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.     

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.     

The GHP conformai Killing equations and their integrability conditions,with application to twisting type N vacuum spacetimes

The conformai Killing equations in resolved form and their first and second integrability conditions are obtained in the compact spin coefficient formalism for arbitrary spacetimes. To facilitate calculations an operatorL is introduced which agrees with the Lie derivative only when operating on quantities with GHP weights (0,0). The resulting equations are used to find the conditions for the existence of a two dimensional non-Abelian group of homothetic motions in a twisting typeN vacuum spacetime. The equivalence of two such sets of metrics is established, metrics that were recently the subject of independent investigations by Herlt on the one hand and by Ludwig and Yu on the other.     

Expanding algebraically degenerate vacuum spaces with homothetic motions

Halford and Kerr [3] and Halford [4] discuss metrics which arise as solutions of Einstein's vacuum field equations and which have an algebraically degenerate Weyl tensor and an expanding repeated principal null direction. A number of the metrics which are explicitly listed appeared to be new. Here all of these explicit metrics are identified, either as well-known metrics or as particular cases of known metrics with arbitrary functions when these functions take various specific forms. Corrections are made to a number of results in these two papers. An overview of the status of homothetic motions in expanding algebraically degenerate vacuum spacetimes is also given.     

Review: Kerr-Schild and Generalized Metric Motions

In this work we study in detail new kinds of motions of the metric tensor. The work is divided into two main parts. In the first part we study the general existence of Kerr-Schild motions — a recently introduced metric motion. We show that generically, Kerr-Schild motions give rise to finite dimensional Lie algebras and are isometrizable, i.e., they are in a one-to-one correspondence with a subset of isometries of a (usually different) spacetime. This is similar to conformal motions. There are however some exceptions that yield infinite dimensional algebras in any dimension of the manifold. We also show that Kerr-Schild motions may be interpreted as some kind of metric symmetries in the sense of having associated some geometrical invariants. In the second part, we suggest a scheme able to cope with other new candidates of metric motions from a geometrical viewpoint. We solve a set of new candidates which may be interpreted as the seeds of further developments and relate them with known methods of finding new solutions to Einstein's field equations. The results are similar to those of Kerr-Schild motions, yet a richer algebraic structure appears. In conclusion, even though several points still remain open, the wealth of results shows that the proposed concept of generalized metric motions is meaningful and likely to have a spin-off in gravitational physics. We end by listing and analyzing some of those open points.     

Conformal groups and conformally equivalent isometry groups

It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper conformal gradient every (local) conformal group is conformally homothetic. In the Riemannian case there is always a conformal scalar unless the metric is conformally Euclidean. In the case of a Lorentzian 4-manifold it is proved that the only metrics with no conformal scalars (and hence the only ones admitting a (local) conformal group not conformally isometric) are either conformal to the plane wave metric with parallel rays or conformally Minkowskian.     

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.     

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.     

Single kerr-schild metrics: A double view

Real-vacuum single Kerr-Schild (ISKS) metrics are discussed and new results proved. It is shown that if the Weyl tensor of such a metric has a twist-free expanding principal null direction, then it belongs to the Schwarzschild family of metrics — there are no Petrov type-II Robinson-Trautman metrics of Kerr-Schild type. If such a metric has twist then it belongs either to the Kerr family or else its Weyl tensor is of Petrov type II. The main part of the paper is concerned with complexified versions of Kerr-Schild metrics. The general real ISKS metric is written in double Kerr-Schild (IDKS) form. TheH andl potentials which generate IDKS metrics are determined for the general vacuum ISKS metric and given explicitly for the Schwarzschild and Kerr families of metrics.     

Finding Collineations of Kimura Metrics

Kimura investigated static spherically symmetric metrics and found several to have quadratic first integrals. We use REDUCE and the package Dimsym to seek collineations for these metrics. For one metric we find that three proper projective collineations exist, two of which are associated with the two irreducible quadratic first integrals found by Kimura. The third projective collineation is found to have a reducible quadratic first integral. We also find that this metric admits two conformal motions and that the resulting reducible conformal Killing tensors also lead to Kimura's quadratic integrals. We demonstrate that when a Killing tensor is known for a metric we can seek an associated collineation by solving first order equations that give the Killing tensor in terms of the collineation rather than the second order determining equations for collineations. We report less interesting results for other Kimura metrics.     

