Towards a theory of macroscopic gravity

By averaging out Cartan's structure equations for a four-dimensional Riemannian space over space regions, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space. The averaged space is characterized by a metric, Riemannian and non-Rimannian curvature 2-forms, and correlation 2-, 3- and 4-forms, an affine deformation 1-form being due to the non-metricity of one of two connection 1-forms. Using the procedure for the space-time averaging of the Einstein equations produces the averaged ones with the terms of geometric correction by the correlation tensors. The equations of motion for averaged energy momentum, obtained by averaging out the contracted Bianchi identities, also include such terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (the non-Riemannian one is then the field tensor), a theorem is proved which relates the algebraic structure of the averaged microscopic metric to that of the induction tensor. It is shown that the averaged Einstein equations can be put in the form of the Einstein equations with the conserved macroscopic energy-momentum tensor of a definite structure including the correlation functions. By using the high-frequency approximation of Isaacson with second-order correction to the microscopic metric, the self-consistency and compatibility of the equations and relations obtained are shown. Macrovacuum turns out to be Ricci non-flat, the macrovacuum source being defined in terms of the correlation functions. In the high-frequency limit the equations are shown to become Isaacson's ones with the macrovauum source becoming Isaacson's stress tensor for gravitational waves.     

Integrability conditions for a gravitational theory with nonmetric connection

In the framework of a new gravitational theory with nonmetric connection recently introduced by one of us (J. K.), it is shown that the matter stress tensors satisfy a certain identity, which, via the contracted Bianchi identities, turns out to be a formal integrability condition for the gravitational field equations. The conservation law for the Hilbert tensor is also discussed.     

Geometric Quantization and the Metric Dependence of the Self-Dual Field Theory

We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2, 0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.     

Cosmological solutions in macroscopic gravity

In the macroscopic gravity approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We present exact cosmological solutions to the equations of macroscopic gravity for a spatially homogeneous and isotropic macroscopic space-time and find that the correlation tensor is of the form of a spatial curvature term. We briefly discuss the physical consequences of these results.     

Variational formalism and field equations of tetrad gravitation theory in the Weyl-Cartan space

A variational formalism of tetrad gravitation theory is developed in the Weyl-Cartan space with independent variations in the tetrad coefficients, metric tensor components, and affine connectivity coefficients that considers the Weyl condition imposed on the nonmetricity based on the method of undetermined Lagrange multipliers. The gravitational field equations are derived for the Lagrangian comprising all possible quadratic convolutions of curvature, torsion, and nonmetricity tensors in addition to the linear component. Differential identities are obtained for the general gravitational Lagrangian in the Weyl-Cartan space. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 56–59, June, 2006.     

Über mögliche Grenzen einer Gravitationstheorie

On Possible Limits of a Gravitation Theory This work intends to answer the question why gravitational phenomena are described just by a theory which make use of a Riemannian metric tensor that has to be a solution of the known Einstein equations. The answer is thought of as a link from Maupertuis's principle to Einstein's equations, which appear as a transformed integrability condition. The metric tensor is first formally introduced and direction dependent (a Finslerian one) and then reduced to a Riemannian one by means of the condition of local Lorentz invariance. The construction seems to show that the gravity concept could not have a sense at the atomic level as a consequence of the central role which plays the particle-motion in the whole.     

Linearized gravitation theory in macroscopic media

The refraction of gravitational waves is discussed by developing a macroscopic theory of gravitation along the lines of classical electromagnetism. It is shown that the linearized Bianchi identities may be expressed in a form which is suggestive of Maxwell's equations with magnetic monopoles. The medium is then assumed to be corpuscular in structure and it is shown how, on performing an averaging process on the field quantities, the Bianchi identities must be modified by the inclusion of polarization terms resulting from the induction of quadrupole moments on the individual “molecules”. A model of a medium whose molecules are harmonic oscillators is discussed and constitutive equations are derived. Gravitational waves are demonstrated to slow down in such a medium.     

Nonlinear gravito-electromagnetism

There is a non-linear and covariant electromagnetic analogy for gravity, in which the full Bianchi identities are Maxwell-type equations for the free gravitational field, encoded in the Weyl tensor. This tensor gravito-electromagnetism is based on a covariant generalization of spatial vector algebra and calculus to spatial tensor fields, and includes all non-linear effects from the gravitational field and matter sources. The non-linear vacuum Bianchi equations are invariant under spatial duality rotation of the gravito-electric and gravito-magnetic tensor fields. The super-energy density and super-Poynting vector of the gravitational field are natural duality invariants, and satisfy a super-energy conservation equation.     

