special case of inheriting geodesic flows: there exist no proper conformal Killing vectors (ψ _;ab ≠ 0) for perfect fluids except for Robertson-Walker space-times. For a nonhomothetic vector field the propagation of the quantity ln (R _ab u ^a u ^b ) along the integral curves of the symmetry vector is homogeneous.     

Classification of static spherically symmetric perfect fluid space-times via conformal vector fields in f(T) gravity

In this paper, we classify static spherically symmetric (SS) perfect fluid space-times via conformal vector fields (CVFs) in f(T) gravity. For this analysis, we first explore static SS solutions by solving the Einstein field equations in f(T) gravity. Secondly, we implement a direct integration technique to classify the resulting solutions. During the classification, there arose 20 cases. Studying each case thoroughly, we came to know that in three cases the space-times under consideration admit proper CVFs in f(T) gravity. In one case, the space-time admits proper homothetic vector fields, whereas in the remaining 16 cases either the space-times become conformally flat or they admit Killing vector fields.     

Homothetic maps of the space-time distance function and differentiability

Let (M ₁,g ₁) and (M ₂,g ₁) be time-oriented space-times. Letd _i(p,q) be the supremum of lengths of future directed causal curves inM _i fromp toq. Ifq is not in the future ofp, thend _i (p, q)=0. A distance homothetic mapf is a function fromM ₁ ontoM ₂ which is not assumed to be continuous, but which satisfiesd ₂(f(p),f(q))=cd ₁(p,q) for allp,q M ₁. IfM ₁ is strongly causal, then the distance homothetic mapf is a diffeomorphism which mapsg ₁ to a scalar multiple ofg ₂. Thus for strongly causal space-times, distance homothetic maps are homothetic in the usual sense. WhenM ₁ is not strongly causal, distance homothetic maps are not necessarily differentiable nor even continuous. An example is given of a space-time which has discontinuous maps which are one to one, onto, and distance preserving.     

Homothety groups in space-time

This paper investigates space-times which admit anr-dimensional Lie algebra of homothetic vector fields at least one of which is proer homothetic. The situation is resolved completely ifr 6 and some general results are given whenr 5. Some applications to the study of affine vector fields in space-times are also given and the maximum dimension of this latter algebra is computed and corrects an earlier error.     

A note on proper homothetic motions

It is shown that the only Ricci-flat space-time to support a time-like propert homothetic motion with hypersurface orthogonal trajectories is flat space.     

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.     

A Self-Similar Dynamics in Viscous Spheres

We study the evolution of radiating and viscous fluid spheres assuming an additional homothetic symmetry on the spherically symmetric space-time. We match a very simple solution to the symmetry equations with the exterior one (Vaidya). We then obtain a system of two ordinary differential equations which rule the dynamics, and find a self-similar collapse which is shear-free and with a barotropic equation of state. Considering a huge set of initial self-similar dynamics states, we work out a model with an acceptable physical behavior.     

T4 gauge theory of gravity and new general relativity

We formulate a space-time translationT ₄ gauge theory of gravity on the Minkowski space-time with appropriate choice of the Lagrangian. By comparing the energy-momentum law of this theory with that of new general relativity constructed on the Weitzenböck space-time we find that in the classical limit the gauge potentials correspond to the parallel vector fields in the Weitzenböck space-time and the gauge field equation coincides with the field equation of gravity in new general relativity in the linearized version. Thus we conclude that in the classical limit theT ₄ gauge theory of gravity leads to the new general relativity.     

