Abstract
Geometric methods for constructing exact solutions of equations of motion with first order alpha ^{prime } corrections to the heterotic supergravity action implying a nontrivial Yang–Mills sector and six-dimensional, 6-d, almost-Kähler internal spaces are studied. In 10-d spacetimes, general parametrizations for generic off-diagonal metrics, nonlinear and linear connections, and matter sources, when the equations of motion decouple in very general forms are considered. This allows us to construct a variety of exact solutions when the coefficients of fundamental geometric/physical objects depend on all higher-dimensional spacetime coordinates via corresponding classes of generating and integration functions, generalized effective sources and integration constants. Such generalized solutions are determined by generic off-diagonal metrics and nonlinear and/or linear connections; in particular, as configurations which are warped/compactified to lower dimensions and for Levi-Civita connections. The corresponding metrics can have (non-) Killing and/or Lie algebra symmetries and/or describe (1+2)-d and/or (1+3)-d domain wall configurations, with possible warping nearly almost-Kähler manifolds, with gravitational and gauge instantons for nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants encoding string gravity effects. A series of examples of exact solutions describing generic off-diagonal supergravity modifications to black hole/ellipsoid and solitonic configurations are provided and analyzed. We prove that it is possible to reproduce the Kerr and other type black solutions in general relativity (with certain types of string corrections) in the 4-d case and to generalize the solutions to non-vacuum configurations in (super-) gravity/string theories.
Highlights
The problem of constructing exact solutions of equations of motion in ten-dimensional, 10-d, string and gravity is of great importance
Various generalizations of well-known and physically important exact solutions for the Schwarzschild, Kerr, Friedman–Lemaître–Robertson– Walker (FLRW), wormhole spacetimes etc were constructed. These classes of diagonalizable metrics are generated by a certain ansatz where equations of motion are transformed into certain systems of nonlinear second order ordinary equations (ODEs), 2-d solitonic equations etc
We prove that using the AFDM extended to models with almost-Kähler internal spaces, the Kerr solution can be constructed as a particular case by prescribing a corresponding class of generating and integration functions
Summary
The problem of constructing exact solutions (in particular, with parametric dependence on some deformation parameters) of equations of motion in ten-dimensional, 10-d, (super-) string and gravity is of great importance. Various generalizations of well-known and physically important exact solutions for the Schwarzschild, Kerr, Friedman–Lemaître–Robertson– Walker (FLRW), wormhole spacetimes etc were constructed These classes of diagonalizable metrics (the off-diagonal terms in the Kerr solutions are determined by rotations and respective frame/coordinate systems) are generated by a certain ansatz where equations of motion are transformed into certain systems of nonlinear second order ordinary equations (ODEs), 2-d solitonic equations etc. In Einstein gravity, a d-connection is considered auxiliary, which in certain canonical forms can be uniquely defined by the metric structure following the conditions of metric compatibility and some other geometric conditions (for instance, that certain zero values for “pure” horizontal and vertical components contain nonholonomically induced torsion fields) Such a canonical d-connection allows us to decouple the equations of motion in general forms and generate various classes of exact solutions in generalized/modified string and gravity theories.
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