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.     

Canonical superenergy tensors in general relativity

We present the concept of superenergy tensors in the framework of general relativity (GR). These tensors were introduced constructively by the author years ago and they were obtained by a suitable averaging of the energy-momentum tensors or pseudotensors. Because in GR the Einstein canonical energy-momentum pseudotensor_E t _i ^k of the gravitational field and the canonical energy-momentum complex , matter and gravitation,are physically distinguished, we confine this paper to thecanonical superenergy tensor ^g S _i ^k of the gravitational field Γ _kl ⁱ and to the canonical total superenergy tensorS _i ^k = ^g S _i ^k + ^m S _i ^k of matter and gravitation only. These superenergy tensors can be obtained by the above-mentioned averaging of the pseudotensor_E t _i ^k and complex_E K _i ^k . We give the analytic forms of these two canonical superenergy tensors and show some of their possible applications in GR. The canonical superenergy tensor ^g S _i ^k of the gravitational field Γ _kl ⁱ can be used as asubstitute for the nonexisting energy-momentum tensor of this field.     

Comparison of the DeWitt metric in general relativity with the fourth-rank constitutive tensors in electrodynamics and in elasticity theory

We perform a short comparison between the local and linear constitutive tensor \(\chi ^{\lambda \nu \sigma \kappa }\) in four-dimensional electrodynamics, the elasticity tensor \(c^{ijkl}\) in three-dimensional elasticity theory, and the DeWitt metric \(G^{abcd}\) in general relativity, with \({a,b,\ldots =1,2,3}\). We find that the DeWitt metric has only six independent components.     

General relativity without tensors in a central field

For static and spherically symmetric gravitational fields in the general theory of relativity, it is found possible completely to avoid tensor analysis. The principle of equivalence, illustrated by Einstein's elevator, is used to obtain Schwarzschild's equation, on which the three well-known tests of the general theory are usually based. The derivation is guided, as with Einstein, by Poisson's (Laplace's, in empty space) equation, which here can be solved by simple calculus.     

General relativity with a background metric

An attempt is made to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe. This is done by introducing a background metric γ_μν (in addition to g_μν) describing a spacetime of constant curvature with positive spatial curvature. The additional terms in the field equations are negligible for the solar system but important for intense fields. Cosmological models are obtained without singular states but simulating the “big bang.” The field of a particle differs from the Schwarzschild field only very close to, and inside, the Schwarzschild sphere. The interior of this sphere is unphysical and impenetrable. A star undergoing gravitational collapse reaches a state in which it fills the Schwarzschild sphere with uniform density (and pressure) and has the geometry of a closed Einstein universe. Any charge present is on the surface of the sphere. Elementary particles may have similar structures.     

Space tensors in general relativity II: Physical applications

The general theory of space tensors is applied to the study of a space-time manifoldsV ₄ carrying a distinguished time-like congruence Γ. The problem is to determine a physically relevant spatial tensor analysis over (V ₄, Γ), in order to proceed to a correct formulation of Relative Kinematics and Dynamics. This is achieved by showing that each choice of gives rise to a corresponding notion of ‘frame of reference’ associated with the congruence Γ. In particular, the frame of reference (Γ, ∇*) determined by the standard spatial tensor analysis is shown to provide the most natural generalization of the concept of frame of reference in Classical Physics. The previous arguments are finally applied to the study of geodesic motion inV ₄. As a result, the general structure of the gravitational fields in the frame of reference (Γ, ∇*) is established. This work was assisted by funds from the C.N.R. under the aegis of the activity of the National Group for Mathematical Physics.     

Quartic equations and classification of Riemann tensors in general relativity

For space-times in general relativity, the Petrov classification of the Weyl conformai curvature and the Plebaski or Segre classification of the Ricci tensor each depend on the properties of the roots of quartic equations. The coefficients in these quartic equations are in general complicated functions of the space-time coordinates. We review the general theory of quartic equations, and discuss algorithms for determining the existence and values of multiple roots. We consider practical implementation of an algorithm and the consequent Petrov classification. Tests of programs embodying this algorithm, using the computer algebra system CLASSI based on SHEEP, are reported.     

