Einstein's theory in a three-dimensional space-time

As a preparation for studying quantum models, we analyze unusual features of Einstein's theory of gravitation in a three-dimensional space-time. In three dimensions, matter curves space-time only locally and the gravitational field has no dynamical degrees of freedom. The standard correspondence of Einstein's theory with Newton's theory breaks down. A dust distribution moves without any geodesic deviation between the particles. The cosmological models and relativistic stars behave in a qualitatively different way from their Newtonian counterparts. These features are important for the correct understanding of mini-superspace models.     

Newtonian versus relativistic nonlinear cosmology

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order. Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the post-Newtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume.     

Complementary aspects of gravitation and electromagnetism

A convention with regard to geometry, accepting nonholonomic aether motion and coordinate-dependent units, is always valid as an alternative to Einstein's convention. Choosing flat spacetime, Newtonian gravitation is extended, step by step, until equations closely analogous to those of Einstein's theory are obtained. The first step, demanded by considerations of inertia, is the introduction of a vector potential. Treating the electromagnetic and gravitational fields as real and imaginary components of a complex field (gravitational mass being treated as imaginary charge), the Maxwell stress-momentum-energy tensor for the complex field is then used as the source for both fields. The spherically symmetric solution of these unified field equations describes the electron. Third, effects arising from motion of aether fluid with respect to the artificial reference systems of flat spacetime are included. On the grounds that attraction between likes and repulsion between likes are, a priori, equally possible, it is suggested that gravitational and electromagnetic phenomena should enjoy equal status. This can be achieved on the scale of an infinite cosmos by introducing a hierarchy of isolated systems, each of which is a universe when viewed internally and an elementary particle when viewed externally. A universe (defined by the Hubble radius), an electron, and a neutrino are three consecutive isolated systems of the hierarchy. Implied is the existence of antiuniverses where gravitational mass has opposite sign and antimatter predominates. Remarkable relationships between physical constants emerge.     

The gravitational interaction: Spin,rotation, and quantum effects-a review

Previous work on spin, rotation, and quantum effects in gravitation is surveyed, with particular emphasis on the gravitational two-body interaction, both for elementary particles and for macroscopic bodies. Applications considered include (a) the precession of a gyroscope, (b) rotational effects on the equations of motion for the orbit, (c) binary systems, particularly the binary pulsar PSR 1913+16, and (d) the prospects of measuring spin-orbit and spin-spin forces in the laboratory. In addition, we discuss quantum effects that arise in the interaction between elementary particles. In particular, we point out the potentially decisive role of these forces in high-density matter, with emphasis on the fact that repulsive forces arise that may prevent gravitational collapse. All of the above considerations are within the framework of Einstein's theory of general relativity, albeit extended to treat spin-dependent and quantum forces. Finally, we consider the additional quantum terms that are present if one works with a generalization of Einstein's theory, the Einstein-Cartan-Sciama-Kibble theory of gravitation, in which the spin of matter, as well as its mass, plays a dynamical role.     

Letter: Epicyclic Orbital Oscillations in Newton's and Einstein's Gravity from the Geodesic Deviation Equation

In a recent paper Abramowicz and Kluniak [1] have discussed the problem of epicyclic oscillations in Newton's and Einstein's dynamics and have shown that Newton's dynamics in a properly curved three-dimensional space is identical to test-body dynamics in the three-dimensional optical geometry of Schwarzschild space-time. One of the main results of this paper was the proof that different behaviour of radial epicyclic frequency and Keplerian frequency in Newtonian and General Relativistic regimes had purely geometric origin contrary to claims that nonlinearity of Einstein's theory was responsible for this effect. In this paper we obtain the same result from another perspective: by representing these two distinct problems (Newtonian and Einstein's test body motion in central gravitational field) in a uniform way — as a geodesic motion. The solution of geodesic deviation equation reproduces the well known results concerning epicyclic frequencies and clearly demonstrates geometric origin of the difference between Newtonian and Einstein's problems.     

A Generally Covariant Wave Equation for Grand Unified Field Theory

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrödinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.     

