Abstract
We find general parameterizations for generic off-diagonal spacetime metrics and matter sources in general relativity (GR) and modified gravity theories when the field equations decouple with respect to certain types of nonholonomic frames of reference. This allows us to construct various classes of exact solutions when the coefficients of the fundamental geometric/ physical objects depend on all spacetime coordinates via corresponding classes of generating and integration functions and/or constants. Such (modified) spacetimes display Killing and non-Killing symmetries, describe nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants. Our method can be extended to higher dimensions which simplifies some proofs for embedded and nonholonomically constrained four-dimensional configurations. We reproduce the Kerr solution and show how to deform it nonholonomically into new classes of generic off-diagonal solutions depending on 3-8 spacetime coordinates. Certain examples of exact solutions are analyzed and that are determined by contributions of new type of interactions with sources in massive gravity and/or modified f(R,T) gravity. We conclude that by considering generic off-diagonal nonlinear parametric interactions in GR it is possible to mimic various effects in massive and/or modified gravity, or to distinguish certain classes of "generic" modified gravity solutions which cannot be encoded in GR.
Highlights
The gravitational field equations in general relativity, GR, and modified gravity theories, MGT, are very sophisticated systems of nonlinear partial differential equations (PDEs)
A number of examples of exact solutions are summarized in the monographs [1,2] where the coefficients of the fundamental geometric/physical objects depend on one and/or two coordinates in four-dimensional (4-d) spacetimes and when the diagonalization of the metrics is possible via coordinate transformations
(in Sect. 5), we provide our conclusions and speculate on the physical meaning of the exact solutions constructed using the AFDM for massive modified gravity theories and how such effects can be modeled by nonlinear off-diagonal interactions in Einstein gravity
Summary
The gravitational field equations in general relativity, GR, and modified gravity theories, MGT, are very sophisticated systems of nonlinear partial differential equations (PDEs). To prove the decoupling property in the simplest way we have to consider spacetime fibrations with splitting of dimensions, 2(or3) + 2 + 2 + · · · , introduce certain adapted frames of reference, consider formal extensions/embeddings of 4-d spacetimes into higher-dimensional ones and work with necessary types of linear connections Such an (auxiliary, in Einstein gravity) adapted connection is completely defined in a compatible form by the metric structure and contains a nonholonomically induced torsion field. 5), we provide our conclusions and speculate on the physical meaning of the exact solutions constructed using the AFDM for massive modified gravity theories and how such effects can be modeled by nonlinear off-diagonal interactions in Einstein gravity. We state the geometric conventions and outline the formalism which are necessary for decoupling and integrating the gravitational field equations in GR and MGTs; see relevant details in [5,6,7,8]
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