A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel‐Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.

Curvature Collineations: A Fundamental Symmetry Property of the Space‐Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

The nature and uses of self-similarity in general relativity are discussed. A spacetime may be self-similar (homothetic) along surfaces of any dimensionality, from 1 to 4. A geometric construction is given for all self-similar spacetimes. As an important special case, the “spatially self-similar cosmological models” are introduced, and their dynamical properties are studied in some detail: The initial-value problem is posed, the ADM formulation is established (when applicable), and it is shown that the evolution equations preserve a self-similarity of initial data. The existence of a conserved quantity is deduced from self-similarity. Possible applications to cosmology and singularities are mentioned.

/pdf/self-similar-spacetimes-geometry-and-dynamics-1jbl6bjgzs.pdf

Self-similar spacetimes: Geometry and dynamics

We study the consequences of the existence of a one‐parameter group of conformal motions for anisotropic matter, in the context of general relativity. It is shown that for a class of conformal motions (special conformal motions), the equation of state is uniquely determined by the Einstein equations. For spherically symmetric and static distributions of matter we found two analytical solutions of the Einstein equations which correspond to isotropic and anisotropic matter, respectively. Both solutions can be matched to the Schwarzschild exterior metric and possesses positive energy density larger than the stresses, everywhere within the sphere.

Anisotropic fluids and conformal motions in general relativity

The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models of modern cosmology in order to derive exact solutions and deal with the problem of dark matter/energy. The key role in this approach are the first integrals of the field equations. We start with the Lie point symmetries and the first integrals defined by them, that is, the Hojman integrals. Subsequently, we discuss the Noether point symmetries and the well-known method for deriving the Noether integrals. By means of the Inverse Noether Theorem, we show that, to every Hojman quadratic first integral, it is possible to associate a Noether symmetry whose Noether integral is the original Hojman integral. It is emphasized that the point transformation generating this Noether symmetry need not coincide with the point transformation defining the Lie symmetry which produces the Hojman integral. We discuss the close connection between the Lie point and the Noether point symmetries with the collineations of the metric defined by the kinetic energy of the Lagrangian. In particular, the generators of Noether point symmetries are elements of the homothetic algebra of that metric. The key point in the current study of cosmological models is the introduction of the mini superspace, which is the space that is defined by the physical variables of the model, which is not the spacetime where the model evolves. The metric in the mini superspace is found from the kinematic part of the Lagrangian and we call it the kinetic metric. The rest part of the Lagrangian is the effective potential. We consider coordinate transformations of the original mini superspace metric in order to bring it to a form where we know its collineations, that is, the Killing vectors, the homothetic vector, etc. Then, we write the field equations of the cosmological model and we use the connection of these equations with the collineations of the mini superspace metric to compute the first integrals and subsequently to obtain analytic solutions for various allowable potentials and finally draw conclusions about the problem of dark energy. We consider the ΛCDM cosmological model, the scalar field cosmology, the Brans–Dicke cosmology, the f(R) gravity, the two scalar fields cosmology with interacting scalar fields and the Galilean cosmology. In each case, we present the relevant results in the form of tables for easy reference. Finally, we discuss briefly the higher order symmetries (the contact symmetries) and show how they are applied in the cases of scalar field cosmology and in the f(R) gravity.

Symmetries of Differential Equations in Cosmology

The consequences of taking the total active gravitational mass of the universe phasewise constant together with a decaying vacuum energy in the background of Robertson-Walker space-time are investigated. The model so determined admits a contracted Ricci-collineation along the fluid flow vectorν
 i. It is geometrically closed but ever-expanding and does not possess the initial singularity, horizon, entropy, monopole or cosmological constant problems of the standard big bang cosmology. Estimates of the present matter; radiation and vacuum energy densities, the age of the universe and the present values of the deceleration parameter and the scale factor are also obtained.

A Model of the universe with decaying vacuum energy

The intimate relation of symmetry properties describable by groups of motions and the consequent conservation laws realizable for a particle in classical mechanics and in the mechanics of restricted and general theories of relativity are discussed in detail using elementary results of the theory of continuous groups. In addition, for the case of special relativistic mechanics the finite form of the groups of motions, underlying all of the possible constants of the motion of the form of first integrals linear in the momenta, are constructed and shown to be the inhomogeneous (i.e., including displacements) proper Lorentz transformations.

Mechanical Conservation Laws and the Physical Properties of Groups of Motions in Flat and Curved Space-Times

By definition, a Riemannian space Vn admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξi of the Riemann curvature tensor vanishes. It is shown that if a Vn admits a parallel vector field, then it will admit groups of CC's. It follows that every space‐time with an expansion‐free, shear‐free, rotation‐free, geodesic congruence admits groups of CC's, and hence gravitational pp waves admit such groups of symmetries.

Groups of Curvature Collineations in Riemannian Space‐Times Which Admit Fields of Parallel Vectors

The problem of determining explicitly the forces of holonomic constraints in particle mechanics formulated in Riemannian space is carried through in a general manner. It is shown that the so-called projection operator hji ≡ δji — ninj enters quite naturally in problems involving one equation of constraint constituting a Vn − 1 in the given Vn. In addition, the results of this paper go beyond the early work of Synge in this area, in that their detailed connections with certain elementary results of the mathematical theory of subspaces Vn − 1 in Vn are given, which provide further physical insight.

William R. Davis

Papers

Curvature Collineations: A Fundamental Symmetry Property of the Space‐Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

Mechanical Conservation Laws and the Physical Properties of Groups of Motions in Flat and Curved Space-Times

Groups of Curvature Collineations in Riemannian Space‐Times Which Admit Fields of Parallel Vectors

Constraints in Particle Mechanics and Geometry