On the effective lagrangian in spinor electrodynamics with added violation of Lorentz and <Emphasis Type="Italic">CPT</Emphasis>-symmetries

We consider quantum electrodynamics with additional coupling of spinor fields to the space-time independent axial vector violating both Lorentz and CPT-symmetries. The Fock-Schwinger proper-time method is used to calculate the one-loop effective action up to the second order in the axial vector and to all orders in the space-time independent electromagnetic field strength. We find that the Chern-Simons term is not radiatively induced and that the effective action is CPT-invariant in the given approximation. Received: 29 January 2003 / Published online: 24 March 2003 RID="a" ID="a" e-mail: sitenko@itp.unibe.ch RID="b" ID="b" e-mail: rulik@to.infn.it     

Real Kasner and related complex “windmill” vacuum spacetime metrics

The Kasner family of vacuum solutions of Einstein's field equations admits a simply-transitiveH ₄, a four-parameter local homothetic group of motions which has an AbelianG ₃ subgroup. It is shown that a complex transformation of coordinates and constants exists which maps this family from the normal Kasner form into a form of vacuum metrics whose Weyl tensors are each Petrov type I and which were published in 1932 by Lewis. These metrics also admit a similarH ₄; however for one particular metric (for one parameter value) theH ₄ becomes aG ₄ and the resultant metric is one which was rediscovered by Petrov in 1962. These Lewis metrics are thus shown to be Kasner metrics over complex fields. Here they are calledwindmill metrics because of the rotating relationship between the coordinates and the Killing vector fields admitted. The principal null directions of thereal Kasner and the windmill metrics are discussed; the two families then provide illustrations of two degenerate classes of spacetime metrics whose Weyl tensors are of Petrov type I, as discussed elsewhere by Arianrhod and McIntosh. An extension of the windmill-type generation of metrics to some other families of metrics is also discussed.     

Conformal Killing vectors and FRW spacetimes

Fluid space-times which admit a conformal Killing vector (CKV) are studied. It is shown that even in a perfect fluid space-time a conformal motion will not, in general, map the fluid flow lines onto fluid flow lines; consequently, perfect fluid space-times and, in particular, the simplest perfect fluid space-times known to admit a CKV, namely the Friedmann-Robertson-Walker (FRW) space-times, are studied. A direct proof that there do not exist any special CKV in FRW space-times will be given, thereby motivating the study of the physically more relevant proper CKV. Indeed, one of the principal motivations of the present work is the study of the symmetry inheritance problem for proper CKV. Since the FRW metric can, in general, satisfy the Einstein field equations for a non-comoving imperfect fluid, the relationship between the FRW models (and in particular the standard comoving perfect fluid models) and the conditions under which conformal motions (and in addition homothetic motions) map fluid flow lines onto fluid flow lines are investigated. Finally, further properties of fluid space-times which admit a proper CKV, and in particular space-times in which the CKV is parallel to the fluid four-velocity, are discussed.     

Massive vector fields and black holes

A massive vector field inside the event horizon created by the static sources located outside the black hole is investigated. It is shown that the back-reaction of such a field on the metric nearr=0 cannot be neglected. The possibility of the space-time structure changing nearr=0 due to the external massive field is discussed.On leave of absence from P. N. Lebedev Physical Institute, Moscow, USSR.     

Dual models and the geometry of space-time

The zero slope limit of the closed string sector of a dual model yields a Lagrangian theory in which space-time has a non-Riemannian geometry. We find that there is torsion but no homothetic curvature.     

A class of field theories in two-dimensional space-time with aU 1×U 1 symmetry

The derivative coupling of massless pseudoscalar neutral particles with a charged spinor field in two-dimensional space-time is reduced to a self-interacting spinor field and a free pseudoscalar field.More generally, it is shown that any given local field theory with a conserved vector current and without massless particles can be extended to a local theory with an additional pseudoscalar field and with aU ₁×U ₁ symmetry.     

Topological invariants and the dynamics of an axial vector torsion field

A generalized theory of gravitation is discussed which is based on a Riemann-Cartan space-time,U ₄, with an axial vector torsion field. Besides Einstein's equations determining the metric of theU ₄, a system of nonlinear field equations is established coupling an axial vector source current to the axial vector torsion field. The properties of the solutions of these equations are discussed assuming a London-type condition relating the axial current and torsion field. To characterize the solutions use is made of the Euler and Pontrjagin forms and the associated quadratic curvature invariants for theU ₄ space-time. It is found that there exists for a Riemann-Cartan space-time a relation between the zeros of the axial vector torsion field and the singularities of the Pontrjagin invariant, which is analogous to the well-known Hopf relation between the zeros of vector fields and the Euler characteristic.