A compact Hamiltonian formulation of metric gravity is developed, investigated and applied to some relevant problems. This new Hamiltonian formulation is closely related (by some canonical transformations of the dynamical variables) to the two known and only successful Hamiltonian formulations of the metric gravity proposed earlier (by Dirac and by the author’s group). In reality, our set of natural and canonical dynamical variables in metric gravity allows one to reduce both canonical and total Hamiltonians to very compact expressions, each being a pure quadratic form in the spatial momenta p^mn . Analytical formulas for the primary and secondary constraints are also compact and physically transparent. In general, our new approach transforms the traditional Hamiltonian formulations of metric gravity into a simple procedure which allows one to solve a number of long-standing problems.
We perform a comprehensive analysis of thermodynamic fluid space-time. First we describe a 4D relativistic thermodynamic fluid space-time. Next we explore a new class of semi-Riemannian manifolds which are Ricci-symmetric and derive Codazzi type Ricci tensor on sequential warped product thermodynamic fluid space-time. Furthermore, we show that the warping functions for a sequential warped product thermodynamic fluid space-time are constants under specific conditions. In the latter part, specific curvature aspects in relativistic thermodynamic fluid space-time are studied. We conclude with an elucidation of 4D relativistic thermodynamic fluid space-time through an example.
Bażański’s approach is extended and applied in the framework of Parametrized Absolute Parallelism (PAP) geometry and Einstein–Maxwell geometry. Two novel path deviation equations are deduced. The first equation is derived in the PAP geometry, and it incorporates two parameters. It could be used to investigate the deviation from charged spinning test particles’ trajectories in unified field theories. Three special cases are considered to confirm the viability of this equation. The second equation is deduced in Einstein–Maxwell geometry. It can be employed to investigate the deviation of charged particles from their trajectories in Einstein–Maxwell theory. Both new path deviation equations tend to the geodesic deviation equation under certain conditions.
This article explores the dynamics and stability of cosmological models within the framework of Parameterized Absolute Parallelism (PAP) geometry, with particular emphasis on the effects of spin-torsion interactions. Utilizing the Parameterized Raychaudhuri Equation, we derive stability conditions for fixed points and examine their influence on the evolution of the universe’s expansion scalar. Our findings indicate that the parameter b associated with spin-torsion coupling plays a pivotal role: when it fosters stability, it prevents singularities and facilitates smooth exponential expansion reminiscent of inflation. Conversely, when b<1 , instability and divergence occur. This framework also addresses singularity avoidance through nonsingular bounces and establishes a connection between spin-torsion effects and accelerated expansion, effectively mimicking dark energy. Numerical simulations support these conclusions, illustrating evolution of the scale factor, the expansion scalar, and stability conditions. The results highlight the potential of the PAP geometry in tackling significant cosmological challenges, including singularity resolution and the mechanisms driving accelerated expansion. We apply these findings to a comprehensive model encompassing the Big Bang, Big Rip, and Big Crunch scenarios.
This research paper studies almost Riemann solitons and almost gradient Riemann solitons on space-times obeying the f(r,T) -gravity field equations. First, we obtain the divergence of the potential vector field of an almost Riemann soliton in terms of the scalar curvature on a perfect-fluid space-time in f(r,T) -gravity. We then discuss almost Riemann solitons on special perfect fluids (dark fluid, dust, radiation). As a natural generalization, we extend the framework to anisotropic-fluid and field-based matter sources (e.g., scalar and electromagnetic fields), indicating how the soliton structure equations adapt in f(r,T) -gravity beyond the perfect-fluid ansatz.
