Decoupling of Field Equations in Einstein and Modified Gravity

Abstract

Abstract. This paper is concerned with giving the proof that there is a general decoupling property of vacuum and nonvacuum gravitational field equations in Einstein gravity and f (R, T )-modifications. The constructions are possible in terms of geometric and physical objects adapted to certain nonholonomic 2+2 splitting with local fibred structure. This allows us to generate exact and/or parametric depending solutions with generic off-diagonal metrics and generalized, or Levi-Civita, connections. Different classes of modified spacetimes are determined by corresponding generating and integration functions depending, in general, on all space and time coordinates and may possess, or not, Killing symmetries. The initial data sets for the Cauchy problem and their global properties are analyzed. There are formulated the criteria of evolution with spacetime splitting and decoupling of fundamental field equations. Examples of exact solutions defining ellipsoid deformations of black hole metrics and solitonic configurations are provided. IntroductionThe issue of constructing exact and approximate solutions of (modified) gravitational and matter field equations is of interest in mathematical physics and for various applications in modern cosmology and astrophysics. Mathematically, the fundamental field equations in gravitational physics are defined by sophisticate systems of nonlinear partial differential equations (PDE) which are very difficult to be integrated and studied in general forms.In this work, we address again and develop a geometric approach (the so-called anholonomic frame deformation method, AFDM, see [20,21,22]) to constructing exact solutions in gravity described by generic off-diagonal and nonlinear gravitational, gauge and scalar field interactions. Such a geometric method allows us to generate exact solutions when the coefficients of generic off-diagonal metrics and various classes of connections depend on all spacetime coordinates. Our goal is to provide new results on the decoupling property 1 of modified gravity equations and formal integration of such nonlinear systems of PDEs. Certain issues on the Cauchy problem and decoupling of gravitational equations and the initial data sets and nonholonomic evolution (with non-integrable constraints and/or with respect to anholonomic frames) will be analyzed. Examples of off-diagonal solutions for modified black holes/ellpsoids and solitonic configurations will be given.

Ghost-free massive $$f(R)$$ f ( R ) theories modeled as effective Einstein spaces and cosmic acceleration

“…The generalized gravitational field equations written with respect to N-adapted frames (6) are similar to those studied in our works [28][29][30][31][32][33][34][35][36]. The main difference of such MTGs is determined by a corresponding source which in this work is considered in the form (7).…”

“…a nonlinear connection (N-connection) structure; see details in [28][29][30][31][32][33][34]. The local coefficients {N …”

“…This requires a more advanced and cumbersome geometric techniques; see the examples in [28][29][30][31][32][33][34]36] and references therein. 6 Such metrics cannot be diagonalized by coordinate transforms because for general N-connections the anholonomy coefficients, W γ αβ , are not zero.…”

“…For simplicity, we do not study in this paper d-metrics for which h a = 0 and/or ϒ μ = 0 (such solutions in vacuum MGFT can be constructed, for instance, for f -and/or μ-modifications of black hole solutions; see the examples in [28][29][30][31][32][33][34]). Here we note that the relevant equations (12), (13) …”

“…It is the point of this paper to apply in MGFT the so-called anholonomic frame deformation method, AFDM, [28][29][30][31][32][33][34] for constructing generic off-diagonal exact solutions. Such a method provides geometric techniques, which allows us to integrate systems of partial differential equations, PDEs, with functional and parametric dependencies for the Levi-Civita (zero torsion) and nontrivial torsion configurations.…”