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.