仿射广义引力理论中宇宙解和静力解的统一描述：矢量标量对偶及其应用
Unified description of cosmological and static solutions in affine generalized theories of gravity: vecton - scalaron duality and its applications
摘要：
本文简要地描述了具有对称连接的多维时空中最简单的一类仿射引力理论，以及它们到二维非对称矢量引力场论（DVG）的约化。这些理论的显著特点是存在一个绝对中性的质量（或快子）矢量场（vecton），它与强子引力（DG）具有本质上的非线性耦合。我们表明，在DVG的矢量场可以始终取代一个有效的大规模标量场（标量）与一个不寻常的耦合到标量引力。有了这种矢量-标子对偶性，人们可以将标准DG耦合到普通标量（DGS）的方法和结果应用到更复杂的DSG中，等价于DVG。我们给出了由多维仿射理论约化导出的DVG模型，得到了在任意规范下同时描述宇宙态和静态的一维动力学系统。我们的方法是完全适用于研究静态和宇宙学的多维理论，以及在一般的一维DGS模型的解决方案。我们专注于模型的全局性质，寻找积分和分析的解空间的结构。在可积的情况下，它可以有效地可视化绘制一个“拓扑画像”类似的动态系统的相图，并简单地暴露静态和宇宙学的解决方案，包括地平线，奇点等的全球性的解析近似，我们还提出了一个积分方程非常适合迭代。
英文摘要：
We briefly describe the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections and their reductions to two-dimensional dilaton - vecton gravity field theories (DVG). The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with essentially nonlinear coupling to the dilaton gravity (DG). We show that in DVG the vecton field can be consistently replaced by an effectively massive scalar field (scalaron) with an unusual coupling to dilaton gravity. With this vecton - scalaron duality, one can use methods and results of the standard DG coupled to usual scalars (DGS) in more complex dilaton - scalaron gravity theories (DSG) equivalent to DVG. We present the DVG models derived by reductions of multidimensional affine theories and obtain one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories as well as in general one-dimensional DGS models. We focus on global properties of the models, look for integrals and analyze the structure of the solution spaces. In integrable cases, it can be usefully visualized by drawing a `topological portrait' resembling phase portraits of dynamical systems and simply exposing global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations we also propose an integral equation well suited for iterations.
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