Curvature collineations for type-N Robinson-Trautman space-times

The metric of type-N Robinson-Trautman space-times is generated by a real functionP satisfying certain field equations. Canonical forms forP are obtained under the assumption that at least one curvature collineation exists. In order to give an example of the improper subgroup structure of a group of curvature collineations all the curvature collineations are determined for the space-times corresponding to one of the two canonical forms.     

Killing-Yano tensors in general relativity

It is shown that space-times admitting more than one independent Killing-Yano tensor belong to a small collection of highly idealised space-times. A new characterization of Robertson-Walker space-times arises as a corollary of the main theorem.     

Occurrence of whimper singularities

HOmogeneous space-times (i.e. those admitting a three-parameter group of isometries) are studied using the Newman Penrose formalism. It is found that solutions containing horizons depend on two fewer parameters than the most general solution, so that horizons and the associated whimper singularities are not stable features of homogeneous space-times. In the vacuum case, there are just three two-parameter families with horizons, two of which are the NUT solutions and certain plane waves.     

Spherically symmetric space-times transparent to scalar multipole waves

Using nonscattering potentials of Chang and Janis, a large class of spherically symmetric space-times is constructed on which all multipole solutions to the minimally coupled scalar wave equation are expressible in terms of characteristic data functions in essentially as simple a fashion as for flat space-time. The space-times are transparent to multipole waves in the same sense that flat space-time is. Both conformally flat and not conformally flat space-times are obtained. Some examples are discussed which show that the variety of transparent space-times is large even within the class of Robertson-Walker spaces.     

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.     

On the fundamental groups of space-times in general relativity

It is shown that there are upper bounds on the first and second betti numbers of compact space-times or space-times with Cauchy surfaces whose fundamental groups are abelian. Homological classifications of compact space-times and space-times with compact Cauchy surfaces are given.     

Discrete symmetries of Petrov type-I space-times

The arbitrariness in the definition of the canonical frame in a Petrov type-I space-time is examined and shown to imply that the Newman-Penrose formalism is invariant under a 24-element discrete symmetry group. The transformations of the N-P scalars under the operations of the group are obtained, and conditions are formulated which characterize space-times that maintain this symmetry globally.     

Ghost neutrinos as test fields in curved space-time

Without restricting to empty space-times, it is shown that ghost neutrinos (their energy-momentum tensor vanishes) can only be found in algebraically special space-times with a neutrino flux vector parallel to one of the principal null vectors of the conformal tensor. The optical properties are studied. There are no ghost neutrinos in the Kerr-Newman and in spherically symmetric space-times. The example of a non-vacuum gravitational pp-wave accompagnied by a ghost neutrino pp-wave is discussed.     

Construction of Hadamard States by Pseudo-Differential Calculus

We give a new construction based on pseudo-differential calculus of quasi-free Hadamard states for Klein–Gordon equations on a class of space-times whose metric is well-behaved at spatial infinity. In particular on this class of space-times, we construct all pure Hadamard states whose two-point function (expressed in terms of Cauchy data on a Cauchy surface) is a matrix of pseudo-differential operators. We also study their covariance under symplectic transformations. As an aside, we give a new construction of Hadamard states on arbitrary globally hyperbolic space-times which is an alternative to the classical construction by Fulling, Narcowich and Wald.     

Bianchi types of a three-parameter group of curvature collineations

The Bianchi types of the three-parameter group of curvature collineations admitted by a previously discussed family of typeN Robinson-Trautman empty space-times are obtained.     

Perfect fluid space-times with a two-dimensional orthogonally transitive group of homothetic motions

We use theghp formalism to obtain perfect fluid space-times with a two-dimensional and orthogonally transitive group of proper homothetic motionsH ₂, with the additional condition that the four-velocity of the fluid either lies on the group orbits or is orthogonal to them. In the first case the orbits of theH ₂ are timelike and all possible solutions are explicitly given. They comprise (i) space-times of Petrov type I that admit a groupH ₃ containing two hypersurface orthogonal and commuting Killing vectors (when theH ₂ is abelian, the fluid has a stiff equation of state and the space-time is of type D), and (ii) a class of type D static space-times with a maximalH ₂ in which the two-spaces orthogonal to the group orbits have constant curvature. When the orbits of theH ₂ are spacelike, the fluid is necessarily stiff and different classes of solutions admitting maximalH ₂ andH ₃ are identified.     

Grand Canonical Ensembles in General Relativity

We develop a formalism for general relativistic, grand canonical ensembles in space-times with timelike Killing fields. Using that, we derive ideal gas laws, and show how they depend on the geometry of the particular space-times. A systematic method for calculating Newtonian limits is given for a class of these space-times, which is illustrated for Kerr space-time. In addition, we prove uniqueness of the infinite volume Gibbs measure, and absence of phase transitions for a class of interaction potentials in anti-de Sitter space.     

Characterization of locally rotationally symmetric space-times

We give a simple characterization of locally rotationally symmetric space-times in terms of the existence of a canonical null tetrad or canonical orthonormal tetrad. The result is applied to space-times which satisfy the Einstein field equations with a perfect fluid or electromagnetic field as source.     

An Entropy Functional for Riemann-Cartan Space-Times

By viewing space-time as a continuum elastic medium and introducing an entropy functional for its elastic deformations, T. Padmanabhan has shown that general relativity emerges from varying the functional and that the latter suggests holography for gravity and yields the Bekenstein-Hawking entropy formula. In this paper we extend this idea to Riemann-Cartan space-times by constructing an entropy functional for the elastic deformations of space-times with torsion. We show that varying this generalized entropy functional permits to recover the full set of field equations of the Cartan-Sciama-Kibble theory. Our generalized functional shows that the contributions to the on-shell entropy of a bulk region in Riemann-Cartan space-times come from the boundary as well as the bulk and hence does not suggest that holography would also apply for gravity with spin in space-times with torsion. It is nevertheless shown that for the specific cases of Dirac fields and spin fluids the system does become holographic. The entropy of a black hole with spin is evaluated and found to be in agreement with Bekenstein-Hawking formula.     

Relation between quasirigidity andL-rigidity in space-times of constant curvature and weak fields

The relation between quasirigidity andL-rigidity in space-times of constant nonzero curvature and in space-times with small curvature (weak fields) is studied. The covariant expansion of bitensors about a point is considered. We obtain an increase in the order of magnitude, underL-rigidity conditions, of the rate of change with respect to a comoving orthonormal frame of the linear momentum, angular momentum, and reduced multipole moments of the energy-momentum tensor. Thus,L-rigidity leads to quasirigidity in such space-times.     

Ricci recurrent space-times with torsion

A Ricci recurrent space-time with covariantly constant stress tensor is an Einstein space-time. We extend this result to Ricci recurrent space-times with torsion. The result is applied to the case of Riemann-Cartan space-times with spin density.     

Space-times admitting a three-dimensional conformal group

Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general tilting case.     

Killing vector fields and the Einstein-Maxwell field equations in general relativity

A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector F_ij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor F_ij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient α_ij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,_i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case F_ij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.