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.     

Dynamic field theory and equations of motion in cosmology

We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays the role of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmological manifold defined as a solution of Einstein’s equations in the form of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations with an effective matter density contrast δρ/ρ≤1$\delta \rho /\rho \le 1$. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ?1$\delta \rho /\rho ?1$. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equations while the small scale perturbations are described by a particular solution of these equations with the bare stress–energy tensor of the baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars, galaxies and their clusters.     

First-order invariant Einstein-Cartan variational structures

A first-order invariant Einstein-Cartan structure is a Lagrangian structure on a differential manifold defined by a generally invariant Lagrangian depending on a metric field, a connection field, and the first derivatives of these fields. Moreover, it is assumed that the metric and connection fields satisfy the so-called compatibility condition. In this paper the problem of finding all such invariant Einstein-Cartan structures is discussed. It is shown that each Lagrangian of these structures depends only on certain tensors constructed from the metric and the connection fields, which means that all the Lagrangians can be described within the framework of the classical theory of invariants. The maximal number of functionally independent Lagrangians is determined as a function of the dimension of the underlying manifold.     

Metric‐free definition of the energy momentum tensor: Gravitation with symmetric connection

A metric‐free definition of the energy momentum tensor is presented. That definition can be used with any Lagrangian based gravitational theory where the independent variable is a symmetric connection. The case of Hilbert‐Einstein gravitation is treated as an example of such a theory.     

The generalized Einstein-Maxwell theory of gravitation

A new Lagrangian theory of gravitation in which the metric and the arbitrary affine connection are regarded as independent field variables has been considered. Making use of the pure geometrical objects only from the variational principle the empty field equations are derived. It is shown that the metric obeys the ordinary Einstein equations of general relativity. However, the covariant derivative of the metric tensor does not vanish, so that the vector's length is generally nonintegrable under the parallel displacement. The torsion trace vector turns out to be the natural dynamical variable, satisfying the Maxwell-like equations with tensor of homothetic curvature as the Maxwell tensor. The equations of motion are explored; they are shown to be identical to the motion of electric charge under the Lorentz force. The conservation laws are discussed.     

Gravitation,Electromagnetism and Cosmological Constant in Purely Affine Gravity

The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the purely affine formulation of gravity by the Eddington Lagrangian. In this paper we combine the electromagnetic field and the cosmological constant in the purely affine formulation. We show that the sum of the two affine (Eddington and Ferraris-Kijowski) Lagrangians is dynamically inequivalent to the sum of the analogous (ΛCDM and Einstein-Maxwell) Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid, like the affine Einstein-Born-Infeld formulation, only for weak electromagnetic fields, on the order of the magnetic field in outer space of the Solar System. Therefore the purely affine formulation that combines gravity, electromagnetism and cosmological constant cannot be a simple sum of affine terms corresponding separately to these fields. A quite complicated form of the affine equivalent of the metric Einstein-Maxwell-Λ Lagrangian suggests that Nature can be described by a simpler affine Lagrangian, leading to modifications of the Einstein-Maxwell-ΛCDM theory for electromagnetic fields that contribute to the spacetime curvature on the same order as the cosmological constant.     

Geometrization of String Theory Gravity

It is shown that the low energy string theory Lagrangian can be interpreted as pure gravity. In particular, it is shown that the Lagrangian is simply R, the curvature scalar of spacetime with torsion, and unlike in previous work, the covariant derivative of the metric tensor vanishes. As a consequence, it is shown that the physical origin of the scalar field results from the pseudoscalar invariant. This yields, for the first time, a definite physical origin of the dilaton field in four dimensions.     

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.     

Method for estimating local lattice distortions near a magnetic ion based on the parameters of the ligand hyperfine interaction: Ce3+ in the fluorite homologous series

A method is suggested for estimating local lattice distortions near a paramagnetic center by using the parameters determined from experimental tensors of the ligand hyperfine interaction with surrounding nuclei through the use of a simple mathematical procedure. In general, a tensor with nine independent components can be unambiguously reduced to nine independent parameters characterizing the hyperfine interaction: the isotropic part, deviations from the symmetric and axially symmetric form, the axially symmetric component (containing dipole and pseudodipole contributions), and the angles defining the bond direction. Under certain physical assumptions, these parameters are used to estimate distortions of the first and second coordination shells of Ce³⁺ in the homologous series CaF₂, SrF₂, and BaF₂.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.     

Quadratic torsion-metric model consistent with then-dimensional gravitational theories of einstein and weyl

We examine a gauge model with a Lagrangian, quadratic in the Riemann-Cardan curvature, without a separate Hilbert Lagrangian. It is argued that torsions must be massive particles, the torsion field does not act on world fields, and the orthogonal components of the contorsion tensor must not vary with variations of the metric. It is shown that a new torsion-metric interaction arises in this case, which generates the gravitational theories of Einstein and Weyl. Kazan State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 21–25, June, 1998.     

Scalar concomitants of a metric and a curvature form

In an Einstein-Yang-Mills field theory the field equations can be obtained by applying a variational principle to a Langrangian minimally coupled to a Lagrangian concomitant of a curvature form and the metric tensor. As a step in the discussion of the uniqueness of these equations, we find the general form of such a Lagrangian.     

