Rigid motions relative to an observer:L-rigidity

A new definition of rigidity,L-rigidity, in general relativity is proposed. This concept is a special class of pseudorigid motions and therefore it depends on the chosen curveL. It is shown that, for slow-rotation steady motions in Minkowski space, weak rigidity andL-rigidity are equivalent. The methods of the PPN approximation are considered. In this formalism, the equations that characterizeL-rigidity are expressed. As a consequence, the baryon mass density is constant in first order, the stress tensor is constant in the comoving system, the Newtonian potential is constant along the lineL, and the gravitational field is constant along the lineL in the comoving system.     

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.     

Mappings of empty space-times leaving the curvature tensor invariant

It is shown that if in some local coordinate system the componentsR ⁱ _jkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorR ⁱ _jkl invariant are found for each class of these space-times.     

Energy and Momentum of Robinson-Trautman Space-Times

Robinson and Trautman space-times are studied in the context of teleparallel equivalent of general relativity (TEGR). These space-times are the simplest class of asymptotically flat geometries admitting gravitational waves. We calculate the total energy for such space-times using two methods, the gravitational energy-momentum and the translational momentum 2-form. The two methods give equal results of these calculations. We show that the value of energy depends on the gravitational mass M, the Gaussian curvature of the surfaces λ(u,θ) and on the function K(u,θ). The total energy reduces to the energies of Schwarzschild’s and Bondi’s space-times under specific forms of the function K(u,θ).     

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.     

Nonscalar singularities in spatially homogeneous cosmologies

Singularities in vacuum spatially homogeneous cosmological models are investigated. It is shown that in general the curvature scalarR _* ^abcd R^*abcddiverge and that the only solutions which have curvature singularities at which this scalar does not diverge describe certain plane-wave space-times. It is argued that with matter present these nonscalar singularities are even less likely to occur. The exceptional case of Bianchi type VI_–1/9 is not considered.     

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.     

The nonlinear graviton in interaction with a photon

It is shown that Einstein-Maxwell complex space-times with self-(anti-self-) dual Weyl tensor and algebraically general anti-self-(self-) dual Maxwell tensor are completely characterized as quasi-Kählerian space-times with vanishing scalar curvature. Following Penrose's interpretation ofH-spaces, we propose that an electrifiedH-space be interpreted as a nonlinear graviton in interaction with a photon. Two families of exact solutions are presented as examples.Supported in part by NSF Grant #MPS74-15246.     

Stability of two-dimensional quasisingular space-times

We study the stability of a class of two-dimensional cylindrical space-times with quasiregular singularities using massless scalar waves. The fact that the stress-energy scalarT _v T ^v diverges indicates the instability of the singularity toward formation of a scalar curvature singularity. In special cases a nonscalar curvature singularity results.     

Letter: Comment on Ricci Collineations for Spherically Symmetric Space-Times

It is shown that the results of the paper Contreras, G., Nunez, L. A., Percoco, U. Ricci Collineations for Non-degenerate, Diagonal and Spherically Symmetric Ricci Tensors (2000). Gen. Rel. Grav. 32, 285-294 concerning the Ricci Collineations in spherically symmetric space-times with non-degenerate and diagonal Ricci tensor do not cover all possible cases. Furthermore the complete algebra of Ricci Collineations of certain Robertson-Walker metrics of vanishing spatial curvature are given.     

Stationary Riemannian space-times with self-dual curvature

Riemannian space-times with self-dual curvature and which admit at least one Killing vector field (stationary) are examined. Such space-times can be classified according to whether a certain scalar field (which is the difference between the Newtonian and NUT potentials) reduces to a constant or not. In the former category (called here KSD) are the multi-TaubNUT and multi-instanton space-times. Nontrivial examples of the latter category have yet to be discovered. It is proved here that the static self-dual metrics are flat. It is also proved that each stationary metric for which the Newtonian and nut potentials are functionally related admits a Killing vector field relative to which the metric is KSD. It has also been proved that the regularity of the field everywhere implies that the metric is KSD. Finally it is proved that for non-KSD space-times every regular compact level surface of the field encloses the total NUT charge, which must be proportional to the Euler number of the surface.The research reported here was done while the author was an NSERC Postdoctoral Fellow at Simon Fraser University.The author is also a member of the Theoretical Science Institute at Simon Fraser University, and preparation for publication was partially assisted NSERC Research Grant No. 3993.     

Momentum and angular momentum in theH-space of asymptotically flat,Einstein-Maxwell space-times

We propose new definitions for the momentum and angular momentum of Einstein-Maxwell fields that overcome the deficiencies of earlier definitions of these terms and are appropriate to the newH-space formulations of space-time. We make our definitions in terms of the Winicour-Tamburino linkages applied to the good cuts of CI⁺. Our transformations between good cuts then correspond to the translations and Lorentz transformations at points inH-space. For the special case of Robinson-Trautman typeII space-times, we show that our definitions of momentum and angular momentum yield the previously published results of Ludvigsen.Part of this work was completed while William Hallidy was supported by a fellowship from the Alexander von Humboldt Foundation.     

Black holes in higher dimensional space-times

Black hole solutions to Einstein's equations are examined in asymptotically flat N + 1 dimensional space-times. First generalizations of Schwarzschild and Reissner-Nordstrøm solutions are examined in a discussion of static black holes in N + 1 dimensions. Then a new family of solutions is found which describe spinning black holes in higher dimensional space-times. In many respects these new solutions are similar to the familiar Kerr and Schwarzschild metrics which are recovered for N = 3. One exceptional case though is that for N ≥ 5, black holes with a fixed mass may have arbitrarily large angular momentum.     

Initial Data Engineering

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.Partially supported by a Polish Research Committee grant 2 P03B 073 24Partially supported by the NSF under Grants PHY-0099373 and PHY-0354659Partially supported by the NSF under Grant DMS-0305048 and the UW Royalty Research Fund     

Uniqueness of Operators Generated by Partial Isometries

The pair of operators P = VpV+ and Q = VqV+ where p and q are momentum and position and V is a partial isometry if of Hilbert space is shown to be essentially unique and unitarily equivalent to angular momentum L₃ and angle φ.     

Curvature collineations and the determination of the metric from the curvature in general relativity

It is shown that for a very general class of space-times, the componentsR _bcd ^a of the curvature tensor determine the metric components up to a constant conformal factor. This general class contains most of those cases which are usually considered to be interesting from the point of view of Einstein's general relativity theory. The connection between the above result and the existence of proper curvature collineations is given.     

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.     

Born’s Reciprocal Gravity in Curved Phase-Spaces and the Cosmological Constant

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of the curvature of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in space is very large, namely the small size of P is of the order of . Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the Quaplectic group that is given by the semi-direct product of U(1,3) with the (noncommutative) Weyl-Heisenberg group involving four coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented.