A background-dependent approach to the theory of gravitation II. Relativistic gravitational equations

On the basis of the results of Paper I and guided by a Machian view of nature, we find new gravitational equations which are background dependent. Such equations describe a purely geometrical theory of gravitation, and their dependence on the background structure is through the total energy-momentum tensor on the past sheet of the light cone of each space-time pointx [θ^μν x, say], i.e., through the integral on the past sheet of the light cone ofx of the parallel transport of the energy-momentum tensor from the space-time point in which it is defined tox along the geodesic connecting the two space-time points. Following Gürsey, we assume that the source of the De Sitter metric is not the cosmological term, but, rather, the energy-momentum tensor of a “uniform distribution of mass scintillations” [T ^μν x, say].T ^μν x, indeed, turns out to be equal to the metric tensor times a constant factor. As a consequence, in any local inhomogeneity A of a space-time whose background structure is determined by the Perfect Cosmological Principle,θ ^μν turns out to be approximately equal to the metric tensor times a constant factor, providedT=g _αβ T ^αβ is sufficiently small and the structure of the past sheet of the light cones of the space-time points belonging to Λ is not too much perturbed by the local gravitational field. As a consequence, in Λ the new equations approximately reduce to Einstein's equations. If one considers a “superuniverse model” in which our universe is considered as a local inhomogeneity in a De Sitter background, then from the above result there follows a fortiori the agreement of the new gravitational equations with the classical tests of gravitation. Furthermore, the dependence on the background structure is such that the new equations (i) incorporate the idea that the frame has to be fixeddirectly in connection with cosmological observations, and (ii) are singular in the absence of matter in the whole space-time. Moreover, (iii) the coupling constant turns out to be dimensionless in natural units (c=1=?), and (iv) a local inertial frame in a De Sitter background is determined by the condition that with respect to it the background structure is homogeneous in space and in time and is Lorentz invariant.     

Lorentz covariant approach in the gravitation theory

The gravitation theory is constructed in the framework of the flat Minkowski space-time. The approach is found that provides unambiguity of the construction. The derived formulas coincide in form with the formulas of general relativity theory.     

A possible approach to the description of topological transitions in the quantum theory of gravitation

Russian Physics Journal - One of the problems generated in attempts of describing topological transitions in the quantum theory of gravitation is the introduction of variables describing the...     

A bimetric machian approach to gravitation

We propose a bimetric machian approach to gravitation with a mathematical structure much simpler than the one of Rosen's bimetric theories. We obtain two cosmological models based on the simplest assumption that the Universe be filled of pure dust matter. One of the two cosmological models is compatible with the currently observed value of the density of dust matter, and provides an age of the Universe which is of the order of the inverse of the present Hubble parameter. One also obtains the Schwarzschild-like solution and its Newtonian limit together with the modified three Kepler laws which allow us to find that presently 0.5×10^?10 (yr)^?1 δ(A/A)δ 0.625×10^?10 (yr)^?1, a denoting the semimajor axis of the orbit of the test particle. Lastly we obtain the Newtonian limit of the theory.     

A bi-metric theory of gravitation

A bi-metric theory of gravitation is proposed, satisfying the covariance and equivalence principles. It is based on a simple form of Lagrangian and has a simpler mathematical structure than that of the general theory of relativity. The theory agrees with general relativity up to the accuracy of the observations made up to now. The static spherically symmetric solution of the present field equations does not involve any 'black hole'.     

A scalar field theory of gravitation

A scalar theory of gravitation is developed from a variational principle. The speed of light is taken to be a function of the potential of the gravitational field. The predictions of the light deflection and the advancement of the perihelion agree with those made by Einstein's theory. The gravitational (active) mass differs from the inertial (passive) mass and both are dependent on the gravitational potential.     

Gauge gravitation theory

The particularity of the gauge gravitation theory is that Dirac fermion fields possess only Lorentz exact symmetries. It follows that different tetrad gravitational fieldsh define nonisomorphic representations _h of cotangent vectors to a space-time manifoldX ⁴ by Dirac's-matrices on fermion fields. One needs these representations in order to construct the Dirac operator defined in terms of jet spaces. As a consequence, gravitational fieldsh fail to form an affine space modeled after any vector space of deviationsh'–h of some background fieldh. They therefore fail to be quantized in accordance with the familiar quantum field theory. At the same time, deformations of representation _h describe deviations ofh such thath + is not a gravitational field. These deviations form a vector space, i.e., satisfy the superposition principle. Their Lagrangian, however, differs from familiar Lagrangians of gravitation theory. For instance, it contains masslike terms.     

A non-covariant theory of gravitation,I

Homogeneous isotropic models of the universe, based on the general theory of relativity, lead to the existence of a preferred frame of reference, which is similar to the absolute space of, Newton, and a preferred time coordinate, which resembles the absolute time of Newton. These concepts seem to be in contradiction to the principle of covariance on which the general relativity theory is based. A theory of gravitation is therefore proposed which uses the world picture of general relativity but is not covariant. In the three crucial tests, the proposed theory gives the same results as the general relativity theory. However, in contrast to general relativity, the present theory predicts the emission of gravitational waves by spherically symmetric systems, and gravitational waves are found, in general, to have both transverse and longitudinal components.     

A bi-metric theory of gravitation. II

The bi-metric theory of gravitation proposed previously is simplified in that the auxiliary conditions are discarded, the two metric tensors being tied together only by means of the boundary conditions. Some of the properties of the field of a particle are investigated; there is no black hole, and it appears that no gravitational collapse can take place. Although the proposed theory and general relativity are at present observationally indistinguishable, some differences are pointed out which may some day be susceptible of observation. An alternative bimetric theory is considered which gives for the precession of the perihelion 5/6 of the value given by general relativity; it seems less satisfactory than the present theory from the aesthetic point of view.     

A non-covariant theory of gravitation,II

This is a continuation of a previous paper, in which a theory of gravitation was developed based on the existence of a preferred frame of reference and a preferred time coordinate in the universe. The gravitational field equations are derived with the help of a variational principle containing three constants. Two relations among the constants are introduced, leaving one of them arbitrary. This constant does not affect the precession of the perihelion of Mercury but does affect the behaviour of gravitational waves. By changing one of the relations among the constants, one can account for the discrepancy in the precession of the perihelion associated with the oblateness of the sun, as found by Dicke and Goldenberg.     

A coupled theory of electromagnetism and gravitation

A coupling electromagnetism with a previously developed scalar theory of gravitation is presented. The principle features of this coupling are: (1) a slight alteration to the Maxwell equations, (2) the motion of a charged particle satisfies an equation with the Lorentz force-appearing on the right side in place of zero, and (3) the energy density of the electromagnetic field appears in the gravitational field equation in a manner similar to the mass term in the Klein-Gordonequation. The field of a static, spherically symmetric charged particle is computed. The electromagnetic field gives rise to l/r ² terms in the gravitational potential.     

On the Goldstonic gravitation theory

The physical specificity of gravity as a Goldstone-type field responsible for spontaneous breaking of space-time symmetries is investigated and extended up to supergravity. Problems of the Higgs gravitation vacuum and its matter sources are discussed. A particular “dislocation” structure of a space-time due to Poincaré translation gauge fields and the corresponding modification of Newton’s gravitational potential are predicted.     

Dynamical space and gravitation within a purely geometrical framework: A new approach to gravitation

We state a purely geometrical framework apparently implementing Machian ideas on inertia. Only coupling constants dimensionless in natural units have been introduced in the theory. In anynonvacuum cosmos the field equations describing the gravitational phenomena in cosmological units turn out to be identical to Einstein's equations, with the Einstein gravitational coupling expressed in terms of the parameters defining the cosmological structure. This dependence, however, is not detectable. Indeed, such equations do not need to incorporate the standard Machian requirements (apart from the requirement that they are not conceivable in the total absence of matter) in order to be Machian, since, just on the basis of Mach's principle, one cannot expect to be able to detect Machian effects in Nature by using a system of units based on gravitational phenomena. On the contrary, the equations describing the gravitational field in local atomic units are Machian in the standard sense and, in particular, they incorporate the ideas that the frame has to be fixeddirectly in connection with the observed distribution and motion of matter and that there does not exist any kind of space-time in the total absence of matter. Finally, to reconcile, at least in the weak-field approximation, Einstein's equations (considered as equations describing the gravitational phenomena in local atomic units) with Mach's principle and to be in agreement with cosmological observations, we suggest that our cosmos be identified with a superuniverse model in which the background structure is homogeneous (in space and in time) and isotropic, while our universe is represented by one of the local inhomogeneities of the background. Then we prove that in any region of our universe in which the gravitational field issufficiently weak and smooth the equations, describing the gravitational field in local atomic units, are expected to approximate Einstein's equations all the better, the more the dimensions of our universe are negligible with respect to the dimensions of the background and the background curvature is small. As regards the experimental predictions of the present approach, any prediction for experiments involving only purely gravitational measurements is identical to that of Einstein's theory and the above result also guaranteesa fortiori the agreement with the available experimental data, also asnonpurely gravitational experiments are concerned.This paper appeared as Istituto Matematico L. Tonelli, preprint 78–10 (April 1978) (unpublished).     

New approach to relativity and gravitation

A theory of gravitation, where the stress-energy tensor of the gravitational field is part of the source term of the geometric curvatures, is presented. The theory predicts correctly all the Newtonian and post-Newtonian effects of gravity currently considered as crucial. For extremely strong and nonstatic fields, such as are presumed to exist in quasars and exploding stars, new predictions differing from those of the conventional theory are implied. In the correspondence limit of special relativity the theory recovers the conservation of energy-momentum properly and seems to represent a more satisfactory extension of special theory of relativity.     

Minicompact objects according to the bimetric theory of gravitation

It is shown that, similarly to general relativity, the bimetric theory of gravitation predicts the formation of primordial minicompact objects. Contrary to general relativity, however, it predicts that such objects are stable. Observational consequences are discussed.In partial fulfillment of the requirements for the D.Sc. degree.     

Lorentz-Covariant theory of gravitation

A Lorentz-covariant theory of gravitation is proposed. It is based on a simple form of the Lagrangian for the gravitational field. The field equations have a simple mathematical structure where the energy-momentum tensor of matter and of gravitational field is the source of the field. The theory agrees with general relativity for the three well-known effects, i.e., red shift, deflection of light, and perihelion.     

Matrix theory of gravitation

A new classical theory of gravitation within the framework of general relativity is presented. It is based on a matrix formulation of four-dimensional Riemann-spaces and uses no artificial fields or adjustable parameters. The geometrical stress-energy tensor is derived from a matrix-trace Lagrangian, which is not equivalent to the curvature scalar R. To enable a direct comparison with the Einstein-theory a tetrad formalism is utilized, which shows similarities to teleparallel gravitation theories, but uses complex tetrads. Matrix theory might solve a 27-year-old, fundamental problem of those theories (Sect. 4.1). For the standard test cases (PPN scheme, Schwarz schild-solution) no differences to the Einstein-theory are found. However, the matrix theory exhibits novel, interesting vacuum solutions.     

On the bimetric theory of gravitation

Rosen's bimetric theory is analyzed anew and is shown to have deficiencies if the space is assumed to be Riemannian. The problems are due mainly to the introduction of the flat metric , and the identification of the stress-energy tensor,T . It is indicated that if the Riemannian interpretation could be avoided the theory still holds promise as a viable theory of gravitation.     

A geometrical property of action in gravitation theory

It is shown that in regard to the special spacial foliation associated with the gas of a standard clock, the action of gravitation theory is proportional to the time parameter, while the coefficient of proportionality is equal to the energy of the gravitational field and other fields in the reference system formed by the gas of the standard clock.St. Petersburg State Technical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 81–84, November, 1993.