Dimensional regularization of gauge theories in spherical spacetime: Free field trace anomalies

The O(n + 1) covariant formulation of massless quantum electrodynamics in spherical spacetime is further developed to allow a calculation of the energy-momentum tensor trace anomalies for the free Dirac, electromagnetic, and SU(2) gauge fields. The principal technical development is the construction of the Faddeev-Popov ghosts for electrodynamics and SU(2) Yang-Mills theory. This construction is unconventional first in that the gauge fixing term in the Lagrangian is not a perfect square, and second because it is necessary to remove radial as well as gauge degrees of freedom from the measure of the functional integral. The ghost fields are shown to satisfy a minimal scalar field equation. The free field effective action is found to be divergent in four dimensions, and is renormalized by the inclusion in the Lagrangian of a counterterm local in the gravitational fields. The energy-momentum tensor calculated from this renormalized effective action is shown to have a trace anomaly.     

Complete integrability of geodesic motion in general higher-dimensional rotating black-hole spacetimes

We explicitly exhibit n-1=[D/2]-1 constants of motion for geodesics in the general D-dimensional Kerr-NUT-AdS rotating black hole spacetime, arising from contractions of even powers of the 2-form obtained by contracting the geodesic velocity with the dual of the contraction of the velocity with the (D-2)-dimensional Killing-Yano tensor. These constants of motion are functionally independent of each other and of the D-n+1 constants of motion that arise from the metric and the D-n=[(D+1)/2] Killing vectors, making a total of D independent constants of motion in all dimensions D. The Poisson brackets of all pairs of these D constants are zero, so geodesic motion in these spacetimes is completely integrable.     

Quantization of the Schwarzschild Black Hole: A Noether Symmetry Approach

We study the canonical formalism of a spherically symmetric space-time. In the context of the 3+1 decomposition with respect to the radial coordinate r, we set up an effective Lagrangian in which a couple of metric functions play the role of independent variables. We show that the resulting r-Hamiltonian yields the correct classical solutions which can be identified with the space-time of a Schwarzschild black hole. The Noether symmetry of the model is then investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. According to the Noether symmetry approach, we also quantize the model and show that the existence of a Noether symmetry yields a general solution to the Wheeler-DeWitt equation which exhibits a good correlation with the classical regime. We use the resulting wave function in order to (qualitatively) investigate the possibility of the avoidance of classical singularities.     

Two theorems on energy-momentum conservation

The mathematical meaning of the law of conservation of energy-momentum is examined. A distinction is made between the intrinsic properties of the metric tensor (i.e., those properties that are independent of the coordinate system), and the nonintrinsic properties of this tensor (i.e., those properties that depend upon the coordinate system). The covariance of the energy-momentum law is used to demonstrate that if one is given (a) any analytic contravariant energy-momentum tensor density in a given coordinate systemx and (b) an analytic specification of the intrinsic properties of the metric tensor, no matter what these properties may be, one can always choose the nonintrinsic properties of the metric tensor in such manner as to satisfy the law of conservation of energy-momentum in the coordinate systemx and thereby in every coordinate system. This result is proved only in the case where the contravariant components of the energy-momentum tensor density are given. Neither the covariant, nor the mixed energy-momentum tensor densities are considered. Other theorems similar to that described above are also derived. Many of the results obtained are nontrivial even when space-time is flat.     

Nonlinear massive spin-2 field generated by higher derivative gravity

We present a systematic exposition of the Lagrangian field theory for the massive spin-2 field generated in higher-derivative gravity upon reduction to a second-order theory by means of the appropriate Legendre transformation. It has been noticed by various authors that this nonlinear field overcomes the well-known inconsistency of the theory for a linear massive spin-2 field interacting with Einstein’s gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second-order pictures usually called “(Helmholtz-)Jordan frame” and “Einstein frame.” In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin-2 field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame (each frame therefore provides a self-contained theory). The full equations of motion and the (variational) energy-momentum tensor for the spin-2 field in Einstein frame are given, and a simple but non-trivial exact solution to these equations is found. The comparison of the energy-momentum tensors for the spin-2 field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy-momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost-like nature of the nonlinear spin-2 field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected.     

A GROUP OF EXACT (3+1) DIMENSIONAL CYLINDRICALLY SYMMETRIC WAVE SOLUTIONS TO THE EINSTEIN GRAVITATIONAL FIELD EQUATION

Starting with the diagonal spacetime metric tensor, the Einstein gravitational field equation is solved, and a set of exact (3+1) dimensional cylindrically symmetric wave solutions with two arbitrary functions are found. In these solutions all nonvanishing components of spacetime metric tensor are varying with the same propagating factor (ct-z) while the waves are travelling along z axis. The physical picture and the condition of positive energy density of the wave solutions are discussed.     

Fermion Fields,Boson Fields,and Gravitational Field

If a tetrad theory is derivable from a variational principle with a Lagrangian ?? of the form ?? = ??_F+??_M 6 tetrad components will be defined by the vacuum equations if the energy momentum tensor is symmetric. Therefore, we look for a realisation of a programme proposed in a little different way by TREDER according to which the 16 tetrad field equations should degenerate to 10 equations for the Riemannian metric if boson fields are the only source of the gravitational field.