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 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.     

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.     

The gauge in general relativity. III

A noncustomary gauge theory of general relativity, developed in detail in the preceding paper (II), is here applied to cosmology. A universe that is homogeneous and isotropic in the customary gauge, is considered-first generally, and then in more detail for the case where the noncustomary universe is matter dominated and static. With a particular choice of equation, this model is solved and a new relation between customary mass density, Hubble constant, and deceleration parameter is found. For a customary deceleration parameter of 1.98, this relation yields a customary mass density of 3.1×10^–31 g/cm³-in good agreement with experiment. Finally, the age of the universe in this model is found to be>6.6× 10⁹ yr, again in agreement with other estimates.     

Nonlinear spinor equations in general relativity

General relativistic nonlinear spinor equations are proposed which reduce in the linear approximation to the Dirac equations, and in the slightly nonlinear approximation reduce to the Ivanenko - Heisenberg equations. When written in a vector form, the nonlinear spinor equations take the form of the Einstein equations, in which matter is produced by spinor fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 121–125, March, 1977.The author thanks professor D. D. Ivanenko for his support and a number of useful observations.     

Quasi-local conservation equations in general relativity

A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza–Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner–Sharp energy. Moreover, on the event horizon of a general dynamical black hole, the quasi-local energy conservation equation coincides with the conservation equation studied by Thorne et al. We discuss the remaining quasi-local conservation equations briefly.     

Non-local equations for general relativity

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.

Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be derived. This single equation encodes the entire conformal structure of a vacuum space—time. The same perturbation technique yields, via quadratures, solutions to the vacuum Einstein equations to any order.     

The equations of motion of macroscopic bodies in general relativity

A new approach to the problem of motion in General Relativity, based upon the systematic approximation procedure of Synge, is presented. The equations of transnational motion for a system of spherical bodies moving under their mutual gravitational attractions are derived. Approximations are based upon the weakness of the field and on the distance between any two of the bodies being considered large by comparision with their radii. The most general stress distribution consistent with maintaining the symmetry of the bodies throughout the motion is chosen. The use of controlled errors enables us to derive equations of motion applicable to a wider class of physical systems than the original equations of Einstein, Infeld and Hoffmann and Fock-Papapetrou.     

Some solutions of the Einstein-Maxwell equations in general relativity

A solution of the Einstein-Maxwell equations corresponding to source-free electromagnetic field plus pure radiation is obtained. The solution is algebraically special. A particular case of the solution is considered which encompasses many known solutions. Among them is a radiating Ruban metric.     

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.     

The general class of the vacuum spherically symmetric equations of the general relativity theory

The system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered. The general solution to a problem representing two classes of line elements with arbitrary functions g ₀₀ and g ₂₂ is obtained. The properties of the found solutions are analyzed.     

Spin and torsion in general relativity II: Geometry and field equations

From physical arguments space-time is assumed to possess a connection is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above, in this series of articles we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing in a unified way gravitation and a very weakspin-spin contact interaction. In Part I of this work† we discussed the foundations of the theory. In this Part II we present in section 3 the geometrical apparatus necessary for the formulation of the theory. In section 4 we take the curvature scalar (or rather its density) as Lagrangian density of the field. In this way we obtain in subsection 4.1 the field equations in their explicit form. In particular it turns out that torsion is essentially proportional to spin. We then derive the angular momentum and the energy-momentum theorems (subsections 4.2-4); the latter yields a force proportional to curvature, acting on any matter with spin. In subsection 4.5 we compare the theory so far developed with GR. Torsion leads to a universal spin-spin contact     

The gauge in general relativity

The view is taken that the field equations of General Relativity, without a definition of congruence of length and time intervals at different events, are without physical content. The possibility is explored that the customary Einstein field equations are to be used but with a different congruence definition than is customary. When these resulting equations are, in turn, expressed with the customary congruence, they comprise a new set of field equations physically not equivalent to either Einstein's or Brans-Dicke's formulations of general relativity. Similarities with Einstein's and Brans-Dicke's formulations are discussed, and the possibility of experimental confirmation of these new equations is also briefly considered.     

Comments on models of scalar-tensorial field equations in general relativity

In this paper we consider some of the proposed models for introducing the long-range scalar interaction in Riemannian space-times. The relationship among these models is discussed. Particular emphasis is placed on the introduction of the scalar interaction via a conformal mapping on the original Riemannian geometry. Following this method we introduce a spinorial model for the coupled system. We also discuss the meaning of the identities satisfied by the left-hand side of the coupled field equations, which are present for any model derivable from an action principle.     

Space-time symmetries and the gravitational-neutrino field equations in general relativity

We consider a neutrino field with geodesic rays in interaction with a gravitational field admitting a Killing vector field n^μ. It is found that for solutions of the Einstein-Weyl field equations the neutrino field ξ^A and the neutrino flux vector l^μ are restricted by the equations: $\text{L}{}_{\mathrm{n}}\text{ξ}{}^{\mathrm{<mtext>A\; =\; ?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{is}\text{ξ}{}^{\mathrm{A}}$ and $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= 0}$, whereas s is a real constant. In the case of pure radiation neutrino fields these equations become: $\text{L}\text{ξ}{}^{\mathrm{A}}\text{=}\text{case1}\text{2}\text{(p ? is)ξ}{}^{\mathrm{A}}$, $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= pl}{}^{\mathrm{\mu }}$, where p and s are in general real functions of the coordinates.     

Extended class of metric tensors in relativity

An extended metric tensor that is a function of an internal vectory ^a(x) leads to a spin-1 massless field of gravitational origin. It is shown that this new field vanishes in the linear approximation for the extended metric.On leave of absence from the Universidade de Brasilia.     

Macroscopic Maxwell's equations of the general theory of relativity

Covariant Maxwell's equations of the general theory of relativity for a system of electromagnetically and gravitationally interacting particles of the form

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.