SPHERICALLY SYMMETRIC AND STATIC FIELDS IN LINEARIZED GAUGE THEORIES OF GRAVITATION

In this paper Newtonian limit in the Poincare gauge field theory of gravitation is investigated. In spherically symmetric and static cases interior and exterior solutions of the linearized field equations with gravitational sourtion are obtained by maens of Green's function for the five Lagrangians with out ghosts and tachyons. In cases of four Lagrangians,the space-time metrics outside gravitational source are the usual Schwarzschild one of the first-older, while in the case of the fifth hagrangian the space-time metric differs from the Schwarzschild one. Under both,Newtonian and-weak gravitational field approximations,the motion of a test particle without span should therefore be different from Newton's second law. As a result of the exchanged particles of spin o⁺ the deviation from Newton's second law is a Yukawa term which is attractive. A distance-dependent gravitational "constant" G(r) can be defined according to the new result. The difference between G(r) and Newton's gravitational constant G_∞ is due to a nonzero component of torsion tensor, the effect of which can be tested by measuring G(r).     

Newtonian gravitational field theory: Two-body problem

The effects of a dual force which appears in a consistent field theory of Newtonian gravitation are explored by a study of the motion of two bodies which interact with each other through the gravitational field. The equations of motion are solved exactly. Among the results obtained, we find that the present theory formulated in accordance with the Special Theory of Relativity leads to the same analytical result for the precession of the perihelion of the orbit as does Einstein's General Theory of Relativity. Another result is that classical particles are endowed with an intrinsic angular momentum of constant magnitude—a helicity of classical origin. Other results, such as the period of revolution, are similar to Kepler's law, except for relativistic corrections. A slight deviation from the planar orbit of classical theory results, and may be observable. This deviation is related to the magnitude of the precession of the perihelion of the orbit. The significance of these results for charged particles, viewed classically or quantum mechanically, are discussed.     

New Weyl theory: Geometrization of electromagnetism and gravitation: Motivations and classical results

A geometrical unified field theory of electromagnetism and gravitation is developed in a Weyl space-time. The integrability conditions of the field equations cast the laws of classical perfect fluids under electromagnetic interactions. The purely gravitational limit of the theory is Einstein's General Relativity and the purely electromagnetic case coincides with the predictions of Maxwell's theory.     

Isotropic Space with Discrete Gravitational-Field Sources. On the Theory of a Nonhomogeneous Isotropic Universe

Existing cosmological theories are based on Einstein's law of gravitation (7). In this equation the average is taken only in the right-hand side by a substitution of the energy momentum tensor corresponding to uniform and continuous distribution of matter. In this paper a new cosmological equation (48), which is more correct from the physical and mathematical point of view, is obtained by space-time averaging of all the terms of Eq. (7) and taking into account the fluctuations of the gravitational field due to nonuniformities in the distribution of matter. An estimate of these fluctuations within the framework of Newton's approximation leads to the cosmological equations (51), (52) and (53) for flat space and positive and negative curvature. The solutions of these equations, in distinction from all the variants of Friedman's theory, do not have a singular point for some initial moment of time with an infinitely large density of matter. However, this result follows when the relations obtained are extrapolated beyond the range of their applicability, and therefore final conclusions can be made on the basis of the solutions of the new cosmological equations (48) when we go beyond the Newtonian approximation.     

A background-dependent approach to the theory of gravitation

On the basis of a Machian view of nature we find a covariant formulation of Newton's gravitational equation in a general frame which satisfies the requirements (i) of being singular if the density of mass is zero everywhere and (ii) of depending on the parallel transport of the four-momentum density of matter (from the three-space point in which it is defined to any other three-space point, at any fixed time) in such a way that it incorporates the idea that the frame has to be fixeddirectly in connection with the distribution and motion of matter. In Paper II we will use such an equation as starting point in order to find relativistic gravitational equations which are supposed to hold in any conceivable universe, describe a purely geometrical theory of gravitation, and explicitly incorporate Mach's principle.     

Inflationary universe without GUTs

The existence of a primordial inflationary era is unavoidable due to the puzzling nature of semiclassical gravitation, regulated by Einstein's equations and the laws of quantum mechanics. This interaction appears to be controlled by a mass-dependent effective gravitational coupling constant. The latter undergoes an unexpected transition from a classical gravitational attractive to an antigravitational repulsive regime when the corresponding mass of a quantum matter field passes through a definite threshold. This induces in turn a gravitational, spontaneously broken symmetry phenomenon responsible for the presence of an unusual non-Minkowskian ground state: the inflationary de Sitter space-time. This then acquires the status of the primordial cosmological vacuum, the generic configuration of our cosmological history.     