Space tensors in general relativity III: The structural equations

A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), d_T (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis \((\tilde \nabla ,\tilde \nabla T)\) (M,Γ). The structural equations of \((\tilde \nabla ,\tilde \nabla T)\) and the corresponding spatial Bianchi identities are discussed. The special case \((\tilde \nabla ,\tilde \nabla T) = (\tilde \nabla *,\tilde \nabla T*)\) is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν₄,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.     

Doubly metric formalism and observables in general relativity theory

The gravitational field is described as a standard physical field in two-dimensional space (a radical variant of doubly metric formalism). By a systematic execution of this approach, one is able to clarify certain difficulties inherent in doubly metric theories. The analysis of properties of a gravitational field in small spatial regions shows a close analogy to electrodynamics, which enables one to reduce a number of problems of system mechanics in the gravitational field to similar problems in electrodynamics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 69–74, March, 1977.     

Generating potential for the nut metric in general relativity

It is shown that the Newtonian potential that generates the static Schwarschild and Reissner-Nordstrom metrics also generates the NUT space metric from the class of Papapetrou stationary fields.     

Symmetry constraints in general relativity and the Schwarzchild interior metric

Einstein's equations with a perfect fluid source are subjected to compatibility conditions in the context of a space-time that contains symmetric subspaces. These conditions constitute, in some cases, a powerful tool for exhibiting the solutions to a given problem. The Schwarzchild interior metric in conformally flat coordinates is derived using these methods.     

Space tensors in general relativity I: Spatial tensor algebra and analysis

A pair (M, Γ) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence Γ. The spatial tensor algebraD associated with the pair (M, Γ) is discussed. A general definition of the concept of spatial tensor analysis over (M, Γ) is then proposed. Basically, this includes a spatial covariant differentiation and a time-derivative , both acting onD and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M, Γ) is finally established, and the resulting mathematical structure is examined in detail. This work was assisted by funds from the C.N.R. under the aegis of the activity of the National Group for Mathematical Physics.     

Algebraic determination of the metric from the curvature in general relativity

The general solution for a symmetric second-order tensorX of the equationX _e(a R ^e _b cd=0 whereR is the Riemann tensor of a space-time manifold, andX is obtained in terms of the curvature 2-form structure ofR by a straightforward geometrical technique, and agrees with that given by McIntosh and Halford using a different procedure. Two results of earlier authors are derived as simple corollaries of the general theorem.     

Extended relativity: Mass and the fifth dimension

A self-consistent relativistic formalism is presented which postulates that mass is the eigenvalue of a fifth momentum operator component. Lorentz covariance is generalized so that a systematic program for covariant wave equations can be formed. The fifth dimension is identified with cosmic time, resulting in a bias toward matter over antimatter for the universe. A distinction between μ ande also seems possible through the space-time extension.     

Spinors,algebraic geometry,and the classification of second-order symmetric tensors in general relativity

Two approaches to the problem of classifying second-order symmetric tensors in space-time given by Ludwig and Scanlon and by Penrose are discussed. Ludwig and Scanlon use both spinor and tensor algebra in their approach, whereas Penrose uses spinors and the properties of certain curves in complex projective 3-space. These approaches yield essentially identical classifications, and this paper points out the connections between them in detail and tabulates the results.     

Space times of embedding class one in general relativity

The algebraic structures of both the Ricci and Weyl tensors, given that of the second fundamental tensor, are tabulated. In particular, the Weyl tensor is algebraically special if and only if the second fundamental tensor is algebraically special. A class one perfect fluid is found to possess at least one of the following properties: (a) conformai flatness; (b) the flow is geodesic; (c) it admits a three-dimensional group of isometries with two-dimensional space-like trajectories. All solutions with property (b) are obtained explicitly.     

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.     

A class of static charged dust spheres in general relativity

Using static spherically symmetric space-times with associated 3-spaces obtained as hypersurfacest= const as 3-spheroidal, a class of physically viable relativistic models for spherical distributions of uniformly charged dust in equilibrium is obtained. The charged analog of Schwarzschild interior solution given by Cooperstock and de la Cruz follows as a particular case of this class.