Dislocations and internal length measurement in continuized crystals. I. Riemannian material space

Distributions of dislocations creating point defects are considered. These point defects are described by a metric tensor, which supplements a Burgers field responsible for dislocations. The metric tensor depends on the distribution of dislocations and defines a Riemannian geometry of the material space of a continuized crystal and thus an internal length measurement in this crystal. The dependence of the distribution of dislocations on the existence of point defects created by these dislocations is modeled by treating the Burgers field as a field defined on the Riemannian material space. Field equations, following from geometric identities, are formulated as balance equations on this Riemannian space and their source terms, responsible for interactions of dislocations and point defects, are identified.     

A Finsler generalisation of Einstein's vacuum field equations

This paper gives a generalisation of Einstein's vacuum field equations for Finsler metrics. The given generalised field equation reproduces the Einstein equations for Riemannian metrics, and also admits non-Riemannian solutions. This is shown in detail by deriving a first order Finsler perturbation, solving the new field equation, of the Schwarzschild metric. This perturbation turns out to be time independent. The effects of the perturbation on the three Classical Tests of General Relativity are derived, and used to give limits on the size of the perturbation parameter involved.     

Variational principle for gravitational equations of the Bianchi identity type

It is shown that a square invariant of the Weyl conformal curvature tensor can lead to a Lagrangian in a variational principle for a gravitational equation in vacuum of the Bianchi identity type which is compatible with the Einstein equation. Moreover we show that such a Lagrangian implicitly includes a conformally invariant theory characterized by two gauge fields and the metric tensor.     

Electrodynamics from a metric

A metric is given that produces a space in which the geodesic equation is identical with the Lorentz equation of motion for a charged particle. The gravitational field equations in the same space indicate a geometric origin for the electromagnetic energy-momentum tensor. A comparison is made with Kaluza-Klein theories and it is determined that the present theory is distinct from them because it corresponds to a timelike, noncompact fifth dimension. Since the metric is velocity-dependent, it is actually a Finsler space rather than a Riemannian space metric. Its special form, however, allows computations to be done in terms of Riemannian geometry.     

Basic gravitational currents and Killing–Yano forms

It has been shown that for each Killing–Yano (KY)-form accepted by an n-dimensional (pseudo)Riemannian manifold of arbitrary signature, two different gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general geometrical facts implied by these conservation laws are also elucidated. In particular, the conservation of the one-form currents implies that the scalar curvature of the manifold is a flow invariant for all of its Killing vector fields. It also directly follows that, while all KY-forms and their Hodge duals on a constant curvature manifold are the eigenforms of the Laplace–Beltrami operator, for an Einstein manifold this is certain only for KY 1-forms, (n − 1)-forms and their Hodge duals.     

Curvature tensors unified field equations on SEXn

We study the curvature tensors and field equations in then-dimensional SE manifold SEX_n. We obtain several basic properties of the vectorsS andU and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field equations is discussed in SEX_n and one of its particular solutions is constructed and displayed.     

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

A Review About Invariance Induced Gravity: Gravity and Spin from Local Conformal-Affine Symmetry

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics.     

Macroscopic Einstein equations to second order in the interaction constant

The present paper is a direct continuation of an earlier paper [JETP 83, 1 (1996)] devoted to the derivation of the macroscopic Einstein equations to within terms of second order in the interaction constant. Ensemble averaging of the microscopic Einstein equations and the Liouville equation for the random functions leads to a closed system of macroscopic Einstein equations and kinetic equations for one-particle distribution functions. The macroscopic Einstein equations differ from the classical equations in that their left-hand side contains additional terms due to particle interaction. The terms are traceless tensors with zero divergence. An explicit covariant expression for these terms is given in the form of momentum-space integrals of expressions dependent on one-particle distribution functions of the interacting particles of the medium. The given expressions are proportional to the cube of the Einstein constant and the square of the particle number density. The latter relationship implies that interaction effects manifest themselves in systems of very high density (the universe in the early stages of its evolution, dense objects close to gravitational collapse, etc.) Zh. éksp. Teor. Fiz. 112, 1153–1166 (October 1997)     

A Berger-type theorem for metric connections with skew-symmetric torsion

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement of a well-known theorem of Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.     

A Finslerian extension of general relativity

A Finslerian extension of general relativity is examined with particular emphasis on the Finslerian generalization of the equation of motion in a gravitational field. The construction of a gravitational Lagrangian density by substituting the osculating Riemannian metric tensor in the Einstein density is studied. Attention is drawn to an interesting possibility for developing the theory of test bodies against the Finslerian background.