Quantization of contact manifolds and thermodynamics

The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this ‘quantization’ describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.     

Motion of photons in a gravitational wave background

Photon motion in a Michelson interferometer is re-analyzed in terms of both geometrical optics and wave optics.The classical paths of the photons in the background of a gravitational wave are derived from the Fermat principle,which is the same as the null geodesics in general relativity.The deformed Maxwell equations and the wave equations of electric fields in the background of a gravitational wave are presented in a flat-space approximation.Both methods show that even the envelope of the response of an interferometer depends on the frequency of a gravitational wave,but it is almost independent of the frequency of the mirror's vibrations.     

Locally Anisotropic Kinetic Processes and Thermodynamics in Curved Spaces

The kinetic theory is formulated with respect to anholonomic frames of reference on curved spacetimes. By using the concept of nonlinear connection we develop an approach to modelling locally anisotropic kinetic processes and, in corresponding limits, the relativistic nonequilibrium thermodynamics with local anisotropy. This leads to a unified formulation of the kinetic equations on (pseudo) Riemannian spaces and in various higher dimensional models of Kaluza–Klein type and/or generalized Lagrange and Finsler spaces. The transition rate considered for the locally anisotropic transport equations is related to the differential cross section and spacetime parameters of anisotropy. The equations of states for pressure and energy in locally anisotropic thermodynamics are derived. The obtained general expressions for heat conductivity, shear, and volume viscosity coefficients are applied to determine the transport coefficients of cosmic fluids in spacetimes with generic local anisotropy. We emphasize that such local anisotropic structures are induced also in general relativity if we are modelling physical processes with respect to frames with mixed sets of holonomic and anholonomic basis vectors which naturally admits an associated nonlinear connection structure.     

Quantum Surfaces,Special Functions,and the Tunneling Effect

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. This is interpreted as a tunneling effect in the quantum geometry.     

Many-time hamiltonian gravitation theory

It is shown that given any “good” coordinate condition in Hamiltonian general relativity one can construct an associated many-time formulation in which the constraints can be solved for some of the momenta as functionals of the remaining canonical variables. Since good coordinate conditions appear to be available for both open and closed spaces it follows that the functional wave equation for general relativity can be always put in a Tomonaga-Schwinger form. The implications of this result and some open problems are briefly discussed.     

Extension of loop quantum gravity to f(R) theories

The four-dimensional metric f(R) theories of gravity are cast into connection-dynamical formalism with real su(2) connections as configuration variables. Through this formalism, the classical metric f(R) theories are quantized by extending the loop quantization scheme of general relativity. Our results imply that the nonperturbative quantization procedure of loop quantum gravity is valid not only for general relativity but also for a rather general class of four-dimensional metric theories of gravity.     

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.     

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.     

Strong gravitational waves in semiclosed electromagnetic universes. Statement and analysis of the problem in terms of the method of spin coefficients

The problem of integration is discussed for a complete system of Newman-Penrose equations for electrovacuum spaces of the general theory of relativity with nonzero cosmological constant. In terms of the method of spin coefficients, we formulate conditions on the electromagnetic and gravitational field variables, which distinguish a special class of Riemann spaces corresponding to strong gravitational waves in semiclosed Universes of Bertotti-Robinson type.Translated from Izvestiya Vysshikh Uchebnykh Zavederiii, Fizika, No. 11, pp. 74–78, November, 1987.     

Fundamentals of the nonholonomically covariant formulation of the general theory of relativity

The physical foundations of the nonholonomic formulation of general relativity are determined, and the role of the Fock-Ivanenko coefficients in setting up and developing the tetrad formalism in general relativity is discussed. The physical and geometrical meaning of the nonholonomic transformations used in general relativity is determined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 55–60, December, 1974.     

A Geometric Approach to the Classification of the Equilibrium Shapes of Self-Gravitating Fluids

The classification of the equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in Mathematical Physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of them are also obtained, e.g. classification and existence for some Riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold. Our approach to the problem is geometrical and allows to study the equilibrium shapes on general Riemannian spaces.     

Strong gravitational waves in semiclosed electromagnetic universes. Complete system of Newman-Penrose equations and radial dependence of the field variables

We consider the process of integration of a complete system of Newman-Penrose equations for electrovacuum spaces of the general theory of relativity with nonzero cosmological constant. The restrictions used for the field variables correspond to strong gravitational waves in semiclosed Universes of Bertotti-Robinson type.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 29–33, December, 1987.     

AUTOMORPHISMS OF CURVES

The article deals with local symmetries of the infinite-order jet space of C^∞-smooth curves in ?^m+1 (m ≥ 1). Transformations under consideration are the most general possible: they need not preserve the distinction between dependent and independent variables and the order of derivatives may be arbitrarily changed. Unlike the common prolonged point and Lie's contact transformations, they destroy the finite-order jet spaces.     

Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's “geometrical” method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.     

An operator approach to Poisson brackets

The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments.     

Scalar field cosmology in phase space

Using dynamical systems methods, we describe the evolution of a minimally coupled scalar field and a Friedmann-Lemaître-Robertson-Walker universe in the context of general relativity, which is relevant for inflation and late-time quintessence eras. Focussing on the spatially flat case, we examine the geometrical structure of the phase space, locate the equilibrium points of the system (de Sitter spaces with a constant scalar field), study their stability through both a third-order perturbation analysis and Lyapunov functions, and discuss the late-time asymptotics. As we do not specify the scalar field’s origin or its potential, the results are independent of the high-energy model.     

Equations of motion in unified five-dimensional geometrized theory

The equations of motion in unified five-dimensional theory of gravitation, electromagnetism, and scalar field are considered. It is shown that some of the equations of the theory follow from the rest as equations of motion. In the classical limit of the theory, the equations of motion are found, which coincide with the related equations of general relativity. The similarity of the classical limit of the five-dimensional theory and of the Brans-Dicke theory is noted.     

Quadratic Poincaré gauge theory of gravity: A comparison with the general relativity theory

The classical dynamics of the gravitational field in the Poincaré gauge theory is studied. The most general Lagrangian quadratic in curvature and torsion is considered. The relevant field equations and their solutions are analyzed in detail, with particular emphasis on the comparison of the Poincaré gauge models with the general relativity theory. We investigate correspondence between the spaces of exact solutions of these theories, both in the presence and absence of material sources, and with or without torsion. Some new exact solutions are obtained without the use of the double duality ansatz. The weak-field approximation is discussed, and gravitational radiation is considered.     

Geometrothermodynamics for black holes and de Sitter space

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald’s method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald’s method. The new method is applied to Schwarzschild, Kerr, and Kerr–Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.