Dynamical symmetries in relativistic kinetic theory

This paper is part of a program investigating symmetries that are defined at a physical or observational level rather than purely geometrically. Here we generalize previous work on dynamical matter symmetries of relativistic gases. If the matter symmetry vector is surface-forming with the dynamical Liouville vector, then Einstein's equations reduce it to a Killing symmetry of the metric. We show that this conclusion is unaltered if the gas particles are subject to a nongravitational force (including the electromagnetic force on charged particles) or if the gravitational field obeys higher-order field equations. In the Brans-Dicke theory, the matter symmetry reduces to a homothetic symmetry of the metric. This is also the case for a generalized conformal symmetry in Einstein's theory. We consider the problem of relaxing the surface-forming assumption in an attempt to determine whether there are dynamical symmetries that do not necessarily reduce to geometrical symmetries of the metric.     

Kosmologische Lösungen der lorentzinvarianten Gravitationstheorie

In the framework of the Lorentz invariant theory of gravitation a cosmology in the flat space-time is investigated. As in the Newtonian cosmology we start from an infinitely extended system of incoherent matter under the influence of its own gravitational field. The field equations, the equations of motion and the world postulate of homogenity and isotropy for geodetic observes lead then to the Friedman equation. In order to handle the coupled system of equations for the gravitational field and the matter a conveniant approximation method is developed. The calculations are carried out in the second order of this method. The Einstein theory, which is in some respect equivalent to the Lorentz invariant theory of gravitation, serves as a guiding principle for our formal developements. On the other hand the flat space-time cosmology presented here, gives rise to a new interpretation of the Einstein Cosmology.     

A Non-geometric Relativistic Theory of Gravitation

A non-geometric relativistic theory of gravitation is developed by defining a semi-metric to replace the metric tensor as gravitational vector potential. The theory show that the energy-momentum tensor of the gravitational field belong to the gravitational source, gravitational radiation is contained in Einstein’s field equations that including the contribution of gravitational field, the real physical singularity in the gravitational field can be eliminated, and the dark matter in the universe is interpreted as the matter of pure gravitational field.     

Einstein gravitation from scalar nonsingular sources alone

In the light of Einstein's equations a system only containing two scalar fields is considered: One is of long range and attractive, the other is of short range and repulsive. The sources of these fields are taken to be nonsingular and spherically symmetric. All components of the energy-momentum tensor are continuous. A static solution of the equations is obtained in the weak-field approximation. The source of the gravitational field shows a finite concentration on the center of symmetry and dilutes monotonically to zero outwards. A Schwarzschild-type gravitation is found at infinity.     

The simplest black holes

A relativistic gravitational theory in (1 + 1) dimensions is presented which exhibits many of the qualitative features of (3 + 1)-dimensional general relativity. The field equations are simple enough for undergraduates to solve yet rich enough in structure to form a useful pedagogical example for exploring the qualitative features of relativistic gravitation. Black hole solutions to the field equations of the theory are derived and its relationship to Newtonian gravity is discussed in detail.     

Determination of the Lagrangian of the tetrad gravitational field

By requiring correspondence with Newtonian gravitational theory and the Lorentz covariant theory of nongravitational matter and by establishing the simplest possible form of the linear approximation of the field equations, the gravitational Lagrangian of the tetrad theory of gravitation is determined uniquely. It contains two characteristic constants: Einstein's gravitational constant and the specific dimensionless “teleparallel” constant ω ≈ 1.     

Classical Gravitational Interactions and Gravitational Lorentz Force

In quantum gauge theory of gravity, the gravitational field is represented by gravitational gauge field. The field strength of gravitational gauge field has both gravitoelectric component and gravitomagnetic component. In classical level, gauge theory of gravity gives classical Newtonian gravitational interactions in a relativistic form. Besides, it gives gravitational Lorentz force, which is the gravitational force on a moving object in gravitomagnetic field. The direction of gravitational Lorentz force is not the same as that of classical gravitational Newtonian force. Effects of gravitational Lorentz force should be detectable, and these effects can be used to discriminate gravitomagnetic field from ordinary electromagnetic magnetic field.     

Classical Gravitational Interactions and Gravitational Lorentz Force

In quantum gauge theory of gravity, the gravitational field is represented by gravitational gauge field.The field strength of gravitational gauge field has both gravitoelectric component and gravitomagnetic component. In classical level, gauge theory of gravity gives classical Newtonian gravitational interactions in a relativistic form. Besides,it gives gravitational Lorentz force, which is the gravitational force on a moving object in gravitomagnetic field The direction of gravitational Lorentz force is not the same as that of classical gravitational Newtonian force. Effects of gravitational Lorentz force should be detectable, and these effects can be used to discriminate gravitomagnetic field from ordinary electromagnetic magnetic field.