Eur. Phys. J. Plus (2020) 135: 148
https://doi.org/10.1140/epjp/s13360-019-00043-w

Regular Article

Quasi-stationary solutions in gravity theories with modified dispersion relations and Finsler–Lagrange–Hamilton Geometry

¹ TVR Iaşi, 33 Lascǎr Catargi Street, Iaşi, Romania
² University Apollonia, 2 Muzicii Street, 700107, Iaşi, Romania
³ Physics Department, California State University Fresno, Fresno, CA, 93740, USA
⁴ Department of Theoretical Physics and Computer Modelling, Yuriy Fedkovych Chernivtsi National University, 101 Storozhynetska street, Chernivtsi, 58029, Ukraine

^* e-mail: sergiu.vacaru@gmail.com
^** e-mail: sergiuvacaru@mail.fresnostate.edu

Received: 19 February 2019
Accepted: 9 October 2019
Published online: 28 January 2020

Abstract

Modified gravity theories, MGTs, with modified (nonlinear) dispersion relations, MDRs, encode via indicator functionals possible modifications and effects of quantum gravity; in string/brane, noncommutative, and/or nonassociative gravity theories etc. MDRs can be with global and/or local Lorentz invariance violations, LIVs, determined by quantum fluctuations, random, kinetic, statistical, and/or thermodynamical processes, etc. Such MGTs with MDRs and corresponding models of locally anisotropic spacetime and curved phase spaces can be geometrized in an axiomatic form for theories constructed on (co) tangent bundles with base spacetime Lorentz manifolds. In certain canonical nonholonomic variables, the geometric/physical objects are defined equivalently as in generalized Einstein–Finsler and/or Lagrange–Hamilton spaces. In such Finsler like MGTs, the coefficients of metrics and connections depend both on local Lorentz spacetime coordinates and, additionally, on (co) fiber velocity and/or momentum-type variables. The main goal of this work is to elaborate on a nonholonomic diadic “shell by shell” formulation of MGTs with MDRs, with a conventional splitting of total phase space dimensions, when the (dual) modified Einstein–Hamilton equations can be decoupled in general forms. We show how this geometric formalism allows us to construct various classes of exact and parametric solutions determined by generating and integration functions and effective sources depending, in principle, on all phase space coordinates. There are derived certain most general and important formulas for nonlinear quadratic elements and studied the main geometric and physical properties of quasi-stationary generic off-diagonal and diagonalizable phase spaces. This work provides a self-consistent geometric and analytic method for constructing in our further partner papers different types of black hole solutions for theories with MDRs and LIVs, and elaborating various applications in modern cosmology and astrophysics.