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.     

Post-post-Newtonian equations of motion for point particles in the general theory of relativity

The post-post-Newtonian equations of motion for point particles are derived from the Einstein gravitational field equations by using the Einstein-Infeld-Hoffmann method with the help of the energy-momentum tensor proposed by Infeld and Plebanski [5, 6]. The obtained equations of motion coincide with the equations derived by Kopeikin [10] by using the Fock method.     

Nongeodesic motion in general relativity

The equations of motion of a spinning body in the gravitational field of a much larger mass are found using both the Corinaldesi-Papapetrou spin supplementary condition (SSC) and the Pirani SSC. These equations of motion are compared with our previous result derived from Gupta's quantum theory of Gravitation. It is found that the spin-dependent terms differ in each of the above three results due to a different location of the center of mass of the spinning body. As expected, these terms are not affected by the choice of either Schwarzschild or isotropic coordinates. Finally, for the presently planned Stanford gyroscope experiment, we find the maximum secular displacement of the orbit of the gyro with respect to the orbit of its non-rotating housing to be of the order of (10⁻⁷ cm/year)t, a result much smaller than Schiff's result which is proportional to time squared.     

The motion of charged test particles in general relativity

We derive, from the Einstein-Maxwell field equations, the Lorentz equations of motion with radiation reaction for a charged mass particle moving in a background gravitational and electromagnetic field by utilizing a line element for the background space-time in a coordinate system specially adapted to the world line of the particle. The particle is introduced via perturbations of the background space-time (and electromagnetic field) which are singular only on the source world line.     

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.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

Heisenberg equations of motion for spin-1/2 wave equation in general relativity

The Heisenberg equations of motion for the spin-1/2 wave equation in general relativity are obtained by a covariant procedure. They are found to be similar to the equations of motion for a classical pole-dipole test-particle in general relativity. The identification is complete when the Heisenberg equations are taken to be satisfied by the respective expectation values.     

Riemannian-Maxwellian invertible structures in general relativity

It is shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2-form and the exterior derivative of a related 2-form are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group with nonsingular orbits.On sabbatical leave from the Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada.     

The optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons

In many metrics of physical interest, the gravitational field can be represented as an optical medium with an effective index of refraction. We show that, in such a metric, the orbits of both massive and massless particles are governed by a variational principle which involves the index of refraction and which assumes the form of Fermat's principle or of Maupertuis's principle. From this variational principle we derive exact equations of motion of Newtonian form which govern both massless and massive particles. These equations of motion are applied to some problems of physical interest.     

Conformal and projective structures in general relativity

It is proposed that compatible conformal and projective structures be taken as the basic space-time structures in general relativity, with the symmetry group restricted to unimodular diffeomorphisms. Models of classical massless fields, such as the Maxwell field, interact directly with the conformal structure; while classical bodies composed of massive particles, such as solids and fluids, interact directly with the projective structure. It is suggested that this difference is the classical limit of the respective quantum-gravitational interactions, which should reflect the differing approaches to the quantization of fields and particles when gravity is neglected. Models of general relativity based on compatible conformal and projective structures should be the basis for the exploration of ideal measurement procedures, both classical and quantum.     

Lorentzian framed structures in general relativity

We study a class of four-dimensional real Lorentzian manifolds generated by a global operator (satisfying a cubic equation). We characterize empty, Einstein, conformally flat spaces and construct mathematical models of two physical space-times. Locally, the solutions presented have been identified with known solutions. Kruskal space-time is generated by using the warped product technique [11].     

Exotic differentiable structures and general relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (fake or exotic) differentiable structures on topologically simple manifolds such asS ⁷, ⁴ andS ³ X ¹. Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.     

Gravitational radiation and the motion of bodies in general relativity

We present a selective review of the problem of the general relativistic prediction for the gravitational radiation emitted by gravitationally interacting systems with particular emphasis upon the two-body problem.On leave from the Department of Physics, University of Victoria, Victoria, British Columbia, Canada V8W 2Y2.     

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.     

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.     

A linear theory of resonance structures in spiral galaxies

The resonance mechanism for the formation of galactic spirals is considered. Expressions are derived for the resonance responses of disks with circular and nearly circular stellar orbits. The spiral responses produced by the central oval-shaped structures (bars) available in many galaxies are shown to have the characteristic properties of the spirals observed in these galaxies. In the most interesting case of a quasi-steady state, the spiral responses possess a similarity property: the spiral thickness and inclination are proportional to the mean size of an epicycle (an analog of the Larmor circle in plasma).     

Local and global algebraic structures in general relativity

The extent to which the well-known pointwise algebraic canonical forms used for the energy-momentum tensor, the Weyl tensor, etc., can be regarded as smooth relations over some open subset of (possibly the whole of) space-time is investigated.     

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.     

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.     

Analogs in general relativity of Ferraro's law of isorotation

Two relativistic analogs of Ferraro's law of isorotation for a fluid with infinite electrical conductivity are given. In the first case it is assumed that the flow is shear-free withH ^a =0 and it is shown that £(H) ^a =0. In the second case it is assumed that there exists a space-like Killing vector field parallel toH _a , and it is proved that, _a H ^a =0.