Energy conditions and conservation laws in LTB metric via Noether symmetries

In this paper, we have investigated Noether symmetries in Lemaitre–Tolman–Bondi (LTB) metric. Using the Lagrangian associated with the LTB metric, the set of determining equations for Noether symmetries is obtained and then integrated in several cases. It is shown that the LTB metric can be classified in to eight distinct classes corresponding to Noether algebra of dimension 4, 5, 6, 7, 8, 9, 11 and 17. The obtained Noether symmetries are compared with Killing and homothetic vectors. The well known Noether’s theorem is used to find the expressions for conservation laws in each case. Moreover, it is shown that most of the obtained metrics are anisotropic or perfect fluid models which satisfy certain energy conditions and the equation of state.     

Energy Conditions and Conservation Laws in LRS Bianchi Type I Spacetimes via Noether Symmetries

In this paper, we have completely classified the locally rotationally symmetric (LRS) Bianchi type I spacetimes via Noether symmetries (NS). The usual Lagrangian corresponding to LRS Bianchi type I metric is used to find the set of determining equations. To achieve a complete classification, these determining equations are generally integrated to find the components of NS vector field and the metric coefficients. During this procedure, several cases arise which give different Noether algebras of dimension 5,..., 9, 11, and 17. A comparison is established between the obtained NS and the Killing and homothetic vectors. Corresponding to all NS generators, the conservation laws are stated by using Noether's theorem. The metrics which we have obtained as a result of our classification are shown to be anisotropic or perfect fluids which satisfy certain energy conditions.     

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.     

Diagonalizing cohomogeneity-one Einstein metrics

We investigate when it is possible to globally diagonalize an Einstein metric of cohomogeneity-one under the action of a compact connected Lie group G with isotropy subgroup K. Restricting our attention to the non-monotypic case in which cohomogeneity-one metrics are not automatically diagonalizable, we make use of a subset of the ODEs of the Einstein condition to globally diagonalize the Einstein metric in certain cases. The program we use is inspired by a result well-known to relativists which states that Einstein metrics must be diagonalizable in the Bianchi IX case. We present a full classification of all the cohomogeneity-one metrics on which this method can be used.     

The cornerstone role of the torsion in Finslerian physical worlds

It is shown that there are no metric-compatible connections with zero torsion onproperly Finslerian, i.e. post-Riemannian, metrics. Since Finslerian connections exist on Riemannian metrics, the torsion rather than the metric becomes the object which determines whether the geometry is properly Finslerian or not. On the other hand, the solder forms and connection are determined by the torsion if the affine curvature is zero, the torsion then containing all the information about the geometric reality of spacetime. Since the metric curvature may still be Riemannian, the question arises of whether its present central role in spacetime physics is but a consequence of requiring that all the geometric content of spacetime be contained in the metric.     

Killing vectors in empty space algebraically special metrics. I

Empty space algebraically special metrics possessing an expanding degenerate principal null vector and a Killing vector are investigated. It is shown that the Killing vector falls into one of two classes. The class containing all asymptotically timelike Killing vectors is investigated in detail and the associated metrics are identified. Several theorems concerning these metrics are given, among which is a proof that if the metric is regular and possesses an asymptotically timelike Killing vector, then it must be typeD. In addition some relations between Killing vectors in general spaces are developed along with a set of tetrad symmetry equations stronger than those of Killing.     

Proper homothetic maps and fixed points

Let f be a proper homothetic map of the pseudo-Riemannian manifold M and assume f has a fixed point p. If all of the eigenvalues of either f* _p or f ^-1*_p have absolute values less than unity, then M is topologically R _n and M has a flat metric. This yields three characterizations of Minkowski spacetime. In general, a homothetic map of a complete pseudo-Riemannian manifold need not have fixed points. Furthermore, an example shows the existence of a proper homothetic map with a fixed point does not imply M is flat. The scalar curvature vanishes at a fixed point, but some of the sectional curvatures may be nonzero.     

Homothetic motions and vacuum Robinson-Trautman solutions

The approach to symmetries given by Kerr and Debney (1970) is applied to the class of vacuum Robinson-Trautman solutions. The coordinate freedom is extended and used to define the pull-backs of local diffeomorphisms which are then shown to be homothetic motions. The resulting homothetic Killing's equations are solved to give a form of all homothety groups for this class.     

Weighted Relative Group Entropies and Associated Fisher Metrics

A large family of new α-weighted group entropy functionals is defined and associated Fisher-like metrics are considered. All these notions are well-suited semi-Riemannian tools for the geometrization of entropy-related statistical models, where they may act as sensitive controlling invariants. The main result of the paper establishes a link between such a metric and a canonical one. A sufficient condition is found, in order that the two metrics be conformal (or homothetic). In particular, we recover a recent result, established for α=1 and for non-weighted relative group entropies. Our conformality condition is “universal”, in the sense that it does not depend on the group exponential.     

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.