International Journal of Geometric Methods in Modern Physics

In this updated review of canonical ADM tetrad gravity in a family of globally hyperbolic asymptotically Minkowskian space-times without super-translations I show which is the status-of-the-art in the search of a canonical basis adapted to the first-class Dirac constraints and of the Dirac observables of general relativity (GR) describing the tidal degrees of freedom of the gravitational field. In these space-times the asymptotic ADM Poincaré group replaces the Poincaré group of particle physics, there is a York canonical basis diagonalizing the York–Lichnerowicz approach and a post-Minkowskian linearization is possible with the associated description of gravitational waves in the family of non-harmonic 3-orthogonal Schwinger time gauges. Moreover I show that every fixation of the inertial gauge variables (i.e. the choice of a non-inertial frame) of every generally covariant formulation of GR is equivalent to a set of conventions for the metrology of the space-time (like the GPS ones near the Earth): for instance the freedom in clock synchronization is described by the inertial gauge variable York time (the trace of the extrinsic curvature of the instantaneous 3-spaces). This inertial gauge freedom and the non-Euclidean nature of the instantaneous 3-spaces required by the equivalence principle are connected with the dark side of the universe and could explain the presence of dark matter or at least part of it by means of the adoption of suitable metrical conventions for the ICRS celestial reference system. Also some comments on a canonical quantization of GR coherent with this viewpoint are done.

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ² and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

With acceptance of the Dirac's observation that the canonical quantization entails using Cartesian coordinates, we examine the operator e_rP_r rather than P_r itself and demonstrate that there is a decomposition of e_rP_r into a difference of two self-adjoint but noncommutative operators, in which one is the total momentum and another is the transverse one. This study renders the operator P_r indirectly measurable and physically meaningful, offering an explanation of why the mean value of P_r over a quantum mechanical state makes sense and supporting Dirac's claim that P_r "is real and is the true momentum conjugate to r".

In the presence of sources, the usual Maxwell equations are neither symmetric nor invariant with respect to the duality transformation between electric and magnetic fields. Dirac proposed the existence of magnetic monopoles for symmetrizing the Maxwell equations. In the present work, we obtain the fully symmetric Dirac–Maxwell's equations (i.e. with electric and magnetic charges and currents) as a single equation by using 4 × 4 matrix presentation of fields and derivative operators. This matrix representation has been derived with the help of the algebraic properties of quaternions and octonions. Such description gives a compact representation of electric and magnetic counterparts of the field in a single equation.

Given a compact symplectic manifold M, with integral symplectic form, we prequantize a certain class of functions on the path space for M. The functions in question are induced by functions on M. We apply our construction to study the symplectic structure on the solution space of Klein–Gordon equation.

We introduce new tetrads that manifestly and covariantly diagonalize the stress–energy tensor for a perfect fluid with vorticity at every spacetime point. This new tetrad can be applied to introduce simplification in the analysis of astrophysical relativistic problems where vorticity is present through the Carter–Lichnerowicz equation. We also discuss the origin of inertia in this special case from the standpoint of our new local tetrads.

We present a new class of elliptic-like strings on two-dimensional manifolds of constant curvature. Our solutions are related to a class of identities between Jacobi theta functions and to the geometry of the light-cone in one dimension more. We show in particular that two well-known fundamental identities among theta functions have a natural interpretation as expressing the Virasoro constraints of dS or AdS strings.

In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

We provide a detailed and rigorous proof of (a generalized version of) the Ehlers–Geroch theorem on geodesic motion in metric theories of gravity: we assume that (M, g) is a spacetime satisfying an averaged form of the dominant energy condition and some further technical assumptions indicated in the bulk of this paper. Then, a small body which is allowed to deform the original spacetime metric g moves, nonetheless, along a geodesic of (M, g).

In this paper we present the induced representation of SO(2N) canonical transformation group and introduce coset variables. We give a derivation of the time-dependent Hartree–Bogoliubov (TDHB) equation on the Kähler coset space from the Euler–Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati–Hartree–Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO(2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.

Using the new approach on higher-dimensional Kaluza–Klein theories developed by the first author, we obtain the 4D Einstein equations on a (4 + n)D relativistic gauge Kaluza–Klein space. Adapted frame and coframe fields, adapted tensor fields, and the Riemannian adapted connection, have a fundamental role in the study. The high level of generality of the study, enables us to recover several results from earlier papers on this matter.

A Clifford-gravity-based model is exploited to build a generalized action (beyond the current ones used in the literature) and arrive at relevant numerical results which are consistent with the presently-observed de Sitter accelerating expansion of the universe driven by a very small vacuum energy density ρ_obs ~ 10^-120(M_P)⁴ (M_P is the Planck mass) and provide promising dark energy/matter candidates in terms of the 16 scalars corresponding to the degrees of freedom associated with a Cl(3, 1)-algebra-valued scalar field Φ in four dimensions.

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on C^N × R^N is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.